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[Yasuaki Takeda](https://orcid.org/0000-0001-7217-9853), Yuji Tsuchiya, [Gen Nishijima](https://orcid.org/0000-0001-7493-0559)

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[Interpretation of time-dependent current and resistance of HTS closed loop with superconducting joint considering flux creep](https://mdr.nims.go.jp/datasets/b066de70-6c50-46a1-b353-493cb54db32c)

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Interpretation of time-dependent current and resistance of HTS closed loop with superconducting joint considering flux creepApplied Physics Express     LETTER • OPEN ACCESSInterpretation of time-dependent current andresistance of HTS closed loop withsuperconducting joint considering flux creepTo cite this article: Yasuaki Takeda et al 2023 Appl. Phys. Express 16 093002 View the article online for updates and enhancements.You may also likeDevelopment of a persistent-mode NMRmagnet with superconducting jointsbetween high-temperaturesuperconductorsY Yanagisawa, R Piao, Y Suetomi et al.-Development of a persistentsuperconducting joint between Bi-2212/Ag-alloy multifilamentary round wiresPeng Chen, Ulf P Trociewitz, Daniel SDavis et al.-An efficient approach for superconductingjoint of YBCO coated conductorsDaxing Huang, Hongjing Shang, BoweiXie et al.-This content was downloaded from IP address 111.238.231.10 on 20/09/2023 at 01:36https://doi.org/10.35848/1882-0786/acf7a9/article/10.1088/1361-6668/ac2120/article/10.1088/1361-6668/ac2120/article/10.1088/1361-6668/ac2120/article/10.1088/1361-6668/ac2120/article/10.1088/1361-6668/30/2/025020/article/10.1088/1361-6668/30/2/025020/article/10.1088/1361-6668/30/2/025020/article/10.1088/1361-6668/ac6bcb/article/10.1088/1361-6668/ac6bcbInterpretation of time-dependent current and resistance of HTS closed loop withsuperconducting joint considering flux creepYasuaki Takeda1* , Yuji Tsuchiya2, and Gen Nishijima11National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan2Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan*E-mail: TAKEDA.Yasuaki@nims.go.jpReceived July 3, 2023; revised August 10, 2023; accepted September 6, 2023; published online September 19, 2023A low circuit resistance is required for a superconducting magnet operated in persistent mode using superconducting joints. We performed currentdecay measurements on a high-temperature superconducting (HTS) closed loop with a superconducting joint to evaluate the time dependence ofthe current and resistance. The results have been quantitatively explained by considering current sharing and flux creep. After the elapse of 105 s,current sharing was suppressed and a circuit resistance of less than 10−13 Ω was observed. The main finding is that joint resistance of an HTSclosed loop is inversely proportional to time, contributing to low circuit resistance. © 2023 The Author(s). Published on behalf of The Japan Societyof Applied Physics by IOP Publishing LtdNMR spectroscopy requires a temporally stable mag-netic field. A 400MHz (9.4 T) NMR magnet exhibitsfield stability of less than 10 ppb h−1. Such a stablefield is typically generated by a superconducting magnetoperated in persistent mode. Because a magnet consists of adozen or more superconducting coils, a correspondingnumber of inter-coil joints are expected to be present.Assuming self-inductance (L) of several tens of Henry, theresistance of the joints should be less than 10−13 Ω to achievestability of less than 10 ppb h−1. A joint between super-conducting wires/tapes that exhibits such a low joint resis-tance (Rj) is called a superconducting (or persistent) joint.1,2)This low Rj value is too small to be evaluated usingtransport measurements. It is typically evaluated by thecurrent decay method, i.e. by measuring the time (t)dependence of the current flowing in a closed loop (Iloop)made of a superconducting wire/tape with a superconductingjoint.1–6) The following equation is used to deduce the circuitresistance (R) of the closed loop:⎛⎝⎞⎠= -( ) ( ) ( )I t IRLt0 exp . 