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

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.

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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 ${d_{{\text{eff}}}} = \frac{3}{4}\pi {{\Delta }}M/{J_{c,t}}$ (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, ${J_c} \propto {H^{ - {\alpha _t}}}$ and $\Delta M \propto {H^{ - {\alpha _m}}}$ 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 (${d_{{\text{eff}}}} \propto {{\Delta }}M/{J_c} \propto {H^{ - \left( {{\alpha _m} - {\alpha _t}} \right)}}$). 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).

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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.

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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 ${\chi ^{{\prime}}\!\!\!}_1$ shows some evidence of double diamagnetic transitions that become more obvious with increasing DC field, whereas the dissipative peaks ${\chi ^{{\prime} ^{\prime}}_1}$ 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 ${\chi ^{{\prime} ^{\prime}}_1}$ 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 intra-grain (at higher temperature) and INTER-grain (at lower temperature) contributions. From the ${\chi ^{{\prime} ^{\prime}}_1}$ 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 ${\chi ^{{\prime} ^{\prime}}_1}$ 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 U₀ to be evaluated by the relation $f/{f_0} = \exp \left( { - {U_0}/{k_B}{T_p}} \right)$ by using the Arrhenius plots ln(f) versus 1/T_p (f₀ 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.

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3.3.1. 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 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).

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Table 2. Heat treatment temperature, J_c, intra- and INTER-grain irreversibility temperature at 9 T, and fitting parameters (anisotropy and T_c) 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].

Tmax °C	Jc (5 T, 4.2 K) A mm−2	TIrr,intra(9 T) K	TIrr,INTER(9 T) K	γfit	Tc ,INTER fit K
885	7115	29.9	25.7	36.21	74.4
887.5	5565	29.4	25.1	37.28	73.5
896	5085	28.6	25.1	37.10	73.1
886 (RF exp.) [19]	5236			37.12	69.8
886 (RF exp.) [19]	4274			38.5	69.4

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 ${B_m} = \frac{{c_L^4\phi _0^5}}{{4\pi \mu _0^2\lambda _0^4{\gamma ^2}}}\frac{1}{{k_B^2{T^2}}}{( {1 - \frac{{{T^2}}}{{T_c^2}}} )^2}$ by using only the anisotropy γ and T_c as free parameters (${\phi _0}$ is the magnetic flux quantum, c_L ∼ 0.2 the Lindemann number, λ₀ ∼ 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 anti-correlation 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 anti-correlation. 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].

3.3.2. Pinning energies.

The pinning energies U₀ for both the INTER- and intra-grain contribution were separately obtained from the Arrhenius by varying the frequency. The U₀ 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 U₀(H_DC ) and U₀(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-to-sample difference (figure 8(a)). On the contrary U_0,intra(H_DC ) 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.

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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 α-value) 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 ${\gamma _2} = 8\left( {{d_{{\text{eff}}}} - {d_{{\text{Bundle}}}}} \right)/3\pi L$ 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 ${\gamma _{{\text{2,Bundle}}}} = 8\left( {{d_{{\text{eff}}}} - {d_{{\text{filament}}}}} \right)/3\pi L$ 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% (${\gamma _{{\text{2,Bundle}}}}$ data at 5 T). Since ${\gamma _{{\text{2,Bundle}}}}$ is an intrinsic property (i.e. independent of the sample length), it allows a more direct sample-to-sample comparison than d_eff in case of significant length variation.

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.

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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]).

In the case of the pinning energies the most obvious sample-to-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, U₀ 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 U₀ 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 U₀ 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 U₀(T_peak,H_DC ) surface), so we cannot clearly establish whether an INTER-grain/intra-grain U₀ cross-over occurs. However, in the lower-J_c samples, differently from figure 10, there is no cross-over in the U₀(H_DC ) projection and in the U₀(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).

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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.