Variation of effective filament diameter, irreversibility field, anisotropy, and pinning efficiency in Bi-2212 round wires

In recent years Bi2Sr2CaCu2O x (Bi-2212) received increasing attention due to its round wire multifilamentary architecture, a unique feature in high-Tc superconductor. In fact, round wires are preferable for magnet designs, including solenoids for nuclear magnetic resonance (NMR) or research purpose and accelerator magnets. However, due to the narrow over-pressure heat treatment conditions necessary to obtain high Jc and to the peculiar microstructure of Bi-2212 wires, a full understanding of the correlations between the different properties has not yet been developed. In this paper we investigate the effect of a vital part of Bi-2212 optimization, the maximum heat-treatment temperature T max in the range of 885 °C–896 °C, on the variations of Jc , effective filament diameter d eff, anisotropy γ, INTER- and intra-grain irreversibility fields and pinning energies U 0, all critical parameters in unravelling the complex mix of vortex pinning and connectivity that ultimately determines the critical current density. We found that d eff of the higher Jc wires heat-treated at lower temperature is much smaller than for the lower Jc wires. Moreover, a systematic increase of the irreversibility field and a decrease of the intrinsic Bi-2212 anisotropy underpins the higher Jc . The analysis of the pinning energies reveals that there is little sample-to-sample variation in the INTER-grain pinning, whereas in all samples the intra-grain pinning has an enhancement below ∼40–45 K becoming more and more evident with increasing Jc . These results suggest that the overall Jc performance are not only related to the wire microstructure and connectivity, which obviously affect the INTER-grain properties, but they are also intimately related to the intrinsic and intra-grain properties such as γ and U 0.

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Bi 2 Sr 2 CaCu 2 O x (Bi-2212) is one of the most promising high temperature superconductors (HTSs) for high field applications [1].In fact, Bi-2212 wire is the only HTS round conductor.It is also made in kilometre-piece length with a relatively simple powder-in-tube (PIT) multifilamentary design [2,3] suitable for the realization of accelerator and research magnets at fields larger than 20 T [4][5][6][7].To achieve its best performance, Bi-2212 wire has to undergo an overpressure heat treatment (OPHT) with multiple steps and multiple parameters, which affect both the microstructure and transport critical current density (J c ) [1,8,9].It was recently showed that the quality of the precursor powder has a significant effect on the wire performance and on the narrow processing window on Bi-2212 [10,11].An important factor in understanding J c defined by I c /A fil is that in unreacted wires the filaments are not fully dense, with the powders typically occupying 2/3 of the filament cross-section and the balance being trapped gas, typically air.Although mechanical deformation can reduce the volume of gas trapped in the PIT wire [12,13], OPHT processing hydrostatically shrinks the gas bubbles in the wire to almost zero, greatly preventing the bubbles from compromising the filament connectivity [1,14,15].In OPHT wires, the filament cross-section is estimated after densification, making J c = I c /A fil ∼ I c /A Bi-2212 .Another important factor is that heat treating short Bi-2212 round samples without overpressure leads to large effective filament diameter, d eff , exceeding the bundle diameter, d Bundle [16].
The microstructure of reacted Bi-2212 wire has a few peculiar and distinctive features.The first is a vital local biaxial texture in which Bi-2212 grains with a strongly aligned crystallographic a-axis develops along the wire longitudinal axis [17].This assures a relatively small grain-to-grain misorientation (FWHM < 10 • in high-J c wires) with high grain boundary connectivity essential to large J c .Since the wire is made of more than thousand filaments and because of the rotation along the filament length of the local c-axis direction, the macroscopic Bi-2212 wire properties are isotropic despite the high anisotropy (γ) of the individual Bi-2212 grains.A second feature is the presence of Bi-2212 filament 'bridges' through the silver matrix [18]: these Bi-2212 bridges outgrow the initial filaments during the partial melt connecting some of the filaments in the bundle.Depending on the wire architecture (e.g.filament distance) and heat treatment (HT) conditions, these connections can actually develop from random angle impingement into actual filament merging with common grains.The benefits and detriments of these inter-filament connections is not fully clear: filament merging seems to be mostly detrimental causing a worsening of the biaxial texture, whereas conceptually some filament bridging could be beneficial because it generates additional paths for supercurrent flow, which might bypass current blockages.
One of the key OPHT parameters is the maximum HT temperature T max (typically set in the 885 • C-896 • C range), which also directly affects the time in the melt t melt .In fact t melt is defined as t 872 • C↓ -t 884 • C↑ , where t 884 • C↑ and t 872 • C↓ are respectively the times at which Bi-2212 starts melting on heating and starts solidifying on cooling [8,9,19].In this step, together with Bi-2212 powder particles melting, there is also some dissolution of Ag from the sheath and penetration of liquid along Ag grain boundaries, which allow filament bridging to occur.This makes t melt a vital parameter in evaluating the final J c .Jiang et al demonstrated that, despite property variations caused by wire design or diameter, and quality of the precursor powers, J c generally reaches its maximum for the lowest possible T max that allows full melting.At higher T max there is a lower-J c plateau or one with a moderate decrease up to ∼896 • C [8].At the lowest T max having maximum J c there is only limited filament bridging, but with increasing T max significant inter-filament connective bridges develop.These are the features of the wire investigated here.The cross-sections of the samples in the present paper were recently characterized by scanning electron microscopy (similar to that in [8]) and by extensive x-ray diffraction to evaluate their grain texture [20], which revealed a systematic worsening of the [100] Bi-2212 texture with increasing T max .Some of the driving questions for this study of Bi-2212 wires are: Does 2212 show signs of granularity despite its good overall J c performance?Is there evidence of multiple scales of current flow?How does the irreversibility line vary, and can we tune it?How are variations of vortex pinning affecting the overall J c performance?Answers to these questions are vital to understanding what really controls current flow and thus the macroscopic J c .In our previous study [21] we demonstrated that detailed magnetic characterizations can provide useful information for answering some of these questions.For instance, using AC susceptibility we demonstrated that in moderate and low-J c samples (∼5200-4200 A mm −2 at 5 T and 4.2 K, obtained by varying the HT cooling rate) there was evidence of separate intra-and INTER-grain currents with different irreversibility lines and evidence for variable vortex pinning performance.In this paper we are now investigating the properties of higher-J c samples (∼7100-5100 A mm −2 at 5 T and 4.2 K) in order to understand which are the key features that lead to the best performance.

