## 3 Results and Discussion  
### AC Susceptibility as a Function of AC Field Amplitude: \(\chi_{1}^{\prime\prime}(h_{AC})\)  
We want to determine the pinning potential U values of the sample. It is known that there are several methods to accomplish this task: One of them is by performing DC relaxation measurements, which consist of measuring the DC magnetization as a function of time at a fixed temperature and DC field [38]. In particular, by taking the slope of the normalized magnetization as a function of the natural logarithm of time, it is possible to directly obtain the creep rate and thus the pinning energy values [39]. Another method to determine the pinning potential values is by considering the slope of the Arrhenius plots, which allows for directly determining the pinning potential values as a function of the DC and AC fields [40]. Another way to find the pinning potential in superconducting materials by means of AC measurements is to measure the frequency-dependent out-of-phase (imaginary) AC susceptibility \(\chi_{1}^{\prime\prime}\) as a function of the AC field amplitude h\({}_{\text{AC}}\) at fixed temperatures and fixed DC fields [41; 42]. The critical current density of the sample can be correlated with the \(\chi_{1}^{\prime\prime}(h_{AC})\) dependence at a given temperature for different H\({}_{\text{DC}}\) values. This dependence may show a peak in the experimental window of measurements at a given temperature, with its position \(h^{\star}\) representing the AC field of full penetration of the perturbation in the center of the sample [43]. We are going to use this last method in order to obtain the pinning potential values as a function of the DC field at a fixed temperature equal to 77 K, which is of particular interest in the framework of the power application of superconductivity. In Figure 1, at H\({}_{\text{DC}}\) = 0 T, \(\chi_{1}^{\prime\prime}(h_{AC})\) shows an evident peak at all the AC frequencies ascribable to the full penetration of the AC field in the center of the sample. Then, it is clear that by increasing the DC field (up to 2 Tesla), there is a disappearance of the peak, which would induce one to think that the \(\chi_{1}^{\prime\prime}\) values are going toward zero with the increasing in the DC field due to the transition of the sample to the normal state. On the contrary, by further increasing the DC field, the \(\chi_{1}^{\prime\prime}\) values start to increase again with \(h_{AC}\). Figure 1: _Cont._  
This could be due to a granular conformation of the sample and therefore to inter-grain and intra-grain contributions to the whole signal [44; 45; 46]. In particular, the first peak at low DC fields, which quickly disappears, is associated with the inter-grain link among the grains, while the second peak, which develops at higher DC fields, can be ascribed to the contribution deriving from the single grains. We will look at this in more detail in the next subsections. Now, even after the discovery of the granularity in the sample, we want to focus on the possibility of extracting the pinning energy values starting from the curves of Figure 1 and, in particular, by taking the \(h^{\star}\) values at the different AC frequencies and DC fields. Because the thickness of our sample is much smaller than the other two dimensions, it is possible to use the Brandt approach [47], which links the location of the maximum \(h^{\star}\) in the \(\chi_{1}^{\prime\prime}(h_{AC})\) dependency to the critical current density:  
\[\mathrm{J_{c}=h^{\star}/\gamma d} \tag{1}\]  
where \(\gamma\approx 0.9\) is a coefficient that varies slightly with the geometry and d is the thickness of the sample equal to 4 \(\upmu\)m. So, the critical current densities at different frequencies, at 77 K, and in multiple DC applied fields have been estimated from the \(h^{\star}\). The dependence of \(\mathrm{J_{c}}\) (in logarithmic scale) on \(\ln(\mathrm{f_{0}/f})\) is displayed in Figure 2, where \(\mathrm{f_{0}}\) is the macroscopic attempt frequency, which is approximately \(10^{6}\) Hz.