Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions

Chiral magnets are an emerging class of topological matter harboring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as an entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets can not only host skyrmions but also antiskyrmions as least energy configurations over all non-trivial homotopy classes. We derive practical criteria for their occurrence and coexistence with skyrmions that can be fulfilled by (110)-oriented interfaces depending on the electronic structure. Relating the electronic structure to an atomistic spin-lattice model by means of density functional calculations and minimizing the energy on a mesoscopic scale by applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultra-thin magnetic films grown on semiconductor or heavy metal substrates with C 2v symmetry are prototype classes of materials hosting magnetic antiskyrmions.

This paper does not seem appropriate for Nature Comm., which seeks to publish works of general interest. The paper may be accessible only to a small community of people which are well familiar withe the particular topic of skyrmion patterns in magnetic media. The text is quite verbose, but it is not written in a form that would be friendly to the potentially broad readership. In particular, while the central topic is the use of the Dzialoshinskii -Moriya interaction, it is not even explicitly defined in the paper, assuming that it is known to everyone, which cannot be correct. Further, while the paper is a fully theoretical one, and its stated objective is to demonstrate that antiskyrmions may realize a physically relevant stable state, the text only briefly mentions that the analysis is carried out by means of a variational method, but no details of the analysis -which is, obviously, the core part of the work -are displayed. Instead, readers are suggested to look for "mathematical details" in some supplementary material, which is not a part of the paper. As a result, the entire text seems as an extended abstract (a lengthy one, which would, nevertheless, be quite obscure to non-specialists), rather than a systematically written article.
It may be that the prediction of the stability of the antiskyrmion, announced in the paper as its main result, is an essential prediction, that may stimulate new experiments (magnetic media, in which the authors expect a possibility to realize the prediction, are briefly mentioned in the paper). However, if the paper is intended for the publication in a broad-interest journal, such as Nature Comm., the text should be made accessible to the broad readership, and it should definitely be made self-contained, providing explicit definitions of the basic concepts in the narrow topic under the consideration, rather than assuming that they are known to everyone (which may sometimes be acceptable in papers submitted to a specialized journal).
It may be recommended to rewrite the paper in the self-consistent form, including the core proof of the antiskyrmion's stability, rather than concealing it in supplementary files, and resubmit the work to one of leading solid-state-physics journals, such as Phys. Rev. B or J. Phys. Cond. Matt.
Reviewer #2 (Remarks to the Author): This paper shows that anisotropic Dzyaloshinskii-Moriya interaction at a low-symmetry surface can stabilize antiskyrmions relative to skyrmions. A model is presented and described in several different ways (analytical theory, micromagnetic and atomistic simulations), and a specific system (Fe bilayer on W(110)) is predicted using first-principles calculations. The message is clear and timely, and I recommend this paper for publication. The authors may want to cite the experimental paper on arXiv (which has appeared after the present paper) reporting the observation of antiskyrmions: https://arxiv.org/abs/1703.01017.
I have a few comments that the authors may consider before publication: [19] pointed out that antiskyrmions are unstable in a more symmetric system, whereas the present manuscript have carried out systematic theoretical and numerical studies of antiskyrmion stabilities in chiral magnets. This might be an important research advance of antiskyrmions since it has some novel properties as compared to the conventional skyrmions which have been intensively studied in the literature. For example, it may add a new dimension and new degree of freedom in skyrmionic devices, such as manipulating its helicity etc for spin logic applications [see, Scientific Reports 5, 9400 (2015)]. In a recent arxiv paper on frustrated skyrmion [arXiv:1703.07501], it is shown that frustrated antiskyrmions is stable solution and the antiskyrmion can interact with skrymion in a fascinating way. So it would be interesting to compare this manuscript with the arxiv paper briefly. In addition, I hope the authors can provide a brief discussion on the practical application of antiskyrmions, which might be beneficial for a wide readership.
In my view, the main results of interface stabilized antiskyrmions will contribute to the skyrmonic research by adding new dimension to this exciting field. It presents something very important and will motivate further research efforts towards its experimental realization. Therefore I support the publication of this paper in Nature Communications provided the above comments can be addressed.

