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[Yasuaki Takeda](https://orcid.org/0000-0001-7217-9853), [Gen Nishijima](https://orcid.org/0000-0001-7493-0559), [Hitoshi Kitaguchi](https://orcid.org/0000-0002-5998-2649)

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[Resistance and Voltage–Current Characteristics of REBCO Superconducting Joint](https://mdr.nims.go.jp/datasets/d7ce1c37-8ad9-41e2-93a0-b5286d76ae72)

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1 4MPo1D-01  Resistance and Voltage–Current Characteristics of REBCO Superconducting Joint  Yasuaki Takeda, Gen Nishijima, and Hitoshi Kitaguchi    Abstract— We discuss resistance and voltage–current characteristics of a REBa2Cu3Oy (REBCO, RE = rare earth) intermediate grown superconducting (iGS) joint. These characteristics are evaluated based on the current decay measurements for a REBCO closed loop. The temperature and magnetic field dependencies of the n value for the iGS joint, assessed using an empirical power-law model, are similar to those observed for REBCO tapes. The percolation model describes the voltage–current characteristics of the iGS joint more accurately. Using the critical current and n values at 10−8 V, we can approximately estimate the upper limit of the current for a low target resistance.  Index Terms—2G HTS conductors, coated conductors, resistance measurement, critical current  I. INTRODUCTION UPERCONDUCTING joints are essential for a superconducting magnet operated in the persistent mode [1][2]. Recently, significant progress has been made in developing superconducting joints between high-temperature superconductors (HTSs) [3]–[12]. The persistent mode operation of closed HTS coils and loops with HTS joints has been successfully demonstrated. For REBCO HTS tapes, Ohki et al. developed an intermediate grown superconducting (iGS) joint [5]. In this design, REBCO thin films of the joined tapes are connected via an epitaxially grown REBCO intermediate layer. The iGS joints exhibit high critical current and low resistance. A nuclear magnetic resonance (NMR) magnet was successfully operated in the persistent mode using the REBCO insert coil closed with the iGS joints [13]. The iGS joint is considered the most promising solution for achieving a superconducting joint between REBCO tapes. We have evaluated joint critical current (Icj) and joint resistance (Rj) characteristics for both REBCO and (Bi,Pb)2Sr2Ca2Cu3Oy (Bi-2223) superconducting joints [14]–[21]. The temperature, magnetic field, and field angular dependencies of Icj have been evaluated. The Icj characteristics are similar to those of the HTS tapes. We have also evaluated  Submitted for review September 25, 2024. This work was supported by JST-Mirai Program Grant Number JPMJMI17A2 and JSPS KAKENHI Grant Number JP22K14482, Japan. (Corresponding author: Yasuaki Takeda.)  Yasuaki Takeda, Gen Nishijima, and Hitoshi Kitaguchi are with National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan (e-mail: TAKEDA.Yasuaki@nims.go.jp).  Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org the dependence of Rj not only on the temperature, field, and field angle but also on the time and current. The Rj value is primarily determined by the load factor, defined as the ratio of the current flowing in the joint to Icj. The time dependence of Rj can be explained by considering flux creep within the joint. Our studies have suggested that the Icj and Rj characteristics can be understood using the conventional models for HTS tapes. For an HTS magnet containing superconducting joints operated in the persistent mode, Rj at the operating current must be sufficiently low. Evaluating resistance below 10−12 Ω using common transport measurement methods is challenging. Low Rj values are usually evaluated by the current decay measurements [22][23], but the measurements are time-consuming. The Rj evaluation system we have developed enables more efficient measurements than before [24]. However, it still takes a long time to evaluate Rj under various measurement conditions, such as currents, magnetic fields, and temperatures. It would be useful if such a low Rj, that is, a low voltage at a given current, could be estimated from the voltage–current characteristics in the voltage range that can be evaluated using common methods.  In this study, we evaluate and discuss Rj and voltage–current characteristics of the REBCO iGS joint. We measure herein the decay of the current flowing in a REBCO closed-loop sample with the iGS joint. Assuming that the critical current of the tape outside the joint is sufficiently higher than Icj, the joint characteristics can be evaluated from the current decay data. We propose a method for estimating the current that can flow in the joint while maintaining a certain Rj. This method extrapolates the Icj and n values at 10−8 V to the low-voltage range using an empirical power-law model. Additionally, we discuss the voltage–current characteristics using the percolation model.  