1loop loopIn current decay measurements, an initial fast decay is usuallyobserved after the introduction of the Iloop. This is due tocurrent sharing, i.e. an inhomogeneous current distribution inthe superconducting wire/tape or joint.1,3,4,6) After settlingthe fast decay, a subsequent slow decay can be observed.Assuming that Rj is equivalent to the circuit resistance R andis constant during slow decay, Rj value is obtained by fittingthe data points to Eq. (1).With recent developments in superconducting joint tech-nology for high-temperature superconducting (HTS)tapes/wires,2,7–13) studies have been published on the evalua-tion of Rj for an HTS closed loop using the current decaymethod. However, the time dependence of Iloop (or themagnetic field trapped in the loop) in these studies did notfit well to Eq. (1).8,9,12–15) This implies that Rj for an HTSclosed loop is time dependent and not uniquely determinedusing Eq. (1).It is well known that an electric field is generated inside asuperconductor by flux creep.16–21) This causes a decay in themagnetization current, which is experimentally observed asmagnetic relaxation. Ohki et al. reported that a logarithmiccurrent decay was observed in a REBa2Cu3Oy HTS closed-loop sample, which suggested voltage generation by fluxcreep at the joint.8) Even though other studies have alsomentioned the flux-creep effect to explain the decay of thecurrent flowing in a closed loop,1,4,5) the flux-creep effect hasnot yet been clarified sufficiently.In this study, we propose an interpretation of the time-dependent current and resistance of an HTS closed loop witha superconducting joint observed in current decay measure-ments. This interpretation, which considers current sharingand flux creep, quantitatively explains the time dependenceof Iloop and Rj.Current decay measurements were performed on a three-turnclosed-loop sample with a superconducting joint, as shown inFig. 1. The sample was made of a commercially available Ag-sheathed multifilamentary (Bi,Pb)2Sr2Ca2Cu3Oy (Bi-2223) HTStape (DI-BSCCO® Type H).22,23) Both ends of the 1.6m longBi-2223 tape were joined by a hot-pressing process developedby us to form a praying-hands-type superconducting joint.12,24)The procedure involved the synthesis of a Bi-2223 intermediatelayer of the joint. The diameter of each loop was 100mm. Theself-inductance L of the sample was estimated to be about1.4 μH.Measurements at a temperature (T) of 4.2 K were madeusing the developed joint resistance evaluation system.25)Iloop was induced by magnetic induction using a copper coillocated at the center of the loop. The measurements werecarried out using a current transformer consisting of a splitcore made of laminated electromagnetic steel and a Hallsensor.15) A magnetic field (B) of 1 T was applied to the joint,as shown in Fig. 1. Inside the joint, the Iloop has a componentorthogonal to the direction of B, which causes flux creep atthe joint.The time dependence of Iloop obtained from the experi-ment, Iloop–t, is shown in Fig. 2. We defined t = 0 when theinduced Iloop reached its maximum value, as indicated in theinset. A fast decay until about t = 103 s and subsequent slowdecay was observed. A decay in Iloop was observed even fort > 2 × 105 s and the slope of Iloop–t gradually approachedContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of thiswork must maintain attribution to the author(s) and the title of the work, journal citation and DOI.093002-1© 2023 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdApplied Physics Express 16, 093002 (2023) LETTERhttps://doi.org/10.35848/1882-0786/acf7a9https://crossmark.crossref.org/dialog/?doi=10.35848/1882-0786/acf7a9&domain=pdf&date_stamp=2023-09-19https://orcid.org/0000-0001-7217-9853https://orcid.org/0000-0001-7217-9853mailto:TAKEDA.Yasuaki@nims.go.jphttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1882-0786/acf7a9zero. The three dashed lines were obtained by fitting theexperimentally obtained data points at different time intervalsto Eq. (1). The three lines indicate that Eq. (1) can only fit asmall portion of Iloop–t and not the entire region. Thissuggests that circuit resistance R is time dependent.Figure 3 shows the time dependence of R. The value of Rwas obtained by dividing the data points of Iloop–t into timeintervals and fitting each interval to Eq. (1) using the leastsquares method. Each interval was chosen such that thecoefficient of determination (r2) was higher than 0.96, asshown in Fig. 2. Up to t = 102 s, the time variation of R wassmall, decreasing in the same order of magnitude rangingfrom 9 to 2 × 10−10 Ω. At t > 102 s, R decreases significantlywith increasing t. At t = 2.3 × 105 s, R reached the lowestvalue of 8.4 × 10−14 Ω within the measurement time.Let us interpret the time dependence of Iloop and R byconsidering flux creep at the joint. Assuming flux creep underthe conditions of T = 4.2 K and B = 1 T, we used theAnderson-Kim model.16,17) From this model, the generationof an electric field (E) by flux creep is described as⎜ ⎟⎜ ⎟⎛⎝⎛⎝⎞⎠⎞⎠µ - - ( )E BUk TJJexp 1 , 20B c0where J, U0, and Jc0 are the current density, pinning potential(U) at J = 0, and critical current density at U = 0,respectively.20) Based on Eq. (2), the voltage generated atthe joint Vj(t) can be expressed as follows:⎜ ⎟⎛⎝⎜⎛⎝⎞⎠⎞⎠⎟= - -( ) ( )( )( )V t VUk TI tI0 exp 1 , 3j j0Bjcj0where Ij(t) and Icj0 are the current and characteristic criticalcurrent (Ic) of the joint, respectively.We also considered current sharing at the joint. Theequivalent circuit model of the closed-loop sample shown inFig. 