Methods
The Bi-2212 conductor used in this work is an 85 × 18 restack design (pmm180410) constituted of 18 bundles with 85 filaments each, for a total of 1530 filaments.The Bi-2212 precursor is made of Engi-Mat powder (LXB116).The wire diameter was 1 mm, reaching an averaged densified filament diameter prior to melting of ∼11.2 µm.Three samples underwent the typical OPHT described for instance in [9,19] reaching maximum temperatures, T max , of 885 • C, 887.5 • C and 896 • C.
The critical current was measured in transport at 4.2 K up to 15 T as reported in [20], finding that the engineering current density J E and so the critical current density J c have a clear variation with T max , similarly to what previously reported on the same conductor in [8].The maximum J c is obtained at the lowest T max = 885 • C reaching 7115 A mm −2 at 5 T and 4.2 K.With a small increase of T max (887.5  1 and, in the following the samples will be identified by T max and/or their J c (5 T).All magnetic characterisations were performed with field perpendicular to the wire axis, which is the most relevant to investigating properties related to the transport performance, using samples about 4 mm in length.The superconducting transitions were obtained by DC magnetization in a Quantum Design (QD) MPMS-5 SQUID magnetometer applying 1 mT to estimate the average bundle size [21].Hysteresis loops were evaluated in a QD 16 T PPMS using a vibrating sample magnetometer (VSM), with a ramp rate of 0.462 T min −1 , in a temperature range between 4.2 and 45 K to evaluate the effective filament diameter d eff at 4.2 K and to obtain the Kramer plots and their extrapolations, H K .AC susceptibility was performed in a QD 9 T PPMS varying µ 0 H ac between 0.1 and 1.5 mT, µ 0 H DC between 0 and 9 T and the frequency from 13 to 211 Hz [22,23].This characterization allows us to separate the intra and INTER-grain contributions and, analysing the AC amplitude and frequency dependences of the first harmonic χ ′ ′ 1 peaks, to determine the corresponding irreversibility lines and pinning energies.The details of this approach are described in [21] and references within.

