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Takumi Ichikawa, [Gen Nishijima](https://orcid.org/0000-0001-7493-0559), Yuya Hattori, Yoshikazu Mizuguchi

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[Enhancement of a transport critical current by flux trapping in Sn-Pb superconducting solder wire](https://mdr.nims.go.jp/datasets/7257fce8-8e85-4699-878c-7f5309984af1)

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Enhancement of a transport critical current by flux trapping in Sn-Pb superconducting solder wireJapanese Journal ofApplied Physics      LETTER • OPEN ACCESSEnhancement of a transport critical current by fluxtrapping in Sn-Pb superconducting solder wireTo cite this article: Takumi Ichikawa et al 2026 Jpn. J. Appl. Phys. 65 100904 View the article online for updates and enhancements.You may also likeInfluence of trapped magnetic field of Sn-Pb solders on electrical resistivitymeasurement: an example ofsuperconducting transition of SnTakumi Ichikawa, Yuto Watanabe, TakumiMurakami et al.-Large self-heating by trapped-fluxreduction in Sn-Pb soldersYoshikazu Mizuguchi, Takumi Murakami,Md. Riad Kasem et al.-The effect of the return fields ofmagnetized grains on flux trapping in typeII superconductorsMoh’d Rezeq, S Celebi, C Gigault et al.-This content was downloaded from IP address 124.35.78.14 on 01/06/2026 at 22:05https://doi.org/10.35848/1347-4065/ae6ea6/article/10.35848/1347-4065/ad879b/article/10.35848/1347-4065/ad879b/article/10.35848/1347-4065/ad879b/article/10.35848/1347-4065/ad879b/article/10.1209/0295-5075/ad6802/article/10.1209/0295-5075/ad6802/article/10.1088/0953-2048/22/12/125018/article/10.1088/0953-2048/22/12/125018/article/10.1088/0953-2048/22/12/125018aaaEnhancement of a transport critical current by flux trapping in Sn-Pbsuperconducting solder wireTakumi Ichikawa1, Gen Nishijima2, Yuya Hattori1, and Yoshikazu Mizuguchi1*1Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan2National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan*E-mail: mizugu@tmu.ac.jpReceived March 18, 2026; revised April 18, 2026; accepted May 14, 2026; published online May 27, 2026We investigate the magnetic-field hysteresis effect on a critical current (Ic) of Sn-Pb solders by focusing on flux trapping states. A clear differencein Ic is observed between the zero-field-cooled (ZFC) and field-cooled (FC) states. The Ic for the FC sample is 127.7 A which is larger thanIc = 107.8 A for the ZFC sample at H = 300 Oe. Through investigation of Ic and magnetization, we propose that the phase-separatedsuperconducting alloys can show clearly enhanced Ic by flux trapping, and the exotic states are potentially available for designing superconductingapplications. © 2026 The Author(s). Published on behalf of The Japan Society of Applied Physics by IOP Publishing LtdA critical current (Ic) is the maximum current that asuperconductor can carry while maintaining itssuperconducting state. Ic improvement of practicalsuperconductors has been an important research objective fora long time.1–5) However, recently, nonreciprocal Ic char-acteristics have been of interest as superconducting diodeeffect.6–9) Furthermore, Ic can be essential for memory-device applications using superconductivity.10) To accom-modate these emerging applications, further development oftechniques for controlling and modifying Ic characteristics isnecessary. In this work, we examine magnetic-field (H)sensitivity to transport Ic of well-known phase-separatedsuperconducting alloys, Sn-Pb solders.The Sn-Pb solders are phase-separated alloys with Sn andPb microstructures, and magnetic fluxes are trapped afterfield experience.11–14) Using the flux-trapping states belowthe transition temperature (Tc), nonvolatility of magneto-thermal switching is achieved.13,15) In particular, a largerfield is trapped in Sn-poor concentrations including a Sn10-Pb90 solder, which is the studied material of this work. Thelower-Tc Sn regions are surrounded by the higher-Tc Pbregions, and magnetic fluxes are strongly trapped in the Snregions by the supercurrents in the Pb regions. In ourprevious study, the effects of flux-trapped Sn-Pb solders onlow-temperature electrical and specific heat measurementshave been discussed.14,16) The previous studies reportednegative effects, such as the broadening of superconductingtransition of Sn wire due to flux trapped in Sn-Pb solderjoints. In contrast, in this study, we explore positiveutilization of the flux-trapping states of Sn-Pb solders todevelop superconducting applications. Here, we demonstratethat the Ic of Sn-Pb solders can be tuned by flux-trappingamount, which will be useful for developing supercon-ducting diodes or memory devices based on new strategy.We used a Sn10-Pb90 (wt.