Small grains: a key to high-field applications of granular Ba-122 superconductors?

Round multifilamentary superconducting wires are the preferred choice for building high-field magnets such as those used in particle accelerators and fusion reactors [1]. Currently classic compounds, i.e. Nb–Ti and Nb₃Sn, are used for these applications since wires and tapes based on high-temperature superconductor (HTS) materials are not competitive so far. Their development is closely related to the optimization of the growth process and the grain boundary (GB) properties. Great efforts have been made to reduce the misalignment between the crystallographic axes of the grains in HTSs to prevent the weak-link character of large-angle GBs and thereby increase the maximum transport current. This has been successfully achieved for HTS tapes [2], but their geometry is less preferable than the round cross section of wires and their fabrication process is complicated and expensive.

The discovery of iron-based superconductors excited much interest because of their relatively high transition temperatures, small anisotropy, and high upper critical fields $({H}_{{\rm{c}}2}\gtrsim 100\;{\rm{T}})$ [3]. Unfortunately, measurements of the critical current density across [001]-tilt GBs in thin-film Co-doped Ba-122 bicrystals [4, 5], as well as magnetization and critical current measurements on polycrystalline iron-based compounds [6–8], indicate weakly linked GBs in these superconductors [9, 10], but their limiting effect on ${J}_{{\rm{c}}}$ is less pronounced compared to other HTSs [4]. Recently, results on Bi-2212 wires [11] have been promising for future HTS magnet applications. The large ${J}_{{\rm{c}}}$ of these wires is ascribed to a local biaxial texture of the grains [12]. The iron-based HTSs also have the potential for the production of superconducting wires. In contrast to Bi-2212, the wires based on Ba-122 in [13] are untextured, nonetheless, the achieved performance is much better compared to other untextured HTS wires. Still, major improvements of the iron-based materials are needed in order to become competitive with the long established Nb–Ti and Nb₃Sn wires currently used in superconducting magnets.

A feature of the weak-link character of the GBs is the hysteresis of the inter-grain critical current density ${J}_{{\rm{c}}}$ [14–18], which appears as an asymmetry in magnetization loops (see section 5 for details). The values of ${J}_{{\rm{c}}}$ are found to be smaller when the absolute value of the external field, $| {H}_{\mathrm{ext}}|$, is increased (referred to as the increasing field branch) compared to the case when $| {H}_{\mathrm{ext}}|$ is decreased (the decreasing field branch). The commonly accepted explanation of this effect is based on the reverse field ${H}_{\mathrm{return}}$ at the GBs arising from the intra-grain current density ${J}^{{\rm{G}}}$ [15, 19, 20]. The local magnetic induction B at the GBs is given by the vector sum of ${H}_{\mathrm{ext}}$ and ${H}_{\mathrm{return}}$. As a result the magnetization curve is shifted by the value of ${H}_{\mathrm{return}}$ relative to ${H}_{\mathrm{ext}}$. Therefore, ${H}_{\mathrm{return}}$ can be estimated from the field necessary to compensate this shift [20]. However, as shown later, the ${J}^{{\rm{G}}}$ necessary to explain the observed shift in a K-doped Ba-122 bulk with a grain radius of approximately $0.1\;\mu {\rm{m}}$ is about ${10}^{12}\;{\rm{A}}\;{{\rm{m}}}^{-2}$, which is far larger than the maximum current density found in K-doped Ba-122 single crystals [21]. Although ${H}_{\mathrm{return}}$ contributes to the hysteresis of ${J}_{{\rm{c}}}$ to some extent, another effect has to be responsible for the observed shift in the magnetization curve.

