Title: Josephson Vortex Loops in Nanostructured Josephson Junctions

Manuscript N: NCOMMS-13-09832

Authors: G. R. Berdiyorov et al.

Title: Josephson vortex loops in nanostructured Josephson junctions

Dear Editor, We thank you for your prompt handling of our manuscript and for finding clearly adequate reviewers. We also thank the reviewers for their positive opinions about our work and for their very useful comments and suggestions. We took the time to carefully consider all the remarks, and perform some additional calculations to accommodate those remarks. On the following pages we give the point-by-point response to reviewers’ recommendations, and we list the changes made in the new version of the manuscript (and Supplementary Material). One of the main concerns of both referees is the effect of the superconducting pillars (which connect the two parallel layers of the Josephson junction) on the current distribution in the system. As we have shown below, our calculation includes this effect - the pillars indeed affect the current distribution in the system, but this does not affect the main findings in our paper. The fact that applied current is concentrated inside the pillars brings more interesting physics in itself, and we now describe an additional possibility to nucleate loops from within the pillar, not only by curling the Josephson vortex around the pillar. This connects to the other important question of the Referees, about the mechanism of the formation of the vortex loops in the system. We now directly show that in our original concept loops are formed solely due to energetic interplay of the elongation energy of the Josephson vortex and the barrier formed by the pillars. We also made this finding accessible to wider audience using simple arguments. We further describe the new concept, where loops are induced by the high current applied only through pillars, but we render this scenario difficult to realize in experiment. Besides the need for selectively injected current across the sample, superconductivity inside the pillars will be suppressed by the needed high current for loop formation, and dissipation/heating will become an issue in this case. Having thus completed this study from several angles, and improved the presentation/appeal of the results, we expect that our revisions are satisfactory and that the manuscript will be accepted for publication in Nature Communications. Sincerely yours, Authors

Responses to Recommendation and Comments of the First Referee

Recommendation

This paper nicely describes interesting phenomena caused spontaneously by the interplay between topology and dynamics of the system to be observable in superconducting Josephson junctions, which are analogous to the phenomena widely seen in the area of electromagnetism, plasma, liquid crystals and classical as well as quantum fluids in nature. Although in the latter areas good examples have been found, no direct experimental evidence has been observed in the field of superconductivity to date. In this sense, this paper brings a novel idea to the field of superconductivity and made a breakthrough in the conventional vortex physics by theoretical model calculations. Their calculated results seem to be sufficiently realistic except for one point (see below), not simply imaginary calculations, and the phenomena predicted also match the physical intuitions generally acceptable for the experts in the field of superconductivity, although the detailed calculation processes cannot be followed. Judging from these, this paper certainly contributes to the development of the community and, therefore, deserves publishing in one of the prestigious journals such as Nature or Nature Communications or even other Nature related journals. Our response

We thank the referee for his/her positive opinion about our work and for the recommendation.

Comment 1

However, I have some difficulty to understand one point, the model they used in their calculations, although the concept may be all right. The point is that when the external current is fed into the Josephson junction as shown in Fig. 1, where the current flows exactly? Since the pillars are assumed to be made of the same superconducting material (Nb) as given in the figure caption of Fig. 1, it seems that the current can flow only in the pillars and does not flow uniformly in the Josephson junction. Is it possible to make a uniform current flow with such superconducting short cutting pillars in the junctions?

