Soldered joints—an essential component of demountable high temperature superconducting fusion magnets

Demountable superconducting magnet coils would offer significant benefits to commercial nuclear fusion power plants. Whether large pressed joints or large soldered joints provide the solution for demountable fusion magnets, a critical component or building block for both will be the many, smaller-scale joints that enable the supercurrent to leave the superconducting layer, cross the superconducting tape and pass into the solder that lies between the tape and the conductor that eventually provides one of the demountable surfaces. This paper considers the electrical and thermal properties of this essential component part of demountable high temperature superconducting (HTS) joints by considering the fabrication and properties of jointed HTSs consisting of a thin layer of solder (In52Sn48 or Pb38Sn62) sandwiched between two rare-earth-Ba2Cu3O7 (REBCO) second generation HTS coated conductors (CCs). The HTS joints are analysed using numerical modelling, critical current and resistivity measurements on the joints from 300 to 4.2 K in applied magnetic fields up to 12 T, as well as scanning electron microscopy studies. Our results show that the copper/silver layers significantly reduce the heating in the joints to less than a few hundred mK. When the REBCO alone is superconducting, the joint resistivity (RJ) predominantly has two sources, the solder layer and an interfacial resistivity at the REBCO/silver interface (∼25 nΩ cm2) in the as-supplied CCs which together have a very weak magnetoresistance in fields up to 12 T. We achieved excellent reproducibility in the RJ of the In52Sn48 soldered joints of better than 10% at temperatures below Tc of the REBCO layer which can be compared to variations of more than two orders of magnitude in the literature. We also show that demountable joints in fusion energy magnets are viable and need only add a few percent to the total cryogenic cost for a fusion tokamak.


Introduction
Recent improvements [1] in the current density of coated conductors (CCs) with high temperature superconductors (HTS) using self-assembling pinning sites have driven it up to values as high as a few percent of its theoretical limit in high magnetic fields. Together with the very high upper critical Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
fields and critical temperature of these materials, this has encouraged development of high-field HTS magnets and brightened the prospects of higher field fusion machines [2]. Since there are still no ductile high-field superconductors that can produce magnet fields above about ∼10 T, toroidal field (TF) magnets in future tokamaks will still have to include one of the many brittle high-field superconductors, and the lifecycle of a commercial fusion machine must include economic options for replacing these coils. In this paper we investigate whether the properties of HTS materials can help mitigate the costs of replacing TF coils, by enabling demountable joints [3][4][5][6]. Demountable joints will: eliminate the single point failure of a monolithic TF coil structure; enable the modular construction of very large complex superconducting magnets; improve reactor maintenance by improving access; increase the availability of a fusion reactor; enable simplified materials component testing; and open the possibility of novel topologies for fusion reactors which are not otherwise practical. We have studied small simple soldered joints consisting of a thin layer of solder sandwiched between two rare-earth-Ba 2 Cu 3 O 7 (REBCO) second generation (2G) HTS CCs. Although these hand-made joints will probably not provide the solution for large-scale commercial joints in a tokamak by themselves, understanding this basic building block of joints is essential for evaluating and optimising larger mechanically pressed or soldered joints because transfer of current out of an HTS layer and across solder into a normal metal is a process that is unavoidable in most practical large tokamak-relevant joint designs. Our results can also provide useful information for the design of general high-field magnets which involve joining HTS CCs [7,8]. We present numerical analysis, resistivity measurements, critical current measurements from 300 to 4 K and in applied magnetic fields up to 12 T as well as scanning electron microscopy (SEM) images of the HTS soldered joints fabricated using In 52 Sn 48 and Pb 38 Sn 62 solders. The contents of this paper are arranged as follows: background information is included in section 2; the description of the fabrication, structure and components of the joints is in section 3; section 4 includes the finite element analysis (FEA); section 5 describes the cryogenic measurements and results; the last two sections use our results and understanding of joints to calculate the expected cryogenic cost of demountable joints in fusion applications, summarises the important conclusions and briefly discusses our on-going and future work.