An alternative description of the critical current hysteresis was suggested by Svistunov and D'yachenko [22], which is based on the results of the Josephson current density of a single Josephson junction [23]:

$$\begin{eqnarray}&&{J}_{{\rm{J}}}(B)={J}_{0}\left|\displaystyle \frac{\mathrm{sin}(k(B)s)}{k(B)s}\right|.\end{eqnarray} \tag{ 1 }$$

Equation (1) is a different formulation for the Fraunhofer pattern, where the (normalized) flux through the junction ($\pi \phi /{\phi }_{0}$ with ${\phi }_{0}$ the magnetic flux quantum) is replaced by $k(B)s$, which explicitly considers the length of the junction: $2\;s$. $| {J}_{0}|$ defines the maximum Josephson current density across the junction. Equation (1) already suggests the favorable property of a small junction length. The envelope of the Josephson current density (${J}_{{\rm{J}}}\sim {J}_{0}/| {ks}|$) indicates that a small value of s (and also k) leads to higher values of ${J}_{{\rm{J}}}$ [24].

The field dependent variable k is derived by carrying out an integration over a closed path parallel to and across the GB [23], as visualized in figure 1. It defines the variation of the phase φ of the order parameter [23–25]:

$$\begin{eqnarray}&&k(B)\equiv \displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}y}=\displaystyle \frac{4\pi {\mu }_{0}}{{\phi }_{0}}\left({\lambda }^{2}{J}^{{\rm{G}}}(B)+\displaystyle \frac{d}{{\mu }_{0}}B\right).\end{eqnarray} \tag{ 2 }$$

Here, B is the magnetic induction which is parallel to the z-axis while the integration path is in the xy-plane, d is half of the thickness of the junction, and λ denotes the magnetic penetration depth of the material.

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Svistunov and D'yachenko proposed that the intra-grain current density ${J}^{{\rm{G}}}$ is composed of a reversible contribution ${J}_{\mathrm{rev}}^{{\rm{G}}}$, stemming from the reversible magnetization of the grain, and of an irreversible contribution ${J}_{{\rm{P}}}^{{\rm{G}}}$, stemming from the pinning of the flux-lines (e.g. surface and bulk pinning)[22]:

$$\begin{eqnarray}&&{J}^{{\rm{G}}}={J}_{\mathrm{rev}}^{{\rm{G}}}+{J}_{{\rm{P}}}^{{\rm{G}}}.\end{eqnarray} \tag{ 3 }$$

If the local magnetic induction at the GB is smaller than the lower critical field ($| B| /{\mu }_{0}\lt {H}_{{\rm{c}}1}$) and no flux-lines are present inside the grains (${J}_{{\rm{P}}}^{{\rm{G}}}=0$), the current density at the GB is the Meissner shielding current density: ${J}^{{\rm{G}}}\sim B/({\mu }_{0}\lambda )$ (London model), and thus ${ks}=4\pi (\lambda +d){Bs}/{\phi }_{0}$ is equivalent to the term $\pi \phi /{\phi }_{0}$, which is a more common representation for ks in the literature [23]. Equation (2) allows the extension of the description of Josephson junctions to the mixed state (${H}_{{\rm{c}}1}\lt | B| /{\mu }_{0}\lt {H}_{{\rm{c}}2}$), where flux-lines have penetrated the superconductor [24, 25]. ${J}_{\mathrm{rev}}^{{\rm{G}}}$ is defined by the distribution of the flux-lines when no pinning is present inside the grain. With pinning, this flux-line configuration is modified compared to the ideal case, i.e. a field gradient develops, which is described by ${J}_{{\rm{P}}}^{{\rm{G}}}$. The value of ${J}_{{\rm{P}}}^{{\rm{G}}}$ depends on the history of the external field, while ${J}_{\mathrm{rev}}^{{\rm{G}}}$ is determined by the external field itself (see section 2). These different dependencies are responsible for the hysteresis of ${J}_{{\rm{c}}}$ [22].

