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Skyrmion-vortex hybrid and spin wave solutions in superconducting ferromagnets
by Shantonu Mukherjee, Amitabha Lahiri
Submission summary
| Authors (as registered SciPost users): | Amitabha Lahiri · Shantonu Mukherjee |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2407.00405v3 (pdf) |
| Date submitted: | May 8, 2025, 8:06 a.m. |
| Submitted by: | Mukherjee, Shantonu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The coexistence of Ferromagnetism and superconductivity in so called ferromagnetic superconductors is an intriguing phenomenon which may lead to novel physical effects as well as applications. Here in this work we have explored the interplay of topological excitations, namely vortices and skyrmions, in ferromagnetic superconductors using a field theoretic description of such systems. In particular, numerical solutions for the continuous spin field compatible to a given vortex profile are determined in absence and presence of a Dzyaloshinskii-Moriya interaction (DMI) term. The solutions show that the spin configuration is like a skyrmion but intertwined with the vortex structure -- the radius of the the skyrmion-like solution depends on the penetration depth and also the polarity of the skyrmion depends on the sign of the winding number. Thus our solution describes a novel topological structure -- namely a skyrmion-vortex composite. We have also determined the spin wave solutions in such systems in presence and absence of a vortex. In absence of vortex frequency and wave vector satisfy a cubic equation which leads to various interesting features. In particular, we have shown that in the low frequency regime the minimum in dispersion relation shifts from $k=0$ to a non zero $k$ value depending on the parameters. We also discuss the nature of spin wave dispersion in the $\omega \sim \Tilde{m}$ regime which shows a similar pattern in the dispersion curve. The group velocity of the spin wave would change it's sign across such a minimum which is unique to FMSC. Also, the spin wave modes around the local minimum looks like roton mode in superfluid and hence called a magnetic roton. In presence of a vortex, the spin wave amplitude is shown to vary spatially such that the profile looks like that of a N\'eel Skyrmion. Possible experimental signature of both solutions are also discussed.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We would like to thank SciPost Physics authority for considering our manuscript for review. We also thank the anonymous referee for a very thorough and thoughtful review. We have considered each points raised by the anonymous referee carefully and modified our previous version of the manuscript accordingly. With this letter we are enclosing the revised manuscript. We hope this revised manuscript will convince the concerned authority and it will be considered for publication with no further delay.
Yours Sincerely
Shantonu Mukherjee
Amitabha Lahiri
List of changes
Details of the changes are given in the reply to reviewers comments. Please have a look.
Current status:
Reports on this Submission
Strengths
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The authors discuss a timely topic, since there has been interest recently on such composite excitations due to their potential role in creating Majorana excitations and implementing quantum computing.
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The authors investigate an interesting problem concerning superconducting electrons which are under the influence of the exchange field generated by localized moments. Such a study may find experimental relevance for rare-earth superconductors and reveals how magnetic skyrmions can be engineered in superconducting vortices.
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One other novelty of the authors’ approach is that they explored the magnon dispersion in ferromagnetic superconductors with Dzyaloshinskii-Moriya interaction.
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The technical approach appears correct.
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The results are reasonable and interesting.
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Overall this manuscript reads well.
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This work can inspire further studies in this field and motivate revisiting the possibility of Majorana zero modes in rare-earth superconductors. Some early ideas when it comes to Majorana modes in such systems were first discussed by Martin and Morpurgo in a PRB of 2012. There, the Majorana zero modes formed flat bands. Here, instead, the authors propose a route to realize localized Majorana modes, which can be advantageous for quantum computing applications.
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This version is substantially improved compared to the previous one.
Weaknesses
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There are still a few spots where the presentation needs to be improved. I raise these issues later on.
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The authors have added the minimization of the energy in order to find the vorticity value which becomes stabilized for the superconducting vortex of the composite skyrmion-vortex excitation. However, the authors minimize the energy by not properly accounting for the integer character of the vorticity. A suitable amendment is required here.