Our response

Indeed, the current distribution in the system changes considerably when the pillars are introduced. This is shown in Figure 1 below where we present the vector plots of the supercurrent density for different states. When the system is the Meissner state, most of the current flows through the pillars, as anticipated by the referee (see panel 1 of Fig. 1). Our 3D calculations also account for the inhomogeneous current distribution in the presence of Josephson vortices. This is visible in panel 2 of Fig. 1, where we superimposed the vector plot of the current distribution onto the contourplot of the phase of the order parameter (for facilitated view of vortex locations). Panels 3 and 4 in Fig. 1 show the current density distribution for the other possible arrangements of the fluxons, including the vortex loop state (see panel 4, corresponding to the inset in the main panel). Thus, we always consider full 3D distribution of the current across the Josephson junctions, and the effect of that distribution is accounted for in our results. We made this discussion available to readers as supplemental material to the new version of the manuscript. Figure 1. Main panel: An example of the voltage vs. time characteristics of the sample (of size L=100ξ, w=100ξ, d=5ξ and δ = 1ξ) for in- plane magnetic ﬁeld H = 0.01Hc2 and applied current j = 0.052j0. The size of the pillars is R = 9ξ. Inset shows the isosurface plot of the 2 Cooper-pair density at time 4 (taken isovalue is 30% of |ψ| max , i.e. dark blue color outlines the vortex cores). The shaded z-y plane in the inset is chosen to plot the current distribution and the phase of the order parameter in panels 1-4. Panels 1-4: Supercurrent distribution inside the sample at different times (indicated in the V(t) curve). In panels 2-4 the supercurrent distribution is superimposed onto the contourplot of the phase of the order parameter. Comment 2

It may be possible to generate Josephson vortex loops even only by the super current flowing in the pillars, not introducing by the external magnetic field. Please make the physical picture more transparent on this point.

Our response

To accommodate this comment of the referee, we conducted new simulations where current was applied only through the pillars [i.e., contacts are attached only on the pillars, not entire plane of the junction (see panel 1b in Fig. 2)]. The current is then focused through the pillars (see panel 1a in Fig. 2), with consequently suppressed superconductivity there (panel 1b in Fig. 2). The increased current density will eventually lead to the formation of vortex loops around each pillar, as shown in panel 2 of Fig. 2. With time, these loops expand under the action of the Lorentz force (panels 3 and 4) until they reach each other and/or sample boundaries (panel 5). Subsequently, vortex loops around the pillars recombine into vortex loops between the pillars (panel 6), with different chirality, and more importantly, with nothing to prevent their shrinkage and collapse under further action of the Lorentz force (panel 7). In summary, vortex loops can be created by the supercurrent flowing only through the pillars, and this is a completely different scenario from our original proposal. However, for that one would need spatially nanoengineered current leads, very large applied current density (in the shown example, 5 times larger compared to the original proposal), and carefully chosen parameters of the pillars so that superconductivity there is not destroyed by the current (otherwise, there is nothing remaining to stabilize loops if current is switched off, plus thermal quench is very likely). We therefore render this scenario for loop formation very experimentally challenging, but we do mention it in the manuscript and provide Fig. 2 as supplemental material to the new version of the manuscript. Figure 2. Main panel: Example of V(t) curve of the sample (size L=100ξ, w=100ξ, d=5ξ and δ = 1ξ) for applied current j = 0.28j 0 mainly through the pillars, in the absence of applied magnetic field. The size of the pillars is R = 9ξ. Panel 1a: Supercurrent distribution inside the sample at t=24t0. Panel 1b: Contourplot of the Cooper-pair density inside the sample at t=24t 0. Panels 2-7: Isosurface plots of the 2 Cooper-pair density at times indicated on the V(t) curve (taken isovalue is 30% of |ψ| max , i.e. dark blue color outlines the vortex cores). Comment 3

I found some errors in the text.

1. In page 2 in the right column, the 12th lines from the top, ..."in a local minimum in the voltage curve (point 6)" ... should be ..."in a local maximum in the voltage curve" ...

2. In the list of the references, There are several mistakes in cited pages, which made me very time consumed to find out the correct ones. Please check them more carefully. References [2], [6], [7], [8], [10] and [11] should have page numbers 118, 1083, 139, 62, 27, and 28, respectively.

3. Furthermore, the citations of the Journal are not correct in ref. [5] and [11], which must be Plasma Phys. Control. Fusion 41B 167 (1999) and Phys. Today 49(12) 28 (1996).