Response to Referees
We would like to thank all reviewers for the careful reading of our manuscript and for the many insightful comments and observations. In the following we present our detailed replies to all the points that have been raised. Please notice the blue color indicates changes of the manuscript. To all reviewers: (1) Briefly after the submission of our paper we got aware of some papers uploaded to arXiv after our paper was submitted. Some of those have already been identified by you and we include them and comment on those case by case at the individual comments to the reviewers. Among them was a paper on Co/W(110) (https:// arxiv.org/abs/1701.05062). This has been added now to our paper: We included the following sentence in the Discussion section: llllllllllllllllllllllllIIll Other members of this class are certainly those with (110) oriented interfaces between 3d and 5d transition--metals. In fact, an anisotropic DMI was recently measured in a thin epitaxial Au/Co film on W(110) \cite{Camosi:arXiv}, but this system favored elliptical skyrmions instead of anti--skyrmions. Other candidates are 3d metals on semiconductor (100) and (110) surfaces, e.g., Fe on Ge or GaAs, exhibiting C2v and Cs symmetry, respectively. (2) During the March Meeting of the American Physical Society and the German Physical Society (the largest March meeting in Europe) we noticed that the topic of antiskyrmions comes up, but there is much confusion in the field of antiskyrmions. To clarify this, we introduce a proper distinction and classification of different antiskyrmion carrying material systems: the isotropic rank--three--and rank--two--DMI materials, the anisotropic rank--two--DMI materials and the rank--one materials systems that have antiskyrmions of different properties. We introduce in the Discussion section: It serves also as classification scheme of chiral magnets into isotropic rank--three--DMI bulk and rank--two--DMI film magnets, with a DMI described by a single constant, the spiralization, for which antiskyrmions are stable only for bulk crystals with certain point group symmetries. Then, we have the anisotropic rank--two--DMI film magnets, were skyrmions and antiskyrmions can coexist, and the sign of $\det \matt{D}$ determines which of the two has the lower energy. Zero determinant determines then rank--one--DMI material, for which skyrmions and antiskyrmions have the same energy. (3) In the third--last paragraph of the section before the start of the section: Results, we introduced the terms Bloch--type and Néel type skyrmions clearer: We speak of right--(left--) handed Bloch--type skyrmions if $\vc{c}_\chi(\hat{\vc{R}}_{ij})\cdot \hat{\vc{R}}_{ij} >0\, (<0)$, and of N\'eel-type skyrmions if $\vc{c}_\chi(\hat{\vc{R}}_{ij})\cdot \hat{\vc{R}}^\perp_{ij} >0\, (<0)$, where $\hat{\vc{R}}^\perp $ is an arbitrary vector of the positive quadrant of the orthogonal complement of $\hat{\vc{R}}$,} \eg formed by the vectors $\hat{\vc{e}}_z\times \hat{\vc{R}}$ and $\hat{\vc{e}}_z$.
(4) Re--reading our manuscript we found a few minor spelling errors that we changed without further notice in the manuscript (cannot only --> can not only; possesses --> possess; offers --> offer; whereas --> where; on --> of). (5) In the section: Discussion, paragraph 2, we corrected a mistake in the following mathematical expression from BJ ≥ D^2 to BJ > D^2. (6) We have moved the following two sentences: The handedness of the skyrmion texture is determined by the tensorial components of $\matt{D}$. Right--(left--) handed N\'eel--type skyrmions typical for interfaces are obtained if $\sign\left(\mathcal{D}_{21} --\mathcal{D}_{12}\right) = 1\ (--1)$ provided $\mathcal{D}_{21} \ne \mathcal{D}_{12}$. from the second paragraph of the section Discussion to the first one in order to improve the flow of reading. This is not further indicated in the text. (7) Re--reading our manuscript, we realized that it can be improved by additional words or we found a few more elegant formulations: • … is typically of minor importance … • changes to become --> is replaced by • The index v is also referred to as… • Note $v$ does not depend on --> Being independent of • skymions and antiskyrmions are also distinct in their handedness properties • monochiral skyrmions of positive $\mathbb{S}^1$ winding number • which, when applied to these high--symmetry surfaces,… • …, as for instance … --> Examples are … Reviewer #1: This paper does not seem appropriate for Nature Comm., which seeks to publish works of general interest. The paper may be accessible only to a small community of people which are well familiar with the particular topic of skyrmion patterns in magnetic media. The text is quite verbose, but it is not written in a form that would be friendly to the potentially broad readership. Reply: We agree with the referee that a Nature paper should address subjects as well as presentations in such a way that it is accessible to broad readership as much as possible. But in Nature Communications we should be also balance with the mission statement of this journal which states: Papers published by Nature Communication represent important advances of significance to specialists within each field. We do not quite agree with the statement that the paper is only accessible to a small community. Currently the field of magnetic skyrmions is a field which experiences a strong expansion and spread of interest, in which quite different communities meet: Mathematics, magnetism, electronic properties, transport, dynamics, Terahertz