## 3 Results and Discussion  
To draw the complete sample vortex phase diagram, the lower critical field H\({}_{\text{cl}}\) has been the first studied critical parameter. In particular, the first magnetization was measured versus the field at several temperatures. In Figure 1, the m(H) curves have been reported up to 0.1 T at 5 K, 10 K, 15 K, 20 K, and 25 K. It is visible the initial linear behavior of the magnetic moment due to the Meissner state and the reduction of the superconducting signal due to the temperature increase. In the main panel of Figure 2, H\({}_{\text{cl}}\) has been determined for T = 10 K. The dashed red line and the red arrow individuate the linear behavior due to the Meissner state and the H\({}_{\text{cl}}\) value, respectively. Considering all the temperatures, the H\({}_{\text{cl}}\)(T) behavior has been obtained and fitted with the power law [29]:  
\[\text{H}_{\text{cl}}(\text{T})=\text{H}_{\text{cl}}(0)(1-\text{T}/\text{T}^{* })^{\text{n}} \tag{1}\]  
where H\({}_{\text{cl}}\)(0) is the value of HCl at T = 0 K and T\({}^{*}\) indicates when the phenomenon is undetectable. The H\({}_{\text{cl}}\)(T) curve together with its fit is reported in the inset of Figure 2. From the fit procedure, H\({}_{\text{cl}}\)(0) \(\approx\) 120 Oe, T\({}^{*}\)\(\approx\) 27.7 K, n \(\approx\) 0.67 have been obtained. The study of the H\({}_{\text{cl}}\)(T) line is interesting since below it the sample is in the Meissner state while above it the vortices start to enter the sample causing dissipation processes. From our previous work [15], we already know that a strong pinning regime characterizes our sample. Now, we want to determine how the vortex interact with each other by means of magnetic relaxation measurements. At this aim, m(t) measurements have been made at \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measurement** & **Minimum Value** & **Maximum Value** & **Rate** \\ \hline m(T) & 2.5 K & 300 K & 0.5 K/min \\ m(H) & \(-\)9 T & +9 T & 0.01 T/s \\ m(t) & 100 s & 10,800 s & 1 datapoint/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement parameters values. Figure 1: Field dependence of the magnetic moment at different temperatures in the range of low magnetic fields. several temperatures and fields. Figure 3 shows relaxation measurements at 2.5 K between 0.1 T and 1 T.  
From Figure 3, the pinning energy versus field U(H) values have been extracted by means of U = \(-\)Tdln(t)/dln( 1 M1) [30; 31]. In Figure 4, the U(H) for T = 2.5 K has been shown. It can be noted how the U(H) behavior changes for increasing fields. When the field is low, U(H) decrease is slow while the U(H) decrease is faster when the field is increased. These curve behaviors were fitted with U(H) \(\propto\) H\({}^{-\alpha}\) where \(\alpha\) specifies the pinning regime of the sample. Figure 3: Time dependence of magnetization at different magnetic fields at T = 2.5 K.  
Figure 2: Field dependence of the magnetic moment curve for T = 10 K. The red dashed line corresponds to the linear behavior due to the Meissner state while the red arrow indicates the H\({}_{\rm cl}\) value. Inset: H\({}_{\rm cl}\) versus T data are fitted with H\({}_{\rm cl}\) = H\({}_{\rm cl}\)(0) (1 \(-\) T/T\({}^{*}\))\({}^{\rm n}\). Fit results are reported in the text. Specifically, a single vortex pinning regime is obtained when \(\alpha\approx 0\)[32] while a collective pinning regime is individuated by \(\alpha>0.5\)[33]. From the fitting procedure, the field where the U(H) behavior changes was obtained, i.e., \(\rm H_{cross}\approx 0.53\) T. When H \(<\)\(\rm H_{cross}\), \(\alpha\approx 0.09\) indicates that the vortices do not interact with each other (single vortex regime). When H \(>\)\(\rm H_{cross}\), \(\alpha\approx 0.7\) indicates a collective pinning behavior. The same fitting procedure can be repeated at different temperatures.