# Fileset

[Supplemental_Material.pdf](https://mdr.nims.go.jp/filesets/971b1fac-143c-4419-b416-c59bfb8d226f/download)

## Creator

[S. Ooi](https://orcid.org/0000-0003-2129-0310), [M. Tachiki](https://orcid.org/0000-0002-6033-3515), [T. Mochiku](https://orcid.org/0000-0003-2208-4279), H. Ito, T. Kubo, [A. Kikuchi](https://orcid.org/0000-0002-5044-7156), [S. Arisawa](https://orcid.org/0000-0001-8155-9401), K. Umemori

## Rights

©2025 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Dynamical visualization of attractively interacting single vortices in type-II/1 superconducting Nb by magneto-optical imaging](https://mdr.nims.go.jp/datasets/d133338c-4f16-4a9f-ba57-72a42c74d225)

## Fulltext

Supplemental Material for ”Dynamical visualization ofattractively interacting single vortices in type-II/1superconducting Nb by magneto-optical imaging”S. Ooi1, M. Tachiki1, T. Mochiku1, H. Ito2, T. Kubo2,3,A. Kikuchi4, S. Arisawa4, K. Umemori2,31International Center for Materials Nanoarchitectonics, NationalInstitute for Materials Science, Sengen 1-2-1, Tsukuba 305-0047, Japan2High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba 305-0801, Japan3SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193, Japan4Research Center for Energy and Environmental Materials (GREEN), NationalInstitute for Materials Science, Sengen 1-2-1, Tsukuba 305-0047, Japan1I. MOLECULAR DYNAMICS SIMULATION ON CLUSTERING VORTICESAs a demonstration to see how the vortices form clusters in the case of Eq. (A1) and (A2),we performed a two-dimensional MD simulation with COMSOL Multiphysics®[1] based onthe force balance equations for each massless vortex∑j,j 6=i Fv−vi,j = ηvvi, where vi is thevelocity of the i-th vortex and ηv is the viscous coefficient. In the present simulations, ηvis set to 10−6 dyn·s/cm2, and the values of λL and κ are the same as in Appendix A. N isthe number of vortices confined in a circular area of 30 µm diameter. An initial state wherethe vortices are almost uniformly separated is prepared using the repulsive vortex-vortex(d) (e) (f)(a) (b) (c)FIG. S1. Vortex configurations of (a) initial and (b) final states in the simulations for N=1000.(c) Histograms of the number of vortices as a function of intervortex distance for (a) and (b).(d,e,f) Configurations and histograms for N=700. The three peaks in the histogram of the finalstates correspond to the first, second and third nearest neighbors of the vortex lattice, from left toright. The temporal evolution of the clustering can be observed in the videos ’MD1 N1000.mp4’and ’MD2 N700.mp4’ in the Supplemental Material.2interaction for the London limit of type-II superconductors. In the following, the positions rof the first and second local minima and the local maximum between them of the potentialdepicted in Fig. 6 are denoted as rmin1 , rmin2 , and rmax1 , respectively.The vortex configurations of the initial state and the final state after sufficient timeelapsed are shown in Fig. S1(a,b) and (d,e) for N=1000 and 700, respectively. The his-tograms of the number of vortices by intervortex distance are shown in Fig. S1(c) and (f).In both final states, the vortices form clusters or larger domains of hexagonal lattice with alattice constant a0 of 0.6 µm, nearly the same value as rmin1 . The size of the clusters tends tobe larger for larger N , corresponding to larger B, which is consistent with our observations.When N is sufficiently small, clusters formed by the interaction force from the firstpotential minimum do not appear. However, vortices can be aggregated by the force fromthe second potential minimum after a longer elapsed time as shown in Fig. S2, where a0 ofthe initial state is larger than rmin2 . It seems that this situation is rarely observed becausethe other forces, i.e. pinning force, driving force by the screening current, etc., exceed thesecond attractive interaction force in the actual experimental conditions.II. VIDEOS OF MO OBSERVATIONSIn the Supplemental Material, Video1 to 3 are provided for Figs. 2-4 in the main text.In addition, Video4 to 6 are also provided for reference, obtained in three other fields of(a) (b) (c)FIG. S2. Vortex configurations of (a) initial and (b) final states in the simulation for N=200.The corresponding video is ’MD3 N200.mp4’ in the Supplemental Material. (c) Histograms of thenumber of vortices by intervortex distance for the initial and final states.3-4.3, 10.7, and -14.3 Oe, respectively. The images are taken during field cooling underthe condition of an exposure time of 500 msec and no interval between successive images.The elapsed time from the first image is shown at the top left of the videos. For imageprocessing, Fiji [2] is used for background subtraction, filtering, resizing, and AVI conversion,and MATLAB [3] is used for image registration.Although we have used the eclipse PLD method to improve the quality of the MO films,the films still contain visible defects, probably precipitates or tiny droplets, which interferewith the vortex observation in the images. As can be seen, even after the image registrationand the background subtraction, the processed images still show the traces of the defects.REFERENCES[1] COMSOL Multiphysics® www.comsol.com. COMSOL AB, Stockholm, Sweden.[2] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch,C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri,P. Tomancak, and A. Cardona, Fiji: an open-source platform for biological-image analysis,Nature Methods 9, 676 (2012).[3] MATLAB version: 9.11.0 (R2021b), Natick, Massachusetts: The MathWorks Inc.; 2021.4