Inverted-Parabolic and Weak strain dependencies for the Critical Current in Practical <110> and <100> Oriented REBCO Tapes

Two of the types of practical REBCO (REBa 2 Cu 3 O 7-d , RE = Y and Gd) tapes that have been commercialized, have different twinned structures characterised as <100> (from SuperPower) and <110> orientations (from Fujikura) along the tape axis. In order to elucidate the effect of these different twinned structures on the critical current of these practical REBCO tapes, complementary critical current measurements and synchrotron radiation diffraction measurements were carried out to characterise them. For the <100> oriented tapes, inverted parabolic strain behavior was observed for the uniaxial strain dependence of the critical current. In contrast, the <110> oriented REBCO tape does not give any maximum in the strain dependence of the critical current and shows a weak strain behaviour. The different uniaxial strain dependencies observed for the critical current of the <100> and <110> oriented REBCO tapes were analyzed using a one-dimensional twin


Introduction
Despite the huge effort directed at developing practical composite superconductors, there are currently only five different materials that have been comprehensively commercialised [1]. These practical superconducting wires and tapes include multicomponent structures designed to meet optimized engineering requirements. Because the constituent components have different coefficients of thermal expansion (CTE) and different moduli of elasticity (E), it is difficult to determine the local stress and strain exerted on the superconductive component and how it influences the electromagnetic properties under operation [2]. Several analyses of the local stress and strain have been reported in the literature for the commercialized superconductors that are brittle, MgB2 [3], BSCCO [4,5], REBCO [6,7,8] and Nb3Sn [9,10,11,12]. The nature and origin of the stress and strain behavior on the electromagnetic properties depends on the details of the architecture of these composite systems, so it is still essential to carry out quantitative studies of these materials in order to get a full understanding and to realize the best available performance under operational stress and strain conditions. It has been well-established that a maximum in the critical current appears in the strain dependence for practical Nb3Sn wires [13,14,15] at strain values close to zero deviatoric strain [16]. For practical BSCCO-2223 tapes, it is less clear whether there exists a maximum in the reversible region. The maximum in the strain dependence of the critical current that has been observed, very probably has its origin in degradation due to compressive fracture of superconducting filaments [17]. Hence the elastic regime for BSCCO-2223 tapes has a weak strain behavior that is qualitatively quite different to the inverted parabolic behavior of Nb3Sn.
Ever since REBCO composite tapes were developed for applications, their mechanical and electromagnetic properties have been investigated widely [18,19]. Two practical REBCO (REBa2Cu3O7-d, RE = Y and Gd) tapes have been widely commercialized and are characterized as having <100> and <110> orientations along the tape axis. Recently we have shown [20] that the <100> oriented REBCO tapes show an inverted parabolic strain dependence with a clear maximum in critical current, which can be associated with the supercurrent repeatedly passing through A-and B-domains. The focus of the present study is to determine how the strain dependence of the critical current behaves for the <110> oriented (Fujikura) REBCO tape. In particular we investigate whether the <110> oriented REBCO tapes give any maximum in the strain dependence of critical current or not. We also compare the results for the <100> and <110> oriented REBCO tapes and show that the different strain dependencies can be explained within the context of the one-dimensional twin (or chain) model [20].
Complementary transport and synchrotron studies are reported here on the two different tapes. The strain dependence of critical current over a wide range of uniaxial 3 external strain has been investigated using two well-established techniques -the freestanding sample holder and the springboard [21]. Furthermore, in order to get direct information about the local strain exerted on the REBCO superconductive layer, a synchrotron diffraction study using  has also been completed.