II. EXPERIMENTAL We conducted current decay measurements for a single-turn REBCO closed-loop sample containing the iGS joint. The diameter of the loop was 10 cm. The self-inductance (L) of the sample was 0.47 μH. For the current decay measurements, we utilized the joint resistance evaluation system previously developed [16][24]. In this setup, a loop current (Iloop) was induced in the sample through magnetic induction using a copper coil positioned at the center of the loop. The decay of Iloop, representing its time (t) dependence, was recorded at a sampling rate of 1 Hz. Measurements were performed at temperatures (T) ranging from 30 to 85 K and magnetic fields S 2 4MPo1D-01  (B) between 0.1 and 1 T, with the field aligned parallel to the c-axis of the REBCO iGS joint, as shown in Fig. 1(a). Fig. 1(b) illustrates the time dependence of the Iloop at 40 K and 1 T, which is the typical result of the current decay measurements. We introduced the initial Iloop higher than joint critical current (Icj). This is because the decay of Iloop is clearly observed, as shown in Fig. 1(b). To evaluate joint resistance (Rj), data points of the time dependence of Iloop were fitted to the following equation using the least-squares method:  𝐼loop(𝑡) = 𝐼loop(0) exp (−𝑅j𝐿𝑡).           (1)  Here, the initial Iloop was introduced at t = 0. Our previous study demonstrated that Rj is time-dependent [18]. Since Iloop is also time-dependent, we determined the Rj value corresponding to each Iloop by fitting the Iloop–t data to (1) across various time intervals. To evaluate Icj, the voltage across the joint (Vj) was calculated from the time dependence of Iloop using the following equation:  𝑉j = −𝐿Δ𝐼loopΔ𝑡.             (2)  The Iloop dependence of Vj was smoothed using a 25-point moving average. From the smoothed Vj–Iloop curve within the voltage range of 0.5–2 × 10−8 V, we determined Icj from Iloop based on the voltage criterion (Vc) of 10−8 V [19]. For this sample, the Icj value at 77 K in the self-field was evaluated to be 80 A.  As shown in the inset of Fig. 1(b), a magnified view of Iloop–t at 7.00–7.01 × 105 s, ΔIloop decreases as time progresses, leading to a lower signal-to-noise ratio. This reduced signal-to-noise ratio made it challenging to evaluate Vj values below 10−8 V using (2). To address this, we used the equation Vj = RjIloop along with the Rj value at each Iloop obtained from (1) to derive the Vj–Iloop curves at voltages lower than 10−8 V.   III. RESULTS AND DISCUSSION A. Voltage–current characteristics Fig. 2 shows the Vj–Iloop curves obtained from the current decay measurements at temperatures ranging from 30 to 85 K and magnetic fields between 0.1 and 1 T. All curves exhibit upward concavity or a power-law dependence, suggesting that the vortex glass-liquid transition [25] was not observed under the given measurement conditions (T ≤ TGL and B ≤ BGL). Considering that the slope of the Vj–Iloop curves is almost constant at the low-voltage range, we applied the following empirical power-law model to the curves:    Fig. 2. Vj–Iloop curves obtained from the current decay measurements (a) in 0.5 T and at temperatures ranging from 30 to 85 K, and (b) in magnetic fields ranging from 0.1 to 1 T and at 40 and 77 K. All curves exhibit upward concavity or a power-law dependence. An empirical power-law model is applicable to the low-voltage range.        Fig. 1. (a) Schematic of single-turn REBCO closed-loop sample containing the iGS joint. Magnetic field (B) parallel to the c-axis is applied to the joint. (b) Time dependence of Iloop at 40 K and 1 T. The inset shows a magnified view at 7.00–7.01 × 105 s. As time progresses, ΔIloop decreases, leading to a lower signal-to-noise ratio.   3 4MPo1D-01  𝑉j = 𝑉c (𝐼loop𝐼cj)𝑛.             (3)  We obtained n values using (3) by fitting the Vj–Iloop curves at the voltage range of 0.2–2 × 10−10 V. Fig. 3 presents the obtained n values over the temperature range of 30–85 K and field range of 0.1–1 T. We also show the n values in the range of 0.5–2 × 10−8 V calculated from the smoothed Vj–Iloop curve to determine Icj with Vc = 10−8 V. With a decrease in the voltage from 10−8 to 10−10 V, the n values were increased. The maximum increase in the n value was 8.4.  In the temperature dependence plot of n values at 0.5 T, shown in Fig. 3(a), a plateau exists between 30 and 65 K for n values obtained within the range of 0.5–2 × 10−8 V. Additionally, the variation of the n value obtained at 0.2–2 × 10−10 V appears to be small in the range of 30–65 K. In contrast, the n values were decreased from 65 to 85 K. Similar temperature dependence is reported in the transport measurements for REBCO tapes, where the electric field criterion is 10−6 V cm−1 [26][27]. As shown in Fig. 3(b), n values at 40 and 77 K decreased as the fields increased from 0.1 to 0.5 T. At 40 K, the n value remained nearly constant from 0.5 to 1 T. Similar field dependence is also reported for the transport measurements of a REBCO tape [28] and iGS joints [13]. These findings suggest that the field and temperature dependencies of the n value at the low-voltage criterion for the REBCO iGS joint are similar to those for REBCO tapes.  