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[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

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[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)

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Dynamical visualization of attractively interacting single vortices in type-II/1superconducting Nb by magneto-optical imagingS. 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, Japan(Dated: March 26, 2025)In type-II/1 superconductors, the intervortex interaction has a potential minimum, resulting inan attractive force at certain distances. To better understand this unique vortex system, in-situobservations in high-purity Nb have been performed using magneto-optical imaging with single-vortex resolution. We have dynamically visualized the behavior of the vortices during cooling,including clustering of attractively interacting single vortices, the hopping motion of vortices, theformation of an Abrikosov triangular lattice, and various degrees of disordered states with or withoutclusters. From the direct observations of clustering vortices, a conversion temperature for theboundary between type-II/2 and type-II/1 states is evaluated. The diverse vortex configurationsarise from the interplay of the vortex-vortex interaction, the screening current, and the pinningpotential landscape.I. INTRODUCTIONThe Ginzburg-Landau (GL) theory, basically validnear the critical temperature Tc, classifies supercon-ductors into type-I and type-II. Beyond this classifica-tion, theories applicable well below Tc[1–6] further dis-tinguish type-II superconductors into type-II/1 and type-II/2. Type-II/1 superconductors, with a GL parameterκ < 1.1, exhibit attractive interactions of vortices at in-termediate distances, in contrast to the repulsive forcesseen in type-II/2 (normal type-II) superconductors. Theattraction is linked to a potential minimum of the vortex-vortex interaction at a length scale of several times of themagnetic penetration depth λ [4, 7]. Initial research inthe 1970s on materials such as Nb, V, TaN, and PbTl re-vealed these phenomena [8–12]. The origin of the attrac-tive interaction is qualitatively understood by the con-densation energy mechanism, i.e., the order parameteroverlap [13]. Recently, advanced experimental studies onthis issue have rekindled interest in the type-II/1 super-conductors [14–17]. While there are various supercon-ductors that have been expected to manifest attractivelyinteracting vortices, due to different mechanisms, i.e.,magnetic superconductors [18], multi-band superconduc-tors [19–21], and topological superconductors [22], type-II/1 superconductors can provide a tractable platform tostudy such vortex system with less influence of pinningsince the pinning effect would often mask the behaviororiginating from the vortex-vortex interaction.The boundary between Type-II/1 and Type-II/2 hasbeen evaluated numerically by solving the Eilenbergerequations without approximation [4] (for the 2D case,ref.[23]). The experimentally determined boundaries ofthe type-II/1 in the κ-T diagram have been quantita-tively compared with several theories in ref.[24]. Here weshow the schematic κ-T diagram in Fig. 1(a) [11, 25], andFig. 1(b) is, as an example, the numerically calculatedvortex-vortex interaction energy expected in the type-II/1 superconductivity, where the London penetrationdepth λL and κ are set to 100 nm and 0.77, respectively.The details are described in Appendix A. The first poten-tial minimum appears at ∼ 6λL. Interestingly, the inter-action potential oscillates with reducing the amplitude[26]. The repulsive force caused by the non-monotonicpotential may prevent the vortices from coalescing intoa single domain [27]. A molecular dynamics simulationusing this interaction potential is demonstrated in theSummplement Material [28] (see also references [29–31]therein).High-purity Nb of a representative type-II/1 supercon-ductor undergoes a transition from type-II/2 to type-II/1behavior at the conversion temperature