Background
The value of demountable joints has long been appreciated [9][10][11][12]. Designs of magnets with demountable joints for tokamak [13] and heliotron [14] reactors were proposed in 1980s. However, the heat generated by these joints in low temperature superconductor (LTS) were considered so high as to undermine the stability of the magnets [15] and as a result, the demountable option was not included in fusion reactor designs such as the ITER tokamak [16]. The ITER TF systems consist of 18 TF coils, each of which comprise of seven double pancakes sub-coils connected by six praying-hands joints and two shaking hands joints connected to the feeder bus bars. These joints carry 68 kA and have resistances of ∼1 nΩ to ensure the static heat dissipation is within the 5 W design limits [17,18], but use a wind and react type process so are permanent [19,20]. Techniques for making permanent joints with LTS are well established [21] and include socalled persistent joints with extremely low joint resistance (10 −13 Ω) for MRI applications [22]. Recently, there has also been success in fabricating persistent joints using HTS [23]. However, because it is very time-consuming and demanding to fabricate hand-made persistent joints at elevated temperatures, they are not useful for commercial fusion applications. Although the optimum design of the joints for fusion energy is currently an open question, any practical solutions will avoid making several million demountable tape-to-tape or strand-to-strand joints sequentially.
There are essentially two types of resistive joints reported in the HTS literature, the lap (or bridge) joint which maximises contact area to achieve low resistance and the butt (or edge) joint where the priority is to minimise the separation of the superconductors. HTS lap joints are most successful to date because the demountable interfaces are metallic whereas butt joints are susceptible to mechanical damage [3,7,24,25]. There are many groups working on connecting multiple strands or tapes using soldered joints. Hartwig et al have proposed a demountable TF coil design for a small tokamak [3]. Each joint in the TF coil is made out of a stack REBCO CCs which are joined mechanically. Kawai et al have proposed a demountable helical coils design for a force free helical reactor at NIFS [24][25][26] and have produced total joint resistances in the nΩ range. In this design, two conductors are joined together mechanically by the ends or edges that may also be soldered using a bridging unit. A cooling system that uses bronze sintered porous media with subcooled liquid nitrogen was proposed to remove the heat generated at these joints under operating conditions to make the demountable helical reactor design more feasible [15]. HTS power transmission cable projects [27][28][29] may also offer fusion relevant solutions for joints [28,29]. For example the 275 kV-3 kA, 30 m HTS cable system from Yagi et al [28] demonstrates that soldered joints remain robust in a demanding AC environment. There is also a creative cable termination design reported by Takayasu et al that has been developed for an HTS cable made of stacked REBCO and BiSCCO tapes [27]. The asymmetric electric property of REBCO tapes is overcome by soldering REBCO tapes onto symmetric BiSCCO tapes.
Large TF coils are designed to carry very high currents of 10 kA-100 kA to produce high magnetic fields without generating high voltages. In standard power engineering, conventional joints between cables carrying high currents are ubiquitous. Experiments using current pulses of ∼250 kA include those at lightning test laboratories [30] and those on circuit breaker devices for use in the power generation and distribution industry [31]. Industrial areas that use continuous high currents up to 400 kA include electrolysis plants [32] and industrial furnaces [33]. Although there is not a 'standard' high current joint design, the vast majority of joints in the power engineering sector are straightforward, mechanically pressed, welded or bolted joints which offer design options for fusion energy applications.
There is also work on a smaller scale, looking at joining a few tapes and the detail of how the supercurrent leaves the HTS superconductor, crosses the normal metal layers and reenters another HTS superconductor. This includes investigating how to design and fabricate such joints for use in HTS laboratory high-field magnets [7,8]. In such work, HTS CCs are simply soldered directly together. Soldering has been widely used for joining 2G HTS CCs because of the simplicity of the fabrication technique, the reasonably low joint resistivity (R J ) achieved and the lack of additional structures needed to ensure mechanical joints remain intact under pressure. However, because of the multilayer structure of the 2G CCs, and the sensitivity of the brittle tapes to mechanical stress and elevated processing temperature, technical problems remain in soldering 2G CCs. The range of R J values reported in the literature is large. The R J of similar soldered joints with the same CCs can vary from ∼4 to over 1000 nΩ cm 2 at 77 K [7,[34][35][36]. Inconsistent results are also found in the magnetic field dependence of R J . Some studies claimed that there was no observable field dependence of R J [7,36]. Other studies demonstrated that R J of the samples fabricated using the same kind of CCs changed significantly with an applied magnetic field [37]. If resistive demountable joints are to help enable commercial fusion, they must be robust and reproducible and their resistances must be low enough under all operating conditions (e.g. in-field) to avoid large cryogenic costs or quenching the superconductor. There are also only limited studies on the temperature dependence of R J [37][38][39] which to date has meant there is no model for describing R J accurately nor is there a reliable estimate of the joint resistivity expected under operating conditions for different applications. Small scale, soldered joints will be an integral component of any large fusion joint between superconductors and are the focus of this paper. We fabricate the small joints, and investigate their resistive and thermal properties. Our numerical results allow us to calculate the heating in these resistive soldered joints when they are subjected to an electric current. Our experimental results explain the inconsistent findings in R J in the literature and allow us to propose a simple but accurate model for R J .

