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Gapful electrons in a vortex core in granular superconductors
by Dmitry E. Kiselov, Mikhail A. Skvortsov, Mikhail V. Feigel'man
This Submission thread is now published as
|Authors (as registered SciPost users):||Mikhail Feigel'man · Dmitry Kiselov|
|Preprint Link:||https://arxiv.org/abs/2212.01862v2 (pdf)|
|Date submitted:||2023-02-14 20:37|
|Submitted by:||Kiselov, Dmitry|
|Submitted to:||SciPost Physics|
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 is the main ingredient needed to understand strong suppression of microwave dissipation recently observed in a mixed state of granular Al.
Published as SciPost Phys. 15, 008 (2023)
Author comments upon resubmission
Dear Editor and Referees,
please find resubmitted version of our manuscript “Gapful electrons in a vortex core in granular superconductors”. The manuscript is considerably modified according to the reports of the referees 2 and 3 (referee 1 did not asked for any changes and supported publication in the original form). Detailed list of changes made is located in the end of the present file.
Below you find point-to-point answers to the points raised by the referees.
We hope the resubmitted version of the manuscript solves all the issues raised and will be accepted for publication. Please note that the subject of high-inductance and low-dissipation superconducting materials currently attracts high interest from many groups around the World. Please also note that two (out of all three) referees supported univocally the publication of our manuscript in SciPost – Physics.
Sincerely yours, D. Kiselov, M. Feigel’man and M. Skvortsov
Answers to the issues raised by the referees:
Referee 2 comments:
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. We added, to avoid any further misunderstanding, an extensive discussion concerning the conditions for the use of the mean-field (self-consistent) approach. This discussion is mainly located in the Section V, Discussions.
2. 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.. We reformulated the corresponding piece of the text, to avoid any further misunderstanding.
3. With these assumptions, the answer has all chances to be numerically incorrect and even misleading.
As shown above, this statement has no ground.
4. 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. We added detailed discussion of this issue into the Section II and Section V.
5. 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.
6. Delta under assumptions always dissapears at one of the granula
This is an explicit incorrect statement; vortex center is located between the grains, of course. We added additional text to explain this point once again; it is located in the end of the 2nd paragraph in the left column of page 2.
- Periodic arrangement. How sensitive is the answer to the lattice type/random arrangement
We added detailed discussion of this issue to the Sec. V. In short, randomness makes our results even more stable.
8. The "core" of superconducting phases does not have to be in the same position as the "core" of \chi phases.
This incorrect statement was already commented above. The phases \chi_i(\epsilon) and \phi_i coinside due to the tunnel nature of junctions between grains, as we reiterated again in the paragraph of Sec.II that is situated above Eq.(5).
Referee 3 comments:
To replace “vortex size” by “coherence length” -
Done, in 2 locations.
To tone down the claim about explaining low dissipation in granular Al films -
Done, in the Abstract
List of changes
1. a. Abstract: stylistic changes aimed to explain actual outcome of the paper
more precisely were made.
b. The term “vortex size” was replaced by “coherence length”
2. First section: comment was added with the general explanation on the
position of the vortex in the end of the 2nd paragraph in the left column
on page 2.
3. Second section:
a. comment was included on usability of the model employed in real 3D material - in the end of the 1st paragraph in the right column of page2.
b. The paragraph (the one just before Eq.(5)) about approximations used for the phase variables was rewritten. Also, some explanation was added in the previous paragraph about validity of the saddle-point approximation.
4. Third section: minor rework was done to clarify the numerical approach
employed. Also, in the caption to Fig.2 the term “vortex size” was
replaced by “coherence length”
5. Section V: an additional discussion was provided about the role of disorder
in the inter-grain couplings. Major outcome of this qualitative consideration
is as follows: the region of validity of our main result becomes wider due to
inhomogeniety of parameters in real granular Al films. This conclusion is
supported by the analysis of experimental data from Ref..
6. Section VI: comment on another effect that can contribute to the suppression
of the microwave losses was added.
7. Bibliography: A citation  of the recent experiment on optical response,
superconductivity and resistance in granular Al is added.
Submission & Refereeing History
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- Report 4 submitted on 2023-04-03 13:20 by Anonymous
- Report 3 submitted on 2023-02-06 01:04 by Anonymous
- Report 2 submitted on 2023-01-23 18:45 by Anonymous
- Report 1 submitted on 2023-01-16 18:11 by Anonymous