T ∗ during cool-ing. The superconducting properties of high-purity Nbsamples have been studied by magnetization measure-ments [8, 9, 32]. Without any demagnetization effect(demagnetization factor D = 0), a magnetization jumpin Hc1 appears in the M -H curves, indicating a first-order transition due to the attractive interaction (the or-ange line in Fig. 1(c)). For D ̸= 0 (Fig. 1(d)), insteadof the first-order transition, the intermediate mixed state(IMS) appears between Meissner and mixed states belowT ∗. While real-space observations of the IMS have beenrealized by the Bitter decoration technique [12, 33] , thedynamical behavior of the vortices, e.g. the emergenceprocess of the IMS, has not been revealed due to the lackof real-time observations.From an application perspective, the behavior of vor-tices in pure Nb is closely related to the performance ofthe superconducting radio frequency (SRF) cavity, sinceremanent vortices that cannot be expelled from the cav-2MeissnerMixed stateT (K)Hc2H (Oe)IMST (K) Tc0 T (K)B (G)𝜅t=T/Tc𝜅 =c 1/ 2Type IType II/1Type II/201.00.5 0.7 0.9 1.10.5Hc1T* T*Tc0MeissnerMixed stateIMSMixed stateTc0T*Bc2D=0 D≠0T-scanD≠01st min(a) (b)(c) (d) (e)2.52.01.51.00.50.0Ev-v  (erg/cm)x10-6 2.01.51.00.50.0r (μm)-1.0-0.50.00.51.0Ev-v  (erg/cm)x10-8 2.01.51.00.50.0r (μm)FIG. 1. (a) Schematic t-κ diagram in the vicinity of κ = 1/√2based on previous papers [11, 25]. Type-II/1 superconduc-tivity exists between the type-I and type-II/2 (type-II). (b)Calculated vortex-vortex interaction energy as a function ofintervortex distance. The inset is an enlarged plot of the zeroenergy region. A potential minimum, indicated by the arrow,exists at an intermediate distance. (c,d) SchematicH-T phasediagrams of type-II/1 superconductors with demagnetizationfactors D = 0 and D ̸= 0 for the type-II side κ > 1/√2. T ∗is the conversion temperature which separates type-II/2 andtype-II/1 behavior. (e) B-T phase diagrams for the type-IIside of type-II/1 superconductors.ity during cooling degrade the Q-factor [34–36]. In thiscontext, understanding the mechanism of flux expulsion,where vortex clusters may be relevant, is of significantimportance to improve the Q-factor.To study the dynamic behavior of vortices, magneto-optical imaging (MOI) is a powerful technique that allowsreal-time real-space observation of magnetic field distri-butions in superconductors [37]. Individual vortices havealready been resolved by MOI in several groups [38–40]and even manipulated with the help of a laser beam [40].Recently, we have visualized the formation of IMS dur-ing field cooling in cavity grade Nb using the MOI [16].However, in that study, the behavior of vortices in lowerfields could not be investigated because the resolution ofthe MOI at that time was not sufficient to resolve indi-vidual vortices.In this paper, we present the dynamical visualizationof attractively interacting single vortices in high-puritycavity-grade Nb using a MOI with single vortex reso-lution. In our Nb samples, the conversion temperatureT ∗ was found to be around 8.5 K. In addition, variouspatterns of vortex configurations emerged depending ontemperature and field due to the interplay of multiplefactors, i.e., the vortex-vortex interaction, the Lorentzforce from a screening current, and the disordered pin-ning potential landscape.II. EXPERIMENTSSingle-domain Nb samples with a square-cuboid shapeof 7×7×3 mm3 were cut from high-purity large-grainplates (Tokyo Denkai Co.) originally prepared for SRFcavities. The samples were annealed at 900◦C for 3h in a vacuum of ∼10−4 Pa to dehydrogenate, afterone side of the square surfaces was mirror-polished bychemical-mechanical polishing. The residual resistanceratio (RRR) of the plates is 496 according to the datasheet provided by the company. We expected the valueof κ of our Nb samples to be in the range of 0.73–0.81[16].The MO observations were performed using a conven-tional polarizing microscope setup. A schematic drawingof our low-temperature MO imaging setup is shown inFig. 1 of ref.