4 was used. We assumed a constant resistance (R0)parallel to the joint. R0 probably corresponds to the normalresistance of Ag around the joint. We adopted the variableresistance of the joint, Rj(Ij) = Vj/Ij. The resistance of the loopis neglected owing to the significantly higher Ic of thesuperconducting tape than that of the joint,22,23) which leadsto the generation of a negligibly small E by the flux creep atthe loop.Vj(t) can also be described as follows:= -( ) ( ( ) ( )) ( )V t R I t I t . 4j 0 loop j= -( )( )( )V t LdI tdt. 5jloopUsing Eqs. (3)–(5), the time development of Iloop, Vj, and Rare calculated. The calculated R–t and Iloop–t curves areshown in Figs. 3 and 5, respectively. The parameters U0, Icj0,and R0 used for the calculation were 412 K (35.5 meV),Fig. 1. Schematic of three-turn closed-loop sample with superconductingjoint. Both ends of Bi-2223 tape were joined by a praying-hands-typesuperconducting joint. Directions of Iloop and B in current decay measure-ments are also shown.Fig. 2. Iloop–t obtained by current decay measurements at 4.2 K and 1 T.Inset shows magnified view at approximately t = 0. Three dashed lines werededuced by fitting data points with different time intervals to Eq. (1).Equation (1) can only fit small portion of Iloop–t and not entire region,suggesting that circuit resistance R is time dependent.Fig. 3. Time dependence of experimentally obtained circuit resistance R incurrent decay measurements at 4.2 K and 1 T. Value of R was obtained byfitting data points to Eq. (1). R–t and Ij/Iloop–t curves obtained by calculationare also displayed. Calculated R–t curve agrees well with experimentallyobtained R–t plots. At t > 104 s, Ij/Iloop–t shows that current sharing is almostcompletely suppressed, and R is inversely proportional to time.Fig. 4. Schematic of an equivalent circuit model of closed loop samplewith superconducting joint. We assumed constant resistance (R0) in parallelto the joint. We adopted variable resistance of joint, Rj(Ij) = Vj/Ij.093002-2© 2023 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 16, 093002 (2023) Y. Takeda et al.65.5 A, and 40 nΩ, respectively. The experimentally obtainedIloop–t, shown in Fig. 2, is also displayed in Fig. 5. Thecalculated R–t and Iloop–t curves agreed well with theexperimental results.The time dependence of Ij/Iloop was calculated, as shown inFig. 3. Ij/Iloop is ranging from 0.97 to 0.99 at t < 102 s andhigher than 0.9999 at t > 104 s. Up to t = 102 s, currentsharing with R0 ranging from 3% to 1% is estimated, whichleads to the small variation of R shown in Fig. 3. In contrast,current sharing was almost completely suppressed at t >104 s. This means that the current decay is dominated by fluxcreep at the joint. Thus, the time dependence of Iloop and Rcan be quantitatively explained by considering currentsharing and flux creep at the joint.As noted above, the current sharing is negligible at t >104 s. This indicates that R and Iloop are equivalent to Rj andIj, respectively, as Rj becomes considerably lower than R0 (R Rj 0). By assuming R = Rj and Iloop = Ij and fromEqs. (3) and (5), we obtain⎜ ⎟⎛⎝⎞⎠= - +( ) ( )I t Ik TIUttCln , 6loop cj0B cj00 0where t0 = LIcj0kBT/U0Vj(0), and C is a constant. An Iloop–tcorresponding to Eq. (6) with C = 1 and t > 101 s is shown inFig. 5, which agrees well with the experimentally obtaineddata at t > 104 s. This approximation provides an analyticalsolution for the current decay dominated by flux creep at thejoint. The decrease in the Iloop was proportional to thelogarithm of time.From Eq. (4) and the calculation results, Vj(0) is estimatedto be 6.6 × 10−8 V, thereby resulting in t0 of 1.4 × 101 s. Att > 104 s, where the current decay is dominated by flux creep,C will be neglected compared with t/t0 (>7.1 × 102). Underthe assumption that + @/ /t t C t t ,0 0 we obtain the approx-imation of an analytical solution for Rj using Eqs. (5) and (6)as follows:@-( ( ))( )/ /RLU k T t t tln. 7j0 B 0Here, we show that the circuit resistance R is inverselyproportional to time at t > 104 s, which is suggested in Fig. 3.Using Eq. (7), we obtain⎜ ⎟⎡⎣⎢⎛⎝⎞⎠⎤⎦⎥= - + --( )( )( )d Rd tUk Tttlnln1 ln . 