How resistive must grain boundaries in polycrystalline superconductors be, to limit J_c?

Samples with two different compositions were made for this work—Y1: YBa₂Cu₃O_7−x and Y2: 75 wt% YBa₂Cu₃O_7−x + 25 wt% Y₂BaCuO₅ to which an additional 1 wt% CeO₂ was added [38, 39]. Commercial YBa₂Cu₃O_7−x, Y₂BaCuO₅ (99.98%, Toshima) and CeO₂ powders (99.99%, Alfa Aesar) were used to fabricate the samples. The Y1 samples were produced from the commercial powders directly. The Y2 composition was chosen because of its high ${J}_{{\rm{c}}}$ in bulk single crystal form [40]. Powders were first mixed together by shaking the starting powders for 30 min in a stainless steel vial using a SPEX 8000D high-energy shaker mill. Next, samples were milled using the miller and tungsten carbide (WC 94/Co 6) milling media in an argon atmosphere. In an earlier pilot study, we used copper milling media [35]. Although it is expected that copper is less detrimental to the superconducting properties of YBCO than WC or Co, we choose not to use Cu milling media in this work because it is too soft. The samples were milled in batches of 10 g, with a ball-to-powder mass ratio of 3:1, for a total of 30 h. The milling vial and balls were scraped with a tungsten carbide rod regularly, in argon, to increase yield and improve homogeneity. The powders were placed into small niobium foil packets (0.025 mm thick, 99.8%, Alfa Aesar), which acted as a diffusion barrier and then consolidated using a hot isostatic press (HIP). The Nb packets were sealed into stainless steel tubes (type 316, 1 mm thickness) and HIP'ed at a temperature of 400 °C and pressure of 2000 atm for 5 h. Many samples were subsequently annealed in pure flowing oxygen atmosphere in a dedicated oxygen furnace to optimise oxygen content and restore some crystallinity. In this paper, the letters 'P', 'M', 'H' and 'A' denote that a sample has been processed through a combination of powder or pellet Pressing, Milling, HIP'ing, or Annealing respectively. The letters are added after the label for composition in the order that they occurred during processing. Table 2 lists the microcrystalline and nanocrystalline samples where the superconducting properties have been studied in detail.

Table 2.  The fabrication process, transport and magnetic properties of the microcrystalline and nanocrystalline samples in this paper. 'Y1' and 'Y2' represent Y123 and Y123 + Y211 + CeO₂ compositions respectively. The letters 'P', 'M', 'H', and 'A' stand for pressed powders, milled, HIP'ed and annealed respectively. Milled samples (M) were milled for 30 h. HIP processing (H) was at 400 °C and 2000 atm for 5 h. Letter 'A' denotes the standard annealing heat treatment used, which includes a dwell at 750 °C for 20 h followed by 450 °C for 60 h. Ramping between temperatures was completed at 600 °C h^–1. A* denotes using heat treatment A, but with a ramp rate of 60 °C h^–1. B denotes a dwell at 450 °C for 20 h, followed by heat treatment A. A × 2 and A × 3 were heat treated using heat treatment A, twice and three times respectively. ${T}_{{\rm{c}}}$ was determined from the onset of ACMS data. 'Para' indicates a sample behaves paramagnetically and that no ${T}_{{\rm{c}}}$ was measured. ${B}_{{\rm{Irr}}}(0)$ was determined by extrapolation from variable temperature susceptibility data (figure 13) and equation (8). μ₀ΔM/ΔB is from d.c. magnetisation hysteresis measurements. ${J}_{{\rm{c}}}^{{\rm{m}}}(0,4.2)$ is the magnetisation critical current density at zero field and 4.2 K unless otherwise stated, calculated using the grain dimensions of the samples. ${J}_{{\rm{c}}}^{{\rm{t}}}$ is the transport critical current density at a 1 mV m^–1 criterion. ${\rho }_{{\rm{N}}}$ (300 K) is the normal state resistivity at 300 K. The symbol '—' denotes that the property was not measured.

Sample	Grain size (nm)	Annealed	${T}_{{\rm{c}}}$ (K)	${B}_{{\rm{Irr}}}(0)$ (T)	$\displaystyle \frac{{\mu }_{0}{\rm{\Delta }}M}{{\rm{\Delta }}B}$	${J}_{{\rm{c}}}^{{\rm{m}}}(0,\,4.2)$ (A m–2)	${J}_{{\rm{c}}}^{{\rm{t}}}$ (A m−2)	${\rho }_{{\rm{N}}}$(300 K) (Ω m)
Y1P	5000	—	81	140	−2 × 10–1	8.3 × 1010	—	—
Y1H	5000	—	53	70	−3 × 10–2	4.1 × 1010	—	—
Y1HA	5000	A	86	163	−2 × 10–1	2.9 × 1011	1.2 × 105 (0.1 T, 4.2 K )	7.1 × 10–5
Y1MH	20	—	Para	—	—	—	—	62
Y1MHA(1)	100	A	Para	—	−4 × 10−4	9.3 × 109	Resistive	2.5 × 10−2
Y1MHA(2)	100	A*	Para	—	−6 × 10−4	1.0 × 1010 (10 K)	Resistive	2.0 × 10−2
Y1MHA(3)	100	B	70	66	−3 × 10−3	4.5 × 1010	Resistive	8.9 × 10−3
Y1MPA	25	A	73	40	−2 × 10−3	2.7 × 1010	—	—
Y2P	5000	—	81	119	−1 × 10−1	5.1 × 1010	—	—
Y2H	5000	—	53	62	−2 × 10−2	4.0 × 1010	—	—
Y2HA	5000	A	83	132	−2 × 10−1	1.5 × 1011	—	—
Y2MHA(1)	100	A	Para	—	—	—	—	1.0 × 10−2
Y2MHA(2)	100	A × 2	Para	—	−7 × 10−4	1.7 × 1010 (10 K)	70 (0 T, 2 K)	5.2 × 10−3
Y2MHA(3)	100	A × 3	17	—	—	—	—	—

The phases present and grain sizes of the samples were obtained using powder XRD measurements. Figure 2 shows the evolution of the XRD spectra for the as-supplied powders with the compositions Y1 and Y2, after they were milled for up to 30 h. Both compositions show similar behaviour, namely the peaks broadened with increased milling time. The associated decrease in the grain size of the YBa₂Cu₃O_7−x was calculated using TOPAS Academic software and Rietveld refinement. The insets show the grain size as a function of milling time. The grain size of the as-supplied materials is estimated to be 5 μm from SEM (not shown). Within the first 5 h of milling, the grain size is drastically reduced by 3 orders of magnitude down to the nanometre scale. After 30 h, the reduction in grain size saturates as it reaches <10 nm. Figure 3 shows the XRD spectra of the MP, MH and MHA samples. The additional peaks at 30° in the Y1MHA(1) 30 h milled sample and at 24° in the Y2MHA(1) 30 h milled sample should be interpreted with care. We attribute these peaks predominantly to our samples being ground in air for and prior to XRD measurement itself, and the known high sensitivity of YBa₂Cu₃O_7−x to decomposition to parent oxides and Y₂BaCuO₅ in the presence of water vapour in air, particularly in highly milled samples [41–43]. We do not expect such decomposition to occur in our bulk HIP'ed samples that were not exposed to air. We have not identified the peak at 29° in the Y2MHA(1) sample. The grain size of the MHA samples is approximately 100 nm, with a relatively large uncertainty of a factor of 2, due to the unidentified peaks and high strain in these materials that complicates the refinement process. Trace amounts of WC were found in the XRD and EDX (not reported here) in some milled materials of both Y1 and Y2 compositions. There exist methods in which the oxygen content of YBa₂Cu₃O_7−x can be calculated using an analysis of the c-axis lattice parameter, however we were unable to apply such analysis to our samples because of the very high strain content in these milled materials [44].

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Figure 4 shows the TG and DSC data for the P, MP and MHA samples for both Y1 and Y2 compositions. Data were obtained over two cycles. In each cycle, samples were heated up to 1100 °C and cooled back to room temperature in a pure argon atmosphere at 10 °C min⁻¹. As was the case for the XRD data, one has to be careful interpreting the data for the highly milled samples. Although the DSC/TG samples were not powdered, they were exposed to air when they were transferred into the DSC/TG sample-holder cups prior to measurement. In particular, any significant mass loss or DSC peaks below 200 °C are usually associated with moisture.