Experimental Procedure
Among the tapes provided from the two commercial suppliers, there were two kinds of Superpower tapes investigated -REBCO <100> with and without Cu layers -as well as the Fujikura REBCO <110> with a Cu layer. The general specifications of the tapes are listed in Table 1 as well as on the manufacturer's web-pages [22,23]. In general, the tapes consist of a thin superconducting layer, grown on the substrate via a buffer layer.
The critical current measurements were carried out in open cryostats in liquid nitrogen. The details of the free-standing transport measurements and those made using the springboard have been reported previously [20]. Basically the freestanding sample was held using a gripping jig which was electrically isolated from the universal testing machine ( Fig. 1 (a)). The voltage taps were soldered onto the tape, 25 mm apart, outside the Nyilas type strain gauge (SG). The critical current was determined using a criterion of 1 V/cm. For the measurements on the springboard ( Fig. 1 (b)), the tape sample was soldered onto the springboard, and a strain gauge was glued onto the tape surface. The voltage taps were soldered onto the tape, 25 mm apart, outside the strain gauge. The springboard enables both compressive and tensile strains to be applied to the tapes by pushing or pulling along the load axis and measuring the changes in strain produced using the strain gauge.
The diffraction experiments were carried out at room temperature at the BL28B2 station of SPring-8 [20]. Just as with the critical current measurements, both a freestanding sample and a sample on a springboard were measured. For the freestanding sample, the Nyilas type strain gauge was placed outside the diffraction point of the incident beam to prevent it from absorbing any of the incident beam. The sample on the springboard was installed in a specially designed load frame. The size of the Cu-Be springboard was 78 mm long, 15 mm wide and 2.5 mm thick. In order to reduce the absorption of the incident beam, a blind hole was incorporated into the spring board. In both cases, the diffraction peaks were measured as a function of uniaxial strain. The diffraction geometry ensured that the scattering vector was parallel to the tape axis. In the present study, the spacing of the (200), (020) and (220) planes of the orthorhombic REBCO phase was employed for local strain measurements because their diffraction intensity was strong enough to ensure sufficient statistical accuracy.

Experimental Results
3.1 Mechanical properties at room temperature (RT) and 77 K Stress (R) -strain (Aa) behavior at RT and 77 K is shown in Fig. 2 for the REBCO <100> without Cu tape. The slope in the small strain region increased linearly. At high strains, beyond the shoulder in the response, the tapes did not fracture as a whole, but plastic deformation was observed accompanied by increasing work hardening of the metallic components. The modulus of elasticity was determined in the linear -small strain region according to the method recommended in the international standard IEC 61788-25 [24], where two types of the modulus of elasticity, E0 and Eu are considered, the initial slope of the stress -strain curve and the slope for the partially unloaded curve respectively. Usually it is recommended that Eu as well R0.2 and A0.2 (the stress and strain at the 0.2% proof strength) are recorded as typical values. Table 2 also shows the mechanical properties for all the tapes including the REBCO <100> with Cu and the REBCO <110> with Cu tapes. As expected the modulus of elasticity is lower in the tapes that have a lamination of Cu sheet included.

Tensile load dependence of critical current -free standing measurements
Previous studies [19,20] have made clear that the critical current of REBCO tapes decreases over a small range of tensile stress and that the critical current returns reversibly on reducing the stress back to zero. When the tensile stress is increased beyond a characteristic limit, the critical current decreases rapidly due to the brittle fracture of REBCO layer. This behavior was reconfirmed for the present REBCO tapes as follows: The applied strain (Aa) dependence of the normalized critical currents, Ic/Ic0 and Icr/Ic0 are plotted in Fig. 3 for REBCO <100> without Cu. The normalized critical current Ic/Ic0 decreased gradually from the beginning and then decreased rapidly over the strain region larger than about 0.38% strain. Also, the normalized recovered critical current Icr/Ic0 as shown in Fig. 3 decreased gradually up to about 0.4% and then decreased abruptly. The abrupt decreases are attributed to the brittle fracture of REBCO layer [18]. As shown in Figure 3, the recovered critical current Icr decreased rapidly beyond a certain strain. We have discussed previously that the reversible stress limit is best defined as the stress at Icr/Ic0=0.99 [19] with the reversible stress and strain limits expressed as R99 and Aa99, respectively. As listed in Table 3, R99 was 871 MPa for REBCO <100> without Cu tape.
Similar stress / strain dependence of the normalized critical currents, Ic/Ic0 and Icr/Ic0 was measured for the REBCO <100> with Cu tape and the corresponding critical parameters are summarized in Table 3. Fig.4 shows the normalized critical currents, Ic/Ic0 and Icr/Ic0 as a function of applied strain for REBCO <110> with Cu tape. Compared to the data for the <100> tape given in Fig. 3, the change of both normalized critical currents, Ic/Ic0 and Icr/Ic0 is far less sensitive to strain and stress in the reversible region. A99 was 0.453 %, before decreasing rapidly due to the brittle fracture of REBCO layer as listed in Table 3. In much of the literature [7,16,20], the 95% retained strain (Aa95) convention has also been used to provide the reversible strain limit for both REBCO and BSCCO tapes. Therefore, in the present study, Aa95 and R95 have also been evaluated and are also listed in Table 3. 3.3 Uniaxial strain dependence of critical current -springboard measurements Similar Ic measurements were carried out for all the tapes mounted on the springboard. The results for the REBCO <100> with and without without Cu tape are shown in Fig. 5. For the tape without Cu, over the strain region measured (i.e. -0.4% to 0.6%), the normalized recovered critical current Icr/Ic0 kept almost to unity, which means there was no observable degradation. The normalized critical current Ic/Ic0 indicated a clear maximum. This convex behavior is not symmetric with respect to the applied strain and the peak of the inverted parabola is located in the tensile strain region. In order to discuss the strain dependence, the following parameters were obtained: Aa0.98(-)=-0.219%, Aamax=0.120% and Aa0.98(+)=0.448%. Similar results, obtained for REBCO <100> with Cu are also shown in Fig. 5. In the whole region of strain measured, the values of Icr/Ic0 were always larger than 0.99 which again means that no degradation occurred. The maximum in critical current appeared at approximately zero applied strain. Again, the following parameters were obtained: Aa0.98(-)=-0.300%, Aamax=0.048% and Aa0.98(+)=0.360%. Fig.6 shows the equivalent normalized critical currents, Ic/Ic0 and Icr/Ic0 as a function of applied strain for REBCO <110> with Cu tape. Compared to the two previous cases of the <100> tapes in Fig. 5, the effect of the strain on both the normalized critical currents, Ic/Ic0 and Icr/Ic0 is far smaller in the reversible compressive and tensile strain region. We also found that Aa99 was 0.453 %, before the critical current decreased rapidly due to the brittle fracture of REBCO layer.