The attempt to verify the model proposed by Svistunov and D'yachenko requires a measurement technique which is able to quantify ${J}^{{\rm{G}}}$ and ${J}_{{\rm{c}}}$. Magnetization measurements probe the global magnetic response of a sample. They provide very limited insight into the spatial distribution of the local magnetic field, or the values and interactions of the inter- and intra-granular currents that are essential to test the predictions of the model. We measured the history dependence of ${J}^{{\rm{G}}}$ and its impact on ${J}_{{\rm{c}}}$, i.e. the ${J}_{{\rm{c}}}$-hysteresis, with scanning Hall-probe microscopy (SHPM). This technique has the advantage of allowing the detection of ${J}^{{\rm{G}}}$ and ${J}_{{\rm{c}}}$ simultaneously.

Svistunov and D'yachenko used the equations describing a single Josephson junction to explain the hysteresis of ${J}_{{\rm{c}}}$ [25]. However, in a polycrystal the inter-grain current has to cross multiple junctions (i.e. GBs) whose lengths vary statistically. In order to account for this, the Josephson current density of a single junction (1) has to be integrated over the distribution density of the junction lengths, i.e. the grain radii s, which is approximated by:

$$\begin{eqnarray}&&P(s,{s}_{0},m)=\displaystyle \frac{1}{{\rm{\Gamma }}(m)}\displaystyle \frac{{s}^{m}}{{s}_{0}^{m+1}}\mathrm{exp}\left(-\displaystyle \frac{{ms}}{{s}_{0}}\right),\end{eqnarray} \tag{ 4 }$$

with $s\in [0,\infty )$ and Γ the gamma function. The parameter s₀ denotes the mode of the distribution density function, i.e. the value where P has a maximum, which is referred to as characteristic grain size in the following. A large value of the dimensionless variable m corresponds to a smaller width of the distribution. Following the calculations by Gonzalez et al [26] we obtain an estimate for the inter-granular critical current density of the sample by integrating equation (1) multiplied by $P(s,{s}_{0},m)$ over s:

$$\begin{eqnarray}&&{J}_{{\rm{c}}}{(\alpha )={J}_{0}\displaystyle \frac{{\alpha }^{m+1}}{{\rm{\Gamma }}(m)}(-1)}^{m-1}\displaystyle \frac{{\partial }^{m-1}}{\partial {\alpha }^{m-1}}\displaystyle \frac{\mathrm{coth}\left(\alpha /2\right)}{{\pi }^{2}+{\alpha }^{2}},\end{eqnarray} \tag{ 5 }$$

where $\alpha =m\pi /(| k| {s}_{0})$. Our definition of α is slightly different from that in [26], because we choose a different distribution density function (4) where the value of s₀ is independent of m. In principle m can take any positive value, but it must be an integer in (5). The Fraunhofer pattern from (1) disappears in (5) and is replaced by a smooth function, which can be applied to quantitatively evaluate the current densities in polycrystalline HTS materials. Accordingly, the value of ${J}_{{\rm{c}}}$ is larger if α is large, which means that small values of $| k|$ and s₀ are preferable.

The influence of the current densities inside the grains on ${J}_{{\rm{c}}}$ is sketched in figure 2 based on (2), (3) and (5). Plots A and B show the simplified behavior of ${J}_{{\rm{P}}}^{{\rm{G}}}$ and ${J}_{\mathrm{rev}}^{{\rm{G}}}$ when the external field is ramped from positive to negative values. The sign of ${J}_{{\rm{P}}}^{{\rm{G}}}$ is always negative because the pinning force preserves the field gradient inside the grains, while the sign of ${J}_{\mathrm{rev}}^{{\rm{G}}}$ changes from positive to negative when passing through zero field. The superposition of the two current densities is illustrated by the insets of plot C for the respective field branches. They can be pictured as flowing in opposite directions in the decreasing field branch so that the magnitude of ${J}^{{\rm{G}}}$ is smaller than in the increasing field branch where they flow in the same direction. In the increasing field branch B, ${J}_{{\rm{P}}}^{{\rm{G}}}$ and ${J}_{\mathrm{rev}}^{{\rm{G}}}$ are all negative so that $| k|$ is larger than in the decreasing field branch where ${J}_{\mathrm{rev}}^{{\rm{G}}}$ is positive. Hence, ${J}_{{\rm{c}}}$ is smaller in the increasing field branch than in the decreasing field branch.