Joint fabrication
All joints were fabricated using SuperPower Inc. IBAD-MOCVD 2G REBCO CC tapes [40]. We note that these tapes are not optimised for fusion applications-silver can become strongly active under a neutron flux [41]. The specifications of the tapes are listed in table 1 and a schematic diagram of its multilayer structure shown in figure 1. The tapes are produced by SuperPower Inc. from wider tapes and sliced down to 4 mm wide strips. The length of the joints made were typically about 4 mm. SuperPower Inc. recommends soldering to them at or below 250°C [42]. Table 2, provides properties from our survey and measurement of many low temperature solders. Two solders were selected and fabricated from elemental powders in-house: Pb 38 Sn 62 because it has the lowest resistivity of the alloys at 77 K, it is the eutectic composition (from phase diagram studies [43]) with a melting point of 183°C and has been widely used in the literature to join REBCO CCs and In 52 Sn 48 because of its lower melting point (118°C), low resistivity and because thermal properties are rather well-known compared to other similar choices.
Each joint studied in this work was soldered in air using a temperature controlled soldering iron, preset to 250°C for Pb 38 Sn 62 solder or 190°C for In 52 Sn 48 solder, with the REBCO layers face to face. The solder was first prepared in the form of foil with thickness of about 50 μm and then placed between two pre-fluxed REBCO tapes. The tapes were held down on a flat surface that was covered in an insulating Kapton layer. Kapton tape was used to temporarily fix the tape to the layer and heat was then applied to the tapes to join them together. The joints are characterised by a joint resistivity (R J ) in nΩ cm 2 , which is an intensive property given by the product of the resistance and the area of the joint. Since all solders and joints were prepared in air, we expect some small oxide content in the solder increases their resistivity by a few percent.

The structure of the joints
A Hitachi Tabletop TM 1000 SEM was used to obtain all SEM images. Cross sectional SEM images of joints made using the two different solders are shown in figure 2 (In 52 Sn 48 ) and figure 3 (Pb 38 Sn 62 ). In addition to the solder layer, there is a thin continuous layer of intermetallic compound between the solder layer and the copper stabiliser which indicates good metallurgical bonding [44]. The typical thickness of the solder layer is ∼22 μm for both solders which at 77 K contributes roughly 10 nΩ cm 2 for Pb 38 Sn 62 and 30 nΩ cm 2 for In 52 Sn 48 solder to the joint resistivity.Although thinner solder layers with lower joint resistivities can be achieved by applying larger pressures during fabrication, we chose not to use that approach here, in order to avoid damaging the brittle REBCO and to achieve better reproducibility of results [45,46]. In the SEM image of the Pb 38 Sn 62 joint there is a clear gap of ∼7 μm in or close to the REBCO Table 1. Specifications for the SuperPower Inc. IBAD-MOCVD REBCO coated-conductor superconducting tapes used in this study [40].

Model number SCS4050
Average I c at 77 K in self-magnetic field 98 A Average n-value at 77 K 30 Width 3.99 mm Overall thickness 0.095 mm layer. We suggest this delamination may have occurred because of the higher soldering temperature used with Pb 38 Sn 62 solder and helps explain the greater variability, we find in this work and found in the literature, for joints fabricated using Pb 38 Sn 62 solder.

The superconducting solders Pb 38 Sn 62 and In 52 Sn 48
The normal-state resistivity and superconducting properties in-field of both Pb 38 Sn 62 and In 52 Sn 48 solders used in the joints were measured at different temperatures and magnetic fields. In these measurements the solder served both as the sample and as the material for connecting the instrumentation leads. Figure 4 shows normal state properties below 300 K. Pb 38 Sn 62 solder has a much lower resistivity than In 52 Sn 48 solder at low temperatures. We have also included data for the widely used non-eutectic commercial solder Pb 40 Sn 60 in the figure for comparison. Figure 5 shows low temperature infield data and the superconducting phase diagrams.

