3. Analysing Results

3.1. Square superconductor

In this section we investigate the square geometry for different external magnetic fields. First we last the system till the Energy starts to stabilizes and investigate the influence of the magnetic field. Then we analyse the nucleation of vortices for selected parameters. The table (Tab.3.1) shows the change of the order parameter |𝜓| and the magnetic field in the sample 𝐵 for external magnetic fields. One could note, that for 𝐵𝑎=3 the order parameter starts to vanish. That’s because the field is closing up to the critical field 𝐵𝑐2, which defines the transition to the normal state. In contrast to the field 𝐵𝑎=0.4, where the superconductor is in the Meissner phase. Below the critical field 𝐵𝑐1 the vorticies can’t nucleate at any time.

3.2. Hexagonal superconductor

In analogy to the earlier case we choose the same parameters for this geometry. The procedure is exactly the same. The same as for the square geometry occurs for the hexagonal superconductor. We can distinguish each phase by the order parameter. In the Meissner phase it is uniform in the sample (except the boundaries, where we have fluctuations) and equals |𝜓|=1. The table (Tab. 3.2) presents the order parameter and the field in the sample in each case of the external magnetic field:

As expected the last case presents the external field almost entirely penetrating the sample, which means (in comparison with the order parameter) that we are close the transition point to the normal state.

3.3.Vortices counting

 3.3.1 External field

An important aspect of simulating a type II superconductor is to know the critical fields for one. The Shubnikov phase lies between two critical fields, which where numerically found in this subsection. The phase diagram for a type II superconductor is shown in (fig. 3.1). One can count the number of vortices for a given field, which is done in the figure (fig.3.2). One can note the lower critical field between the Meissner phase and the Shubnikov phase 𝐵𝑎=0,49. We also conclude that the number of vortices is much higher for higher fields. For fields larger than 𝐵𝑎=1,5 the vortices start to combine as there isn’t enough space for them to stay apart [2]. Above a field around 𝐵𝑎=2,55−2,6 the order parameter vanishes on the boarder and stays only nonzero in the middle of the sample. This yields that in fields higher than 𝐵𝑎=2.7 the sample is in the normal state.

 3.3.2 Time nucleation

Here we investigate the generation and movement of the vortexes over time for the defected square geometry. One can note that the vortices stabilize in a square lattice. For the other geometries we can see the triangular lattice for the vortexes. Looking at the overall Energy of the system across the timelapse one can note when the vortices are nucleated and what happens later. The origin of the nucleated vorticies is always the distortion (indentation etc.) The simulation was last for 1000 timesteps. The first vortices nucleated in the 6 second (pic.1), then around the 120 second the lattice started to relocate (pic.2) to find its steady-state (pic.3-4). And finally the system is at a steady-state, which corresponds to the last plot in (fig.3.3)