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Both TG and DSC data for the (as-supplied) Y1P and Y2P samples are in broad agreement with equivalent data from the literature [35]. The mass losses between 400 °C and 800 °C are consistent with oxygen loss of YBa₂Cu₃O_7−x phase from O₇ to O₆ and there are large endothermic melting peaks with onsets at 970 °C [35]. The Y1P, Y1MP and Y2P samples were most stable to mass loss during both cycles. The other three samples showed mass loss over the entire temperature range during both cycles. The only clear exothermic peaks were observed at about 630 °C as indicated by the ♦ symbols for the Y1MP and Y2MP milled samples in the first cycle. We associate these peaks at ∼630 °C with crystallisation of amorphous, and recrystallisation of nanocrystalline phases, to produce larger grain sizes [45]. As expected, such peaks were not present in unmilled samples Y1P or Y2P nor in any of the second cycle data for any of the samples. These results led us to choose a HIP temperature of 400 °C to fabricate the YBCO materials in this work, to prevent excessive grain growth and follow an approach we have successfully used before to make other nanocrystalline materials [32–37]. In the two samples that were milled, HIP'ed, and annealed (Y1MHA(1) and Y2MHA(1)), there was increased and significant mass loss near 850 °C in cycle 1 and coincident large endothermic peaks, both of which are absent in cycle 2. We attribute these peaks to melting and oxygen loss. At the highest temperatures of the cycles, we associate the large endothermic melting peaks in figure 4 as follows: the peaks that occur in both panels near 1000 °C are due to melting of the YBa₂Cu₃O_7−x phase—the exact melting temperature is dependent on oxygen content [46] and expected to be lower in argon atmosphere than in air [47]. The peaks with an onset near 993 °C are due to the reactions Y₂BaCuO₅ + BaCuO₂ → Liquid and YBa₂Cu₃O_7−x +BaCuO₂ → Y₂BaCuO₅ + Liquid [48]. The peaks with an onset near 875 °C are due to the reaction YBa₂Cu₃O_7−x +BaCuO₂ + CuO → Liquid [48]; and the peaks near 839 °C to melting of BaCuO₂ phase [35].

HIP'ed samples were shaped into cuboid bars for transport measurements with typical dimensions of 1 × 1 × 5 mm. The samples were mounted onto a physical property measurement system (PPMS) resistivity puck [49]. Current and voltage leads were connected to the sample using silver paint for standard four-terminal measurements. The voltage taps were typically 2.5 mm apart. Control and measurement of the temperature and the magnetic field were made using the PPMS. To measure $V\mbox{--}I$ traces, the puck was connected to external high-precision voltmeter and current sources. The current was supplied by a Keithley 220 programmable current source. A resistor was added in series to the sample in order to confirm that the current through the sample was equal to the nominal output current in the range of 10 nA–0.1 A. The voltage across the sample taps was measured with a Keithley 2100 6½ digit multimeter, with an additional ×50 000 amplifier [50] when required, to measure extremely small voltages. Figure 5 shows a summary of the resistivity data for the samples in this paper as a function of temperature, measured using excitation currents of typically 5 mA.

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The YBCO microcrystalline sample that was simply HIP'ed (Y1H) has a weak temperature-dependent resistivity with no evidence of superconductivity. Oxygen annealing decreased ${\rho }_{{\rm{N}}}(100\,{\rm{K}})$ by more than a factor of 10³ and a superconducting transition was observed, which can be seen in the in-field data in the upper panel of figure 6 for sample Y1HA. In zero magnetic field, the onset ${T}_{{\rm{c}}}$ is 92 K and zero-resistivity occurs at 60 K. Figure 6 shows that as the applied field was increased, the onset ${T}_{{\rm{c}}},$ that we associate with the grains, does not vary significantly, whereas the zero-resistivity ${T}_{{\rm{c}}},$ likely associated with the grain boundaries, is very significantly decreased. These findings are consistent with those of Dimos et al [51] where the largest suppression of superconductivity in relatively small fields occurs at the grain boundaries. The lower panel compares the values of ${\rho }_{{\rm{N}}}$ for Y1HA to those of single crystals in which current flows either along the c-axis direction or along the ab-plane. Figure 7 shows the equivalent $V\mbox{--}I$ traces for Y1HA. The $V\mbox{--}I$ traces show superconductivity between 0 and 8 T at 4.2 K. Zero field $V\mbox{--}I$ data were also obtained up to 120 K in steps of 10 K, and thereafter up to 300 K in steps 50 K. Using a 1 mVm^–1 criterion, the transport ${J}_{{\rm{c}}}$ is 1.2 × 10⁵ Am^–2 at 0.1 T and 4.2 K. Figure 8 shows the transport ${J}_{{\rm{c}}}$ of Y1HA determined using the same criterion. The inset includes the zero field ${J}_{{\rm{c}}}$ from 40 and 60 K. As shown later in section 3.3, the intragranular magnetisation ${J}_{{\rm{c}}}$ in this sample is of the order of 10¹¹ Am^–2. Hence the transport ${J}_{{\rm{c}}}$ values measured here are six orders of magnitude lower than the intragranular currents.

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As can be seen in figure 5, the nanocrystalline materials have resistivity values typically three or four orders of magnitude higher than microcrystalline materials. Y1MH sample has the highest ${\rho }_{{\rm{N}}}$ of all the samples—60 Ωm at 300 K. For comparison, the values of the resistivity of a good metal like Cu and a good insulator like diamond are 10^–8 Ωm and 10¹⁰–10¹¹ Ωm [52]. After annealing, the resistivity decreased by a factor of approximately 10³ at room temperature. A smaller, further reduction was found by repeating the annealing process as in the case for Y2MHA(1) and Y2MHA (2). The $V\mbox{--}I$ traces of nanocrystalline Y1MHA(1), (2) and (3) were entirely resistive with no signs of percolating supercurrents. Y1MHA(3) shows an inflection in ${\rho }_{{\rm{N}}}$ at 60 K which can also be seen in ac magnetisation discussed in section 3.2. We tried many different annealing procedures to produce supercurrents flowing across grain boundaries. A single nanocrystalline sample showed evidence that it could transport an intergranular supercurrent. Figure 9 shows the in-field resistivity of nanocrystalline materials of the Y2 composition. This sample was annealed twice. The data after the first annealing, Y2MHA(1), is given by solid symbols, and the data after the second annealing, Y2MHA(2), is given by the open symbols. The second annealing decreased the resistivity by at least a factor of 2 over the entire temperature range. The inset shows the $V\mbox{--}I$ trace of the sample after the second annealing, measured at 2 K and 0 T. It provides evidence for very weak superconductivity. The transport ${J}_{{\rm{c}}}$ at 2 K and 0 T was very small, equivalent to about 70 Am⁻² at an electric field criterion of 1 mV m^–1. This is at least 10⁹ times lower than the transport ${J}_{{\rm{c}}}$ of commercial YBCO tapes.

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The ac magnetic susceptibility and dc magnetisation measurements were all taken in our Quantum Design PPMS system [53]. The non-HIP'ed samples were pressed into pellets with a typical size of 3 mm diameter and a height of 2 mm. The HIP'ed samples were shaped into cuboids with fine emery paper, with typical dimensions of 1 × 1 × 1 mm. The ac magnetisation measurements were taken with an excitation field of 0.4 mT and 777 Hz (equivalent to 0.3 T s^–1).

Figure 10 shows the ac magnetisation (and equivalent susceptibility) of the microcrystalline Y1P material with very broad transitions to the superconducting state. The inset shows the onset signal at 91 K, which shows an inflection at ∼80 K. There is a large signal with a second transition centred at ∼46 K. This granular sample is a pressed powder in which one can expect that the electronic powder–powder connections to be weak. We attribute the high temperature transition to the individual grains becoming superconducting and producing a large screening signal. The low temperature transition at 46 K is attributed to stronger coupling across the grains, allowing sufficiently large intergranular currents (flowing on the scale of the sample size) at low temperatures, to produce an additional signal. The signal of –115 Am^–1 from this sample characterises full screening for our experimental conditions at the lowest temperature and is used to normalise susceptibility values to negative unity.

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However, for most of our HIP'ed nanocrystalline samples, large paramagnetic backgrounds with no superconducting transitions were found in the susceptibility data. A small superconducting signal was recovered in the Y1MHA(3) sample after oxygen annealing, as shown in figure 11. This sample has a ${T}_{{\rm{c}}}$ of ∼70 K, but a low susceptibility of –4.0 × 10^–2 at 4.2 K in zero field. Figure 12 shows typical data for nanocrystalline materials with Y2 composition, which show temperature-dependent paramagnetic-like behaviour. The Y2MHA(3) data in the inset did show a superconducting transition at ∼17 K in zero field with a susceptibility of –1.5 × 10^–2 at 4.2 K, although no signals associated with superconductivity were observed in the in-field data. Nevertheless, it is important to realise that while most nanocrystalline samples showed no superconducting ac screening signals (or more accurately, signals below our noise floor), they were in fact superconducting as demonstrated by the very sensitive dc magnetisation measurements shown in the next section. When screening currents are entirely within very small grains, the susceptibility is reduced by a factor ${\chi }_{{\rm{g}}}^{{\prime} }/{\chi }_{{\rm{b}}}^{{\prime} }$ [54, 55] where

$$\begin{eqnarray}&&\tfrac{{\chi }_{{\rm{g}}}^{{\prime} }}{{\chi }_{{\rm{b}}}^{{\prime} }}=\tfrac{1}{15}\left(\tfrac{{a}^{2}}{{\lambda }^{2}}\right)f(a,{\xi }_{{\rm{BCS}}})\,{\rm{for}}\,\lambda \gt {a},\end{eqnarray} \tag{ 4 }$$

where ${\chi }_{{\rm{g}}}^{{\prime} }$ and ${\chi }_{{\rm{b}}}^{{\prime} }$ are the granular and bulk (intergranular) susceptibilities respectively and $a$ is the grain size. The factor $f(a,{\xi }_{{\rm{BCS}}})$ accounts for non-local effects associated with the BCS coherence length $({\xi }_{{\rm{BCS}}}).$ Low values of $f(a,{\xi }_{{\rm{BCS}}})$ occur when the grain size is much smaller than ${\xi }_{{\rm{BCS}}}$ which is about 4–7 nm [56] for YBCO. It has a value of unity when $a\gg {\xi }_{{\rm{BCS}}}.$ The nanocrystalline samples in this work have grain sizes of 100 nm (see table 2) so we assume $f(a,{\xi }_{{\rm{BCS}}})=1.$