B. Estimation of load factor for low-target resistance using the critical current and n values As shown in Fig. 3, the increase in the n value was not significant at decreasing voltages. This implies that Rj values in the range of 10−12–10−13 Ω can be approximately estimated from the Icj and n values at 10−8 V. From the power-law model (3) and the equation Vj = RjIloop, we obtain the following equation,  𝑅j =𝑉c𝐼cj𝐹𝑛−1,             (4)  where F = Iloop/Icj is the load factor. Given an F value, the corresponding Rj values can be calculated using the Icj and n values at 10−8 V. Conversely, given a target Rj, the upper limit of F can be calculated using the following equation obtained from (4),  𝐹 = (𝑅j𝐼cj𝑉c)1𝑛−1.             (5)  Fig. 4(a) shows the temperature dependence of the Icj and n values at 10−8 V and 0.5 T. Using these Icj and n values, the temperature dependence of Rj is calculated for F values in the range of 0.4–0.9, as shown in Fig. 4(b). As suggested in (4), Rj increases with increasing F values.  Fig. 4(b) shows that as the temperature increases, Rj also increases. From 30 to 65 K, Icj decreases exponentially while the n value remains almost constant, as shown in Fig. 4(a).   Fig. 3. Dependencies of the n values on (a) temperature at 0.5 T and (b) magnetic field at 40 and 77 K. The temperature and field dependencies of the n values are similar to those for REBCO tapes evaluated based on transport measurements.     Fig. 4. (a) Temperature dependence of the Icj and n values at 0.5 T and 10−8 V. (b) Temperature dependence of Rj calculated using the Icj and n values shown in (a) with F values in the range of 0.4–0.9 using (4). Rj increases at increasing temperatures; this is due to the decrease in the Icj and n values. (c) Temperature dependence of F at 0.5 T for the target Rj values of 10−12 and 10−13 Ω. The Fcal values show good agreement with the Fexp values.    4 4MPo1D-01  This suggests that the slight increase in Rj at higher temperatures in the range of 30–65 K is due to the exponential decrease in Icj. In contrast, a larger increase in Rj is observed from 65 to 85 K. This is due to the considerable decrease in both the Icj and n values at increasing temperatures, as shown in Fig. 4(a). Fig. 4(c) shows the temperature dependence of the F values at 0.5 T for the target Rj values of 10−12 and 10−13 Ω. The experimentally obtained F (Fexp) values and the calculated F (Fcal) values using (5) are shown. The F values decrease slightly with increasing temperatures up to 65 K and drop rapidly at higher temperatures. The rapid decrease in the F values is due to the significant decrease in both the Icj and n values. This corresponds to the earlier discussion of the increase in Rj in the previous paragraph. The Fcal values show good agreement with the Fexp values. The maximum differences between these F values are 2.9% and 3.7% for the target Rj values of 10−12 and 10−13 Ω, respectively. This indicates that the upper limit of F for target Rj in the range of 10−12–10−13 Ω can be approximately estimated using the Icj and n values at 10−8 V.  We also examined the magnetic field dependence of Rj and F. Fig. 5(a) shows the field dependence of the Icj and n values at 10−8 V at 40 and 77 K. Using these values, Rj is calculated as shown in Fig. 5(b). Rj increases with increasing field, primarily due to the decrease in the Icj and n values. Figs. 5(c) and 5(d) illustrate the field dependence of Fexp and Fcal for the target Rj values of 10−12 and 10−13 Ω, respectively, at 40 and 77 K. The decrease in F values at increasing fields is attributed to the reduction in both Icj and n values. The Fcal values are consistent with the Fexp values, with maximum differences of 2.9% and 4.5% for target Rj values of 10−12 and 10−13 Ω, respectively. These findings demonstrate that the upper limit of F for target Rj values in the range of 10−12–10−13 Ω can be approximately estimated. From Figs. 4(c), 5(c), and 5(d), the difference between Fexp and Fcal for a target Rj value of 10−13 Ω is larger than that for 10−12 Ω. As most of the Vj–Iloop curves exhibit upward concavity, the difference increases at lower Rj values. In the next section, we discuss the relationship between the Vj–Iloop characteristics and the validity of the estimation of the F values for a low target Rj.  