FEA of temperature and voltages in joints
Numerical calculations of current flow through the joints and the resulting temperature profile were made using a commercial FEA package (FlexPDE TM ). During the fabrication of REBCO tapes, they are first cut (or slit) to the required width and then electroplated with copper. The multilayered structure of the joint that was modelled is shown in figure 6 and does  not include copper on the sides of the tape. We have omitted this Cu because it often becomes partially detached from the tapes and by omitting it, our FEA calculations and conclusions are relevant not just for the narrow tapes investigated here, but also for the wider tapes that may be used in large scale fusion devices. The resistivity and thermal conductivity of different components of a joint used for the computations are listed in table 3 and were obtained either from the literature or in-house measurements. The buffer stack has been ignored because it is very thin and we find experimentally that the two copper layers are electrically well-connected. In the FEA, we assume the joint is completely surrounded by liquid helium (4.2 K) or nitrogen (77 K), as is the case for our experiments below, with cryogen properties given in table 4. By solving Laplace's equation: where V is the voltage, together with the boundary conditions necessary to inject and extract current, the electrical field throughout the system is calculated. From the electrical conductivity of the components of the joint, the current density flowing and the power density dissipated throughout the joint are then obtained. From the definition of the thermal conductivity and the conservation of energy, in steady state one can derive:   where J, T, σ n and κ n are the current density, temperature, electrical and thermal conductivity of the solid components in the joint respectively. Solving equation (4.2) then gives the temperature profile throughout the joint. Accurate knowledge of the heat flux across the boundaries of the joint into the cryogen is demanding, because both conduction and convection are important. If we simply assume the edges of the joint are at the temperature of the cryogen, we overestimate the heat transfer into the cryogen. Equally, assuming the cryogen is simply a solid with finite thermal conductivity underestimates the heat transfer into the cryogen by ignoring convection. In this paper we follow standard cryogenic engineering approximations for describing heat flux across a heated surface into a cryogen (see equations (4.3)-(4.10)) [47]. We assume heat transfer is governed by Newton's law of cooling: h is the heat transfer coefficient, T s is the temperature at the surface and T Cryo is the temperature far from the heat sources (i.e. the boiling point of the cryogen). The heat transfer coefficient (h) is written in terms of the Nusselt number ( ) Cryo k Cryo is the thermal conductivity of the cryogen and L is a characteristic length of the geometry where for horizontal surfaces: whereas for vertical surfaces, L is simply the height of the surface. The Nusselt number is written in terms of the Rayleigh number ( ) R a that includes the role of buoyancy (or equivalently the orientation of the solid surface with respect to gravity) and viscosity. For conduction through the upper surface of a heated plate (or the lower surface of a cooled plate) with low heat flux the Nusselt number is given by: and for high heat flux In this work we find R a ∼10 6 and so we use equation (4.6) rather than (4.7). For the lower surface of a heated plate, where buoyancy is less effective at contributing to heat transfer, one finds: and for conduction through the heated side of a vertical plate: ( ) a n = + where the Rayleigh number is given by: , and p are the thermal expansion, kinematic viscosity, thermal diffusivity, density and specific heat capacity of the cryogen respectively.
The FEA calculations were validated by comparison with analytic solutions for the electric field, current density and temperature profile in an infinite vertical cylinder of solder in cryogen carrying 100 A. The equivalent FEA solutions were obtained for an idealised joint which consists of a 20 μm vertical-cylinder of solder sandwiched between two slabs of thermally insulating, electrically conducting superconductor at the top and bottom also carrying 100 A, as shown in figure 7. The two-dimensional analytic solution for the  temperature difference between the centre of the cylinder ( ) T Centre and the bulk of the cryogen ( ) T Cryo is given by: where the term in curly brackets in equation (4.11) tends to unity in the limit that the thermal conductivity of the cryogen is taken to be infinite or that the surface of the joint is at the temperature of the cryogen. Since h is a function of ΔT, this transcendental equation must be solved numerically using equations (4.4)-(4.10). Using the materials properties data listed in table 3 for 77 K, we found the agreement between the FEA calculations of ΔT and the analytic solutions is good, as shown in table 5. The temperature profile predicted from equation (4.11) and the FEA results shown in figure 7 are also in good agreement. The FEA visualisation results also show that the electric field and the resistive current density are quite uniform within the solder and that if there are no voltage taps across the solder itself, there is a discontinuous change in voltage between taps on one superconducting slab from those on the other one, independent of whether they are in the joint region above the solder or not. We also modelled the multilayered soldered joint (see figure 6). FEA shows the current density flow is quite uniform between the superconducting layers through the solder/copper/silver layers (similar to the results in figure 7) and that most of the heat leaves along the copper/silver layers which significantly reduces the temperature inside the joint. Figure 8 shows the temperature profile throughout the joint. The temperature is rather uniform in the solder and in those parts of the tapes attached to the solder. In the approximation that all the heat leaves the joint along the four copper layers, we can derive an analytic equation for the maximum temperature in the joint (DT ): where  q is power along each copper sheet, k Cu is the thermal conductivity of the copper, A Cu is the cross sectional area of each sheet and d Cu is the length, orthogonal to the direction of heat flow for the surface through which the heat passes into the cryogen. For the tapes considered here, where w and t are the width (4 mm) and thickness of each copper layer (20 μm) respectively. At 77 K where the copper layer conducts much more heat than the thin silver layer, there is reasonable agreement between equation (4.12) and the FEA calculations as shown in table 5.