%) round solder wire with adiameter of 1.65 mm (Sasaki Solder Industry). The transportIc was measured using a standard four-probe method[Fig. 1(a)]. Both ends of the solder wire were mechanicallyfixed to the current terminals.17) The voltage taps werebonded using silver paste and protected with adhesive tape.The distance between the voltage taps was about 3.0 cm. Toavoid deformation of the sample wire due to Lorentz force, itwas fixed to the sample holder with glass cloth adhesive tape.The applied external magnetic field (H) was perpendicular tothe current flow (sample length direction). The measure-ments were performed in liquid He, and the Ic was evaluatedusing a criterion of 1.0 μV, which corresponds to0.33 μV cm−1. The primary purpose of this study was theclarification of the effect of flux trapping [see Fig. 1(b)] onthe transport Ic properties.13–16) Therefore, we compared thedata measured after field cooling (FC, after flux trapping)and zero-field cooling (ZFC). To achieve ZFC states, thesample was heated to T > 10 K, which is higher thanTc = 7.2 K, at H = 0 and cooled to 4.2 K. For the FC data,the sample was cooled at H = 3000 Oe or the field ofH = 3000 Oe was applied at T = 4.2 K, where the same flux-trapping states can be obtained. After ZFC or FC processes,the H was fixed to the target fields (H = 300, 400, 500, and600 Oe). For H below 200 Oe, Ic was not measured becausean abrupt sample quench caused a discontinuous voltagejump far beyond the 1.0 μV criterion, making it impossible toidentify the Ic.Figure 1(c) shows the current dependence of voltage (V-I)of the Sn10-Pb90 solder wire measured at T = 4.2 K. The redand blue symbols represent the data taken for the FC andZFC states, respectively. Noticeably, at H = 300 and 400 Oe,the FC and ZFC data exhibit clear differences, while the FCand ZFC data at H > 400 Oe are comparable. The FC sample(with flux trapping) shows a larger Ic than ZFC. The Ic valuesdetermined at 1.0 μV are plotted in Fig. 2 as a function of H.At H = 300 Oe, we obtained Ic = 127.7 A for FC andIc = 107.8 A for ZFC, resulting in a difference of 19.9 A.Similarly, at H = 400 Oe, the Ic is 68.1 A for FC and 57.8 Afor ZFC, showing a difference of 10.3 A. In addition, there isa difference in the sharpness of the transition near the Ic. Inthe ZFC data, a gradual increase in voltage below Ic is seen,which suggests that the intermediate states are present, andthe motion or instability of magnetic flux grows at lowercurrent. In contrast, the FC data shows robustness of thezero-resistivity states up to Ic. The different behavior shouldbe related to the states of the magnetic fluxes.Figure 3(a) shows the H dependence of magnetization(4πM) measured after ZFC at T = 4.2 K. A typical hysteresiscurve with a critical field of ~ 600 Oe was observed. TheContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution ofthis work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.100904-1© 2026 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJapanese Journal of Applied Physics 65, 100904 (2026) LETTERhttps://doi.org/10.35848/1347-4065/ae6ea6https://crossmark.crossref.org/dialog/?doi=10.35848/1347-4065/ae6ea6&domain=pdf&date_stamp=2026-05-27mailto:mizugu@tmu.ac.jphttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1347-4065/ae6ea6blue line is the initial magnetization process from H = 0 to3000 Oe after ZFC, corresponding to the ZFC cases. The redline shows the subsequent demagnetizing process fromH = 3000 Oe to −3000 Oe, which represents the FC(3000 Oe) cases. Black line represents the final upwardsweep to H = 3000 Oe. Figure 3(b) provides a zoomedview of the middle-field region of Fig. 3(a) to highlight thedifference in magnetization profiles between the ZFC and FCcases. The estimated 4πM and Ic are summarized in Table I.Figure 4 shows schematic images of flux in the sampleinterior in each state. In this article, in the state where pinnedflux and unstable intermediate-state flux coexist, we speci-fically define the state of the pinned fluxes as the flux-boundstate (FBS). In the FBS, magnetic fluxes are strongly pinnedwithin the Sn regions and protected by the supercurrents inthe surrounding Pb regions. Figure 4(a) depicts trapped fluxin flux-trapping state. Figure 4(b) illustrates the coexistenceof pinned flux in FBS and mobile flux in intermediate state.Figure 4(c) shows flux in intermediate state. Here, wediscuss the cause of the difference in Ic at H = 300 and400 Oe between the FC and ZFC cases. As shown in Fig. 3,the initial slope of 4πM -H up to around 400 Oe shows theMeissner states of a type-I superconductor, which means, atH = 300 and 400 Oe, the ZFC sample maintains theMeissner state and behaves like a type-I superconductor.As the applied current increases, the superconducting state isgradually weakened mainly because of self-field generatedby the current,18) and intermediate states are formed near thesurface of the wire. The development of surface intermediatestates allows magnetic flux to invade by forming large-scalenormal-conducting intermediate states as schematically illu-strated in Fig. 4(c). These fluxes in Pb regions are typicallymore unstable and mobile than the vortices strongly pinnedin a mixed state. Therefore, the observed broadening of theV-I transitions is related to the formation of the intermediatestates and their motion with increasing current. In the FCcases at H = 300 and 400 Oe, however, we consider thatsuch formation of large-scale intermediate states does notFig. 