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For an anisotropic superconductor, we can find an approximate value for the angular dependence of the G–L penetration depth $(\lambda (\theta ))$ from the angular dependence of the G–L coherence length $(\xi (\theta ))$ derived from upper critical field, and the angular dependence of the G–L constant $(\kappa (\theta ))$ where $\lambda (\theta )=\kappa (\theta )\xi (\theta )$ [57] so that

$$\begin{eqnarray}&&{\xi }^{2}(\theta )={\xi }_{ab}{({\xi }_{ab}^{2}{\cos }^{2}(\theta )+{\xi }_{c}^{2}{\sin }^{2}(\theta ))}^{\tfrac{1}{2}}\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{\lambda }^{2}(\theta )={\lambda }_{ab}^{2}{\lambda }_{c}/{({\lambda }_{c}^{2}{\cos }^{2}(\theta )+{\lambda }_{ab}^{2}{\sin }^{2}(\theta ))}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 6 }$$

By integrating equation (5) or (6) over all solid angles, we obtain an angular average where for example $\langle 1/{{\lambda }}^{2}(\theta )\rangle ,$ the angular average of the inverse of the G–L penetration depth squared, for a collection of random oriented grains, is

$$\begin{eqnarray}&&\phantom{\rule[-0000]{}{2.3ex}}\langle \displaystyle \frac{1}{{\lambda }^{2}(\theta )}\phantom{\rule[-0000]{}{2.3ex}}\rangle =\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{\pi }\displaystyle \frac{1}{{\lambda }^{2}(\theta )}\,\sin \,\theta {\rm{d}}\theta .\end{eqnarray} \tag{ 7 }$$

Numerical integration of equation (7) with values of ${\lambda }_{{c}}$ = 916 nm and ${\lambda }_{ab}$ = 138 nm [19] and using an average grain size of 100 nm, gives $\langle {\chi }_{{\rm{g}}}^{{\prime} }/{\chi }_{{\rm{b}}}^{{\prime} }\rangle$ = 1.8 × 10^–2. This value is similar to that given in figure 11 for Y1MHA(3) and figure 12 inset for Y2MHA(3), consistent with a reversible ac signal entirely from within the nanocrystalline grains. We note that this calculation does not account for the induced moment and the applied field not being parallel or demagnetisation factors [58]. Figure 13 shows the irreversibility field $({B}_{{\rm{Irr}}}(T))$ as a function of temperature for our samples, taken from the onset of the ac susceptibility data. The data were fitted using the equation [31].

$$\begin{eqnarray}&&{B}_{{\rm{Irr}}}(T)={B}_{{\rm{Irr}}}(0){(1-{t}^{0.5})}^{2.1},\end{eqnarray} \tag{ 8 }$$

where $t=T/{T}_{{\rm{c}}}.$ The grains in the Y1HA and Y2HA samples have the highest superconducting critical properties of our samples. Of the microcrystalline materials, Y1H and Y2H have among the lowest ${T}_{{\rm{c}}}$ and ${B}_{{\rm{Irr}}}(0),$ lower than Y1MHA(3) and Y1MPA, which demonstrates the severity of the oxygen loss that the samples suffered during the HIP process. The onset ${T}_{{\rm{c}}}$ and ${B}_{{\rm{Irr}}}(0)$ values derived using equation (8) are listed in table 2.

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Dc magnetisation hysteresis data were also taken with the PPMS. At each temperature, the field was swept from 0 T down to −1.5 T (or –2 T in some cases), then swept up to 8.5 T and back to –1.5 T. This approach meant we could extract values of ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B$ as the magnetisation changed from the upper branch to the lower branch, as well as magnetisation ${J}_{{\rm{c}}}$ values calculated using Bean's model [59], as shown in table 2. For pellets of radius $R$ and volume $V,$

$$\begin{eqnarray}&&{J}_{{\rm{c}}}=3\displaystyle \frac{{\rm{\Delta }}m}{RV},\end{eqnarray} \tag{ 9 }$$

where ${\rm{\Delta }}m$ is the difference in magnetic moment between the increasing and decreasing field branches. For rectangular bars with length $w$ and width $b,$

$$\begin{eqnarray}&&{J}_{{\rm{c}}}=\displaystyle \frac{2{\rm{\Delta }}m}{w\left(1-\tfrac{w}{3b}\right)V}.\end{eqnarray} \tag{ 10 }$$

Typical hysteresis and ${J}_{{\rm{c}}}$ data for microcrystalline materials are shown in figures 14 and 15. In this paper, we assume that the currents flowing are either entirely intergranular or intragranular, or both. We set aside the possibility of clusters of well-connected grains. Given that the measured transport ${J}_{{\rm{c}}}$ is only of the order of 10⁵ Am^–2 in microcrystalline materials, intergranular ${J}_{{\rm{c}}}$ contributes typically less than 1% of the total dc magnetisation signal in-field and can be ignored. Hence we conclude that the dc magnetisation signal comes predominantly from hysteretic screening currents flowing within grains. The typical response for nanocrystalline materials is shown in figure 16. The data show a paramagnetic background with superconducting hysteresis which has been observed in ac susceptibility data in other granular materials in the literature [60, 61]. The lower panel of figure 16 shows the data after the paramagnetic background has been subtracted, showing a typical Type-II superconductor hysteresis curve. Straumal et al [62, 63] have shown that in ZnO, a high density of grain boundaries leads to ferromagnetism even without doping, but also that the solubility of magnetic contaminants such as Co can significantly increase with the density of grain boundaries. To investigate the effect of contamination, the WC/Co vial and balls were milled without any powder (except for that caked onto the surfaces) which yielded mainly WC/Co powder with small amounts of YBCO. The contaminants were pressed into a pellet and measured using the same method as the superconducting samples. These data are shown in the inset of figure 16. The magnetisation of contaminants are temperature-independent around 0 T, which is different to the background from the sample, consistent with the expectation that the extent of WC/Co contamination and its ferromagnetic contribution to the magnetisation are low. Hence, as with the microcrystalline samples, the dc magnetisation signal from the nanocrystalline samples is almost entirely due to screening currents flowing within the grains. Figure 17 shows a compilation of the intragranular magnetisation ${J}_{{\rm{c}}}$ for both the microcrystalline and nanocrystalline samples (we note that the uncertainty in the grain size is typically about a factor of two) and also contains transport ${J}_{{\rm{c}}}$ values for commercial YBCO tape [13]. Given that in our polycrystalline samples the current flows both along the ab-planes and along the c-direction, whereas ${J}_{{\rm{c}}}$ values in commercial tapes only flows along the ab-plane for the two configurations given, figure 17 shows that the intragranular ${J}_{{\rm{c}}}$ values in our polycrystalline samples are high. The best microcrystalline samples have intragranular ${J}_{{\rm{c}}}$ comparable to that of tapes, and strikingly the field dependence for all the samples that have been annealed is very similar to the commercial tapes. The samples that were HIP'ed-only (Y1H and Y2H) show a more drastic decrease in ${J}_{{\rm{c}}}$ with magnetic field compared to other microcrystalline samples, and at 8 T, have ${J}_{{\rm{c}}}$ comparable to that of the nanocrystalline group. We attribute the poorer in-field properties of some of our samples to the decrease in oxygen content during HIP'ing, consistent with the decrease in ${T}_{{\rm{c}}}$ and ${B}_{{\rm{Irr}}}(0)$ seen in the ac susceptibility data. After annealing (Y1HA and Y2HA), ${T}_{{\rm{c}}},$ ${B}_{{\rm{Irr}}}(0)$ and ${J}_{{\rm{c}}}$ have all recovered. Compared to commercial YBCO tape, transport ${J}_{{\rm{c}}}$ of microcrystalline materials is 10⁶ lower, and for nanocrystalline material Y2MHA(2) (not included on this graph) this difference increases to 10⁹.