[16]. In the present study, a sCMOS cam-era (pco.panda 4.2, Excelitas PCO GmbH) was used forimage acquisition, and a high-power green LED chip wasused as a light source to enhance illumination. Exter-nal magnetic fields Hz and Hx were applied perpendic-ular and parallel to the prepared surface, respectively,where Hx was used to roughly compensate for the in-plane ambient field. Regarding the sign of Hz (or Bz),the direction from bottom to top of the paper is definedas positive. The angle of rotation of the analyzer wasslightly shifted from the crossed Nicols configuration by1–1.5◦ for better contrast [38]. Due to this angular off-set, negative/positive Bz appear darker/brighter in theMO image. Therefore, positive vortex and negative anti-vortex can be distinguished.For the present study, we developed our ownMO imaging sensors, which are garnet films of(Lu,Bi)3(Fe,Ga)5O12. The films were fabricated on agadolinium gallium garnet (GGG) substrate by an eclipsePLD method[41, 42], in which a shadow mask was placedbetween the target pellet and the substrate to prevent thedeposition of large particles such as droplets. The MOfilm used in this study shows a zigzag pattern of mag-netic domain walls at low temperatures, indicating thepresence of in-plane magnetization domains. The filmthickness is ∼500 nm estimated from the total deposi-tion time.All images were taken during field cooling (T -scan).Since the MOI signal reflects the local flux density Bl,it is convenient to use a B-T plot to follow the T -scan(Fig. 1(e)), where a typical change in Bl is indicated bythe arrow [16]. A sudden decrease of Bl occurs at Tcdue to the flux expulsion. To enhance the contrast ofthe images, the background image, which is an averageof several dozen images taken above Tc, was subtractedfrom all successive images after image registration to re-3(a) 9.02 K 6.75 K(b)��mFIG. 2. (a) MO image of single vortices at 9.0 K below Tc dur-ing T -scan in -9.3 Oe. Since the MO film senses the magneticfield a few submicrons above the sample surface and there areother factors such as image processing, contrast adjustment,etc., the size of the vortices appears larger. (b) MO image ofclustering vortices below the conversion temperature T ∗. Thevideo is available in the Supplemental Material.move non-negligible drift or vibration, and application ofGaussian filter to reduce noise.III. RESULTS AND DISCUSSIONFigure 2 shows MO images taken during the T -scan in-9.3 Oe. Note that the temperature shown in the imageshas an error about ±0.05 K due to temperature insta-bility in our experimental setup. Just below Tc around9.2 K many uniform dark spots appear (see ‘Video1’in the Supplemental Material), which is the creation of(anti-)vortices. By counting the number of vortices inFig. 2(a), B is estimated to be -5.1 G. As the temper-ature decreases, slight drifts of the vortices from left toright can be seen down to 8.7 K, indicating the influenceof the Lorentz force exerted by the screening current. Inaddition, a hopping motion of the vortices can be seen,suggesting that pinning by some amount of quenched dis-order exists at least at the surface, although the sample issufficiently clean for IMS to appear at low temperatures.With decreasing temperature further, the vortices be-gin to aggregate at ∼8.5 K, then form vortex clusters.The clusters gradually grow by absorbing other vorticesaround them. As shown in Fig. 2(b), not all vortices areinvolved in cluster formation. This is probably causedby the competition of the attractive force between vor-tices and the pinning force. Since the MO image showsthe vortex pattern on the surface, there is a possibilitythat some of the isolated single vortices are pinned at thesurface and may cluster in the body of the sample if theeffective pinning centers exist only near the surface.Recently, the expectation that ideal pure Nb is anintrinsic type-I superconductor has been proposed, inwhich the similarity of magnetic field patterns of the IMSin Nb and the intermediate state in Pb was thought to bea supporting evidence [43]. Since the spatial resolution(a) (b)(c) (d)(e)Intensity (a.u.)50position (μm)(f)Intensity (a.u.)1050position (μm)(f)8.95 K8.78 K 8.32 K8.11 K 7.96 K8.91 K��mABCFIG. 