Variable temperature resistivity measurements
Cryogenic resistivity measurements were performed either in a 9 T Quantum Design PPMS system or in a Janis cryocooler at temperatures from 300 to 3 K. For each resistivity data point, Ohm's law was measured using currents up to 100 mA with the voltage taps typically 10 mm apart. Figure 9 shows data for a joint with the voltage taps distributed along the upper surface. The voltage for all taps on either one of the REBCO CCs was the same, but shows a discontinuity across the soldered joint, confirming the FEA result that the resistive current is restricted within the joint. We have measured the voltages generated across many CC joints under various conditions. If the current leads are too close to the voltage taps (less than a few millimetres), the voltage taps pick up some of the current transfer voltage. The current transfer voltages are particularly pronounced if the current crosses the The gradient in the voltage (i.e. the electric field) is uniform within the solder (i.e. y=0). Lower: The temperature profile is a slice through the centre of the joint (i.e. z=0). The temperature gradient throughout the slice is radial. Heat conduction into the cryogen takes account of buoyancy or equivalently the direction of gravity which in this figure is down the axis of the solder. high resistivity Hastelloy substrate (see table 3) and can lead to a misleading and much higher apparent joint resistivity (R J ). Therefore, in general we put all the electrical leads on the side of the tape closest to the superconducting layer.
Typical temperature dependencies for the R J of joints fabricated using the two different solders at zero applied magnetic field are plotted in figure 10. At temperatures above 90 K, the REBCO tape was not superconducting and large resistivities, associated with all the normal material components between the voltage taps, are found. As the REBCO becomes superconducting, R J drops by a few orders of magnitude. Eventually, the REBCO layer is completely superconducting and the R J (T) measured is due from the resistive parts of the joint, which includes the solder layer, the silver and copper sandwiched above and below the solder, and any interfacial resistivities between different layers of the REBCO CCs. As shown in figures 10-12, we attribute the joint resistivity predominantly to just two sources: a contribution to the joint resistivity from the solder ( ) R solder because of its bulk temperature dependent resistivity and an interfacial resistivity ( ) R i between the REBCO and the silver layers. The joint resistivity, R , J is given by: The contribution from the bulk resistivities of the copper and silver can be considered negligible. The Cu and Ag may dissolve to form a binary alloy layer. The resistivity of Cu/Ag alloys is typically 25% higher than that of pure Ag [48]. Since the Ag layer in a single HTS tape is only 2 μm thick, this sets the upper limit to the contribution of any Cu/Ag alloy to the joint resistivity to be less than 1 nΩ cm 2 which can also be ignored. Figures 11 and 12 display the temperature dependence of R J from 90 K down to the lowest temperature studied (∼4 K) for five In 52 Sn 48 joints and six Pb 38 Sn 62 joints. The width and length of each joint measured are given in the captions. All the In 52 Sn 48 joints in figure 11 show very straightforward, almost text-book, behaviour following equation (5.1)-the average values for the thickness of the solder layers was 22 μm with a standard deviation of 2 μm which are values consistent with the SEM and an interfacial resistivity that is temperature independent with values of 20-27 nΩ cm 2 -similar to those reported in the literature and attributed to the REBCO/Ag interface [49,50]. The average difference between equation (5.1) and the measured values of R J is 2.3% in the temperature range of 8 K-80 K, demonstrating excellent agreement. We have added interfacial resistances into our FEA calculations. Preliminary results show that, consistent with figure 8, the maximum temperature is determined by the total heat generated by the joint and the temperature profile remains reasonably constant in the joint region, broadly unaffected by how the resistance is distributed throughout the joint. The interfacial resistivity values found experimentally here can be compared with the much higher typical values for domain walls in ferromagnetic materials where R i ∼10 −7 Ω cm 2 or domain walls in single crystal antiferromagnetic materials where R i ∼10 −5 Ω cm 2 [51]. We note further reductions in joint resistivity have been achieved by reducing the thickness of the solder layer by applying high pressure during the fabrication process to reduce the solder layer thickness to