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In addition to finding a clear intragranular signal associated with superconductivity for the nanocrystalline materials, not found using standard ac susceptibility measurements, we can use field reversal in the dc magnetisation measurements $({\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B).$ With these data we can address the type of pinning. Using Bean's relation for a cylinder, $| {\rm{\Delta }}B| =2{\mu }_{0}{J}_{{\rm{c}}}R$ where $| {\rm{\Delta }}B|$ is the magnitude of the field required to reverse the magnetisation, equation (9) gives [64]

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{0}{\rm{\Delta }}M}{{\rm{\Delta }}B}=-\displaystyle \frac{\tfrac{1}{3}{J}_{{\rm{c}}}R}{2{J}_{{\rm{c}}}R}=-0.17,\end{eqnarray} \tag{ 11 }$$

where the negative sign comes from Lenz's law. Figure 18 shows minor hysteresis loops taken at 10 K for Y1P. The inset of figure 18 shows that ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B$ is only weakly field dependent. At very low fields, ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B$ increases, associated with the increased role of reversible screening currents flowing at the surface of the sample. The ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B$ values in table 2 were obtained from the field reversal data at –1.5 or –2 T, calculated from the linear region during the initial field reversal. In most microcrystalline materials, typical values of ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B$ are approximately –0.17, consistent with bulk pinning in Bean's model. For nanocrystalline materials, the values of ${\mu }_{0}{\rm{\Delta }}M/{\rm{\Delta }}B,$ derived from data similar to that in figure 16 are typically 3 orders of magnitude smaller. These small values, compiled in table 2, have been found in the work of Shimizu and Ito [65] and cannot be explained by bulk pinning using Bean's model. We attribute the low values to the surface pinning in the grains, consistent with dc magnetisation signals that are predominantly intragranular. Hence, the magnetisation ${J}_{{\rm{c}}}$ we have calculated using grain size dimensions, provides a lower bound for the grain's surface pinning ${J}_{{\rm{c}}}.$

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While fabricating nanocrystalline materials, it is reasonable to ask first, how small grains can be before they can no longer be considered bulk material. Deutscher et al [66] have provided three methods for calculating the minimum size required to sustain superconductivity in LTSs. The first is the condition that superconductivity is quenched when the fluctuations in the order parameter $(\delta {\rm{\Psi }})$ are of the same order as the order parameter $({{\rm{\Psi }}}_{0}),$ which leads to

$$\begin{eqnarray}&&\tfrac{\langle {| \delta {\rm{\Psi }}| }^{2}\rangle }{{| {{\rm{\Psi }}}_{0}| }^{2}}\approx \tfrac{{k}_{{\rm{B}}}T}{2{E}_{{\rm{c}}}{V}_{{\rm{\min }}}},\end{eqnarray} \tag{ 12 }$$

where ${E}_{{\rm{c}}}$ is the condensation energy and ${V}_{{\rm{\min }}}$ is the minimum volume of a grain that still sustains superconductivity. The second is when there is only one Cooper pair per grain so that

$$\begin{eqnarray}&&N(0){\rm{\Delta }}{V}_{{\rm{\min }}}\approx 1,\end{eqnarray} \tag{ 13 }$$

where $N(0)$ is the density of states at $T=0$ and ${\rm{\Delta }}$ is the superconducting energy gap. The third is when the separation of quasi-particle energy levels $\delta$ is of the order of ${\rm{\Delta }},$ which leads to the equation

$$\begin{eqnarray}&&{r}_{{\rm{\min }}}={\left(\tfrac{8\pi }{3\mathrm{\times }0.18}{\xi }_{0}{\lambda }_{{\rm{F}}}^{2}\right)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 14 }$$

where ${r}_{{\rm{\min }}}$ is the minimum radius and ${\lambda }_{{\rm{F}}}$ is the Fermi wavelength. Deutscher made assumptions that are only strictly justified for LTSs. However, if we naively apply these methods to YBCO, then we obtain ${r}_{{\rm{\min }}}$ = 0.3–1 nm (using literature values of ${E}_{{\rm{c}}}$ = 0.063 ${k}_{{\rm{B}}}{T}_{{\rm{c}}}$ per unit cell [67], $N(0)$ = 2.10 × 10²⁸ m^–3 eV^–1 [68], ${\rm{\Delta }}$ = 30 meV [59], ${\lambda }_{{\rm{F}}}$ = 0.3 nm [69] and ${\xi }_{0}$ = 1.5 nm [70]). These calculations suggest even the (100 nm) grains in our nanocrystalline YBCO are sufficiently large to be well within the bulk material regime.

Without understanding why high angle grain boundaries do not support high ${J}_{{\rm{c}}},$ we cannot know why ${J}_{{\rm{c}}}$ is low in polycrystalline materials. The standard explanation for the Dimos results that showed ${J}_{{\rm{c}}}$ decreases with increased misorientation angle in [001] tilt boundaries is that grain boundaries act as 'weak-links'. However, this does not clarify whether the low ${J}_{{\rm{c}}}$ values found by Dimos were due to poor coupling across the grain boundaries or weak flux pinning in the grain boundaries. TDGL calculations suggest that the surface properties at the ends of any junction strongly affect the current the junction can carry as well as the interior of the junction, which undermines comparisons between single junctions and bulk properties. Other possible explanations for low ${J}_{{\rm{c}}}$ values could include the nature of the fundamental mechanism for superconductivity itself or perhaps the underlying symmetry of the $d$-wave order parameter. The low carrier density or specific electronic structure that leads to HTS or the phononic structure may also have been responsible. Despite the range of possibilities, a review of the literature shows that in S-N-S junctions, the effective resistivity of the normal layer in the junction can easily vary from a factor of 10² times higher than the bulk resistivity of normal material in the junction, as found for Pb/Cd/Pb [71], and up to a factor of 10⁴ times higher, as is the case for Nb/Al/Nb [72] [73], Pb/Cu/Pb [74] [75], and YBCO/Au/YBCO [76]. Recent work in our group on YBCO tapes has shown that large interfacial resistances of 2.5 × 10^–8 Ω cm² can even occur between a silver interface and a YBCO layer [77]. Given the potential for highly resistive interfaces and highly resistive grain boundaries in YBCO, in this paper we try to quantify how much the high resistivity of the grain boundaries lowers the critical current density of polycrystalline YBCO. We consider the most simple case, where the grain boundaries are modelled as a highly resistive N-component (i.e. where the normal layer has ${T}_{{\rm{c}}}$ = 0) of an S-N-S junction.

We first calculate the expected resistivity of a randomly aligned polycrystalline YBCO sample with completely transparent grain boundaries (i.e. normal grain boundaries with zero resistivity). The angular resistivity ${\rho }_{{\rm{N}}}(\theta )$ of a YBCO single crystal when the transport current is at angle $\theta$ with the c-axis in spherical coordinates is given by [78]:

$$\begin{eqnarray}&&{\rho }_{{\rm{N}}}(\theta )={\rho }_{c}{\cos }^{2}\theta +{\rho }_{ab}{\sin }^{2}\theta ,\end{eqnarray} \tag{ 15 }$$

where ${\rho }_{c}$ is the resistivity along the c-axis and ${\rho }_{ab}$ is the resistivity along the ab-plane. Integrating ${\rho }_{{\rm{N}}}(\theta )$ through all solid angles, gives the angular averaged resistivity ${\rho }_{{\rm{N}}}$ where:

$$\begin{eqnarray}&&\langle {\rho }_{{\rm{N}}}\rangle =\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{\pi }{\rho }_{{\rm{N}}}(\theta )\,\sin \,\theta \,{\rm{d}}\theta =\displaystyle \frac{1}{3}(2{\rho }_{ab}+{\rho }_{c}).\end{eqnarray} \tag{ 16 }$$

In equating $\langle {\rho }_{{\rm{N}}}\rangle$ to the resistivity of a randomly aligned polycrystalline material, one is assuming that there is no redistribution or preferential percolation of the current along low resistivity paths. We can assess whether this approach is valid by considering polycrystalline graphite. Graphite is a good choice because it has very low resistivity grain boundaries. Single crystal resistivity values for graphite are: ${\rho }_{ab}=6\times {10}^{-5}\,{\rm{\Omega }}{\rm{m}}$ and ${\rho }_{c}=6\times {10}^{-3}\,{\rm{\Omega }}{\rm{m}}$ [79]. Polycrystalline graphite has a resistivity of $2\times {10}^{-3}\,{\rm{\Omega }}{\rm{m}}$ [80] which is consistent with the value of $\langle {\rho }_{{\rm{N}}}\rangle$ from equation (16). A similar calculation for YBCO using the resistivity of single crystals, where ${\rho }_{ab}=6\times {10}^{-7}\,{\rm{\Omega }}{\rm{m}}$ and ${\rho }_{c}=1.5\times {10}^{-4}\,{\rm{\Omega }}{\rm{m}}$ at 100 K [81], gives $\langle {\rho }_{{\rm{N}}}\rangle =5.0\times {10}^{-5}\,{\rm{\Omega }}{\rm{m}}.$ Figure 6 shows the resistivity of sample Y1HA and compares it to values for single crystals and $\langle {\rho }_{{\rm{N}}}\rangle .$ Sample Y1HA has a resistivity about 50% higher than $\langle {\rho }_{{\rm{N}}}\rangle$. Given the very high values of critical parameters for the grains of this material (${T}_{{\rm{c}}},$ ${B}_{{\rm{Irr}}}(0)$ and magnetisation ${J}_{{\rm{c}}}^{{\rm{m}}}$), we attribute the enhanced resistivity (2.4 × 10^–5 Ωm) to the resistivity of grain boundaries. Using a grain size of approximately 5 μm and a grain boundary thickness of approximately 1 nm [82] leads to a large grain boundary resistivity of ${\rho }_{{\rm{GB}}}\approx 0.12\,{\rm{\Omega }}{\rm{m}}.$ Using a similar approach to the resistivity data for the nanocrystalline materials in table 2, a grain size of 100 nm gives a very large grain boundary resistivity of ${\rho }_{{\rm{GB}}}=8.2\,{\rm{\Omega }}{\rm{m}}.$ In terms of contact (areal) resistivity, the micro and nanocrystalline grain boundary resistivities are $1.2\times {10}^{\mbox{--}10}\,{\rm{\Omega }}{{\rm{m}}}^{2}$ and $8.2\times {10}^{\mbox{--}9}\,{\rm{\Omega }}{{\rm{m}}}^{2}$ respectively. These values can be compared to the contact resistivities for some [001] tilt grain boundaries in thin-film oxide bicrystals (including YBCO) [83], which are generally lower and in the range of ${10}^{\mbox{--}14}$– ${10}^{\mbox{--}11}\,{\rm{\Omega }}{{\rm{m}}}^{2}.$ We note that one can expect the resistivity of grain boundaries with misorientation angles that can include all possible angles to be higher than the [001] tilt, strain-free bicrystal grain boundaries. The resistivity data in figure 6 for Y1HA also provide supporting evidence for the additional resistance of the grain boundaries being similar to that of the grains: after the initial onset of the superconducting transition of the grains at 92 K, there is an inflection at ∼83 K, which we attribute to the grain boundaries starting to carry significant current. The lower temperature part of the transition is much more strongly depressed by the magnetic field than the onset transition which is similar to that observed elsewhere [36] and consistent with the in-field properties of grain boundaries [3].