3. MO images taken in -2.3 Oe during T -scan. (a-c)Sequential images of a hopping vortex (in the white box).Black, red, and blue lines in the inset of (c) show line profilesof the vortex in (a), (b), and (c), respectively, along the redarrow in (b). (d-f) Clustering of three vortices (in the yellowbox). The line profiles along the arrows drawn in the figuresare plotted in the inset of (f). The video is available in theSupplemental Material.of our MOI is insufficient to reveal the internal structureof the cluster, we cannot exclude the possibility that the“vortex cluster” here is a “giant vortex”. However, recentsmall-angle neutron scattering studies using Nb samplesof different purities with RRRs of ∼100 (“Nb-lp”), > 300(“Nb-mp”), and > 10k (“Nb-hp”) have observed Braggspots of hexagonal lattice of single vortices below thetemperature of the aggregation in all cases. The purityof our Nb samples (RRR∼500) is probably between Nb-mp and Nb-hp, and the transition temperature to IMS,TIMS, is also between Nb-mp and Nb-hp in fixed mag-netic fields [15, 16], suggesting that the quality of currentsamples does not reach ideal pure Nb. Therefore, the ob-served aggregate structure of vortices is expected to bea “vortex cluster”. Further direct observation studies for4-1000-800-600-400-2000ΔI (arb. unit)9.08.07.06.0T (K)(b)(d)(g)(h) (e)(f)(c)5�m(a) (b)(g)(e)(d)(c)(f) (h)1�m-1FIG. 4. Vortex arrangement during the T -scan in a relatively higher field. (a) Change in averaged light intensity ∆I fromthe value above Tc during T -scan in 15.7 Oe. Labels (b)-(h) indicate the points corresponding to the following images. (b-h)Successive changes of the vortex arrangements: (b) an image just below Tc (9.2 K), (c) ordered vortex lattice (9.1 K), (d)weakly disordered state (9.1 K), (e) strongly disordered state (8.7 K), and (f,g,h) intermediate mixed state (8.4, 7.5, and 6.1 K,respectively). The insets of (c) and (d) show the results of the FFT for each image. All images are displayed with a fixed grayscale, except for the regions surrounded by yellow squares in (b), (c) and (d), where the contrasts are increased for visibility.The video is available in the Supplemental Material.much higher purity Nb with a better spatial resolutionthan several hundreds nm are required to explore thetype-I superconducting Nb.Snapshots during the T -scan in a lower field (-2.3 Oe)are shown in Fig. 3. There is an overall flow towards 2o’clock just below Tc, where the hopping motion of thevortices can be seen again in Fig. 3(a-c). The two darkspots in the white box of Fig. 3(b) are a trace of a hoppingsingle vortex, indicating an instantaneous jump duringthe exposure time of 0.5 sec, whose profiles are shown inthe inset of Fig. 3(c). The ratio of the intensities of twonegative peaks reflects that of the dwell times at the twopositions.In Fig. 3, most of the vortices remain isolated singlevortices even well below 8.5 K, probably because the vor-tices are too far apart to attract each other against thepinning force. However, in the yellow box in Fig. 3(d-f),we can find three vortices, labeled A, B, and C, formingclusters: the vortex A collides with the second vortexB, and then the cluster consisting of A and B becomesattached with C. Thier profiles are shown in the insetof Fig. 3(f). The clusters appear to form and grow bycollisions of vortices during hopping.Figure 4 shows the changes of the vortex arrangementduring the T -scan in a relatively higher field (+15.7 Oe),where single vortices are still resolved. Note that thevortices are represented by bright spots, unlike those inFig. 2 and 3 because their fields are positive. Just af-ter the superconducting transition, a certain amount ofmagnetic field is expelled from the sample, as shown inFig. 