8j 0B 01[U0/kBT − ln(t/t0)]−1, which increases with increasing t, wasless than 1.1 × 10−2 until t = 106 s. This implies that theright side of Eq. (8) is nearly equivalent to −1, resulting in= µ -R R t .j1 Thus, assuming R = Rj and Iloop = Ij owing tonegligible current sharing, we demonstrated that circuitresistance R is inversely proportional to time. This impliesthat a low R can be achieved after a sufficiently long time.Let us now discuss the appropriateness of parameters U0,Icj0, and R0 used for the calculation. U0 of a sintered Bi-2223monofilament tape was reported to be 4.2 × 102 K (3.6 ×10−2 eV) at 5 K and magnetic field of 1 T parallel to theab-plane.26) Because Bi-2223 grains in the intermediate layerof the superconducting joint used in this study are weaklyc-axis-aligned,24,27) magnetic field (B = 1 T) is appliedparallel to the ab-plane mainly. Thus, U0 at 412 K(35.5 meV) is similar to that reported in 26). Considering thatIloop decayed from 65.3 A, Icj0 of 65.5 A would have a similarvalue. The R0 of 40 nΩ is comparable to the normal resistanceof about 3 × 10−8 Ω observed in our previous transportmeasurements at 4.2 K and 1 T using a Bi-2223 super-conducting joint sample.12)Vj can be obtained by multiplying Iloop and R.Experimentally obtained and calculated Vj–Iloop are shown inFig. 6.28) In general, the current dependence of voltage (V–I)can be approximated by the power law model with anexponent n ( µV I n). It is known that this approximation isapplicable to V–I dominated by flux creep.19,20,29) Theexperimentally obtained Vj–Iloop ranging from 10−12 to10−10 V, corresponding to t > 104 s where the decay ofIloop is dominated by flux creep, is well fitted to the powerlaw model with n = 84. This n value is significantly higherFig. 5. Iloop as a function of time obtained by calculation and current decaymeasurements at 4.2 K and 1 T. Calculated Iloop–t curve agrees well withexperimentally obtained Iloop–t data. The missing data points at t = 2–3 ×102 s is due to the fact that voltage measurement device used to evaluate Iloopwas changed during that time period. Iloop–t using Eq. (6) with C = 1 alsoagrees well with experimentally obtained data at t > 104 s.Fig. 6. Iloop dependence of Vj obtained by multiplying Iloop and R.Experimentally obtained Vj–Iloop ranging from 10−12 to 10−10 V corre-sponding to t > 104 s is well fitted to power law model with n = 84. n valueof calculated Vj–Iloop is estimated to be 20 at approximately 10−7 V.093002-3© 2023 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 16, 093002 (2023) Y. Takeda et al.than that observed in our previous transport measurementsdescribed above, which was approximately 20 at Vj rangingfrom 10−8 to 10−7 V.12) As shown in Fig. 6, the n value ofthe calculated Vj–Iloop is estimated to be 20 at approximately10−7 V. This implies that a low n value is observed at high Vjowing to current sharing, whereas an intrinsic and high nvalue is observed at low Vj owing to flux creep.Equations (6) and (7) imply that persistent-mode operationusing superconducting joints at high temperatures is challen-ging. This is because relatively fast current decay and high Rjwill be observed at high temperatures. In contrast, enhancingJc0 of a superconducting joint contributes to increasingU0.30,31) This may be effective in achieving a slower currentdecay and a lower Rj value. It has been reported that a high nvalue in the power law model provides a large U0/kBT.19,20,29)An increase in the n value of a superconducting joint mayalso be effective.In cases with a low Iloop or T > 20 K, the Anderson-Kimmodel is insufficient for describing the current decayphenomena. When Iloop value is lower than the maximumpersistent current, which corresponds to Ic of the jointdetermined by a low criterion of Vj or Rj, Eq. (3) does notcorrectly describe Vj. This is because the relationship of@J Jc0 is usually assumed in flux creep described by Eq. (2).To describe E under conditions of T > 20 K, the collectiveflux creep model is generally used.18–21) This model needs tobe applied to interpret the current decay at high T. However,the current decay and time-dependent Rj will be describedquantitatively considering the flux motion, including fluxcreep.In summary, the time-dependent current and resistance ofthe Bi-2223 closed loop with the superconducting joint havebeen interpreted considering current sharing and flux creep atthe joint. An approximation of the analytical solutions of thecurrent and resistance after a sufficiently long time wasobtained owing to the suppression of current sharing. 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