C. Application of the percolation model to voltage–current characteristics Most of the Vj–Iloop curves shown in Fig. 2 exhibit upward concavity and are steeper in the low-voltage range. This behavior corresponds to the increase in the n values at decreasing voltages, as shown in Fig. 3. The percolation model [29][30], rather than the power-law model, may be more appropriate for describing the Vj–Iloop curves. The following equation, which was modified from an equation in [30], was used:  𝑉j =𝐴𝑚𝐼loop𝑚 (1 −𝐼cjm𝐼loop)𝑚,           (6)  where A and m are the fitting parameters, and Icjm is the minimum Icj of the distribution of critical current. The percolation model describes the relationship between the electric field (E) and current density (J), effectively capturing an upward concave log E–log J curve over a wide current density range of approximately 102–106 A cm−2 [29]–[31]. Determining the cross-sectional area (S) of the current flowing in the iGS joint is challenging because the current path and effective joint area are not fully understood. However, S is less than the tape's area in the joint, estimated to be within the range of 10−1–100 cm2. Assuming the effective joint area is 10% of the apparent joint area, S will be in the range of 10−2–10−1 cm2. For the iGS joint, since S is estimated to be within 10−2–100 cm2, a current range of 102–103 A corresponds to a current density range of 102–105 A cm−2. Thus, the percolation model is expected to describe the Vj–Iloop behavior at 40 K in the current range of 1–4 × 102 A, as shown in Fig. 2(b).   Fig. 5.  (a) Magnetic field dependence of the Icj and n values at 10−8 V, at 40 and 77 K. (b) Field dependence of Rj calculated using the Icj and n values shown in (a) with F values in the range of 0.6–0.9 using (4). Rj increases at increasing fields, which is due to the decrease in Icj and n values. (c) Field dependence of F at 40 and 77 K for target Rj of 10−12 Ω and (d) 10−13 Ω. The Fcal values are consistent with the Fexp values.    5 4MPo1D-01  Fig. 6(a) shows the Vj–Iloop plots at 40 K and 1 T with the experimentally obtained values displayed in Fig. 2(b). The straight dashed line corresponds to the power-law model (3) using the Icj and n values at 10−8 V. The curves obtained using (6) with Icjm values in the range of 80–120 A are also shown. The A and m values are determined from the Icj and n values at 10−8 V. Note that the Icjm value of 120 A is high, as the calculated curve yields an Iloop value of 126 A at 10−13 V, which is comparable to the Icjm. As the Icjm increases, the Vj–Iloop curve obtained using (6) becomes steeper. The experimental data are well-fitted by the calculated curve with Icjm = 90 A. This indicates that the Vj–Iloop behavior of the iGS joint can be described by the percolation model. In the previous section, the upper limit of Fcal was obtained for a target Rj. This means that the upper limit of the calculated current Ical (Fcal = Ical/Icj) was approximately estimated for a target Rj. The resistance lines of 10−12 and 10−13 Ω are also shown in Fig. 6(a). The Iloop value at the intersection of the resistance line and the dashed line agrees with Ical for the target Rj. The intersection of the resistance line and the calculated curve when Icjm = 90 A corresponds to the experimental Iloop values for the target Rj, as this curve describes the experimental Vj–Iloop. If the experimentally obtained Vj–Iloop curve yields higher Icjm, the Iloop value at the intersection will increase.  Fig. 6(b) shows the Icjm dependence of Ical and Fcal for the target Rj values of 10−12 and 10−13 Ω, calculated using the percolation model (6). The dashed lines represent Ical and Fcal derived from the power-law model. We discuss the difference between the Ical values obtained using the percolation and power-law models, denoted as ΔIcal. Fig. 6(b) demonstrates that larger ΔIcal values are observed for 10−13 Ω compared to those for 10−12 Ω. At Icjm = 90 A, ΔIcal is only 6.8 A for 10−13 Ω, corresponding to a difference in Fcal of 4.5%. Considering that the curve with Icjm = 90 A describes the experimental Vj–Iloop shown in Fig. 6(a), the upper limit of the current for the target Rj value of 10−13 Ω could be approximately estimated using the power-law model. Fig. 6(b) also shows that as the Icjm increases, ΔIcal increases as well. At Icjm = 120 A, ΔIcal is 14 A for 10−13 Ω, which corresponds to a difference in Fcal of 8.7%. This suggests that at increasing Icjm, the upper limit of the current for a low target Rj value is estimated less accurately using the power-law model.  D. Discussion From the discussion of ΔIcal in the previous section, the proposed method to estimate the upper limit of the current flowing in the iGS joint while maintaining Rj values in the range of 10−12–10−13 Ω leads to the following conclusions:  - When Icjm is not high, the upper limit of the current can be approximately estimated using the Icj and n values at 10−8 V, as demonstrated in this study. The difference between practical current and Ical values required to maintain a target Rj will be small, corresponding to F values of less than 5%.  - When Icjm is high and comparable to the current at 10−13 V, the difference between the practical current and Ical for a target Rj will be large and will correspond to an F value of approximately 10% for Rj = 10−13 Ω. This implies that the upper limit of the current is estimated less accurately. Note that since Ical is underestimated, Rj values lower than the target value will be achieved at the Ical.  In previous studies, we clarified that the microstructure of the intermediate layer of the Bi-2223 superconducting joint strongly affects the Icj characteristics [20][32]. Regarding the iGS joint, the improvement of the microstructure is expected to increase the critical current and enhance the voltage–current characteristics at higher Icjm values. Previous studies showed the misorientations and secondary phases in the intermediate layer of the iGS joint [5][33]. The microstructure may be improved by controlling the chemical composition and   Fig. 6. (a) Vj–Iloop plots at 40 K and 1 T with the experimentally obtained values displayed in Fig. 2(b). The straight dashed line corresponds to the power-law model using the Icj and n values at 10−8 V. The curves obtained using (6) with Icjm values in the range of 80–120 A are shown. (b) Icjm dependence of Ical and Fcal for the target Rj values of 10−12 and 10−13 Ω. The dashed lines represent the Ical and Fcal values obtained using the power-law model. As the Icjm increases, the difference between the Ical values obtained using the percolation and power-law models (ΔIcal) increases as well.   6 4MPo1D-01  constituent phase of the intermediate layer or optimizing the heat-treatment condition.  Evaluating the Icj and n values at 10−8 V through common transport measurements is challenging. The voltage criterion of 10−6–10−7 V is typically used for transport measurements [2]. Estimating low Rj values using the Icj and n values within this voltage range can be problematic due to the influence of current sharing caused by the inhomogeneous current distribution in the joint [18].  Magnetic relaxation measurements may be applicable for evaluating the Icj and n values at 10−8 V. This is because E approximately equal to 10−8 V cm−1 is observed during magnetic relaxation of a REBCO tape within a few seconds [34]. A technical problem may be the relatively large size of the iGS joint, as shown in Fig. 1(a). It should also be determined whether these measurements can be conducted while the joints are connected to the coil. If magnetic relaxation measurements can be applied to iGS joints implemented in a REBCO coil, the determination of whether the coil joints show sufficiently low Rj values at the operating current may be allowed by evaluating the Icj and n values at 10−8 V. We plan to evaluate the Icj and n values of an iGS joint using magnetic relaxation measurements. We evaluated the joint characteristics assuming that the critical current of the tape outside the joint is sufficiently higher than Icj. However, this assumption is not always correct. It has been reported that the critical current of a REBCO tape can be degraded by applying temperatures and pressures similar to those used to fabricate a superconducting joint [35]. We should consider evaluating the distribution of the critical current throughout the closed-loop sample, including the iGS joint.  IV. CONCLUSION We evaluated the resistance and voltage–current characteristics of the iGS joint based on the current decay measurements for a single-turn REBCO closed loop. The temperature and magnetic field dependencies of the n value for the iGS joint were similar to those observed for REBCO tapes. The percolation model was found to describe the voltage–current characteristics of the iGS joint more accurately than the power-law model. We approximately estimated the upper limit of the current that can flow through the joint to maintain a low Rj value using critical current and n values at 10−8 V. This estimation method is applicable when the minimum critical current of the joint is not high. Although evaluating the critical current and n values at 10−8 V using common transport measurements is challenging, magnetic relaxation measurements may be applicable for iGS joints. This approach may help determine whether iGS joints implemented in a REBCO coil achieve the low resistance required at the operating current.  ACKNOWLEDGMENT The authors would like to thank Dr. Mamoru Hamada of Japan Superconductor Technology, Inc., and Dr. Kotaro Ohki of Sumitomo Electric Industries, Ltd., for providing the samples. REFERENCES [1] G. D. Brittles, T. Mousavi, C.R.M. Grovenor, C. Aksoy, and S.C. Speller, “Persistent current joints between technological superconductors,” Supercond. Sci. Technol., vol. 28, no. 9, Aug. 2015, Art. no. 093001. [2] Y. Takeda, H. Maeda, K. Ohki, and Y. 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