4(a), but there is no sign of vortices in Fig. 4(b),probably because the field modulation by vortices issmall, since λ(T ) should be longer, much closer to Tc.In the next image (Fig. 4(c)), despite a small decreasein temperature, we notice the appearance of an orderedvortex lattice, which is a hexagonal lattice with a latticeconstant of ∼ 1.5µm, corresponding to 11 G, from the re-sult of the fast Fourier transformation (FFT). However,this ordered lattice is broken within small temperaturedecrease (Fig. 4(d)). This change is probably due to theinterplay of the weak intervortex interaction, the pinningeffect and the Lorentz force due to the screening cur-rent. Since the vortex lattice is too soft near Bc2(T ) orin very low fields [44, 45] to maintain the ordering, somevortices move downward due to the Lorentz force whileothers appear to be fixed. With decreasing temperaturethe arrangement is further disordered (Fig. 4(e)). Then,below ∼8.5 K, the vortices suddenly start to form clus-ters as shown in Fig. 4(f), indicating that the attractiveforce between the vortices overcomes the pinning forcearound this temperature. As the temperature continuesto decrease, the clusters gradually absorb the individ-5-20020H (Oe)9.59.08.58.0T (K)Tc2Tcluster TLatticeFIG. 5. Plot of several charactaristic temperatures deter-mined from the observations in T -scan. Red circles, Tcluster,show the temperature at which vortices start to aggregate.Crossed markers, Tlattice, indicate the point where the vor-tex lattice were detected. Superconducting transition tem-perature Tc2 is also shown by black squares with a line ofHc2(T ) = 4040(1− t2)/(1+ t2), where t = T/Tc and Tc is 9.25K [46].ual vortices around them (Fig. 4(g)). At low tempera-tures (Fig. 4(h)), some vortices remain isolated from theclusters, suggesting existence of relatively strong pinningcenters in this sample.From the consective images in T -scan, we extractedsome characteristic temperatures, i.e., the starting tem-perature of the clustering Tcluster, temperatures wherethe vortex lattice is observed Tlattice, and the supercon-ducting transition temperature Tc2, which are illustratedin Fig. 5. Tlattice gradually shifts toward lower temper-atures as the magnetic field increases, because the vor-tex lattice becomes stiffer as the inter-vortex distancedecreases, maintaining the ordered structure at lowertemperatures against the temperature dependent pinningforce. On the other hand, Tcluster seems to be indepen-dent of the magnetic field and concentrate around 8.5 K,indicating that the conversion temperature T ∗ is aroundthis value, although they are somewhat scattered dueto the ambiguity of the manual determination and thelimited experimental stability of the temperature. It ispossible that the influence of the pinning remained in ourNb sample has led to an underestimation of T ∗. However,it is expected that even under the influence of pinning,clustering could occasionally occur as a result of collisionbetween weakly pinned mobile vortices when the attrac-tive force starts to work. Since there is no clear case ofclustering vortices above ∼8.5 K for the different fields,we expect T ∗ to be around 8.5 K in this sample.The conversion temperature is an important parameterto investigate the superconducting property of Nb includ-ing the quality of samples via the relationship betweenT ∗ and κ. In previous experiments T ∗ has been evalu-ated by examining the magnetization jump accompaniedby the first vortex penetration in theM -H curves in sam-ple shapes of D ∼0 [8] or by detecting slope changes bythe differential magnetization using large-D samples [24].However, these macroscopic methods may give an errorin lower fields and higher temperatures due to the re-duced number of vortices involved. Direct observationby MOI can be an alternative means to determine T ∗,especially for low fields as demonstrated in the presentstudy, and applicable to plate-like shapes.IV. CONCLUSIONIn conclusion, we have successfully conducted the dy-namical observation of attractively interacting single vor-tices in low fields (< 20Oe) in high-purity cavity-gradeNb using a MOI with single vortex