Report
Requested changes
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Throughout the text, the authors use the term ferromagnetic superconductors. In contrast, in the title, they use the term superconducting ferromagnets. I suggest that the authors use "ferromagnetic superconductors" also in the title.
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I think that there is a typo in the second affiliation.
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In the previous version, I expressed my reservation concerning using the word "novel" in the text. In the recent version, one finds the following sentence in the abstract: "Thus our solution describes a novel topological structure- namely a Skyrmion-vortex composite." The skyrmion-vortex excitation has been already proposed. Therefore, the authors should also rectify the above sentence. What I would consider novel, is that the authors discussed such a composite excitations in bulk ferromagnetic superconductors.
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In the abstract the authors write: "We also discuss the nature of spin wave dispersion in the ω ∼ \tilde{m} regime which shows a similar pattern in the dispersion curve. " The parameter \tilde{m} is not defined anywhere. Therefore, I suggest the authors define it already in the abstract or provide an alternative form for the above expression.
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I think it is useful to mention the work by Martin and Morpurgo (Phys. Rev. B 85, 144505 (2012)) as the first to discuss the possible emergence of Majorana zero modes in ferromagnetic superconductors. This can help improve the motivation of this work since they discuss similar systems. In fact, the proposal of the authors has the advantage that such composite excitations can potentially harbor Majorana zero modes which are localized, in contrast to the flat bands discussed by Martin and Morpurgo, and this can be more advantageous for quantum computing applications.
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In page 4, the authors introduce the skyrmion polarity, after the definition of reference 34. However, in the latter work, the authors discussed that the polarity can take two values, i.e., +1 and -1. Here, the authors include also the zero value. But that would imply that the magnetic profile is not a skyrmion. I think it would be best to exclude the zero value for clarity.
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In page 4 one finds the sentence "fixed the amplitude M0 of the magnetization". I think this contradicts that later on the amplitude of the magnetization is \mu. The authors can keep the latter symbol throughout.
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The photon mass is introduced after equation 13. However, it is not clearly spelled out but instead it appears in a relation. It is better to clearly define it.
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Between equations 23 and 26, the authors mention that a potential term is required to be introduced to stabilize magnetic skyrmions in the usual case. I think that the way that this discussion is inserted in the text breaks the flow. The authors can maybe transfer this part as a brief appendix that they can refer to when discussing that in the present case the field is sufficient to stabilize magnetic skyrmions in contrast to the more standard case.
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Below equation 25, the authors write "Ginzberg-Landau". I think it is better to write "Ginzburg-Landau" as it appears elsewhere in the text.
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It would help the reader if the authors explicitly mention which equation they use to obtain equation 31.
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Below equation 31, the authors write "As the spin configuration is assumed to be independent of the z direction,". I think it is better to replace "z direction" by "z coordinate".
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It would be helpful for the reader if the authors explicitly define the integrals of equation 33.
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An important issue that has to be resolved concerns the result in equation 40. The vorticity N is an integer and cannot be given by the result of 40, unless the r.h.s. is also an integer. Similar to the Little-Parks effect, the vorticity takes the integer value which is the closest to the r.h.s. of equation 40. The authors, instead of differentiating equation 36, should simply "complete the square".
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Throughout this work, the use of \hat{n} or \bold{n} for the unit vector is not consistent. The authors should make their notation uniform.
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In the discuss between equations 48-50, the authors mention that they employ the Green's function approach to find the magnetic field. This is of course correct but, personally, I find that this discussion unnecessarily complicates the presentation. The authors could simply say that they employ a Fourier transform and obtain the magnetic field.
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Equation 58 is solved by keeping terms which are quadratic in frequency in the r.h.s.. Why not doing the same for equation 54?
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Above equation 60, one finds the sentence: "Abrikosov vortex along the Z axis". I guess "Z axis" should be "z axis".
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In page 17 one finds the phrase: " The small width of the sample". I think it is better to write " The small thickness of the sample".
Recommendation
Ask for minor revision