## 3 Results and Discussion
### AC Susceptibility as a Function of Temperature: \(\chi_{1}^{{}^{\prime\prime}}(T)\)  
Until now, the sample granularity has just been assumed by looking at the \(\chi_{1}^{{}^{\prime\prime}}(\mathrm{h_{AC}})\) and \(\mathrm{J_{c}}\) behaviors. If this hypothesis is true, we expect to find evidence of granularity in the \(\chi_{1}^{{}^{\prime\prime}}(\mathrm{T})\) and \(\chi_{1}^{{}^{\prime\prime}}(\mathrm{H_{DC}})\) measurements as well. In Figure 4, the \(\chi_{1}^{{}^{\prime\prime}}(T)\) curves at an AC frequency \(\
u=508.94\) Hz, an AC amplitude \(\mathrm{h_{AC}}=10\) Oe, and for different DC fields have been reported. The main panel of Figure 4 shows the curves ranging from 10 K to 100 K and it is evident they do not follow the standard behavior predicted by the Bean critical state model [43; 52], in which usually the \(\chi_{1}^{{}^{\prime\prime}}(T)\) has a single peak associated to the full penetration of the AC field in the center of the sample. From these curves, the granular nature of the samples is visible because the presence of a double peak in the imaginary part of the first harmonic is a known feature of a granular magnetic response [53]. In particular, the peaks visible in the inset of Figure 4 can be associated to the contribution given by the individual grains present in the sample (intragranular contribution), which are electrically disconnected from each other. At lower temperatures, the connections between the neighboring grains become superconducting (intergranular contribution) [53]. Because the intergranular part of the \(\chi_{1}^{{}^{\prime\prime}}(\mathrm{T})\) curves is very large, this lets us understand that the connection among the grains is weak. Another indication about the weakness of the grains' connection is the fast disappearance of the intergranular peak with the increase in the superimposed DC field. In the inset of Figure 4, the colored arrows indicate the end of the low-temperature peak and the beginning of the high-temperature one, thus individuating \(\mathrm{T_{g}}\)[54], i.e., the temperature that marks the transition from a behavior primarily controlled by the intergranular component caused by currents flowing across the grains to a behavior primarily controlled by the intra-grain component caused by currents flowing only inside the grains. In Figure 5, the contour plot of the \(\chi_{1}^{{}^{\prime\prime}}\) with the temperature and DC field is shown. In particular, the \(\mathrm{T_{g}}\) values at the different DC fields have been reported together, with \(\mathrm{T_{c}}\) estimated by taking the first non-zero value of the \(\chi_{1}^{{}^{\prime\prime}}(\mathrm{T})\) curves. Both the \(\mathrm{T_{g}}\) and \(\mathrm{T_{c}}\) field dependencies can be fitted with the power law \(H_{DC}(T)=H_{DC}(T=0\ K){\left(1-T/T^{*}(H_{DC}=0)\right)}^{n}\) where \(T^{*}(H_{DC}=0)\) is the characteristic temperature at zero DC field, and \(n\) is the exponent. Specifically, \(\mathrm{T_{g}}(0)=86.27\) K, and \(\mathrm{n}=1.63\). On the other hand, \(\mathrm{T_{c}}(0)=88.91\) K, and \(\mathrm{n}=1.31\). The red line separates the inter-grain region from the intra-grain one while the black line separates the intra-grain region from the normal state of the material. From the contour plot, it is possible to observe how a major part of the dissipation processes occur in the inter-grain region. It is worth noting that from an electric transport point of view, the intergranular region is preferable because the grains are in contact each other and the current is favored to flow inside the grain junctions instead of flowing inside the single grains. Figure 3: Pinning energy as a function of DC field at 77 K. Solid lines indicate a power law fit \(U(H)\propto H^{-\alpha}\). Fit details and results are reported in the text.