resolution. In ourNb sample, the conversion temperature T ∗ was foundto be about 8.5 K. During field cooling, in addition tothe hopping motion of single vortices, the diverse vor-tex configurations - such as hexagonal lattice, weakly orstrongly disordered state, and mixture of isolated vor-tices and clusters - were observed depending on temper-ature and field, which can be explained by the combinedinfluence of multiple factors, i.e., the vortex-vortex inter-action, the Lorentz force from the screening current, andthe disordered landscape of the pinning potential. As afuture prospect, further improvement of the spatial, tem-poral, and magnetic resolutions in the MOI will make itpossible to dynamically study the internal structure ofvortex clusters, vortex lattice structures in higher fields,or the birth of vortices just at Tc.ACKNOWLEDGEMENTSThe authors would like to thank T. Konomi,S. Yoshizawa, E. Kako, H. Sakai, and K. Tsuchiya,K. Hirata for their close collaboration and advice. Thisstudy was supported by JSPS KAKENHI Grant Num-ber 23K03329 and partially by JSPS KAKENHI GrantNumber 21K04145.DATA AVAILABILITY STATEMENTThe data that support the findings of this article areopenly available [47].62nd min1st max1st min0th peak (a) (b) (c)3.02.52.01.51.00.50.0Ev-v  (erg/cm)x10-6 -2 -1 0 1 2r (μm)-1.0-0.50.00.51.0Ev-v  (erg/cm)x10-8 2.01.51.00.50.0r (μm)-1.0-0.50.00.51.0Ev-v  (erg/cm)x10-11 2.01.51.00.50.0r (μm)FIG. 6. Calculated potential energy of the vortex-vortex-interaction: (a) Central peak (’0th peak’). (b) Around the firstminimum. (c) Around the first maximum and the second minimum.Appendix A: Attractive vortex-vortex interactionThe attractive vortex-vortex interaction is not de-scribed by the conventional GL theory. Several ap-proaches have been studied to go beyond the GL theoryto investigate the vortices in a type-II/1 superconduc-tor [1, 3–5]. On the other hand, the boson method isthe quantum field theory suitable to treat the topologicaldefects in an ordered state (e.g. vortices in superconduc-tors) [48, 49], and the theory of superconductivity hasbeen formulated based on this formalism [2, 50]. Herewe use the vortex-vortex interaction formulae by the bo-son method of superconductivity to study the attractivevortex behavior and vortex cluster formation [2, 26, 51].Vortex-vortex interaction potential energy and force(per unit length) are expressed as follows, respectively[2, 51]:Ev−v(r) =ϕ028π2∫ ∞0kc (k) J0 (kr)λ2Lk2 + c (k)dk, (A1)|F v−v(r)| = ϕ028π2∫ ∞0k2c (k) J1 (kr)λ2Lk2 + c (k)dk, (A2)where r is the distance between the axes of the two vor-tices, ϕ0 is the unit flux quantum, λL is the London pene-tration depth, J0, J1 are the zeroth and first order Besselfunctions, respectively. c(k) is the so-called boson charac-teristic function. c(k) at T = 0K has been calculated nu-merically as follows [26]: c (k) = exp {−ν [k ξ0]η} whereξ0 is the coherence length, ν = 0.559 − 0.4257V N(0),η = 2.207− 0.7857V N(0). V is the coupling constant ofthe electron-electron interaction. N(0) is the density ofstate at the Fermi level. We adopted 0.32 for V N(0) tofollow the reference [26].The numerically calculated vortex-vortex interactionenergy as a function of the distance between two vorticeswith the values λL of 100 nm and κ of 0.77 is shown inFig. 6. Ev−v(r) is an oscilatory damped function [26],which can be seen up to the second minimum in Fig.6(c).Boson methods start from the BCS Hamiltonian, butinstead of formulating using Green’s function in the man-ner of the Gor’kov equations, they advance through theoperator formalism to be derived. Although they shouldbe consistent with other theoretical frameworks throughthe BCS Hamiltonian, this remains a topic for future re-search.REFERENCES[1] G. Eilenberger and H. Büttner, The structure of singlevortices in type II superconductors, Z. Phys. 224, 335(1969).[2] L. Leplae, H. Umezawa, and F. Mancini, Derivation andapplication of the boson method in superconductivity,Physics Reports 10, 151 (1974).