Low-field DC magnetization
Figure 1 shows the superconducting transition for the three wires obtained after zero-field cooling.These data reveal that changing T max has small effect on T c and the transition width in this field configuration.Taking into account the demagnetization factor for samples of cylindrical shape, in the top panel we assumed that the relevant diameter is the single Bi-2212 filament.As already demonstrated in [21], this assumption is, however, incorrect leading to an unphysical χ fil value at low temperature of about −1.4.This can be explained only by screening current being induced on volumes larger than the filaments, an effect caused by the significant filament bridging in Bi-2212 wires.Because of the peculiar microstructure, imposing a low-temperature limit of −1 and back-calculating the size, we demonstrated that the relevant diameter is the bundle size.In fact, we estimated for these samples a d Bundle ranging from 158 to 165 µm, compatible with the microstructurally estimated values.These d Bundle values will be used in the following characterizations.The same data re-analysed as χ Bundle , i.e. imposing a value of −1 at the lowest temperature in order to back-calculate the bundle size (see explanation in the main text).

Effective filament diameter and Kramer analysis
In addition to the transport J c (4.2 K) characterizations [20], we also performed hysteresis loop measurements to evaluate the effect of T max on the effective filament diameter and its field dependence.d eff was calculated through the usual relation 4 π ∆M/J c,t (SI units) with △M being the amplitude of the hysteresis loop of the superconducting phase and J c,t the critical current density obtained in transport [16,24].We found that the higher J c wire (T max = 885 • C) has the smallest d eff at 4.2 K, with d eff varying from 59 to 56 µm in the 3-15 T range, which corresponds to 35%-37% of the bundle diameter (figure 2).The lower J c samples have much large d eff (99-88 µm and 98-85 µm for T max of 887.5 • C and 896 • C, respectively), clearly larger than 50% of d Bundle , and with a stronger in-field variation (table 1).Obviously, most of the sample-to-sample difference of d eff is caused by the magnitude of the transport J c .However, we also observed that the d eff field-dependence becomes stronger with decreasing J c .Both magnetization and transport data yielded power laws, J c ∝ H −αt and ∆M ∝ H −αm but we did observe that there is a significant difference in both the absolute values of α and their trends with respect to the wire J c (figure 3(a)).α obtained by magnetization is always larger than that by transport measurements.Moreover, the difference between α m and α t decreases with increasing wire J c , clearly causing the more marked field dependence of d eff for the lower-J c samples as seen in figure 2 (d eff ∝ ∆M/J c ∝ H −(αm−αt) ).Although some of the differences in the α-values can be caused by the different electric field in the transport and magnetization characterizations, most of it is likely related to the different bridging level, microstructure and the consequent current percolation paths involved in the two techniques (see section 4).
Hysteresis loops were also performed at higher temperature to verify the evolution of the Kramer field with temperature (figure 4) and to compare the samples.In Bi-2212 a standard Kramer extrapolation cannot be performed because the plots do not show a linear trend at high field approaching the irreversibility field.This is presumably caused by the presence of multiple pinning mechanisms [25,26] and by the material anisotropy with current percolating through grains with some degree of misorientation with respect to the field.Despite the lack of high-field linearity, in this material a Kramer extrapolation (for instance at 20 K) is frequently performed in the almost linear region at intermediate-field as a way to compare different samples [8].The Kramer extrapolation at 20 K, reported in table 1, shows that H K progressively decreases with decreasing J c . Figure 4 also reveals that, when the Kramer plots at different temperature are normalized to the linear trends used to estimate H k , the high-field tails substantially overlap suggesting the high-field/low-temperature pinning mechanism does not significantly change.Further discussion will be given in section 4.