In this section we calculate the reduction in the local depairing current density $({J}_{{\rm{DN}}})$ in the boundary caused by its high resistivity. Recently we have found analytic solutions to the G–L equations in zero field for the ${J}_{{\rm{DN}}}(T)$ in a 1D S-N-S junction in the clean and dirty limit [8] where:

$$\begin{eqnarray}{J}_{{\rm{DN}}}(T) & \approx & {J}_{{\rm{DSc}}}(T)\sqrt{2}\tfrac{{\rho }_{{\rm{S}}}{\xi }_{{\rm{S}}}}{{\rho }_{{\rm{N}}}{\xi }_{{\rm{N}}}}\left\{\sqrt{{\left(\tfrac{{\xi }_{{\rm{S}}}{\rho }_{{\rm{S}}}}{{\xi }_{{\rm{N}}}{\rho }_{{\rm{N}}}\sqrt{2}}\right)}^{2}+1}\right.\\ & & -{\left.\tfrac{{\xi }_{{\rm{S}}}{\rho }_{{\rm{S}}}}{{\xi }_{{\rm{N}}}{\rho }_{{\rm{N}}}\sqrt{2}}\right\}}^{2}\exp \left(\tfrac{-2d}{{\xi }_{{\rm{N}}}}\right),\end{eqnarray} \tag{ 17 }$$

where ${J}_{{\rm{DSc}}}(T)$ is the depairing current density in the superconducting grain, ${\rho }_{{\rm{S}}}/{\rho }_{{\rm{N}}}$ is the ratio of the resistivity in the grain to the grain boundary, ${\xi }_{{\rm{S}}}/{\xi }_{{\rm{N}}}$ is the ratio of the G–L coherence length in the superconductor to the decay length of the order parameter in the normal grain boundary, and $d$ is the thickness of the grain boundary. Because equation (17) provides zero-field values, we use it to provide upper bound values for ${J}_{{\rm{DN}}}$ in the grain boundaries of polycrystalline YBCO (${\xi }_{{\rm{S}}}$ values in table 1 were used in these calculations). To simplify the analysis for the anisotropic materials, we have only considered angular averaged properties to calculate an angular average for ${J}_{{\rm{DN}}}$ (i.e. ${J}_{{\rm{DN}}}$) in table 3, where we have used

$$\begin{eqnarray}&&\langle {J}_{{\rm{DSc}}}(T)\rangle \approx \displaystyle \frac{{{\rm{\Phi }}}_{0}}{3\sqrt{3}\pi {\mu }_{0}}\left\langle \displaystyle \frac{1}{{\lambda }^{2}(T)\xi (T)}\right\rangle ,\end{eqnarray} \tag{ 18 }$$

where $\langle \lambda \rangle$ and $\langle \xi \rangle$ are the angular average G–L penetration depth and G–L coherence length and can be found in table 1. The superconducting parameters in equation (17) are well established. Microscopic theory gives the clean coherence length as

$$\begin{eqnarray}&&{\xi }_{{\rm{SClean}}}(T)\approx \displaystyle \frac{\hslash {v}_{{\rm{F}}}}{1.76\pi {k}_{{\rm{B}}}{T}_{{\rm{cS}}}^{\tfrac{1}{2}}{({T}_{{\rm{cS}}}-T)}^{\tfrac{1}{2}}}\end{eqnarray} \tag{ 19 }$$

and the dirty coherence length

$$\begin{eqnarray}&&{\xi }_{{\rm{SDirty}}}(T)\approx {\left(\displaystyle \frac{\pi \hslash {D}_{{\rm{S}}}}{8{k}_{{\rm{B}}}({T}_{{\rm{cS}}}-T)}\right)}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 20 }$$

where ${T}_{{\rm{cS}}}$ is the critical temperature of the superconducting layer. We can use the relation for the diffusivity given by:

$$\begin{eqnarray}&&{D}_{{\rm{S}}}\approx \displaystyle \frac{{\pi }^{2}{k}_{{\rm{B}}}^{2}}{3{e}^{2}{\rho }_{{\rm{S}}}\gamma }=\displaystyle \frac{{v}_{{\rm{F}}}l}{3}=\displaystyle \frac{m{v}_{{\rm{F}}}^{2}}{3{e}^{2}{\rho }_{{\rm{S}}}n},\end{eqnarray} \tag{ 21 }$$

where the equivalent forms in equation (21) have been derived using standard relations [84] for resistivity and the Sommerfeld constant $(\gamma ),$ and also for the angular averaged Fermi velocity $({v}_{{\rm{F}}})$ and mean free path $(l)$ in terms of number of valence electrons per unit volume $(n).$ Accurate values of these microscopic parameters are critical to the calculation of ${J}_{{\rm{DN}}}.$ To test the validity of these values listed in table 3, we used Pippard's approach to find a coherence length $({\xi }_{{\rm{SPippard}}}(0))$ from the clean and dirty values [8] using

$$\begin{eqnarray}&&{\xi }_{{\rm{S}}{\rm{Pippard}}}(0)\approx {\left(\displaystyle \frac{1}{{\xi }_{{\rm{SClean}}}(0)}+\displaystyle \frac{1}{{\xi }_{{\rm{SDirty}}}(0)}\right)}^{-1}.\end{eqnarray} \tag{ 22 }$$

We suggest that the microscopic parameters are reasonable values since a comparison between ${\xi }_{{\rm{S}}{\rm{Pippard}}}(0)$ and ${\xi }_{{B}{\rm{c}}2}(0)$ (see table 3), shows they are similar. We have used the Ginzburg–Landau relation (that is strictly only valid close to ${T}_{{\rm{c}}}$) to define ${\xi }_{{B}{\rm{c}}2}(0)$ where

$$\begin{eqnarray}&&{B}_{{\rm{c}}2}(0)\approx \displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {\xi }_{B{\rm{c}}2}^{2}(0)\,}.\end{eqnarray} \tag{ 23 }$$

Within the context of the G–L theory, the temperature dependence of ${\xi }_{{\rm{N}}}(T)$ is given by

$$\begin{eqnarray}&&{\xi }_{{\rm{N}}}(T)\approx {\xi }_{{\rm{N}}}({T}_{{\rm{cS}}}){\left(\displaystyle \frac{{T}_{{\rm{cS}}}-{T}_{{\rm{cN}}}}{T-{T}_{{\rm{cN}}}}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 24 }$$

We have followed Pippard's approach and related ${\xi }_{{\rm{N}}{\rm{P}}{\rm{i}}{\rm{p}}{\rm{p}}{\rm{a}}{\rm{r}}{\rm{d}}}({T}_{{\rm{cS}}})$ to the microscopic clean and dirty limits [8] using

$$\begin{eqnarray}&&{\xi }_{{\rm{NPippard}}}({T}_{{\rm{cS}}})\approx {\left(\displaystyle \frac{1}{{\xi }_{{\rm{NClean}}}({T}_{{\rm{cS}}})}+\displaystyle \frac{1}{{\xi }_{{\rm{NDirty}}}({T}_{{\rm{cS}}})}\right)}^{-1},\end{eqnarray} \tag{ 25 }$$

where

$$\begin{eqnarray}&&{\xi }_{{\rm{NClean}}}({T}_{{\rm{cS}}})\approx \displaystyle \frac{\hslash {v}_{{\rm{F}}}}{1.76\pi {k}_{{\rm{B}}}{T}_{{\rm{cN}}}^{\tfrac{1}{2}}{({T}_{{\rm{cS}}}-{T}_{{\rm{cN}}})}^{\tfrac{1}{2}}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\xi }_{{\rm{NDirty}}}({T}_{{\rm{cS}}})\approx {\left(\displaystyle \frac{\pi \hslash {D}_{{\rm{N}}}}{8{k}_{{\rm{B}}}({T}_{{\rm{cS}}}-{T}_{{\rm{cN}}})}\right)}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 27 }$$

in which ${D}_{{\rm{N}}}$ is the normal layer equivalent of ${D}_{{\rm{S}}}$ as given by equation (21) and ${T}_{{\rm{cN}}}$ is the critical temperature of the normal layer. The largest uncertainties in calculating ${J}_{{\rm{DN}}}$ are associated with the values of the microscopic properties of the grain boundaries. We have simplified the analysis by assuming that grain boundary structures are not superconducting (i.e. ${T}_{{\rm{cN}}}=0$) and that we can use resistivity to characterise the difference between the microscopic properties of the grains and the grain boundaries. Grain boundaries are complex structures and one can expect that when their resistivity changes, their carrier concentration, effective width and scattering time all change. Our assumptions lead to ${\xi }_{{\rm{NPippard}}}({T}_{{\rm{cS}}})={\xi }_{{\rm{NDirty}}}({T}_{{\rm{cS}}})$ and

$$\begin{eqnarray}&&{\xi }_{{\rm{N}}}(T)\approx {\xi }_{{\rm{NDirty}}}({T}_{{\rm{cS}}}){\left(\displaystyle \frac{{T}_{{\rm{cS}}}}{T}\right)}^{\tfrac{1}{2}}={\xi }_{{\rm{SDirty}}}(0){\left(\displaystyle \frac{{\rho }_{{\rm{S}}}}{{\rho }_{{\rm{N}}}}\displaystyle \frac{{T}_{{\rm{cS}}}}{T}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 28 }$$