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According to the model, the shift of the inter-grain magnetization peak near zero field is determined by the values of ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$. This is seen in figure 2C, where ${J}_{{\rm{c}}}(B)$ is calculated from (5) from which we know that the maximum current density J₀ occurs at k = 0. The dotted curve shows the case ${J}_{\mathrm{rev}}^{{\rm{G}}}={J}_{{\rm{P}}}^{{\rm{G}}}=0$, where the peak is located at B = 0. If $| {J}_{\mathrm{rev}}^{{\rm{G}}}| \lt | {J}_{{\rm{P}}}^{{\rm{G}}}|$ (see figure 2A), the maximum of ${J}_{{\rm{c}}}$ occurs in the decreasing field branch, because ${J}^{{\rm{G}}}$ is negative in the decreasing field branch and thus ${\lambda }^{2}| {J}^{{\rm{G}}}|$ subtracts from ${dB}/{\mu }_{0}$ in (2). The larger $| {J}_{{\rm{P}}}^{{\rm{G}}}|$ is, compared to $| {J}_{\mathrm{rev}}^{{\rm{G}}}|$, the further the peak (i.e. k = 0) is shifted to higher fields of the decreasing field branch. If the absolute values of ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$ are exchanged (see figure 2B), the maximum of ${J}_{{\rm{c}}}$ is located at B = 0 but is smaller than J₀ because ${J}^{{\rm{G}}}$ is positive in the decreasing field branch and therefore: $k\gt 0$. The behavior is unchanged in the increasing field branch because ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$ flow in the same direction, thus the value of ${J}^{{\rm{G}}}$ is the same although the values of ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$ are exchanged.

The return field effect [15, 20] and the effect just described always occur together because both originate from ${J}^{{\rm{G}}}$. Nonetheless, for small grains with a moderate aspect ratio, the latter is far more important.

We investigated three optimally K-doped and three optimally Co-doped polycrystalline Ba-122 bulk samples. The K-doped samples were synthesized using a mechano-chemical reaction path described in [27]. The grain size was controlled by applying different temperatures during the final heat treatment. The samples were sintered for 10 hours in a hot isostatic press at $600\;^\circ {\rm{C}}$, $900\;^\circ {\rm{C}}$, and $1000\;^\circ {\rm{C}}$, where lower temperatures result in a smaller grain size.

The Co-doped Ba-122 polycrystalline bulk samples were synthesized using different starting powders and processing techniques [28]. For the large- and medium-grained samples, the starting materials were Ba, FeAs, and CoAs, while elemental powders were used for the small-grained sample. The powder of the large-grained sample was mixed and ground in an agate mortar and the other two samples were mixed by high-energy ball-milling. All samples were heat treated for 48 hours at $900^\circ {\rm{C}}$ for the large- and medium-grained sample and at $600\;^\circ {\rm{C}}$ for the small-grained sample.

The magnetization of all samples was measured in a $7\;{\rm{T}}$ SQUID magnetometer and a $5\;{\rm{T}}$ vibrating sample magnetometer. The field profiles of the samples were recorded using a SHPM set-up with a spatial resolution of approximately $1\;\mu {\rm{m}}$, and a scan range of $3\times 3\;{\mathrm{mm}}^{2}$, which is located in an $8\;{\rm{T}}$ cryostat with an operable temperature range of approximately 3−300 K. The size of the active area of the Hall-probes is $400\times 400\;{\mathrm{nm}}^{2}$.

The size of the grains is statistically distributed, which is described by the distribution density function $P(s,{s}_{0},m)$ (4). The value s₀ defines the grain size where the distribution density function has a maximum, which is approximately 0.1, 1, and $3\;\mu {\rm{m}}$ for the different K- and Co-doped samples. These characteristic grain radii are referred to as small, medium, and large. As an example, figure 3 shows the grain radii distribution density and its fit for the medium-grained K-doped sample, which was evaluated from polarized light images of the sample surface by choosing multiple lines of the image and measuring the distance from one GB to another. The grain size distribution of the large-grained sample was evaluated in the same way while s₀ of the small-grained sample was roughly estimated from transmission electron microscopy images utilizing the intercept technique [13].