Our response

We thank the referee for his/her carefully reading of our manuscript. These mistakes/misprints are corrected in the revised version of the manuscript. Comments 4

4. Finally, I would recommend to improve the quality of the movies given to visualize the formation processes of the formation of the Josephson vortex loops, cutting them and recombination of them, etc. Better quality will make better impression to the readers, although the essential physics may not be changed.

Our response

As referee is probably aware, one has to find a balance between the resolution of the animations and their memory size. The same issue holds for having better quality by recording the data in every simulation grid point and time step. Unfortunately, every meaningful improvement we attempted resulted in files of 400 MB each, which we believe is much too large for an online database. However, as stated in the related text, higher quality animations will be made available to readers upon individual request. Comments 4

I hope that the authors consider these points listed above and revise the manuscript to remove inadequacies for the nature publication.

Our response

We thank the referee once again for his/her recommendation towards publication of our article in Nature Communications, and his/her suggestions clearly intended to improve the quality of the manuscript. We hope that the made revisions are satisfactory. Responses to Recommendation and Comments of the Second Referee

Recommendation

The generation of vortex loops in superconducting systems is the main theme of this manuscript. This work follows previous activity in the field, and the main claim is the possibility to generate a Josephson vortex loop. This prediction is based on well-established time-dependent Ginzburg-Landau (GL) equations, which are applied to a specific Josephson junction, where superconducting nano-structured pillars are embedded and serve as local barriers for the in-plane motion of Josephson vortices.

Theoretical approach is quite standard, but results are potentially interesting and significant.

Our response

We thank the referee for the interest in our work, and we appreciate very much his/her remark on the significance of our results. Comment 1:

Outcomes need, in my opinion, to be properly explained and supported by additional material to really meet the criteria of robustness, broad interest and impact appropriate for the standards of Nature Communications.

Our response

We have improved the manuscript following the comments and the suggestions of the referee (please also see our correspondence with the first referee). Additional material is added in the manuscript to accommodate this remark of the referee, and we give below more detailed responses to all referee’s comments. Comment 2:

The dynamics of the Josephson vortex loop is well discussed, and the pictures and the animations clearly illustrate the ongoing processes. Nevertheless the physical origin of this dynamics is not adequately described. The effects are presented, but there is no clear correlation with the causes and with the underlying simulations. The Authors should more explicitly explain how all ingredients work to build the desired effect, thus suggesting differences and analogies with other systems. These revisions are quite crucial and represent the only way to prove real novelty and conceptual advance of the work.

These explanations would be of some help to clarify a couple of substantial points, which need to be understood.

Our response

We appreciate the remark of the referee, and we attempted to make the mechanism of the loop formation more understandable for the reader. Namely, one can consider two different situations when a vortex line approaches a pillar under a strong drive: 1) the vortex elongates and bends around the pillar (see Fig.3a), or 2) the vortex crosses the pillar barrier (Fig.3b) and travels through the pillar with minimal consequent elongation. In the first case, a loop has to be formed so that vortex can detach from the pillar, while for the second case the vortex passes the barrier without loop formation. Which mechanism wins is simply determined by needed energy.