AC susceptibility
As mentioned above, AC susceptibility can be used to determine sign of granularity or weak connectivity in superconductors [27,28] and we have recently demonstrated that this approach can be used also to evaluate the properties of Bi-2212 round wires (see [21] for detailed explanations about the analysis procedure and contribution identifications).Figure 5 shows an example of AC susceptibility measurements, performed with an AC field of 1 mT at 73 Hz, on one of the Bi-2212 wires investigated here.The real part χ ′ 1 shows some evidence of double diamagnetic transitions that become more obvious with increasing DC field, whereas the dissipative peaks χ ′ ′ 1 reveal unusual shapes with an enhancement on the high-temperature side of the peak.This is due to the overlapping of two peaks as emphasized in figure 6, where we report the χ ′ ′ 1 data at 9 T fitted by a double-Gaussian function above a linear background (the errors in the evaluation of the peak positions are smaller than the symbol size; they are typically less than 0.1 K, sometimes increasing to ∼0.3 K for the wider and smaller high-temperature peaks).The double diamagnetic transition and the double peaks are caused by intragrain (at higher temperature) and INTER-grain (at lower temperature) contributions.From the χ ′ ′ 1 analysis we can determine the peak positions by varying the H ac amplitude and the frequency.Since J c is proportional to H ac , investigating the shift of the χ ′ ′ 1 peak as a function of H ac , the irreversibility line (defined by the condition J c = 0) can be determined as the limit of the peak position for H ac → 0 [29,30].On the other hand, analysing the peak shift with changing frequency enables the pinning energy U 0 to be evaluated by the relation f/f 0 = exp (−U 0 /k B T p ) by using the Arrhenius plots ln(f ) versus 1/T p (f 0 is the characteristic frequency and T p the peak position) [31].Because of these double peaks, we will be able to evaluate both the intra-grain and INTER-grain properties.

Irreversibility lines.
From the INTER-and intra-grain T p versus H ac plots, the irreversibility temperature T Irr was extrapolated for each DC field.The obtained irreversibility lines are reported in figure 7, showing a clear separation between the INTER-and intra-grain irreversibility lines.Although the sample-to-sample differences are not huge in this relatively small field range, the intra-grain irreversibility line systematically improves with increasing J c .As summarized in table 2, T Irr,intra at 9 T increases from 28.6 to 29.9 K.The difference is smaller for the INTER-grain irreversibility lines 1 at 9 T for the three wires, including the analysis with the 2-Gaussian fits, showing an obvious variation of the higher temperature intra-grain peak (Tp,intra).The shown data are taken at µ 0 Hac = 1 mT and 73 Hz. but, although the trend is almost identical for the two lower J c wires with a T Irr,INTER at 9 T of 25.1 K, the higher J c wire is indeed better (25.7 K).
Since the INTER-grain irreversibility lines is related to current percolating through multiple grains of varying orientation within the length of the sample, similar to transport properties on polycrystalline materials, the irreversibility lines are mostly limited by the lower irreversibility properties of Bi-2212 (i.e.H//c-axis).Our INTER-grain irreversibility lines can be well reproduced by the equation for the melting line of materials with a moderately high anisotropy B m = ) 2 by using only the anisotropy γ and T c as free parameters (ϕ 0 is the magnetic flux quantum, c L ∼ 0.2 the Lindemann number, λ 0 ∼ 200 nm the penetration depth) [32].The fits, showed in figure 7 as solid lines, can provide an estimation of γ for the three wires.We obtained similar values, γ ∼ 37.1-37.3,for the wires with J c (4.2 K, 5 T) = 5085-5565 A mm −2 but a sensibly smaller anisotropy of 36.2 for the higher J c case (7115 A mm −2 , see table 2), suggesting an inverse correlation between J c and γ.To verify this anticorrelation on a wider J c range, we similarly fitted the irreversibility lines of the wires in [21]: for a J c (4.2 K, 5 T) of 5236 A mm −2 γ was 37.1, comparable with those reported above for wires of similar J c , whereas the lowest J c wire (4274 A mm −2 ) had the highest γ of 38.5, confirming the anticorrelation.Although these γ values might appear low with respect to optimal single crystals (reported to be 50 or more) [33], we have to take into account that these wires are in the over-doped state (necessary to optimize J c ), a condition that suppresses T c and γ in Bi-2212 [34].