just 1-2 μm [38]. Equally, we expect that the REBCO/Ag interfacial resistivity can also be substantially reduced by annealing REBCO/Ag in oxygen. A reduction in the resistivity of REBCO/Ag contacts to ∼0.1 nΩ cm 2 has been achieved by annealing for 1 h at temperature above ∼500°C which diffuses some of the silver into the REBCO [52]. The data shown in figure 12 for the Pb 38 Sn 62 joints has larger variability than for In 52 Sn 48 -a result we associate with the higher temperature used in fabrication. In our first Pb 38 Sn 62 joints, fabricated at temperatures higher than 270°C (not otherwise reported here), R J was always higher than 100 nΩ cm 2 at 77 K. Although equation (5.1) fits R J (T) well for 2 of the 6 Pb 38 Sn 62 joints (joints 11 and 12) with reasonable values for solder layer thickness (23 and 25 μm) and for R i (24 and 26 nΩ cm 2similar to the In 52 Sn 48 joints), the reproducibility of Pb 38 Sn 62 joints was far worse than the In 52 Sn 48 joints and non-physical values for the solder layer thickness (35-90 μm) needed to fit the joint resistivity data for the other 4 Pb 38 Sn 62 joints. Nevertheless, because Pb 38 Sn 62 solder has better wetting properties [53] and lower resistivity than In 52 Sn 48 solder (see figure 10) although the properties of the joints were unreliable, we found it produced the very lowest resistivity joints reported in this paper.
At temperatures just below the critical temperature of the superconducting solder, there is another sudden drop in R J (T). The In 52 Sn 48 joints continue to obey equation (5.1) below 6 K and R J (T) becomes just the interfacial resistivity (all joints except joint 5) as expected. However the Pb 38 Sn 62 joints are more complex below 7 K. The resistivity of these joints continued to drop to very low values (10 nΩ cm 2 ) as the temperature dropped below 7 K. In particular, at 4 K the  Table 5. Comparison of the electrical and thermal properties of an idealised soldered joint and a multilayered joint calculated using analytic and FEA methods. The idealised soldered joint consists of two slabs of thermally insulating superconductor connected by a cylinder of solder as shown in figure 7. The multilayered joint structure is shown in figure 6. J and E are the current density and the electric field in the solder respectively. ΔT is the temperature difference between the centre of the joint (hottest) and the surrounding cryogen (77 K) when 100 A flows through the Pb 38 Sn 62 joint. The analytic values are derived using equation (4.11) for the idealised joint and equation (4.12) for the multilayered joint.
Method     resistivity of joints 8, 9 and 10 decreased to below our (low current) detection limit of ∼5 nΩ cm 2 . It is well-known that the oxygen content of REBCO affects its critical temperature. It has been reported that the bulk room-temperature resistivity of REBCO films with thickness of 0.6 μm starts to increase after being annealed in air at temperatures above 200°C for 30 min [54] which implies that oxygen is sufficiently mobile to leave the tape at temperatures relevant for solders. A thin (∼1.5 nm) oxygen deficient REBCO layer at the REBCO/Ag interface is probably the origin of the interfacial resistivity observed in the as-supplied tapes and measured using the In 52 Sn 48 joints [55]. However, although the high soldering temperature of 250°C for the Pb 38 Sn 62 joints may often simply degrade the interface by deoxygenating the surface or by promoting delamination if the duration of soldering time was long enough, it is also possible that some oxygen might be able to diffuse into the interface from the bulk and increase both the local oxygen content and local critical temperature [56] leading to the very low resistivity (i.e. low R i ) at T6 K. The relatively high mobility of oxygen at the melting point of commonly used solders provides a straightforward explanation for some of the large variation of R J found here for Pb 38 Sn 62 joints and also found in the literature. We also note the possibility that if the copper and silver layers are very high purity, proximity coupling between the solder and the REBCO may explain the very low joint resistivity measured at the lowest temperatures [57]. The resistivity of most Pb 38 Sn 62 soldered joints (except joint 11) continued to decrease as temperature dropped below the transition temperature of the solder but only one In 52 Sn 48 soldered joint (joint 5) showed such behaviour. This complex behaviour might be explained if the proximity effect in Pb 38 Sn 62 soldered joints is stronger than in In 52 Sn 48 soldered joints, a proposition that can only be tested and quantitative analysis made, with more information about the fundamental superconducting properties of these solders.