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The inter- (${J}_{{\rm{c}}}$) and intra-grain current density (${J}^{{\rm{G}}}$) are extracted from SHPM data, i.e. the field profiles above the samples. We evaluate ${J}_{{\rm{c}}}$ by fitting an analytical function derived in [29] to the global field profile, where the spatially constant parameter ${J}_{{\rm{c}}}$ is the only free parameter. Figure 4A shows an example for a field profile generated by ${J}_{{\rm{c}}}$. ${J}_{{\rm{c}}}$ crosses the GBs as illustrated in figure 4C, leading to the global field profile shown in figure 4A.

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The intra-grain current density ${J}^{{\rm{G}}}$ is confined to the individual grains as indicated in figure 4D by the small arrows. If the distance between the Hall-probe and sample surface is larger than the dimensions of the grains, the fields, which are generated by the intra-grain current densities of neighboring grains, cancel each other in a first order approximation (hatched area), except for a narrow region at the sample edge. Thus, the intra-granular field profile can be approximated by a field profile resulting from a current density, ${J}^{{\rm{G}}}$, flowing in a thin surface layer of thickness s₀ (large dashed arrows). The resulting field profile for this case is plotted in figure 4B. To fit the field profile originating from ${J}^{{\rm{G}}}$ we again utilize the equation from reference [29]. With this technique we are able to evaluate ${J}^{{\rm{G}}}$ from SHPM data quantitatively.

Figure 5 shows examples of field profiles obtained from the medium-grained K-doped sample and fits of the aforementioned analytical function. The field profiles were recorded along a field run from 3 to $-3\;{\rm{T}}$. First the profile was measured in the decreasing field branch at ${\mu }_{0}{H}_{\mathrm{ext}}=30\;\mathrm{mT}$ and then in the increasing field branch at ${\mu }_{0}{H}_{\mathrm{ext}}=-30\;\mathrm{mT}$. ${J}^{{\rm{G}}}$ can be identified as the slopes at the sample edges in SHPM measurements, while ${J}_{{\rm{c}}}$ is proportional to the slopes in the remaining sample space, as indicated in figure 5A.

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In order to make the SHPM results comparable to the magnetometry data, the obtained values for ${J}_{{\rm{c}}}$ and ${J}^{{\rm{G}}}$ are converted to a magnetization M using the standard formula for cubic samples [30] (compare to section 5.2, e.g. figure 8).

5.1. Qualitative discussion

Figure 6A shows the hysteresis of the transport current density (${J}_{{\rm{c}}}$) of a K-doped Ba-122 wire. The radius of the grains is approximately $0.1\;\mu {\rm{m}}$. This hysteresis corresponds to an asymmetry in the magnetization M of a bulk sample with a similar grain size, which is highlighted in figure 6B by the shaded areas. The asymmetry is given by the difference between the decreasing field branch ($| {H}_{\mathrm{ext}}|$ ramped to smaller values) and the increasing field branch ($| {H}_{\mathrm{ext}}|$ ramped to larger values) after mirroring the latter about the reversible magnetization ${M}_{\mathrm{rev}}$, as defined in [31] with $\lambda =190\;\mathrm{nm}$ and $\kappa =100$. The comparison between ${J}_{{\rm{c}}}$ in figure 6A and the ${J}_{{\rm{c}}}$ evaluated from magnetization measurements shows good agreement [13].

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Another distinct feature is the different behavior of the samples in figure 6B. The curve of the large-grained sample (dash-dotted curve) is very similar to the magnetization curve of single crystals with the maximum value of M in the increasing field branch and its asymmetry is small. The asymmetry increases with decreasing grain size and another peak near ${H}_{\mathrm{ext}}=0$ emerges for a grain radius of $1\;\mu {\rm{m}}$ (dashed line). This peak is located in the decreasing field branch and becomes larger and wider when the grain radius is decreased to $0.1\;\mu {\rm{m}}$ (solid line), while the peak located in the increasing field branch disappears.

