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Gapful electrons in a vortex core in granular superconductors
by Dmitry E. Kiselov, Mikhail A. Skvortsov, Mikhail V. Feigel'man
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Submission summary
| Authors (as registered SciPost users): | Mikhail Feigel'man · Dmitry Kiselov |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.01862v1 (pdf) |
| Date submitted: | 2022-12-07 21:45 |
| Submitted by: | Kiselov, Dmitry |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We calculate the quasiparticle density of states (DoS) inside the vortex core in a granular superconductor, generalizing the classical solution applicable for dirty superconductors. A discrete version of the Usadel equation for a vortex is derived and solved numerically for a broad range of parameters. Electron DoS is found to be gapful when the vortex size $\xi$ becomes comparable to the distance between neighboring grains $l$. Minigap magnitude $E_g$ grows from zero at $\xi \approx 1.4 l$ to third of superconducting gap $\Delta_0 $ at $\xi \approx 0.5 l$. The absence of low-energy excitations explains strong suppression of microwave dissipation in a mixed state of granular Al.
Current status:
Reports on this Submission
Report #4 by Anonymous (Referee 1) on 2023-4-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.01862v1, delivered 2023-04-03, doi: 10.21468/SciPost.Report.6991
Report
I have carefully read the manuscript and found that it contains the very interesting physical idea. It is aesthetically appealing and provides a new insight to the possible explanation of the recent experimental results on microwave spectra in granular superconducting films, thus I suggest this paper for a publication in SciPost and second to Referee 3.
I have also read previous referees' comments and authors replies and believe that majority of these comments are properly answered and when needed are incorporated into the revised 2nd version of the paper.
My minor suggestion (which is optional) is to add a small Appendix where authors may show the form of action, when the latter is expressed solely in terms of variational parameters (a coherence length, an energy dependent phase of the order parameter, etc.), because this action was in fact used to (approximately) solve the Usadel equations (2-3) together with the self-consistent equation (4) for the order parameter.
Report #3 by Anonymous (Referee 2) on 2023-2-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.01862v1, delivered 2023-02-06, doi: 10.21468/SciPost.Report.6681
Strengths
1. Clear formulation of the model
2. Reliable methods for the evaluation of the density of states within the formulated model
Weaknesses
1. The use of the term vortex size in the Abstract, together with notation \xi for it may confuse a potential reader. Perhaps it would be better to use superconducting coherence length in the granular medium as a parameter (\xi).
2. Given the limitations on the model parameters set forth by Eq. (12), it is better to tone down the claim about explaining low dissipation in granular Al films. In addition to modifying the density of states, granularity may affect the vortex motion casually mentioned in the manuscript.
Report
This is a publishable manuscript. Addressing the above concerns would not hurt.
Requested changes
Please see the Weaknesses
Report #2 by Anonymous (Referee 3) on 2023-1-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.01862v1, delivered 2023-01-23, doi: 10.21468/SciPost.Report.6595
Strengths
1. Fine model ( though introduced without proper references)
2. Plasuible and interesting answer
3. A numerical approach is adeqate.
4. The model permits an adequate numerical approach
Weaknesses
1. An ad-hoc assumption upon the self-consistent order parameter.
2. An ad-hoc assumtion upon the Green function phases.
3. With these assumptions, the answer has all chances to be numerically incorrect and even misleading.
4. The applicability of the model to mutli-layer film is questionable.
5. The discussion in Ch. 5 is rather confusing on my taste.
Report
Generally, it is an interesting formulation of a research project that, regretably, has not been accomplished yet. It is my impression that the authors did not have either time or computer resource to provide an extensive numerical analysis beyond the ad-hoc assumptions they put forward. I sincerely wish they will find both and provide a better version of the manuscript.
Requested changes
Obviously, the authors should do all the calculations without imposing the ad-hoc assumptions mentioned. Technically, this is straightforward, though may be time-consuming. Let me list the features of the assumptions that I find especially dangerous with respect to obtaining correct and sensible answer. a. Delta under assumptions always dissapears at one of the granula. Does not have to be the case: the vertex core between the granulas is obviously energetically more favourable. b. Periodic arrangement. How sensitive is the answer to the lattice type/random arrangement c. The "core" of superconducting phases does not have to be in the same position as the "core" of \chi phases. There may be an energy gain associated with such separation.
Author: Mikhail Feigel'man on 2023-01-24 [id 3267]
(in reply to Report 2 on 2023-01-23)
Response to the part "Weaknesses" 1. An ad-hoc assumption upon the self-consistent order parameter.
Self-consistent order parameter is the basic theory concept in low-temperature superconductivity. It can be questioned only in very special cases like proximity to some quantum phase transition. Nothing like that is considered in our paper by construction of the model and due to physical content considered
An ad-hoc assumtion upon the Green function phases.
There are no "ad-hoc assumptions" here. The fact that phases of the Green function and of the order parameter coincide is proven for the systems with tunnel junctions; in particular detailed way it is done in the paper we cite as Ref.[16].
With these assumptions, the answer has all chances to be numerically incorrect and even misleading.
As shown above this statement has no ground.
The applicability of the model to mutli-layer film is questionable.
We consider this situation of magnetic field transverse to the film plane, thus the order parameter variations in the field direction are absent. That makes possible to use 2D model we employed for thick films as well.
The discussion in Ch. 5 is rather confusing on my taste.
This statement reflects personal feeling of its author but does not give us any possibility to understand what is wrong or incorrect or misleading in the text; therefore such a comment is not useful for scientific discussion.
Response to the part Requested changes
First statement of this part is based on the statements from the part "weaknesses" which are incorrect, as shown above. Next we comment to specific statements listed as
a) Delta under assumptions always dissapears at one of the granula
Incorrect statement concerning the content of our paper; vortex center is located between the grains, of course.
We will make it more evident in resubmitted version
b) Periodic arrangement. How sensitive is the answer to the lattice type/random arrangement
The answer is twofold: 1) for weak disorder in grain sizes or positions small change in the position of spectrum edge we found is expected; it will be weak since the gap we found is large, it is some valuable fraction of the full superconducting gap; 2) for strong disorder the problem should be considered separately, and this will be the subject of separate publication.
c) The "core" of superconducting phases does not have to be in the same position as the "core" of \chi phases.
This incorrect statement was commented above regarding the part2 of "weaknesses". These phases coincide due to tunnel nature of junctions between grains, see Ref.[16].
Report #1 by Anonymous (Referee 4) on 2023-1-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.01862v1, delivered 2023-01-16, doi: 10.21468/SciPost.Report.6539
Report
This is important and timely contribution to the physics of vortex matter in superconductors. The manuscript presents the results of theoretical study of quasiparticle density of states (DoS) inside the vortex core in granular superconductors. The results generalize the classical solution applicable for dirty superconductors. By solving a discrete version of the Usadel equations the authors show that, in contrast to the clasasival soluition, the electron DoS exhibits an energy gap in the case when the vortex core size is comparable to the distance between neighboring grains . The results provide a mechanism of suppression of microwave dissipation in a mixed state of granular Al.
The manuscript is clearly written and will be of interest to a broad community of researches in the field of superconductivity.
I suggest the manuscript for publication in its present form.
Author: Mikhail Feigel'man on 2023-02-07 [id 3328]
(in reply to Report 3 on 2023-02-06)We agree with the comments made by the Referee and we will make requested small changes in the resubmitted manuscript.