High current measurements
Critical current (I c ) measurements were conducted with the sample in direct contact with either liquid nitrogen (77 K) or helium (4.2 K) and in applied magnetic fields up to 12 T. The resistivity of some samples was measured at 4.2 K in magnetic fields up to 9 T in a 40 mm bore horizontal split-pair superconducting magnet at different angles between the field and the tape surface. The voltage (V) across the joints was measured using a nanovolt amplifier and a digital voltmeter while slowly increasing the current (I) through the joint [58].
The electric field (E) criterion used for I c was 100 μV m −1 and the index of transition or n-value was calculated using the power-law expression E∝I n with E in the range 100-1000 μV m −1 [59]. Typical voltage-current (V-I) traces taken in zero field at 77 K are shown in figure 13. When I was much smaller than I c , the voltage across a soldered joint increased linearly with I, reflecting the resistive nature of the joint. When I was close to I c an additional V-I power law behaviour from the tape is observed, predominantly associated with the (single) tape regions within the voltage taps but outside the joint. The resulting V-I relationship can be Figure 11. Joint resistivity versus temperature for five joints fabricated using In 52 Sn 48 solder. The soldered joint area for joints 1 to 5 were 4 mm×3.9 mm, 4 mm×2.7 mm, 4 mm×3.9 mm, 4 mm×4 mm and 4 mm×2.8 mm respectively. Lines are fits using equation (5.1) which uses the thickness of the solder as a free parameter and assumes the interfacial resistivity is temperature independent. Interfacial resistivity was found to be in the range of 20 to 27 nΩ cm 2 . Measurements were made in zero applied field. written as where R is the joint resistance ( ) = R R Area of the joint , J / n is known as the index or the n-value of the superconductor and V c is the voltage referenced to the 100 μV m −1 E-field criterion. As shown in the figure 13, I c and the n-value across the joint or along a single CC were similar. These results demonstrate that the resistance of the joint is not dependent on the current and that the REBCO layers in the CCs were not damaged during the fabricating process.
V-I traces at different applied magnetic fields (B) at 4.2 K for an In 52 Sn 48 soldered joint are plotted in figure 14. At zero applied field, the V-I trace was linear up to 120 A consistent with the In 52 Sn 48 solder remaining superconducting and R J (0) equalling the interfacial resistivity (∼24 nΩ cm 2 -consistent with resistivity measurements). There was the expected increase in the resistive slope of the V-I traces when the applied field was increased from 0 T to 2 T although the resistance is hysteretic in this field range. The inset to figure 14 shows the field dependence of the joint resistivity, derived from the high current data in decreasing fields, which has very little field dependence at fields down to 1 T.
The field dependence at 4.2 K of R J for an In 52 Sn 48 and a Pb 38 Sn 62 soldered joints is plotted for different orientations of the joints with respect to the field in figures 15 and 16 respectively. As shown in the figures, R J of these joints has zero or very weak dependence on both the magnitude and direction of the applied magnetic field. These results are  The line is a fit using the assumption that joint resistivity is solely due to an 11.4 μm thick In 52 Sn 48 solder layer with a temperature independent interfacial resistivity of 24 nΩ cm 2 . Figure 15. Magnetoresistance of the joint resistivity at 4.2 K as a function of applied magnetic field. The solder used was In 52 Sn 48 and the soldered joint area was 4 mm×18.6 mm and an expected nominal thickness of ∼25 μm. Inset shows the joint resistivity as a function of the angle between the orientation of the tape and the direction of the applied field at 3 T and 4.2 K. consistent with most of the literature findings [7,36] although there is a report of a very large field dependence in R J [37]. However, in that report [37], R J of all the samples that show large magnetoresistance at 77 K is higher than 100 nΩ cm 2 which may suggest it is specific to damaged samples or those with high joint resistivity.