The Co-doped samples exhibit a similar behavior with a smaller magnitude of the magnetization curves. We observe the largest asymmetry in the sample with the smallest grain size, the two peaks in the magnetization curve of the medium-grained sample, and only one peak in the increasing field branch for the large-grained sample. This indicates that the asymmetry effects are governed by the grain size of the individual polycrystals and not by the dopant, therefore we concentrate on the results obtained from the K-doped samples.

Figure 7 displays the field profiles of the K-doped samples measured at various constant external fields along a run from 3 to $-3\;{\rm{T}}$. In the small-grained sample (figure 7A), the field profiles in the decreasing field branch are shaped roughly in accordance with Bean's critical state model, i.e. a constant slope of the field profile corresponding to a spatially uniform ${J}_{{\rm{c}}}$. Hardly any contribution of the intra-grain currents is observed when looking at the sample edges, which indicates that the peak in the decreasing field branch in figure 6B originates from the inter-grain currents. When the field is ramped through zero, the field profile starts to 'collapse' from the sample edges towards the center. The difference in the magnitude of the slopes at comparable positive and negative fields is striking. For instance, the slope at $0.5\;{\rm{T}}$ (decreasing field branch) is 3 times larger than at $-0.5\;{\rm{T}}$ (increasing field branch).

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No Bean-like field profile is observed for the large-grained sample (figure 7C). Instead, a rather flat field distribution across the sample is present, except for a large field gradient at the sample edges. This is characteristic of a field profile dominated by the contribution of the individual grains and thus the peak in the increasing field branch in figure 6B is caused by the intra-granular current of the grains.

The measured field profiles of the sample with the medium-sized grains show an intermediate behavior between the large- and small-grained samples (figure 7B). When the field is ramped from the maximum applied field to zero, an inter-granular field profile builds up, while hardly any contribution of the intra-grain currents is observed. When the field passes through zero, the sample behaves like the small-grained sample, i.e. the slope of the inter-granular field profile becomes smaller. Simultaneously, the intra-grain currents emerge and start to dominate, leading to a flat field distribution across the sample. The hysteresis of ${J}_{{\rm{c}}}$ is clearly visible when a field profile of the increasing field branch is compared to one recorded in the decreasing field branch at the same $| {H}_{\mathrm{ext}}|$. In the decreasing field branch the profiles exhibit a much steeper slope, whereas $| {J}^{{\rm{G}}}|$ is noticeably larger in the increasing field branch.

The SHPM data in figure 7 reveal that the magnetization peak in the decreasing field branch in figure 6B originates from the inter-granular current density ${J}_{{\rm{c}}}$, and the peak in the increasing field branch originates from the intra-granular current density ${J}^{{\rm{G}}}$. Accordingly, the peak of the small-grained sample is determined by ${J}_{{\rm{c}}}$. The position of this peak on the field axis is generally explained in the context of the return field of the grains, ${H}_{\mathrm{return}}$, which results from ${J}^{{\rm{G}}}$. Following [20], the peak occurs when the local magnetic induction at the GBs is zero, i.e. when ${H}_{\mathrm{ext}}+{H}_{\mathrm{return}}=0$, and therefore ${H}_{\mathrm{return}}$ coincides with the field at which the peak is found: ${\mu }_{0}{H}_{\mathrm{return}}\approx 0.1\;{\rm{T}}$. The value of ${J}^{{\rm{G}}}$ necessary to generate this field is: $| {J}^{{\rm{G}}}| \gt | {H}_{\mathrm{return}}| /{s}_{0}\approx {10}^{12}\;{\rm{A}}\;{{\rm{m}}}^{-2}$, for a moderate aspect ratio of the grains (which is true for the investigated small-grained sample). If we compare this value to the maximum current density found in K-doped Ba-122 single crystals[21]: $5\times {10}^{10}{\rm{A}}\;{{\rm{m}}}^{-2}$, the return field on its own appears to be a highly questionable explanation for the observed behavior.

