4. Final remarks

To summarize one can note that each case is characterized by the same unchanged parameters. For instance in each plot we see some small fluctuations on the boundaries which are associated with the London penetration depth 𝜆 which was kept constant the G-L parameter was kept constant, but 𝜅=𝜆/𝜉 ). Th elondon penetration depth gives us information about how much does the field intrude in the sample. On the othe hand we see something similar in the order parameter. The fluctuations on the boundary are here associated by the coherence length.

Most of the Times the Abrikosov lattice is shown to be triangular (or hexagonal). Only in special cases one can see different lattices, for example the square lattice in the middle in the for the square superconductor with slits. If we had analysed a cylindrical geometry one would expect vortices lining on a circle [4].

The simulations in this paper presents the instability of the Shubnikov phase for high magnetic fields. An important fact is that the vortex sources are always the sharp edges [3] in the system. For the square geometry the vortexes nucleate from the slits (for high field above 1.1 they nucleate from the side too) . For the hexagon lattice the vortices originate from the sharp edges of the inside hexagon. This result is rather unphysical, because it takes a relative high vortex to enter the material. One possible solution could be to refine the mesh [1], but it turns out the mesh convergence of this problem is very slow.

Comparing geometries with defects (slits, holes etc) to ones without one can immediately notice that the disturbed superconductor is more resistant to the magnetic field maintaining a non-zero order parameter. This could be caused by the discontinuity of 𝑑𝜓𝑑𝑥 on sharp edges [4]. This yields one could use distortion to maintain a penetrating high magnetic field in the superconductor giving high application possibilities. For instance in newest magnetic resonance devices or SQUID (Superconducting Quantum Interference Device).

REFERENCES

[1] Tobias Bonsen, „Numerical simulations for type II Superconductors”

[2] Isaías G. de Oliveira , „Instability in the magnetic field penetration in type II superconductors”

[3] Tommy Sonne Larsen , Mads Peter Sørensen , „The Ginzburg-Landau Equation Solved by the Finite Element Method”

[4] Tommy Sonne Alstrøm · Mads Peter Sørensen , „Magnetic Flux Lines in Complex Geometry Type-II Superconductors Studied by the Time Dependent Ginzburg-Landau Equation”