[3] E. H. Brandt, Microscopic theory of clean type-II super-conductors in the entire field-temperature plane, physicastatus solidi (b) 77, 105 (1976).[4] U. Klein, Microscopic calculations and on the vortex andstate of and type II and superconductors, Journal ofLow Temperature Physics, Vol. 69, Nos. I, J. Low Temp.7Phys. 69, 1 (1987).[5] T. Koyama and M. Machida, Nonlocal Ginzburg-Landautheory for superconductors, Physica C: Superconductiv-ity 484, 100 (2013).[6] V. D. Neverov, A. E. Lukyanov, A. V. Krasavin, A. A.Shanenko, M. D. Croitoru, and A. Vagov, Microscopicdescription of intermediate mixed state in superconduc-tors between the first and second types, Physical ReviewB 110, 054502 (2024).[7] E. H. Brandt, The flux-line lattice in superconductors,Reports on Progress in Physics 58, 1465 (1995).[8] D. K. Finnemore, J. R. Clem, and T. F. Stromberg, At-tractive interaction between vortices in Nb, Phys. Rev.B 6, 1056 (1972).[9] J. J. Wollan, K. W. Haas, J. R. Clem, and D. K.Finnemore, Phase transition at Hc1 for superconductingNb and V, Phys. Rev. B 10, 1874 (1974).[10] J. Schelten, H. Ullmaier, and W. Schmatz, Neutrondiffraction by vortex lattlices in superconducting Nb andNb0.73Ta0.27, Phys. Status Solidi B 48, 619 (1971).[11] J. Auer and H. Ullmaier, Magnetic behavior of type-II su-perconductors with small Ginzburg-Landau parameters,Phys. Rev. B 7, 136 (1973).[12] U. Krägeloh, Flux line lattices in the intermediate state ofsuperconductors with Ginzburg Landau parameters near12, Phys. Lett. A 28, 657 (1969).[13] U. Klein, L. Kramer, W. Pesch, D. Rainer, and J. Ram-mer, Microscopic calculations of vortex structure andmagnetization curves for type II superconductors, ActaPhys. Hung. 62, 27 (1987).[14] T. Reimann, M. Schulz, D. F. R. Mildner, M. Bleuel,A. Brûlet, R. P. Harti, G. Benka, A. Bauer, P. Böni,and S. Mühlbauer, Domain formation in the type-II/1superconductor niobium: Interplay of pinning, geometry,and attractive vortex-vortex interaction, Phys. Rev. B96, 144506 (2017).[15] A. Backs, M. Schulz, V. Pipich, M. Kleinhans, P. Böni,and S. Mühlbauer, Universal behavior of the interme-diate mixed state domain formation in superconductingniobium, Phys. Rev. B 100, 064503 (2019).[16] S. Ooi, M. Tachiki, T. Konomi, T. Kubo, A. Kikuchi,S. Arisawa, H. Ito, and K. Umemori, Observation of in-termediate mixed state in high-purity cavity-grade Nbby magneto-optical imaging, Phys. Rev. B 104, 064504(2021).[17] X. S. Brems, S. Mühlbauer, and R. Cubitt, Pushing thelimits of accessible length scales via a modified porodanalysis in small-angle neutron scattering on orderedsystems, Journal of Applied Crystallography 57, 1358(2024).[18] M. Tachiki, H. Matsumoto, and H. Umezawa, Mixedstate in magnetic superconductors, Phys. Rev. B 20, 1915(1979).[19] V. Moshchalkov, M. Menghini, T. Nishio, Q. H. Chen,A. V. Silhanek, V. H. Dao, L. F. Chibotaru, N. D.Zhigadlo, and J. Karpinski, Type-1.5 superconductivity,Phys. Rev. Lett. 102, 117001 (2009).[20] S.-Z. Lin and X. Hu, Vortex states and the phase diagramof a multiple-component Ginzburg-Landau theory withcompeting repulsive and attractive vortex interactions,Phys. Rev. B 84, 214505 (2011).[21] E. Babaev, J. Carlström, J. Garaud, M. Silaev, andJ. Speight, Type-1.5 superconductivity in multiband sys-tems: Magnetic response, broken symmetries and micro-scopic theory – a brief overview, Physica C: Supercon-ductivity 479, 2 (2012).[22] M. Tachiki and H. Koizumi, Vortex state of topologi-cal superconductor CuxBi2Se3, Phys. Rev. B 91, 104505(2015).[23] P. Miranović and K. Machida, Thermodynamics andmagnetic field profiles in low-κ type-II superconductors,Phys. Rev. B 67, 092506 (2003).[24] H. W. Weber, M. Botlo, F. M. Sauerzopf, H. Wiesinger,and U. Klein, Phase transitions between type-I, type-II/1and type-II/2 superconductivity, Jpn. J. Appl. Phys. 26,917 (1987).[25] N. Kimura, N. Kabeya, K. Saitoh, K. Satoh, H. Ogi,K. Ohsaki, and H. Aoki, Type II/1 superconductivitywith extremely highhc3in noncentrosymmetric LaRhSi3,J. Phys. Soc. Jpn. 85, 024715 (2016).