Pinning energies.
The pinning energies U 0 for both the INTER-and intra-grain contribution were separately obtained from the Arrhenius by varying the frequency.The U 0 data are usually plotted versus H DC , because it typically follows a power-law relation.However, this can be misleading when plotting both INTER-and intra-grain trends because the two set of data, estimated at different fields, are also in very different temperature range (the peak shift caused by frequency is rather small and figure 6 shows examples of the temperature difference between the two contributions).For this reason, in figure 8 we report U 0,INTER and U 0.intra as 3D plots as a function of DC field and temperature of the peak at 73 Hz, together with their two projections U 0 (H DC ) and U 0 (T peak ).We found that U 0,INTER (H DC ) does roughly follow a power-law for all samples (whereas the temperature dependence is exponential) with a relatively small sample-tosample difference (figure 8(a)).On the contrary U 0,intra (H DC ) Table 2. Heat treatment temperature, Jc, intra-and INTER-grain irreversibility temperature at 9 T, and fitting parameters (anisotropy and Tc) obtained for the INTER-grain irreversibility lines for the 3 investigated Bi-2212 wires.For comparison the fitting parameters of the wires belonging to the cooling rate experiment are also reported [19].follows a power-law only at low field, whereas at high fields ⩾2 T, corresponding to data below 40-45 K, there is an upward deviation that becomes stronger with the increase of J c (figure 8(b)).This suggests the appearance of a stronger pinning mechanism activated at low temperature that is more effective in the higher-J c sample.

Discussion
In the following we divide the discussion in three main topics.Firstly, we take on d eff , with its amplitude and field dependence, explaining the role played by the microstructure and the bridging level.Then, we compare the results obtained by the Kramer extrapolation and the irreversibility lines, providing a simple relation that allows the estimation of the low temperature H Irr , and we discuss the possible effect of the anisotropy on the overall performance.Finally, we compare the field and temperature dependence of the intra-and INTER-grain pinning energies, showing the important contribution of the former on achieving the highest J c .

Effective filament diameter
The effective filament diameter d eff significantly varies with changing T max , and so J c .In particular, the smallest d eff is achieved for the higher J c wire, which also has the shortest time in the melt (because of the lowest T max ).As demonstrated by Jiang et al the low T max generates the least interfilament bridging, almost keeping the original filament structure [8].As shown in figure 2, in the best case d eff (4.2 K) mildly decreases with increasing field (35%-37% of d Bundle ), whereas stronger changes are observed in the other two cases (55%-62% and 51%-59% of d Bundle ).A similar decreasing d eff trend was previously found in a different type of Bi-2212 wires in [35], where the investigation was performed in a wider temperature range.The authors found a strong difference in d eff depending on the wire quality and architectures (d eff (10 K, 4-7 T) ∼70%-80% d Bundle in the best case), but also observed a clear temperature dependence with d eff approaching the filament size at higher temperature (>20-25 K) and increasing toward d Bundle with decreasing temperature.Both our and their results demonstrate that d eff has variations with field and temperature that have to be kept in mind for magnet design and protection.
The decrease of the field-dependence of d eff with increasing J c is obviously related to the difference in the α values obtained by magnetization and transport measurements (figure 3(a)).The larger magnetization α m with respect to the transport α t is, in general, ascribed to the different electric field, which in transport is chosen as a criterion (here 1 µV cm −1 ) whereas in magnetization is mostly induced by the magnetic field ramp rate (here estimated to be 3 or 4 orders of magnitude smaller).Since with increasing the magnetic field the n-value in the E ∼ (J/J c ) n relation typically decreases, the magnetization J c (or △M) is usually more strongly field dependent.This explanation is however not adequate for these Bi-2212 wires where the n-values estimated from the transport data is almost field independent (n ∼ 26).In Bi-2212 we have to take into account the microstructural changes to understand the difference between α m and α t .Although Bi-2212 wires are microscopically isotropic, the filaments are locally bi-axially textured [17] but with an average c-axis orientation changing along the filaments, as schematically represented by the blue stripes in figure 3(b).These filaments can be connected by bridges, some of which have current carrying capability, and these bridges are more numerous in the high-T max , low-J c wires (a bridge is represented in figure 3(b) by a magenta element).Since transport J c is determined by the percolative path, α changes with the density of bridges.In fact, in the higher J c wire with the least bridging, the current is forced to stay within the filaments for longer length with its current being more strongly limited by the filament worst orientation, which has the stronger field dependence (H//c-axis) inducing a large transport α t .In wires with more bridging (lower J c ) the current can more likely bypass segments of the filaments with a less favourable orientation, percolating through bridges into better oriented filaments, as sketched in figure 3(b): this leads to a smaller transport α.The situation is more complex when α is evaluated by magnetization (α m ) because the total moment is determined by two current contributions: a long-range (sample-long) current, percolating through grain-boundaries and bridges, and local shorter-range inter-and intra-grain currents.The former, involving a global path across the entire sample, has likely a field dependence similar to that evaluated by transport measurements, so with a low α-value.On the other hand, the additional contributions by locally induced currents involve randomly oriented Bi-2212 grains, including grains with an high α-value, or short filament segments not involved in the global path because of their unfavourable orientation, so having an high α-value too.Since, in the lower-J c wires, these unfavourably oriented (higher αvalue) parts of the filaments are less involved in the long-range induced current (because of the many bridges that allow the global current to bypass them), they contribute more as local currents, with an increasing contribution to the total moment and so increasing the overall magnetization α m .
In our estimation of d eff we did not take into account the d eff dependence on the sample length as done in [16,36] (our samples are very short, so approaching the zero-length limit).It is however interesting to follow their approach to estimate the connectivity transversal to the wire axis.In their case the effective filament diameter was exceeding d Bundle and the equation γ 2 = 8 (d eff − d Bundle ) /3π L was used to estimate the 'fraction of longitudinal cross-sectional area of the strand which contains bridges' [16].In our case d eff is always smaller than the bundle size, so this equation does not apply.However, since we have no bundle-to-bundle bridging in these OPHTed wires, we can use a modified formula γ 2,Bundle = 8 (d eff − d filament ) /3π L to instead determine the fraction of longitudinal cross-sectional area of the bundle which contains bridges, so an estimation of the intra-bundle connectivity.These values are reported in table 1 and reveal that in the higher-J c wire only 9% of the bundle area contains bridges, where for the lower-J c wires it increases to ∼16% (γ 2,Bundle data at 5 T).Since γ 2,Bundle is an intrinsic property (i.e.independent of the sample length), it allows a more direct sampleto-sample comparison than d eff in case of significant length variation.