1. (a) Configuration of four-terminals method. (b) Schematic images of flux-trapping states (ZFC and FC) of a Sn-Pb solder. (c) Currentdependence of voltage.Fig. 2. H dependence of the estimated Ic of the Sn10-Pb90 soldersample.100904-2© 2026 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 65, 100904 (2026) T. Ichikawa et al.occur because of the presence of flux-line routes already-formed by flux trapping as shown in Fig. 4(a). Even withincreasing I, the formation of intermediate states would belimited to the outer-core regions. Therefore, Ic is morecorrelated to the strength of the flux pinning in the FCcase. Here, the FC state can be considered similar to a type-IIsuperconductor in that magnetic flux is strongly confinedlocally. We note that while typical type-II superconductorshost vortices with a normal-conducting local core, the fluxesare trapped in the μm-scale Sn regions in the FC state of theSn10-Pb90 solder.14) Although the mechanism of flux(vortex) trapping is different, the relationship between Icand flux motion should be a common scenario. Namely, theIc in the FC case is determined by the critical point where theLorentz force acting on the trapped flux exceeds the pinningforce. The convergence of Ic values at H = 500 and 600 Oefurther supports our scenario. At these higher fields, the ZFCsample is out of the initial Meissner slope, indicating thatmagnetic flux has already penetrated the sample interior, andthe intermediate state has been formed before the current wasapplied as shown in Fig. 4(c). For the FC case, we assumethat the unstable flux in intermediate-state regions coexistswith pinned flux in FBS regions before the current applica-tion as shown in Fig. 4(b). We estimate the flux-trappingamount is about 400 G at H = 0 Oe from Fig. 3. In the flux-trapping process, the internal flux density gradually de-creases as H is lowered from Hc of Pb, due to the gradualexpulsion of magnetic flux from the sample surface. At lowerfields, the intermediate states are reduced because theinternal fluxes are mostly pinned in the Sn regions underthe protection of the Pb-region supercurrents. In contrast, athigher fields, the internal flux density exceeds the trappingcapacity, forcing the excess flux to remain in the Pb regionsas unstable and mobile flux in intermediate-state regions.Since intermediate states are established in both the ZFC andFC cases before the current is applied, the V-I curvesconverge. We remeasured Ic of the same Sn10-Pb90 solderwire and confirmed the reproducibility of the large Ic for theFig. 3. (a) H dependence of 4πM of the Sn10-Pb90 solder sample. (b) Zoomed plot at H = 300–600 Oe.Fig. 4. Schematic image of flux in the sample interior in each state. (a) Trapped flux in flux-trapping state. (b) Coexistence of pinned flux in FBS andmobile flux in intermediate-state. (c) Flux in intermediate-state.Table I. 4πM and Ic of the Sn10-Pb90 solder sample at H = 300–600 Oe.H (Oe) 4πM (G) (FC) 4πM (G) (ZFC) Ic (A) (FC) Ic (A) (ZFC)300 104.4 −294.7 127.7 107.8400 22.8 −377.9 68.1 57.8500 1.8 −267.7 16.6 16.2600 10.0 −6.7 3.2 3.0100904-3© 2026 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 65, 100904 (2026) T. Ichikawa et al.FC state. Ic ∼ 70 A was observed at H = 400 Oe after FC at10 kOe.Reactions of superconducting states to the applied H havebeen studied in various topics. In superhydride H3S, strongflux trapping has been reported from field-memory effect onmagnetization.19) Asymmetric field dependence of Ic has alsobeen studied in nano-scale superconducting devices andsuperconducting tapes.20–22) Therefore, the sensitivity of Icto the flux-trapping direction potentially opens new pathwaysfor superconducting applications.In conclusion, we measured transport Ic of the Sn10-Pb90superconducting solder wire under magnetic fields after ZFCand FC (flux trapping). For the data at H = 300 and 400 Oe,clear differences in Ic and the V-I data were observedbetween FC and ZFC. The Ic for FC was larger than thatfor ZFC. The enhancement of Ic and the robustness of thezero-resistivity states with flux-trapping states can be under-stood by the type-II-like strong flux trapping in the Snregions in the phase-separated solder. Our results will beuseful for further understanding of superconductivity physicsof phase-separated alloys and development of exotic super-conducting applications using sensitivity of Ic to the mag-netic-field direction through the flux-trapping phenomenon.Acknowledgments The authors thank F. Ando for discussion. This workwas partly supported by JST-ERATO (No.: JPMJER2201) and TMU researchfund for young scientist.1) J. Bardeen, Rev. Mod. Phys. 34, 667 (1962).2) R. M. Scanlan, A. P. Malozemoff, and D. C. Larbalestier, Proc. IEEE 92,1639 (2004).3) J. L. MacManus-Driscoll and S. C. Wimbush, Nat. Rev. Mater. 6, 587(2021).4) T. Horide and Y. Yoshida, Eur. Phys. J. B 98, 181 (2025).5) S. Abdelhaleem, M. O. Alziyadi, A. Alruwaili, M. J. Alawi, A. Alkabsh, andM. S. Shalaby, Appl. Phys. A 131, 151 (2025).6) F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota,T. Moriyama, Y. Yanase, and T. Ono, Nature 584, 373 (2020).7) M. Nadeem, M. S. Fuhrer, and X. Wang, Nat. Rev. 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