Change of local strain on REBCO layer
Synchrotron diffraction measurements were carried out to get direct information about the local strain exerted on the REBCO superconductive layer and investigate properly its effect on critical current. As shown in Fig. 7 for REBCO <100> with and without Cu, we paid particular attention to the diffraction peaks from the (200) and (020) crystal planes, where the scattering vector was kept along to the tape axis. Fig. 7 provides the diffraction data obtained under several different tensile strains, where the strain was increased by about 0.025 % in steps up to 0.7 %. As expected, the peak positions shifted towards larger lattice spacings with increasing strain. 6 The change of lattice spacing as a function of applied strain is indicated in Fig. 8 together with the stress-strain curve for both REBCO <100> with and without Cu tape. Their (200) and (020) lattice spacings increased linearly with increasing applied tensile strain up to about 0.5%. Here the solid straight line gives the following relation; which suggests that the REBCO layer deforms elastically with increasing applied tensile strain. Beyond this linear part, a shoulder appeared in the stress-strain curve which we associate with prominent plastic deformation. The data in Fig. 7 also show that the intensity profiles of both the (200) and (020) diffraction patterns overlap each other. By fitting a Gaussian curve to each diffraction peak, the observed intensity profiles were separated into two profiles and then their ratio () were evaluated as follows, φ (2) Fig. 9 indicates the ratio () of the integrated intensity of (200) diffraction to that of (020) for both REBCO <100> with and without Cu tape. The ratio remains in the range between 1.11 and 0.96 in the reversible strain region under about 0.5%. From the relation indicated in Eq. (2), the volume fraction f was estimated to be in the range between 0.526 and 0.490. The width of the peaks at their half intensity maximum was calculated from the two Gaussian curves and are shown in Fig. 10. The width for the (200) lattice spacing peak was broader than that of the (020) peak for both <100> tapes, with and without Cu tape.
In Fig. 11, the diffraction profiles observed for REBCO <110> with Cu tape are provided where the strain was increased in steps of about 0.025% up to 0.5 %. Because the <110> direction is parallel to the tape axis, a single diffraction peak for the (220) plane is recorded for the tapes under different tensile strains as shown in the figure. As expected, the peak position shifts towards larger lattice spacing. The change of lattice spacings as a function of applied strain is indicated in Fig. 12 together with the stressstrain curve for REBCO <110> with Cu tape. The lattice spacing increased linearly with increasing applied tensile strain up to about 0.5%. Here the solid straight line drawn along the linear part of observed lattice spacings is described by Eq. (1) which guarantees elastic tensile deformation without any local fracture.
In order to confirm the existence of the twinned structure for REBCO <110> with Cu tape, the diffraction pattern was measured after the tape was rotated by 45 degrees from the scattering vector axis. As shown in Fig. 13, the double peak behavior due to (200) and (020) lattice planes was observed. From Eq. (2), the factor f was evaluated as 0.52. 7