Let E1 be the Josephson vortex energy per unit length and E2 the energy of a vortex per unit length inside the bulk superconductor. Then the energy needed for loop formation is about 2RE1 while the needed energy for vortex crossing through the pillar is close to 2RE2. From comparison of these energies, one concludes that the loop can be formed if E1 < E2. To demonstrate this directly, we conducted additional simulations where we manipulated E2 by changing the critical temperature of the pillars. In the Ginzburg-Landau formalism, the variation of the critical temperature can be conveniently simulated through varied coefficient α(T)= α0(1-T/Tc) in the system free energy functional. Tc- inhomogeneity is then included as α(r)=α(T)f(r), where the spatially dependent parameter f(r) was taken to be f=1 in the superconducting layers and f<1 inside the pillars. Figure 4 below shows the voltage response of the sample with one pillar for two different values of the inhomogeneity coefficient f. For an isotropic sample (i.e., f=1 everywhere), the pillar creates large enough barrier to create a vortex loop as the vortex is driven across the sample (see panels 1-4). Next, we decrease the barrier by reducing the critical temperature of the pillar [see the inset in Fig. 4, where T/T c2=1-f(1-T/Tc1)]. Dashed red curve in Fig. 4 shows the voltage curve of the sample for f=0.25. In this case no loop is formed around the pillar and vortex penetrates the pillar (see panels 7 and 8), with a weak peak in the voltage found when the vortex detaches from the pillar (point 8). In summary, the main mechanism for the formation of the loops in the here considered scenario is the interplay of energy needed for the deformation of the vortex line and the energy barrier to penetrate the pillars. This discussion is now included in the revised version of the manuscript. Figure 4.Main panel: Voltage vs. time curves of the sample (of size L=50ξ, w=50ξ, d=5ξ and δ = 1ξ) for the inhomogeneity coefficient f=1

(solid black curve) and f=0.25 (dashed red curve). The applied current density is j = 0.047j 0 and in-plane magnetic ﬁeld is H = 0.02Hc2. The size of the pillars is R = 8ξ. Panels 1-9: Isosurface plots of the Cooper-pair density at times indicated in the V(t) curves (taken 2 isovalue is 30% of |ψ| max , i.e. dark blue color outlines the vortex cores). Comment 3:

The pillars serve as local barriers for the in-plane motion of Josephson vortices, but in my understanding they would also favor the passage of the current from one side of the junction to the other. How would this affect the general scenario?

Our response

The referee is right to point out this issue. Indeed, the pillars affect the current flow across the junction with most of the current flowing through the pillars (please see Fig. 1 in the above response to other Referee). Nevertheless, such a non-uniform distribution of the current does not affect our conclusions, unless the applied current is very large. In the latter case, Josephson vortex loops are formed around the pillars first and then expand and recombine between the pillars (see the discussions of Fig. 2 above). This is an alternative scenario for loop formation, which we now included in the manuscript, with suitable comments about its feasibility in experiment.

Comment 4:

Do the Authors have a knob to prove unambiguously that a possible observation of the desired effect is uniquely due the presence of a Josephson vortex loop? From this point of view I would suggest the Authors to consider the work on annular junctions, for instance Fistul et al. PRL 91,257004(2003); PRB 68, 132509 (2003); Franz et al. JAP 89, 471 (2001). In other words, it is important to understand if the proposed benchmarks for the observation of this novel kind of vortex matter could be mimicked by other effects.

Our response:

We thank the referee for taking our attention to these experimental works devoted to the dynamics of Josephson vortices. These papers are cited in the present version of the manuscript, as indeed different possible effects must be considered in the interpretation of the experimental data to be able to deduce behavior of Josephson loops from Josephson vortices. However, detailed analysis of our numerical results shows that the reported effects in annular Josephson junctions are not present in our system and the observed features in the voltage response of the system are solely due to formation of loop structures. To increase the chances for the confirmation of our prediction, in the present version of manuscript we propose one more experiment for detection of the vortex loops. This proposal is based on the voltage response of the system to the focused laser light. Namely, if the laser acts away from the pillars (and pinned loops) the voltage response will be minimal; on the other hand, the impact of the laser near the pillars results in the annihilation of the loops in the hotspots, accompanied by a clear peak in the voltage signal. This is the essence of laser scanning microscopy, and we believe Josephson loops can be detected in this way. We have tested this proposal on the sample with same parameters as in Figure 1. The impact of the laser is simulated as local increase of the temperature (hot spot). Figure 5 shows the voltage response of the sample when the laser acts between the pillars (solid-black curve) and on top of the pillars (dashed-red curve). In the former case, only a little increase of the voltage signal is observed after the laser impact (point 1) due to the formation of a hot spot (panel 1). The saturation of the voltage signal is observed at later times (point 2 and panel 2). However, when the laser acts near the loops (in this particular case on top of the pillar, see panel 3), the annihilation of the vortex loop is observed (panel 4), which results in a pronounced voltage peak (points 4 and 5) followed by relaxation (panel 6 and point 6). We believe that such response can not be mimicked by the other effects reported in the above references. Figure 5.Main panel: Voltage vs. time curve of the sample (of size L=100ξ, w=100ξ, d=5ξ and δ = 1ξ) when the laser impact occurs between the pillars (solid black curve) and on top of a pillar (dashed red curve). The applied current density is j = 0.047j0 at zero applied magnetic field. The size of the pillars is R = 9ξ. Panels 1-6: 2 Isosurface plots of the Cooper-pair density at times indicated in the V(t) curve (taken isovalue is 30% of |ψ| max , i.e. dark blue color outlines the vortex cores). Dashed yellow circles indicate the position of the laser impact. Comment 5:

There are also minor points to be fixed. For instance in Fig. 2, what's the difference between thicker blue line rings and light blue lines?

Our response:

We thank the referee for his/her careful reading our manuscript. Topologically, there is no difference between thin rings around the pillars and wider lines across the samples. They both represent Josephson vortices (loops). The difference is that the loops around the pillars are more adjoined to the pillars due to the current, and become thinner in appearance in our isoplots. These loops become visually thicker once the current is switched off (see panel 9 in Fig. 2 of the manuscript).

Comment 6:

There are also minor points to be fixed ... Why in a) the two rings on the right are marked by a blue light?

Our response:

In all 3D isosurface plots the light blue regions represent pillars, whereas light blue regions surrounded by darker blue rings are pillars with vortex loops around them. This has been made clearer in the figure caption in the new version of the manuscript.

Recommendation:

I think that this manuscript needs a substantial revision to be possibly re-considered for publication in Nature Communications.

Our response:

We thank the referee for his/her useful comments and suggestions. We made an effort to accommodate the remarks and we hope that our revision is satisfactory.

Brief list of made changes in the manuscript

 The mechanism of loop formation has been discussed in more detail in the Results section, and has been further substantiated in the Supplementary Material – where we directly demonstrate the effect of barrier strength on the formation of the loops, by changing the critical temperature of the pillars in the simulations.  In the main text (Results section), we point out that our simulations are three-dimensional, and include the effects of non-uniform current distribution in the sample due to the presence of pillars inside the junction. We show a figure of such current distribution (with corresponding discussion) in the Supplementary Material.  We further discuss (in the Results section) that non-uniform current distribution can lead to nucleation of loops from within the pillar (due to very large current there), even in the absence of the applied magnetic field. We show the corresponding data in the Supplementary Material, since it is very informative and useful for understanding of the system. In the main text, we devote a whole paragraph to this phenomenon, but we state that it does not affect the original proposal for the mechanism of loop formation, and that it is more difficult to be realized experimentally (and why).  In the introductory section of the article, end of Results section related to experimental verification, and at the end of the Discussion section, we describe the new found possibility to directly detect Josephson loops by laser scanning microscopy. Particularly in Discussion section, we added an extensive paragraph and a new figure (Fig. 6) devoted to the distinct response of the loops to the focused laser beam, and how this translates to the aforementioned scanning technique.  References 24-27 were added, to point out relevant dynamical effects that should not be mistaken with the appearance of the Josephson loops. Several misprints in the references were corrected, and few more informative references were added with respect to loop nucleation in superconductors under inhomogeneous field, and attraction between Josephson and pancake vortices in layered superconductors.  The overall readability of the paper has been improved, introduction was expanded and better connected, several statements have been made clearer throughout the text. More information has been added to the Methods section (while still keeping it concise). Reference 44 has been added with respect to used method when varying Tc of the pillars.