Heating inside joints
A miniature copper/constantan Type T thermocouple was inserted into the centre of the solder of a In 52 Sn 48 joint with a 410 μm thick layer of solder and used to monitor the increase in temperature as a function of a slowly-increasing current (I) through the joint. As shown in figure 17 the joint temperature initially varied roughly as I 1.7 up to a few hundred mK, which is similar to I 2 in equation (4.11). Above I c , the temperature increased much more rapidly. The experimental data are consistent with FEA results, that show the temperature inside typical (∼25 μm thick solder) joints will increase by less than a few hundred mK at 77 K at currents below I c .

The cryogenic cost of demountable joints in fusion applications
Resistive demountable joints produce an additional cryogenic heat load cost that we estimate for fusion applications as follows: we expect the R J of a soldered joint for two REBCO CCs to be well below 50 nΩ cm 2 (see figures 11 and 12), with a resistance for a single joint below 25 nΩ assuming the overlap area of the CCs is >2 cm 2 . If we take the ITER TF coils as an example, each cable carries 68 KA and has 900 strands. There are 134 turns in each TF coil and 18 coils. This gives about 4½ million joints in the TF coils generating about 600 W at the operating temperature. In ITER there is a cryogenic requirement of ∼50 kW so the additional cost from the demountable joints is small-below 1% of the total cryogenic cost. Commercial fusion tokamaks will require magnets that can be demounted quickly. We have not yet ruled out a practical route for making and breaking many joints in parallel. However it seems probable that demountable magnets will include demountable cables that include several million component (non-demountable) joints. The cable-cable joints (not considered in this paper) will bring an additional cryogenic cost dependent on the demountable cable design but we expect it to be small because it may for example be pressed copper which has a much lower resistivity than solder (see table 3). Developing cable-cable measurements and design is part of our future work, but we expect that the joints necessary to enable demountable magnets in fusion applications should only add a few percent to the total cryogenic cost.
7. Discussion and future work Table 2 shows that there are many alloys that can be used as solders with melting points lower than room temperature. Such low melting points enable a different process for   mounting or demounting joints soldered by these alloys. One can sandwich a thin layer of one of these alloys between two REBCO CCs tapes or the solder can simply be painted onto the superconducting tape without the need to apply any heat at room temperature. At low temperatures the alloy becomes solid and connects the two tapes together mechanically and electrically. At room temperature such joints can be demounted easily because the joint solder becomes liquid. A joint between two YBCO tapes was fabricated using gallinstan (see table 2) which has a viscosity more than twice that of water. The joint resistivity was 45 nΩ cm 2 at 77 K, comparable to that of our Pb 38 Sn 62 soldered joints (see figure 12) and with a thickness of the gallinstan layer estimated to be ∼30 μm. It is expected that this thickness can be reduced by half by applying higher pressure at the joint and achieve a joint resistivity lower than those achieved using Pb 38 Sn 62 solders. Table 7 shows that indium is much more expensive than the other constituent metals for the low temperature solders listed in table 2 and so may need to be avoided for reasons of cost. In light of this we have also made joints with Bi 52.5 Pb 32 Sn 15.5 solder and achieved a joint resistivity of 63 nΩ cm 2 at 77 K. The soldering temperature was 150°C. Further work is needed to identify the best choice of solder for fabricating joints for fusion applications and guidelines for developing the best designs for demountable cables.

Conclusions
The fundamental properties of soldered joints of two REBCO CCs have been investigated using SEM studies, numerical analysis, resistivity and critical current measurements. Numerical results show that resistive current across a joint is broadly uniform and confined within the joint. The temperature at the centre of a soldered joint was found to be less than a few hundred mK when operating at 100 A. Our resistivity and critical current data showed that at temperature between ∼7 K and ∼90 K, the joint resistivity of an In 52 Sn 48 soldered joint predominately had two sources, the solder layer (∼20-30 nΩ cm 2 ) and the temperature independent interfacial resistivity of the REBCO/Ag interface within the CCs (∼25 nΩ cm 2 ) and can be described using a simple mathematical equation. The magnetic field dependence of the joint resistivity was very weak and confirmed to be independent of the angle between the surface of the tapes and the magnetic field. We propose that increased delamination in REBCO tapes and increased mobility of the oxygen in the REBCO at higher temperatures provide an explanation for the large variation of R J in Pb 38 Sn 62 soldered joints we have found here and found in the literature. This work shows that HTS conductors that have both high critical current density and low interfacial resistivity may provide a practical route for demountable magnets inside an HTS commercial tokamak and demonstrates that demountable joints need only add just a few percent to the total cryogenic load in a commercial fusion tokamak.