radiation, materials science, devices, information technology, etc. E.g. The National Science Foundation of Germany (DFG) has just initiated a national priority program on Skyrmionics to support this research, and in America, DARPA is launching a support program under the name texitronics. We would like to underpin our statements above mentioning that at the March meeting of the American Physical Society (APS) we witnessed 2 sessions fully dedicated to Skyrmions and 34 sessions that contained talks about skyrmions, the March Meeting of the German Physical Society (the largest in Europe) had 8 sessions dedicated to skyrmions and 29 sessions which contained talks about skyrmions, and the Web of Science search of citations (left figure) and papers published (right figure) discussing magnetic skyrmions (black) and skyrmions in all fields of science (red) shown here as function of publication year, shows clearly a rapid interest in this field. It is definitely not a small group of people, instead it is part of the topological revolution in condensed matter physics. Reviewer #1: In particular, while the central topic is the use of the Dzialoshinskii --Moriya interaction, it is not even explicitly defined in the paper, assuming that it is known to everyone, which cannot be correct. Reply: Well, we are not sure we can agree on this statement. We introduce the Dzyaloshinskii--Moriya interaction (DMI) in equations (1) to (3) in terms of the spin-lattice model as well as in the micromagnetic model in simple cubic as well as the most general form. We also briefly mention its origin: The DMI results from the spin--orbit interaction and is only non--zero for solids lacking bulk or structure inversion symmetry. We have introduced the interface induced Dzyaloshinskii--Moriya in the following paper: Bode, Heide, von Bergmann, Heinze, Bihlmayer, Kubetzka, Pietzsch, Blügel, Wiesendanger, Nature 447, 190 (2007). Since then, 10 years have passed. We, as well as many other people have calculated the strength of parameters, have explained the quantities on the basis of analytical and micromagnetic models. Therefore, we decided not to repeat this, instead having a steep entrance into the field, focusing on the novelty relative to the past. Reviewer #1: Further, while the paper is a fully theoretical one, and its stated objective is to demonstrate that antiskyrmions may realize a physically relevant stable state, the text only briefly mentions that the analysis is carried out by means of a variational method, but no details of the analysis -- which is, obviously, the core part of the work -are displayed. Instead, readers are suggested to look for "mathematical details" in some supplementary material, which is not a part of the paper. As a result, the entire text seems as an extended abstract (a lengthy one, which would, nevertheless, be quite obscure to non--specialists), rather than a systematically written article. Reply: The Discussion and Methods sections aim to provide a concise and widely understandable account on the mathematical ideas involved. To keep the paper well accessible to the broad readership of Nature Communications and (hence) well citable we have decided to include the mathematical details in the supplement. Everybody has access to these supplementary materials. We are very happy and proud of the mathematical analysis, but equally to the atomistic spindynamics and the first--principles calculations. This interdisciplinary approach of mathematics, computations and simulations make--up the value of our paper and therefore we have chosen a two--level approach. At first, we present in the main text the overall introduction, the results and a discussion of those to the reader, but provide on the second level more technical information so that all readers can reproduce our results. Reviewer #1: It may be that the prediction of the stability of the antiskyrmion, announced in the paper as its main result, is an essential prediction, that may stimulate new experiments (magnetic media, in which the authors expect a possibility to realize the prediction, are briefly mentioned in the paper). However, if the paper is intended for the publication in a broad--interest journal, such as Nature Comm., the text should be made accessible to the broad readership, and it should definitely be made self-contained, providing explicit definitions of the basic concepts in the narrow topic under the consideration, rather than assuming that they are known to everyone (which may sometimes be acceptable in papers submitted to a specialized journal). Reply: Summarizing above arguments, we think the paper is self--contained. It provides all definitions through equations (1) … (3) and together with the supplementary materials, which we consider an important part of the paper, the reader has all necessary information to reproduce the results. Reviewer #1: It may be recommended to rewrite the paper in the self--consistent form, including the core proof of the antiskyrmion's stability, rather than concealing it in supplementary files, and resubmit the work to one of leading solid--state--physics journals, such as Phys. Rev. B or J. Phys. Cond. Matt. Reply: Since we are convinced that one particular value of the paper lies in the combination of different disciplines, and since we are convinced that this multi-disciplinary approach enables the best advancement of science in this field, we refrain from chopping the paper into pieces and publish the results in parallel in different journals.