One-dimensional twin model for REBCO tapes
The two types of REBCO composite tapes investigated here have their crystal orientation along the tape axis which is in one case parallel to <100> and in the other parallel to <110> as shown in Fig. 14. As shown in Fig. 14 (a), the A[100]-and B[010]domains continue repeatedly along the uniaxial strain direction for the <100> oriented tape. On the other hand, both a-and b-axes are inclined at 45 degrees from the tape axis for the <110> oriented tape as shown in Fig. 14 (b). The different orientation of twin structure along the tape axis results in the different strain dependencies for the critical current as discussed below. Following Ref.
[20], we can expect the strain dependence of Ic to be different in the two domains. In each region of Fig. 14(a), the local electric-field -current (E-I) characteristics of a high current superconductor are empirically described by, , where EC is the electric field at which the critical current Ic is defined and n is known as the index of transition. Following the considerations given in the Annex, the critical current in the A[100]-and the B[010]-domains can be described using; where is the applied strain and g100 and g010 are proportionality constants. Given the critical current Ic is defined when the electric field Ec becomes 1 V/cm, the electric field Ec is given by the equation, 1 By using Eqs. 3 -6, the following relationship is derived, 8 As shown in Fig. 14(a)  . (11) where is the angle between the direction of the applied strain and the a-axis. Eq. (11) is consistent with the limited experimental results on REBCO single crystals that show a tensile strain in the a-direction increases the critical temperature whereas a tensile strain in the b-direction decreases . Eq (11) is consistent with angular averaging in a single domain for example if we apply a strain with tensile components in both a-and bdirections. It is also consistent with recent work showing that both a weak, and an inverted parabolic strain behavior, can both be found experimentally in a single <100> tape if the strain is applied at 45 degrees to the axis of the tape, and along the tape respectively [26].
For the <110> tapes investigated here, we use Eq. (8) and find for the special case when 45 , Eq. (11) leads to : . 45 . 45 0 Hence the resultant external strain dependence of Ic for the <110> tape is given by Eq.
(10) where we expect g110 to be small -indeed zero when Eq (8) holds. For the <110> tapes, as required, Eq (10) also has no -dependence since as can be seen in Fig. 14, when the direction the strain is applied is at 45 degrees to both the a-and b-axes, the relative proportions of the two possible domains does not change the overall straindependence of the tape. We conclude that Eqs. (10) and (12) provide a straightforward explanation for the weak uniaxial strain dependence for the <110> oriented tape shown in Fig. 6.

Conclusion
In the present study, two types of practical REBCO tape have been examined, in which the crystal orientation along the tape axis is parallel to <100> or <110>. We have used two different Ic measurement techniques and investigated their uniaxial strain dependence of critical current. One of the techniques had the sample attached directly to the universal testing machine and was pulled by a tensile load. The other used a springboard on which the sample is soldered, was attached the testing machine and then was pushed or pulled in order to apply both tensile and compressive strains to the tape sample. An inverted parabolic behavior was observed in the uniaxial strain dependence for the critical current for the <100> oriented tapes. In contrast, the <110> oriented REBCO tape does not give any maximum on its strain dependence. We have also reported using synchrotron radiation diffraction to measure the local strain exerted on the REBCO layer at room temperature using the same two techniques for straining the tape as for Ic measurements. On the basis of the observed data, the local strain exerted on the REBCO layer at 77 K was evaluated and the elastic regime clearly identified.
The focus for the present study is to explain why the strain dependence of critical current behaves for the <100> and <110> oriented REBCO tapes are different. The results for the <100> and <110> oriented REBCO tapes have been compared using a common structure model, which is one-dimensional twin (or chain) model for REBCO material with fractional lengths of A-domains and B-domains. It has been made clear that the present approach can reproduce all the broad features of the experimental results on both tapes for the uniaxial strain dependence of the critical current.    Table 3 Summary of 99% recovery and 95% retention

REBCO <100> without Cu REBCO <100> with Cu
A a (%)  Fig. 9 Tensile strain dependence of the ratio () of the integrated intensity of (200) diffraction to that of (020) one for both REBCO <100> with and without Cu tape