Clear evidence that ${H}_{\mathrm{return}}$ alone cannot be responsible for the ${J}_{{\rm{c}}}$-hysteresis is observable in figure 7A. Simply put, the return field has to be of the order of the trapped field inside the grains which corresponds to the height of the field step at the sample edges (compare figures 4B and 5A). Since the step height barely depends on the actual geometry of the intra-granular current flow, this argument includes scenarios where the magnetic grain size differs from the crystallographic grain size (s₀), i.e. the formation of grain-clusters, characterized by a number of connected grains with low-angle GBs that do not hinder the current transport from one grain to another and therefore behave like a single grain. However, such a field step is not present in the SHPM measurements in figure 7A. Consequently, another effect has to contribute to the hysteresis of ${J}_{{\rm{c}}}$.

The model presented in this paper can account for this behavior using much smaller values of $| {J}^{{\rm{G}}}|$. As discussed in section 2, the peak position of ${J}_{{\rm{c}}}$ is determined by ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$ ($| {J}_{\mathrm{rev}}^{{\rm{G}}}| \lt | {J}_{{\rm{P}}}^{{\rm{G}}}|$). To shift the peak of the small-grained sample to the experimentally determined $0.1\;{\rm{T}}$, $| {J}^{{\rm{G}}}|$ has to attain a value of approximately $3\times {10}^{9}\;{\rm{A}}\;{{\rm{m}}}^{-2}$ in the decreasing field branch—a much more realistic current density. Our SHPM data directly confirm the predictions of the presented model. We will show this below for the medium-grained sample, which allows the simultaneous detection of inter- and intra-grain currents.

5.2. Quantitative evaluation

We discussed the hysteresis of ${J}_{{\rm{c}}}$ in dependence of ${J}^{{\rm{G}}}$ in figure 2. In the increasing field branch ${J}_{\mathrm{rev}}^{{\rm{G}}}$ and ${J}_{{\rm{P}}}^{{\rm{G}}}$ flow in the same direction and ${J}^{{\rm{G}}}$ is large while ${J}_{{\rm{c}}}$ is small. According to the presented model, ${J}_{{\rm{c}}}$ in the increasing field branch should decrease further if ${J}^{{\rm{G}}}$ becomes larger. To validate this prediction, a bulk sample with an approximate grain radius of $0.1\;\mu {\rm{m}}$ was exposed to fast neutron irradiation. This procedure induces defects in the crystallographic structure of the grains which are capable of pinning flux-lines, thus increasing $| {J}_{{\rm{P}}}^{{\rm{G}}}|$ [34]. Magnetization measurements performed after neutron irradiation confirm the decrease of ${J}_{{\rm{c}}}$ in the increasing field branch. This implies that a smaller value of ${J}^{{\rm{G}}}$ would increase ${J}_{{\rm{c}}}$, i.e. the pinning in polycrystalline HTS with large-angle GBs should be minimized.

The impact of the modal grain radius, s₀, on the inter-grain critical current density, ${J}_{{\rm{c}}}$, is most interesting in view of the production of technical superconductors. In contrast to k, s₀ is independent of the field and allows a more precise interpretation. Figure 9 visualizes the dependence of ${J}_{{\rm{c}}}$ of a Josephson junction network with different s₀ in the decreasing (A) and increasing (B) field branches. A small s₀ in equation (5) entails a diminished field dependence of ${J}_{{\rm{c}}}$. The dashed arrows in the graphs highlight the corresponding increase of ${J}_{{\rm{c}}}$.

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In the framework of the presented model the most advantageous grain size of a weak-linked superconductor is as small as possible. For the examined iron-based superconductors the grain size can be changed by milling the starting materials and by varying the heat treatment of the bulks. The model neglects the influence of the grain size on other superconducting properties, such as the transition temperature, the upper critical field, or the anisotropy. In reality the optimal/minimal grain size will be determined by the deterioration of the superconducting properties of the grains.