Kramer extrapolations, irreversibility lines and anisotropy
Figure 4 shows a good high-field rescaling for the Kramer plots.However, the H K values are obviously an underestimation of the real irreversibility fields.In this case H K was measured up to 45 K and the irreversibility curves were evaluated by AC susceptibility down to ∼25 K (figure 7), which allows us to compare the two datasets (at temperature ⩾ 35 K, a direct comparison of H Irr determined by the closures of the hysteresis loops and by AC susceptibility shows a good agreement; at lower temperature the high-field noise does not allow a reliable estimation of the loop closure).We found that in the 25-45 K range H K and H Irr are proportional with very close H Irr /H K ratio of ∼1.72-1.77 in these three samples.In figure 9 we replot the AC data and fits obtained in figure 7, including the low-temperature extension of the fits (dashed lines), and compare them with the H Irr data obtained by rescaling the VSM H K according to this H Irr /H K ratio.The agreement is reasonably good, giving an irreversibility field at 10 K of ∼60-70 T, which is certainly of interest for application of Bi-2212 in cryocooled machines.
Only small differences are found in the AC-determined irreversibility lines in these samples, with a more obvious variation in the intra-grain lines, indicating that varying the OPHT causes more significant variation in the intra-grain properties.However, the small changes in the INTER-grain lines indicate an inverse correlation between J c and the material anisotropy γ; this trend is also confirmed when including previously measured samples, where the irreversibility line of sample with the lowest J c (4274 A mm −2 ) can be fitted with the largest γ.Since, because of the anisotropy, the c-axis transport J c is much smaller than the ab-plane J c , the J c -γ anti-correlation suggests that lowering γ could facilitate current percolation along the c-axis [37,38] necessary for grain-to-grain connectivity when obstacles (e.g.secondary phases) or excessive grain misorientations along the longitudinal wire direction are present (see the particular Bi-2212 grain structure in [17]).