As a starting point for calculating $\langle {J}_{{\rm{DN}}}(0)\rangle ,$ in table 3 we have assumed that the grain boundary thicknesses in all the materials (i.e. $d$ in equation (17)) is 1 nm. Equations (17) and (28) show that as the resistivity of the grain boundaries increases, ${\xi }_{{\rm{N}}}$ decreases and $\langle {J}_{{\rm{DN}}}(0)\rangle$ is severely depressed. In the microcrystalline YBCO materials of this work, the resistivity is more than three orders of magnitude higher than the grains, which by itself provides a straightforward explanation for the very low transport current densities we have experimentally measured. The calculations of $\langle {J}_{{\rm{DN}}}(0)\rangle \,$are particularly sensitive to the values of $d$ and ${\xi }_{{\rm{N}}}.$ Once ${\xi }_{{\rm{N}}}$ is smaller than $d,$ the exponential term in equation (17) dominates. In this regime, small increases in $d$ produce very large reductions in $\langle {J}_{{\rm{DN}}}(0)\rangle .$ The resistivity of our nanocrystalline materials is even higher than the microcrystalline values and $\langle {J}_{{\rm{DN}}}(0)\rangle$ even lower. We note that these very high values of resistivity are beyond the Ioffe–Regel criterion (i.e. ${k}_{{\rm{F}}}l\lt 1$ [85]). However, we conclude that even the straightforward analysis provided here demonstrates that the values of grain boundary resistivity we have measured in microcrystalline and nanocrystalline materials are sufficient to explain the low values of our transport ${J}_{{\rm{c}}}$ data.

Table 3.  Resistivity of the normal layer ${\rho }_{{\rm{N}}}$ at ${T}_{{\rm{c}}},$ Sommerfeld constant $(\gamma ),$ Fermi velocity $({v}_{{\rm{F}}}),$ the superconducting layer coherence length in the clean and dirty limits (${\xi }_{{\rm{SClean}}}$ and ${\xi }_{{\rm{SDirty}}}$), comparison of the Pippard coherence length $({({\xi }_{{\rm{SPippard}}})}^{-1}={({\xi }_{{\rm{SClean}}})}^{-1}+{({\xi }_{{\rm{SDirty}}})}^{-1})$ with the ${\xi }_{{B}_{{\rm{c}}2}}(0)$ coherence length calculated from ${B}_{{\rm{c}}2}$ where $({\xi }_{{B}_{{\rm{c}}2}}(0)={({{\rm{\Phi }}}_{0}/2\pi {B}_{{\rm{c}}2}(0))}^{\tfrac{1}{2}}),$ the normal layer diffusivity $({D}_{{\rm{N}}})$ calculated using $\gamma$ values, decay length of the normal layer in a Josephson junction $({\xi }_{{\rm{N}}}),$ and the ratio of the local depairing current density of the grain boundary to the depairing current density of the superconductor: ${J}_{{\rm{DN}}}/{J}_{{\rm{DSc}}},$ given by equation (17). For angular averaged (A.A.), microcrystalline and nanocrystalline YBCO, we have taken the resistivity values from measurements in this work and assumed that the grain boundaries are 1 nm thick. For all other materials, we have again assumed that the grain boundaries are 1 nm thick normal grain boundaries, but that the resistivity is equal to the resistivity of the grains. For all materials, we have assumed that the Sommerfeld constant and the Fermi velocity are an angular average in the calculation of ${J}_{{\rm{DN}}}(4.2\,{\rm{K}})/\langle {J}_{{\rm{D}}{\rm{s}}{\rm{c}}}(4.2\,{\rm{K}})\rangle .$

Material	${\rho }_{{\rm{N}}}\,({T}_{{\rm{c}}})$ (Ω m)	$\gamma$ (J m−3 K−2)	${v}_{{\rm{F}}}$ (105 m s−1)	${\xi }_{{\rm{SClean}}}(0)$ (nm)	${\xi }_{{\rm{SDirty}}}(0)$ (nm)	${\xi }_{{\rm{S}}{\rm{Pippard}}}(0)$ ${\xi }_{{B}_{{\rm{c}}2}}(0)$ (nm)	${D}_{{\rm{N}}}$ (m2 s−1)	${\xi }_{{\rm{N}}}(4.2\,{\rm{K}})$ (nm)	$\displaystyle \frac{{J}_{{\rm{DN}}}\,(4.2\,\,{\rm{K}})}{\langle {J}_{{\rm{DSc}}}\,(4.2\,{\rm{K}})\rangle }$
NbTi	4.0 × 10−7	1.1 × 103	2.0	31	4.3	3.8	5.7 × 10−5	6.4	0.28
	[127]	[127]	[59, 127]			4.6
Nb3Sn	8.8 × 10−8	1.2 × 103	0.60	4.6	6.3	2.7	2.4 × 10−4	13	0.21
	[16]	[33]	[128]			3.3
PbMo6S8	7.0 × 10−7	3.8 × 102	0.40	4.0	4.5	2.1	9.2 × 10−5	8.1	0.21
	[34]	[34]	[128]			2.4
MgB2	1.0 × 10−6	1.5 × 102	6.1	22	3.5	3.0	1.6 × 10−4	10	0.28
(A.A.)	[129]	[130]	[131]			4.3
YBa2Cu3O7−x	5.0 × 10−7	2.0 × 102	5.0	7.5	2.8	2.1	2.4 × 10−4	13	0.16
(ab-plane)		[132]	[56]			1.5
YBa2Cu3O7−x	5.0 × 10−5	2.0 × 102	5.0	7.5	0.28	0.27	2.4 × 10−6	1.3	0.077
(A.A., 5 μm)		[132]	[56]			1.5
YBa2Cu3O7−x	1.2 × 10−1	2.0 × 102	5.0	7.5	0.28	0.27	1.0 × 10−9	0.027	10−35
(Micro, 5 μm)		[132]	[56]			1.5
YBa2Cu3O7−x	8.2	2.0 × 102	5.0	7.5	0.28	0.27	1.5 × 10−11	0.0033	10−270
(Nano, 100 nm)		[132]	[56]			1.5

We have found, using a combination of transport measurements and ac magnetic susceptibility measurements, that our nano- and microcrystalline YBCO can be considered to be high quality grains surrounded by non-superconducting highly resistive grain boundaries that limit transport ${J}_{{\rm{c}}}.$ The current densities within the grains are high and those across the grain boundaries low. We have adopted a pragmatic approach to the analysis of the grain boundaries in our samples and used angular averages for this strongly anisotropic material and ignored percolation. We have also adopted a simple two-component description of our materials as grains and grain boundaries and used an S-N-S description that assumes that the pinning is sufficiently strong along the grain boundaries that ${J}_{{\rm{c}}}$ is determined by the local depairing current density in the grain boundaries themselves. One can expect in superconducting materials there are a range of different length scales for the variations in composition, strain and physical structure as well as electronic and phononic structure. Although it is reasonable for us to have assumed that grain boundary structures are not superconducting themselves in bulk form, finding the other characteristic microscopic properties of the grain boundaries $(\gamma ,{v}_{{\rm{F}}},l,n)$ will be a formidable challenge. Characterising bulk materials in thin film form proved difficult enough for the scientific community with the luxury of free-standing samples, or samples on insulators and data for parent bulk samples. Grain boundaries can be considered as internal surfaces. They bring the challenges of characterising a structure that is inhomogenous, is sandwiched between two grains, and is not available in bulk form [86]. Given the large effort it has taken to understand bulk materials, one can expect an even larger effort will be required for the local normal and superconducting properties of grain boundaries.

