Re: LF12287
Title: Confinement effects on the flux patterns of the intermediate state in mesoscopic type-I superconductors
Authors: G.R. Berdiyorov, A.D. Hernandez, and F.M. Peeters
Dear Editor,
Thank you for your prompt handling of our manuscript. We revised the manuscript according to the referees’ comments and suggestions.
Please find below our response to the referees’ reports and the list of the changes made in the revised version of the manuscript.
Sincerely yours,
Authors
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Report of Referee A -- LF12287/Berdiyorov
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Berdiyorov et al. report in their manuscript calculations of the flux structures in the intermediate state of mesoscopic type-I superconductors performed in the framework of the Ginzburg-Landau theory. The authors find that by progressively increasing the magnetic field these structures change first from the
Meissner state to a phase of single-quantized vortices, then to a giant (multiquanta) vortex state and finally to a laminar structure. At higher fields surface superconductivity is also observed prior the transition to the normal state. The transition points between flux structures depend on the geometry of the sample (the authors study both squared and spherical geometries) and the expected hysteretic behavior in type-I superconductors with decreasing field is also investigated. In my opinion, these results seem sound and represent a novelty in the current state of the art of a topic (mesoscopic superconductivity) that, in the last years, has been of interest for the readers of PRL. Moreover, the manuscript is well structured, clearly written and accessible to a broad audience. Thus, I fully recommend its publication in PRL.
We thank the referee for his/her positive assessment of both scientific and technical aspects of our manuscript.
My only criticism to the manuscript is that the authors strongly state in the introduction that `up to now, the ground state flux structures in mesoscopic type-I superconductors have not been investigated'. However,
Misko et al. have previously investigated the flux patterns in infinitely height and long-finite height (much larger than the superconducting coherence length) type-I superconducting prisms with triangular mesoscopic cross section [PRL 90, 147003 (2003)]. In my opinion, these structures are also mesoscopic, although their size only competes with the coherence length in two of the dimensions of the space. The samples studied by Berdiyorov et al. are `fully' mesoscopic, but I consider that a brief discussion and comparison of this previous work with the present one will benefit the readers of PRL and avoid misunderstandings.
We thank the referee for his/her drawing our attention to this reference. Indeed this work is devoted to the study of the stability of singly quantized vortex-antivortex structures in a long type-I superconductor with triangular cross section due to the competition between vortex-antivortex repulsion in type-I superconductors and confinement effects. However, they mostly focused on the low magnetic field regime, where only two vortices (or three vortices and one anti-vortex) are present in the system. The motivation of the paper by Misko et al. is on the vortex-antivortex molecule state and not on the multi-vortex states in type-I superconductors. To our knowledge, a systematic study of the evolution of the intermediate state flux structures has not been conducted yet for mesoscopic type-I superconductors.

In the revised version of the manuscript the above reference is included into the text (Ref. [17]) and our statement regarding ground state flux structures in mesoscopic type-I superconductors is changed accordingly.
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Report of Referee B -- LF12287/Berdiyorov
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This manuscript touches upon a very interesting and timely topic of the intermediate state (IS) in type-I superconductors. Believed to be solved for good for almost a century, new developments show that the situation is not as simple. While I think that the obtained results are of importance and deserve a rapid publication in PRL, the interpretation should be revised.
We thank the referee for his/her positive opinion about our results.
In particular, it seems that the Authors are confused with regard of what is happening in bulk type-I superconductors. Tubular structure represents the equilibrium only at low magnetic fields. At higher fields another phase, termed "suprafroth" (Ref.1) takes over. This phase represents closed topology, but it is neither tubular, nor laminar. Therefore, the italic-emphasized exclamation in page 2 is quite irrelevant. The real question is whether the intermediate state is represented by the closed or open topology of the flux distribution (this can be quantified in a pure mathematical sense). In that regard, the Authors "rings" belong to the same closed-topology class, not to the laminar structure. In fact what they later call "laminar" maybe except for Fig.4(4), can be called "tubular". Moreover, I doubt that laminar topology can be even identified in a mesoscopic sample. (The Authors call it "closed laminar structure" that does not make much sense).
Still, the above remarks are mostly related to semantics rather than to the results of the calculations. The results are really remarkable in a sense that a full 3D problem has been solved and the effects of size confinement for realistic parameters were identified.
We thank the referee for these nice comments and suggestions. We revised the discussion of our numerical results taking into account referee’s comments. In particular, we avoided the use of “closed laminar structure” and used the terminology “laminar-like structures” instead. We hope that this better describes the domain structures in the mesoscopic regime.
Corresponding modification in the description of the intermediate state of bulk type-I superconductors are done in the revised text.
I therefore recommend publishing this paper in Physical Review Letters, but urge the Authors to rephrase the interpretation from tubular-vs-laminar to closed-vs-closed topology of the intermediate state.
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Report of Referee C -- LF12287/Berdiyorov
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The coexistence between the magnetic field and superconducitivty in type I superconductors is an interesting subject of broad interest. The authors obtain patterns of coexistence in case of mesoscopic superconductors. They apply a finite-difference method to study mesoscopic type-I superconductors using the Ginzburg-Landau theory. They take \kappa = 0.4 and do three-dimensional calculation for a 4\xi by 6-
20 \xi rectangular prism and for a 12.4 \xi radius sphere. For a rectangular prism the increase of the field makes the following states appear: single vortices, single + giant vortices, giant vortices + laminar and laminar (see Fig. 1 and 2). They proceed to study a mesoscopic sphere (Fig. 3) of radius 12.4\xi and obtain, by increasing the field, the following sequence of patterns: single vortices/giant vortices/laminar structures/stripes/ring/surface superconducitivity.