Neglecting this deterioration, a magnetic induction ${B}_{\mathrm{max}}$ can be extracted from figure 9B up to which a wire with a certain grain size can be used in typical applications. The value of ${B}_{\mathrm{max}}$ is defined by the point where $| {J}_{{\rm{c}}}|$ drops below a minimum value ${J}_{{\rm{c}}}^{\mathrm{app}}$, which is determined by the required critical current density of a certain application (see figure 9B). ${B}_{\mathrm{max}}$ can be derived from (5). For typical values of λ ($\approx 200\;\mathrm{nm}$), d ($\approx 2\;\mathrm{nm}$), and $| {J}^{{\rm{G}}}|$ ($\approx {10}^{10}\;{\rm{A}}\;{{\rm{m}}}^{-2}$) the field dependence of k is mainly determined by the term $4\pi {dB}/{\phi }_{0}$ for $B\gt 1\;{\rm{T}}$, which is true for high-field applications. In this case $\alpha =m\pi /(| k| {s}_{0})$ is smaller than 1 and (5) can be expanded into a Taylor series at $\alpha =0$, yielding the first order approximation:

$$\begin{eqnarray}&&{B}_{\mathrm{max}}\approx \displaystyle \frac{{\phi }_{0}}{2{\pi }^{2}}\displaystyle \frac{| {J}_{0}| }{{J}_{{\rm{c}}}^{\mathrm{app}}}\displaystyle \frac{1}{{s}_{0}d}.\end{eqnarray} \tag{ 6 }$$

The value ${J}_{{\rm{c}}}^{\mathrm{app}}$ typically lies in the range of 10⁸–${10}^{9}\;{\rm{A}}\;{{\rm{m}}}^{-2}$, depending on the filling factor of the wire and the envisioned application. Equation (6) indicates that ${B}_{\mathrm{max}}$, which defines the operable field range of a wire, can be increased by one order of magnitude by reducing the grain size by one order of magnitude.

We presented a model describing the history dependence of ${J}_{{\rm{c}}}$ found in polycrystalline HTSs. The irreversible currents in the grains reduce ${J}_{{\rm{c}}}$ in the increasing field branch, while they enhance it in the decreasing field branch. We investigated K- and Co-doped Ba-122 polycrystals of three different grain sizes via magnetization and SHPM measurements. With SHPM, both the inter- and intra-granular current density are accessible, which makes SHPM measurements superior to magnetization and transport measurements in order to understand the current flow in granular materials. The model is able to describe the obtained inter-granular current densities based on the intra-granular current densities both qualitatively and quantitatively. It allows the estimation of the field range in which a wire can be operated from the size of the grains inside the wire. This field range increases significantly as the grain size is reduced, a feature which is confirmed experimentally. Our results show that the grain size is a key parameter for increasing the maximum operable field range of polycrystalline HTS materials that are governed by Josephson tunneling. In its essence the model is independent of the material, and therefore our findings are also relevant to other HTS materials.

We are grateful to F Sauerzopf and H W Weber from TU Wien for helpful discussions and wish to thank W L Starch and R B Richardson for technical contributions at the National High Magnetic Field Laboratory. The authors gratefully acknowledge Y Hayashi, J Shimoyama, and K Kishio from The University of Tokyo for experimental assistance and discussions. This work was supported by the Austrian Science Fund (FWF): P22837-N20 and by the European-Japanese collaborative project SUPER-IRON (No. 283204) and was partially supported by JST PRESTO and by Grant-in-Aid for Scientific Research from the JSPS (Grant No. 15H05519). The work at the National High Magnetic Field Laboratory was supported by NSF (No. DMR-1306785) to E E Hellstrom, and the facilities of the NHMFL are supported by the State of Florida and by the NSF through a facility grant (No. DMR-1157490).