Pinning energies
In the case of the pinning energies the most obvious sampleto-sample differences are observed in the intra-grain trend.In fact U 0,intra does not follow the typical power-law field dependence once the DC field has exceeded ∼2 T and the temperature drops below 40-45 K.This enhanced pinning efficiency is likely due to point pinning likely caused by vacancies or atomic defects, a pinning mechanism commonly found at low temperature in HTS materials.Figure 10 compares the INTER-and intra-grain pinning energy for the higher-J c wire.To better understand the meaning of these data we need to take into account that, unless the pinning mechanisms change, U 0 should decrease with increasing temperature or field.Figure 10 shows that the INTER-and intra-grain pinning energies quite clearly diverge once a small field is applied.The data in the red circles are at the same low field (0.25 T) and almost identical temperature (∼58 K) but the INTER-grain U 0 is about 20% large than the intra-grain, indicating that in this temperature/field range the INTER-grain pinning is stronger than the intra-grain.On the contrary, the data in the green circles are at the same high field (9 T) and have similar U 0 values (∼345 K); however, the intra-grain T peak is 4.3 K higher, indicating that the intra-grain pinning has become stronger than the INTER-grain pinning at these low temperatures.This demonstrates a cross-over of intra-grain pinning with respect to the INTER-grain one.Similar plots (not shown) for the lower-J c samples do not provide such a direct comparison between the two pinning energies (we can only evaluate a trend line over an actual U 0 (T peak ,H DC ) surface), so we cannot clearly establish whether an INTER-grain/intra-grain U 0 cross-over occurs.However, in the lower-J c samples, differently from figure 10, there is no cross-over in the U 0 (H DC ) projection and in the U 0 (T peak ) projection the intra-grain line does not approach the INTER-grain one as close as in the higher J c case.This indicates that, despite some additional intra-grain pinning centres becoming effective at low temperature also in the lower-J c samples, they are not as efficient as in the higher-J c case.This could be caused by a reduced density of pins (e.g.produced by a different doping state) and/or a suppressed efficiency, for instance due to their slightly higher anisotropy (higher anisotropy reduces the vortex-defect interaction).
An overall conclusion of these results is that there are relatively small variations in the INTER-grain properties of both the irreversibility line and the pinning energy, whereas greater changes are found in the intra-grain properties.In particular, the low-temperature intra-grain pinning appears to be a key factor, maybe driven by γ, in increasing the J c performance.In fact, the highest J c sample has a U 0,intra at 9 T and ∼27-28 K about 70% higher than the other two samples.

Conclusions
In this paper, we investigated the superconducting properties of a series of Bi-2212 samples whose performance was varied by changing the maximum temperature of the over-pressure HT.This is the simplest parameter to change in order to vary the time in the melt of the Bi-2212 phase, and it has a strong effect on both the wire microstructure and its performance.
We found that the higher J c sample has properties markedly different from the others.Despite all wires have some level of bridging that involves a significant portion of the filament bundle, the best performing wire has a lower d eff of the order of 35%-37% of d Bundle (compared to more than 50% in the lower J c wires) with a weak field dependence caused by a more filament-constrained percolative path.
The INTER-grain properties (irreversibility line and pinning energy) are only weakly affected by the variation of the HT, as also demonstrated by the small change in the Kramer field at 20 K (table 1), but they reveal a sensible change in the material anisotropy with an anti-correlation between J c and γ.This most likely affects the intra-grain properties as well as the grain-to-grain connectivity.In fact, because of the only partial Bi-2212 texturing, c-axis current must occur to allow a global current percolation, and a lower anisotropy enable a larger c-axis J c and a better connectivity.The intra-grain irreversibility line (which is an average of all possible crystal orientations) is more dependent on the HT, but the most pronounced difference is in the intra-grain pinning energy at low temperature (<40-45 K) that in the best wire clearly cross-over the INTER-grain pinning energy, demonstrating a clear change of dominant pinning mechanism.The enhanced connectivity and stronger pinning, likely induced by a mild suppression of γ, appear to be keys to the large difference in transport J c .Our results also indicate that the percolative path in Bi-2212 wire is complex and variable with small variation of the HT.This implies that achieving the best wire performance is challenging because of the narrow HT window.Despite the plateau J c performance are already good for magnet applications (the peak J c -value cannot be reached in magnets because of the spatial temperature inhomogeneity during the HT), there is an ongoing effort to understand how to generate the peak J c -performance in a wider HT temperature range, more suitable for magnet realization.This paper suggests that for further optimization decreasing γ could be beneficial to improve H Irr and J c .This might be obtainable by a fine tune of the powder chemistry, the doping level and/or the HT.