The authors must address the following aspects of the paper:
(1) The superconduting sphere has a nice Meissner solution and the total field is like that of a dipole outside.
Have the authors retrived this behavior form their calculations? They should comment on the demagnetization factors for the rectangular prism and sphere.
Indeed, due to the nonzero demagnetization factors, the magnetic field at the surface of the sample is different from the uniform applied field H . In the case of the sphere, the distribution of magnetic field outside the sample resembles the magnetic field profile of a dipole, as noticed by the referee.
We have included into the revised version of the manuscript the magnetic field distribution in the case of a superconducting sphere (please, see inset 7 in Fig. 4 of the manucript). Due to the lack of space we could not discuss in detail the magnetic field distributions and demagnetization factors of our samples.
(2) The authors must argue how dependent on radius (sphere) is their sequence of states. I believe that beyond some critical regime their sequence of states no longer holds. According to the common lore of type I superconductors very large spheres go straight from the Meissner phase direct to the collapse of the superconducting state.
Since the 3D calculations are very time and memory consuming, we did not calculate the entire phase diagram for the sphere, i.e. dependence of the states on the radius of the sphere. However, our preliminary results show that the phase diagram does not change qualitatively from the one shown in Fig. 1 of the manuscript for the considered range of radius: R/

=3.5 – 12.4.
As for the results for very large spheres, there is no direct transition from the Meissner state to the normal state with increasing magnetic field. We know from the textbooks (see, e.g., M. Tinkham, “Introduction to superconductivity”) that macroscopic spheres are in the intermediate state in the range of field 2/3<H/H c
<1 before the system transits to the normal state. Our calculations also show that for the thermodynamic ground state no direct superconducting/normal transition is found with increasing radius of the sample (see the figure below).
Figure 1. Intermediate state flux structures in a superconducting cube with size a=40

(left) and in a sphere of radius R=24.8

(right) at the magnetic field H=0.7H
c
.

(3) There is no description of the field outside the mesoscopic superconductor. What is the meaning of the average field <H>. For type I significant corrections to M must be missing here.
In the revised version of the paper we have included the magnetic field distribution outside the superconducting sphere. Due to the page limits we restricted ourselves only for one value of the applied magnetic field.
Indeed, the magnetization curve should take into account the demagnetization factor of the sample. In our simulations the magnetization is calculated as the flux expelled from the sample M=(<H>-H)/4π, where the magnetic field is averaged <H> over the superconducting volume V. This definition also gives a good extimate for the response of the superconductor to the external magnetic field.
The definition of the magnetization is explained in the revised version of the manuscript.
(4) Does the 256^3 lattice refer to points inside the superconductor? Do the authors observe the field become constant at some distance away from the mesoscopic superconductor - `infinity`?
The box with dimension 256x256x256 is the total simulation region, were we solved the second GL equation for the 3D distribution of the magnetic field. This simulation area for the vector potential is chosen such that the magnetic field becomes constant away from the superconductor. In our calculations we tried to keep the size of the simulation area for the vector potential at least 4 times as large as the area of the superconductor. We found that this ratio is good enough to have a convergence of the magnetic field away of the superconductor (please see the figure below).
Figure 2. Radial distribution of the magnetic field in the equatorial plane of a superconducting sphere of radius R=12.4

for the size of the simulation area R s
=3R (black curve) and R s
=6R (yellow curve).
(5) Please, explain what is `N', or improve the `N domains` notation.
Corrections are made in the revised text.
(6) Can the authors provide an argument that mesoscopic supercondutors, like bulk superconductors, split into two kinds, type I and II?
There is no clear distinction between type-I and type-II superconductivity when we are in the mesoscopic scale. The reason is that edge effects interfere strongly and modify the properties of the vortex state considerably.

The paper's content is new, interesting and worth to be published.
We thank the referee once again for his/her very useful comments and suggestions.
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List of the changes:
1. The title of the manuscript is changed from “Confinement effects on the flux patterns of the intermediate state in mesoscopic type-I superconductors” to “Confinement effects on intermediate state flux patterns in mesoscopic type-I superconductors”.
2. Abstract: The second sentence of the abstract is removed in the current version of the manuscript. The third sentence is changed to “In addition to well-established tubular and laminar structures, the strong confinement leads to the formation of (i) a phase of singly quantized vortices , which is typical for type-II superconductors and (ii) a ring of a normal domain at equilibrium.”
3. The second paragraph of the introduction part of the manuscript is rewritten taking into account comments of the second referee.
4. Paragraph 3: Sentences 3-5 are included into the revised text to take into account the first referee’s comments. A new reference, Ref. [17], is included.
5. Ref. [20] and related text is removed due to the page limits. However, the information can be found in
Ref. [19].
6. Section “Equilibrium flux structures”, paragraph 1: Sentences 8-10 and last sentences is changed according to the second referee’s comments.
7. Section “Effect of sample geometry”, paragraph 3: In sentence 6 the correction to the definition of the magnetization is given.
8. Fig. 4: Inset 7 is included to show the magnetic field distribution in and around the sphere.
9. Section “Effect of sample geometry”, paragraph 4: Sentence 5 is included to indicate the magnetic field distribution in/around the sphere.