[26] F. Mancini, R. Teshima, and H. Umezawa, Order of phasetransition between the meissner state and the mixedstate, Solid State Commun. 24, 561 (1977).[27] X. B. Xu, H. Fangohr, Z. H. Wang, M. Gu, S. L. Liu,D. Q. Shi, and S. X. Dou, Vortex dynamics for low-κtype-II superconductors, Phys. Rev. B 84, 014515 (2011).[28] See Supplemental Material [url] for molecular dynam-ics simulations using the interaction of Eq. (A2), anddescriptions of the videos provided, which includesRefs.[29–31].[29] COMSOL Multiphysics® www.comsol.com. COMSOLAB, Stockholm, Sweden.[30] 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. Car-dona, Fiji: an open-source platform for biological-imageanalysis, Nature Methods 9, 676 (2012).[31] MATLAB version: 9.11.0 (R2021b), Natick, Mas-sachusetts: The MathWorks Inc.; 2021.[32] H. W. Weber, E. Seidl, M. Botlo, C. Laa, E. Mayerhofer,F. M. Sauerzopf, R. M. Schalk, H. P. Wiesinger, andJ. Rammer, Magnetization of low-k superconductors .1.the phase-transition at hcl, Physica C-Superconductivityand Its Applications 161, 272 (1989).[33] U. Essmann, Observation of the mixed state, Physica 55,83 (1971).[34] A. Gurevich and G. Ciovati, Effect of vortex hotspotson the radio-frequency surface resistance of superconduc-tors, Phys. Rev. B 87, 054502 (2013).[35] S. Huang, T. Kubo, and R. L. Geng, Dependence oftrapped-flux-induced surface resistance of a large-grainNb superconducting radio-frequency cavity on spatialtemperature gradient during cooldown through Tc, Phys.Rev. Accel. Beams. 19, 082001 (2016).[36] S. Posen, M. Checchin, A. C. Crawford, A. Grassellino,M. Martinello, O. S. Melnychuk, A. Romanenko, D. A.Sergatskov, and Y. Trenikhina, Efficient expulsion ofmagnetic flux in superconducting radiofrequency cavi-ties for high q0 applications, J. Appl. Phys. 119, 213903(2016).[37] A. A. Polyanskii, P. J. Lee, A. Gurevich, Z.-H. Sung, andD. C. Larbalestier, Magneto-optical study high-purityniobium for superconducting RF application, in Sympo-sium on the Superconducting Science and Technology ofIngot Niobium (AIP, 2011).[38] P. E. Goa, H. Hauglin, Å. A. F. Olsen, M. Baziljevich,and T. H. Johansen, Magneto-optical imaging setup for8single vortex observation, Rev. Sci. Instrum. 74, 141(2003).[39] Y. Tsuchiya, Y. Nakajima, and T. Tamegai, Develop-ment of surface magneto-optical imaging method, Phys.C Supercond. Its Appl. 470, 1123 (2010).[40] I. S. Veshchunov, W. Magrini, S. V. Mironov, A. G.Godin, J.-B. Trebbia, A. I. Buzdin, P. Tamarat, andB. Lounis, Optical manipulation of single flux quanta,Nature Communications 7, 12801 (2016).[41] K. Kinoshita, H. Ishibashi, and T. Kobayashi, Improvedsurface smoothness of YBa2Cu3Oy films and related mul-tilayers by ArF excimer laser deposition with shadowmask “eclipse method”, Jpn. J. Appl. Phys. 33, L417(1994).[42] M. Tachiki, M. Noda, K. Yamada, and T. Kobayashi,SrTiO3 films epitaxially grown by eclipse pulsed laserdeposition and their electrical characterization, J. Appl.Phys. 83, 5351 (1998).[43] R. Prozorov, M. Zarea, and J. A. Sauls, Niobium in theclean limit: An intrinsic type-I superconductor, Phys.Rev. B 106, L180505 (2022).[44] E. H. Brandt, On the shear modulus of the flux line lat-tice, physica status solidi (b) 77, 551 (1976).[45] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I.Larkin, and V. M. Vinokur, Vortices in high-temperaturesuperconductors, Rev. Mod. Phys. 66, 1125 (1994).[46] D. K. Finnemore, T. F. Stromberg, and C. A. Swen-son, Superconducting properties of high-purity niobium,Phys. Rev. 149, 231 (1966).[47] https://doi.org/10.48505/nims.5348.[48] H. Umezawa, H. Matsumoto, and M. Tachiki, Thermofield dynamics and condensed states (North-Holland,Amsterdam, 1982).[49] H. Umezawa, Advanced Field Theory: Micro, Macro, andThermal Physics (American Institute of Physics Press,New York, 1993).[50] H. Matsumoto and H. Umezawa, A rigorous formula-tion of the boson method in superconductivity, Fortschr.Phys. 24, 357 (1976).[51] M. Fusco-Girard, U. Klein, and F. Mancini, Attractiveinteraction between the surface of a type-II supercon-ductors and a single flux line, Physica B+C 107, 423(1981).