From a technological perspective, we would like to increase ${J}_{{\rm{c}}}$ in polycrystalline YBCO if at all possible. The [001] tilt boundaries in the literature have contact resistivities which vary from $2\times {10}^{-14}\,{\rm{\Omega }}{{\rm{m}}}^{2}$ in the low angle bicrystal data [3], up to ${10}^{-11}\,{\rm{\Omega }}{{\rm{m}}}^{2}$ for 40° boundaries [87]. The average contact resistivities in our microcrystalline samples were ${10}^{-10}\,{\rm{\Omega }}{{\rm{m}}}^{2}$ and in the nanocrystalline samples ${10}^{-8}\,{{\rm{\Omega }}{\rm{m}}}^{2}.$ Higher average contact resistivity values are expected in the polycrystalline samples than bicrystals because the misorientation angles are not limited to [001] tilt grain boundaries. We have modelled grain boundaries, as simple non-superconducting resistive layers. This ignores complexities such as the possibility that very low angle grain boundaries may have non-zero critical temperature and high angle grain boundaries may be ferromagnetic [62, 63], which in the G–L framework can be described using negative local critical temperatures. However, table 3 shows that even if we just apply our simple model, and reduce the grain boundary resistivity to be the same as the grains, the value of ${J}_{{\rm{DN}}}/{J}_{{\rm{DSc}}}$ along the ab-plane is still 0.16, which is similar to that in the best YBCO tapes. For bulk YBCO with equal grain and grain boundary resistivity, ${J}_{{\rm{DN}}}/{J}_{{\rm{DSc}}}$ is further reduced to 0.077. We may hope that faceted grain boundaries [88] may help but given the high grain boundary resistivities found at high angles in bicrystal data, it seems unlikely that grain boundary engineering will enable high ${J}_{{\rm{c}}}$ in polycrystalline YBCO. Grain boundary engineering such as Ca doping has successfully reduced grain boundary contact resistivity by over an order of magnitude and increased ${J}_{{\rm{c}}}$ by a factor of 6–7, however these results were on bicrystals preferentially doped at grain boundaries [5], thin film bicrystals [89] or multilayer films [90]. We consider it unlikely that one can reduce the grain boundary resistivity in polycrystalline materials to be less than that of the grains themselves and therefore that grain boundary engineering is unlikely to be successful in increasing ${J}_{{\rm{c}}}$ to technologically useful values, in randomly aligned polycrystalline YBCO.

The data in the second panel of figure 1 shows that in high fields, all technological superconductors have a large headroom for improvement. In applied fields of half ${B}_{{\rm{c}}2}(4.2\,{\rm{K}}),$ NbTi is only a few percent of its theoretical limit, and Nb₃Sn(IT) produced using the internal tin route is only about 0.3% of its theoretical limit. Materials with the highest ${J}_{{\rm{c}}}/{J}_{{\rm{DSc}}}$ values include commercial YBCO tape [28], thin film Ba(FeCo)₂As₂ [91] and Nb [12] that have been fabricated with no large-angle grain boundaries. Hence one can expect high ${J}_{{\rm{D}}}$ throughout the entire material, and strong pinning to be effective. It is unclear whether adding even more pinning will lead to further improvements, or whether the low angle grain boundaries or channels formed by contiguous strained and degraded material between pinning sites provide channels for flux flow and hence limit ${J}_{{\rm{DN}}}/{J}_{{\rm{DSc}}}$ in these three materials. In addition to the thin film Ba-122 in figure 1, Ba-122 thin films with artificially engineered superlattices with strong pinning along the ab-plane and c-axis and improved ${J}_{{\rm{c}}}$ over a wide angular range [92] have been developed. Although untextured polycrystalline bulks and wires have achieved ${J}_{{\rm{c}}}$ of 10⁹ Am^–2 at 0 T and 4.2 K, which is more than 10 times higher than other round untextured ferropnictide wires, it is still 10 times lower than the thin film materials [93]. The materials with markedly low ${J}_{{\rm{c}}}/{J}_{{\rm{DSc}}}$ values include MgB₂ [94], FeSe_0.5Te_0.5 [95] and the A15 polycrystalline materials [96, 97]. Excellent polycrystalline, untextured MgB₂ can be produced cheaply through solid state reactions [98–101], as can FeSe_0.5Te_0.5 through sintering [102] or powder-in-tube process [103]. However, such FeSe_0.5Te_0.5 materials achieve ${J}_{{\rm{c}}}$ values of 10⁷ Am^–2 at 0 T and 4.2 K, which are two orders of magnitude lower than the tape sample in figure 1. After Nb₃Sn was discovered in 1954 [104], materials for magnets were fabricated with ${J}_{{\rm{c}}}$ values of 1.5 × 10⁹ Am^–2 at 8.8 T in 1961 [105]. By 1990, ${J}_{{\rm{c}}}$ values of 2.2 × 10⁹ Am^–2 at 11 T were achieved [106] and now state-of-the-art values are approximately 8 × 10⁹ Am^–2 at 11 T. Improvements in ${J}_{{\rm{c}}}$ are hard-earned and have enabled technological applications above 10 T including high-field MRI, particle accelerator magnets and fusion energy. However, an improvement in ${J}_{{\rm{c}}}$ by an order of magnitude over 50 years does not seem very large in the context of a headroom of nearly 3 orders of magnitude between ${J}_{{\rm{c}}}$ and the theoretical upper limit ${J}_{{\rm{DSc}}}.$ If flux flow along channels rather than pinning in the grains are the problem, equation (17) suggests high grain boundary resistivity and/or wide grain boundaries are responsible. The resistivity of Nb₃Sn and the other A15 intermetallics has been studied extensively. Very large changes in resistivity have been found in Nb₃Sn and V₃Si with relatively small changes in composition [107]. For binary Nb₃Sn, ${\rho }_{{\rm{N}}}=4\times {10}^{-8}\,{\rm{\Omega }}{\rm{m}}$ for 25 at% Sn and ${\rho }_{{\rm{N}}}=4\,\times {10}^{-7}\,{\rm{\Omega }}{\rm{m}}$ for 24.5 at% Sn. In a series of nanocrystalline HIP'ed Nb₃Sn fabricated in our group, materials with 120 nm grain size and ${T}_{{\rm{c}}}$ = 17.4 K and ${B}_{{\rm{c}}2}(0)$ = 30 T, ${J}_{{\rm{c}}}=3\times {10}^{9}\,{{\rm{Am}}}^{-2}$ at 0 T and 5 K, ${J}_{{\rm{c}}}=7\times {10}^{8}\,{{\rm{Am}}}^{-2}$ at 6 T and 5 K, were found to have resistivity values of ${\rho }_{{\rm{N}}}=6\times {10}^{-7}\,{\rm{\Omega }}{\rm{m}}$ [108]. When the grain size was reduced to 20 nm, ${T}_{{\rm{c}}}$ reduced to 10 K and ${\rho }_{{\rm{N}}}$ increased to $2\times {10}^{-6}\,{\rm{\Omega }}{\rm{m}}.$ When the grain size was below 10 nm, ${T}_{{\rm{c}}}$ was below 2 K and ${\rho }_{{\rm{N}}}$ increased to $4\times {10}^{-6}\,{\rm{\Omega }}{\rm{m}}$ [109]. These nanocrystalline results show that very high resistivity values can occur in Nb₃Sn and that a non-superconducting width for the grain boundaries of 3 nm in Nb₃Sn polycrystalline materials is not unreasonable. The quality of our nanocrystalline Nb₃Sn materials may simply have been poor, but the sustained low ${J}_{{\rm{c}}}/{J}_{{\rm{DSc}}}$ values in state-of-the-art materials may point to a more fundamental grain boundary problem. There are other factors may also help explain why the effective width of non-superconducting grain boundaries are large in A15 compounds. Band structure calculations point to the density of states at the Fermi level being determined by the one-dimensional chains of Nb atoms [107, 110]. Variable strain measurements show that a strain as low as 1% significantly depresses ${T}_{{\rm{c}}}$ [97, 111, 112]. Calculations using equation (17) show that for Nb₃Sn, low values of ${J}_{{\rm{DN}}}(4.2\,{\rm{K}})/{J}_{{\rm{DSc}}}(4.2\,{\rm{K}})$ of just 0.2% can occur, when the grain boundary width is 3 nm and ${\rho }_{{\rm{N}}}=4\times {10}^{-6}\,{\rm{\Omega }}{\rm{m}}.$ It follows that improvement in the ${J}_{{\rm{c}}}$ of Nb₃Sn can be achieved if the grain boundaries are improved or removed. However, the properties of the grains and the grain boundaries cannot be considered completely decoupled. If we improve the connectivity by lowering the resistivity of the grain boundaries, although we increase the local depairing current in the grain boundaries, we can expect to decrease the surface pinning that restrains the fluxons within the superconducting grains. Hence, we may need to add strong pinning sites into the grains themselves. Whether the improvements would be commercially viable is beyond the scope of this paper.