Figure 1 .
Figure 1.(a) Temperature dependence of χ fil estimated by DC magnetization after zero-field cooling, calculated assuming that the relevant size is the filament diameter.(b) The same data re-analysed as χ Bundle , i.e. imposing a value of −1 at the lowest temperature in order to back-calculate the bundle size (see explanation in the main text).

Figure 2 .
Figure 2. (a) Effective filament diameter, d eff , at 4.2 K and (b) d eff normalized to the bundle diameter, d Bundle , as a function of magnetic field.

Figure 3 .
Figure 3. (a) α values obtained from the Jc ∝ H −αt and ∆M ∝ H −αm relations for both transport and magnetization characterizations.In the inset the field dependence of the transport and magnetization data for the highest Jc wire.(b) Sketch representing the transport current transferring from one filament to another in 2212 wires: the blue elements represent the local filament orientation averaged over the grains within the filament, schematically showing the rotation of the averaged c-axis direction along the filaments; the magenta transversal element represents an inter-filament bridge.

Figure 4 .
Figure 4. (a) Kramer plot in the 4.2-45 K range for the wire with Jc(4.2 K,5 T) of 7115 A mm −2 with the linear extrapolations from the intermediate-field regions to determine H K .(b) Same data of above panel normalized to the respective linear trends.

5 .
Temperature dependence of the AC susceptibility evaluated on the bundle size for the higher Jc Bi-2212 wire measured with µ 0 Hac = 1 mT at 73 Hz and with increasing DC-fields up to 9 T as indicated by labels near the dissipative peaks (H⊥wire).χ ′ 1 shows a double diamagnetic transition and χ ′ ′ 1 reveals an anomalous broadening on the high-temperature side of the dissipation peaks.

Figure 6 .
Figure 6.Comparison of the dissipative peaks χ ′ ′1 at 9 T for the three wires, including the analysis with the 2-Gaussian fits, showing an obvious variation of the higher temperature intra-grain peak (Tp,intra).The shown data are taken at µ 0 Hac = 1 mT and 73 Hz.

Figure 7 .
Figure 7. INTER-and intra-grain irreversibility lines for the three samples as determined by AC susceptibility at 73 Hz.The solid lines along the INTER-grain data are fitting curves as described in the text (dotted lines along the intra-grain data are only guides for the eye).

Figure 8 .
Figure 8.(a) INTER-grain and (b) inter-grain 3D plots with (on top) their respective 2D projections for the three Bi-2212 samples.The plots are showing U 0 data as a function of DC field, µ 0 H DC , and temperature of the 73 Hz peak, T peak,73Hz (data obtained with µ 0 Hac = 1 mT).Note that the x-axis of the 2D U 0 versus µ 0 H DC projections is reversed to match the 3D plots.The red dashed line in the first panel is a power-law guide for the eye.

Figure 9 .
Figure 9. Extended irreversibility lines for the three wires.The solid symbols and solid lines are the AC susceptibility data for the INTER-grain contributions and their fits from figure 7. The dashed lines are the low-temperature extensions of the same fitting curves.The open symbols are the H Irr data obtained by multiplying H K for the H Irr /H K ratio as described in the text.

Figure 10 .
Figure 10.3D plot with (on top) its 2D projections for the higher-Jc Bi-2212 wire comparing the INTER-and intra-grain pinning energies.Shown are U 0 data as a function of µ 0 H DC and T peak,73Hz (data obtained with µ 0 Hac = 1 mT).

Table 1 .
Heat treatment temperature, superconducting properties, effective filament diameter also normalized to the bundle size and intra-bundle connectivity for the 3 investigated Bi-2212 wires.Columns reporting ranges of value refer to the variations in the measured field range.
• C) J c