First we consider the G–L expression for the depairing current density of an anisotropic superconducting material such as YBCO or an isotropic superconductor. When the current is in the $ab$ plane, ${J}_{{\rm{DSc}}}^{ab}$ is given by:

$$\begin{eqnarray}&&{J}_{{\rm{DSc}}}^{ab}(T)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{3\sqrt{3}\pi {\mu }_{0}{\lambda }_{ab}^{2}(T){\xi }_{ab}(T)},\end{eqnarray} \tag{ 29 }$$

where ${{\rm{\Phi }}}_{0}$ is the flux quantum, ${\lambda }_{ab}(T)$ is the G–L penetration depth and ${\xi }_{ab}(T)$ is the G–L coherence length. Since there is no general theoretical expressions for ${B}_{{\rm{c}}2}(T),$ we use an empirical equation of the form:

$$\begin{eqnarray}&&{B}_{{\rm{c}}2}^{c}(T)={B}_{{\rm{c}}2}^{c}(0)(1-{(T/{T}_{{\rm{c}}})}^{\nu }).\end{eqnarray} \tag{ 30 }$$

Note in equation (30) that when the applied magnetic field points in the c-axis direction, the relevant length scale is the G–L coherence length in the ab-plane. We can differentiate equation (30) with respect to $T$ to obtain:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {B}_{{\rm{c}}2}^{c}(T)}{\partial T}\right|}_{T\approx {T}_{{\rm{c}}}}=-\displaystyle \frac{\nu }{{T}_{c}}{B}_{{\rm{c}}2}^{c}(0).\end{eqnarray} \tag{ 31 }$$

We use the G–L equation for ${B}_{{\rm{c}}2}(T)$ (correct only for $T\approx {T}_{{\rm{c}}}$) [113]:

$$\begin{eqnarray}&&{B}_{{\rm{c}}2}^{c}({\rm{T}})={\left.\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {\xi }_{ab}^{2}(T)}\right|}_{T\approx {T}_{{\rm{c}}}}\end{eqnarray} \tag{ 32 }$$

and the G–L expression for the G–L coherence length ${\xi }_{ab}(T),$ which is generally taken to be true for all temperatures, of the form:

$$\begin{eqnarray}&&{\xi }_{ab}(T)={\xi }_{ab}(0){(1-(T/{T}_{{\rm{c}}}))}^{-\mathrm{1/2}}.\end{eqnarray} \tag{ 33 }$$

We can differentiate equation (32) with respect to $T$ to obtain:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {B}_{{\rm{c}}2}^{c}(T)}{\partial T}\right|}_{T\approx {T}_{{\rm{c}}}}=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {\xi }_{ab}^{2}(0)}\left(-\displaystyle \frac{1}{{T}_{{\rm{c}}}}\right).\end{eqnarray} \tag{ 34 }$$

From equations (31) and (34) we have:

$$\begin{eqnarray}&&{\xi }_{ab}^{2}(0)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {B}_{{\rm{c}}2}^{c}(0)}\displaystyle \frac{1}{v}.\end{eqnarray} \tag{ 35 }$$

Substituting this equation into equation (33), we have:

$$\begin{eqnarray}&&{\xi }_{ab}^{2}(T)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {B}_{{\rm{c}}2}^{c}(0)}\displaystyle \frac{1}{\nu }\displaystyle \frac{1}{1-(T/{T}_{{\rm{c}}})}\end{eqnarray} \tag{ 36 }$$

Substituting equation (30) into (36) leads to:

$$\begin{eqnarray}&&{\xi }_{ab}^{2}(T)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {B}_{{\rm{c}}2}^{c}(T)}\,\displaystyle \frac{1}{\nu }\,\displaystyle \frac{1-{(T/{T}_{{\rm{c}}})}^{v}}{1-(T/{T}_{{\rm{c}}})}.\end{eqnarray} \tag{ 37 }$$

The temperature dependence of $\lambda$ in the ab-plane can be calculated in a similar way using:

$$\begin{eqnarray}&&{B}_{{\rm{c}}1}^{c}(T)={B}_{{\rm{c}}1}^{c}(0)(1-{(T/{T}_{{\rm{c}}})}^{\eta }),\end{eqnarray} \tag{ 38 }$$

where $\eta$ has been taken to have a value of 2 for all superconductors. This can be differentiated to give:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {B}_{{\rm{c}}1}^{c}(T)}{\partial T}\right|}_{T\approx {T}_{{\rm{c}}}}={B}_{{\rm{c}}1}^{c}(0)\left(-\displaystyle \frac{\eta }{{T}_{{\rm{c}}}}\right).\end{eqnarray} \tag{ 39 }$$

We also use the G–L relation for the G–L penetration depth, valid at all temperatures:

$$\begin{eqnarray}&&{\lambda }_{ab}(T)={\lambda }_{ab}(0){(1-(T/{T}_{{\rm{c}}}))}^{-\mathrm{1/2}}\end{eqnarray} \tag{ 40 }$$

and the G–L relation for ${B}_{{\rm{c}}1}^{c}(T)$ [113]:

$$\begin{eqnarray}{B}_{{\rm{c}}1}^{c}(T) & = & \displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi {\lambda }_{ab}^{2}(T)}\left\{\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{ab}(T)}{{\xi }_{ab}(T)}\right)+0.5\right\}\\ & = & \displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi {\lambda }_{ab}^{2}(T)}\left\{\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{ab}(0)}{{\xi }_{ab}(0)}\right)+0.5\right\},\end{eqnarray} \tag{ 41 }$$

which can be differentiated to give:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {B}_{{\rm{c}}1}^{c}(T)}{\partial T}\right|}_{T\approx {T}_{{\rm{c}}}}=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi {\lambda }_{ab}^{2}(0)}\left(-\displaystyle \frac{1}{{T}_{{\rm{c}}}}\right)\left\{\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{ab}(0)}{{\xi }_{ab}(0)}\right)+0.5\right\}.\end{eqnarray} \tag{ 42 }$$

Combining equations (39) and (42) leads to:

$$\begin{eqnarray}&&{\lambda }_{ab}^{2}(0)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi \eta {B}_{{\rm{c}}1}^{c}(0)}\left\{\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{ab}(0)}{{\xi }_{ab}(0)}\right)+0.5\right\}.\end{eqnarray} \tag{ 43 }$$

Combining equations (38), (40) and (43) gives ${\lambda }_{ab}(T):$

$$\begin{eqnarray}&&{\lambda }_{ab}^{2}(T)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi \eta }\displaystyle \frac{1-{(T/{T}_{{\rm{c}}})}^{\eta }}{{B}_{{\rm{c}}1}^{c}(T)(1-(T/{T}_{{\rm{c}}}))}\left\{\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{ab}(0)}{{\xi }_{ab}(0)}\right)+0.5\right\}.\end{eqnarray} \tag{ 44 }$$

Equations (37) and (44) can be substituted into equation (29) to calculate ${J}_{{\rm{Dsc}}}^{ab}(T).$

A.2. Calculation of ${J}_{{\rm{D}}{\rm{S}}{\rm{c}}}^{{\rm{c}}}$ in zero field

For anisotropic materials, we can also calculate the depairing current density when the current is flowing in the c-axis direction (${J}_{{\rm{DSc}}}^{c}$), given by:

$$\begin{eqnarray}&&{J}_{{\rm{DSc}}}^{c}(T)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{3\sqrt{3}\pi {\mu }_{0}{\lambda }_{c}^{2}(T){\xi }_{c}(T)}.\end{eqnarray} \tag{ 45 }$$

Using the general result for anisotropic superconductors [57]:

$$\begin{eqnarray}&&{\lambda }_{ab}{\xi }_{ab}={\lambda }_{c}{\xi }_{c},\end{eqnarray} \tag{ 46 }$$

we have:

$$\begin{eqnarray}&&{J}_{{\rm{DSc}}}^{c}(T)={J}_{{\rm{DSc}}}^{ab}(T)\displaystyle \frac{{\lambda }_{ab}(0)}{{\lambda }_{c}(0)}={J}_{{\rm{DSc}}}^{ab}(T)\displaystyle \frac{{\xi }_{c}(0)}{{\xi }_{ab}(0)}.\end{eqnarray} \tag{ 47 }$$

The G–L relation for ${B}_{{\rm{c}}2}^{ab}(T)$ is [113]:

$$\begin{eqnarray}&&{B}_{{\rm{c}}2}^{ab}(T)={\left.\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {\xi }_{c}(T){\xi }_{ab}(T)}\right|}_{T\approx {T}_{{\rm{c}}}}\end{eqnarray} \tag{ 48 }$$

and the scaling with temperature is given by:

$$\begin{eqnarray}&&{B}_{{\rm{c}}2}^{ab}(T)={B}_{{\rm{c}}2}^{ab}(0)(1-{(T/{T}_{{\rm{c}}})}^{\nu }).\end{eqnarray} \tag{ 49 }$$

Similar to the methods described above for current flowing in the ab-plane, by combining the derivatives of equations (48) and (49) we find:

$$\begin{eqnarray}&&{\xi }_{c}(0)=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {B}_{{\rm{c}}2}^{ab}(0){\xi }_{ab}(0)}\displaystyle \frac{1}{\nu }=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{2\pi {B}_{{\rm{c}}2}^{ab}(T){\xi }_{ab}(0)}\displaystyle \frac{1}{\nu }\left(1-{\left(\displaystyle \frac{T}{{T}_{{\rm{c}}}}\right)}^{\nu }\right),\end{eqnarray} \tag{ 50 }$$

where ${\xi }_{ab}(0)$ can be obtained from equation (35), and ${\xi }_{c}(T)$ can be found using:

$$\begin{eqnarray}&&{\xi }_{c}(T)={\xi }_{c}(0){(1-(T/{T}_{c}))}^{-\mathrm{1/2}}.\end{eqnarray} \tag{ 51 }$$

Finally, ${\lambda }_{c}(T)$ can be found given ${\xi }_{ab}(T),$ ${\xi }_{{c}}(T)$ and ${\lambda }_{ab}(T)$ using equation (46). Hence we can calculate the values necessary to produce (45) and figure 1 [13–25].