2. COMSOL simulation and Geometry

The simulation was based on the paper published by Tobias Bonsen [1]. Having the time dependent Ginzburg-Landau equations (using ℏ=𝑐=𝑒=1 units):

One has to rewrite those equations to use them in COMSOL. Therefore we define

So the Ginzburg-Landau equations take the form:

where we assumed the boundary conditions:

Those can be simplified defining another variable 𝑢5:

Using the general form of a PDE (seen below)

One can write the coefficients as:

Additionally we assume the boundary condition −𝒏∙𝚪=0, which means that the superconducting current is parallel to the Edge of the sample. The external magnetic field is taken in the z-direction. Because of gauge invariance for the magnetic vector potential, one can use the symmetric gauge defined as:

𝑨=[−0.5𝐵𝑎𝑦,0.5𝐵𝑎𝑥,0]

The simulations were made for different magnetic field strengths, various G-L parameters 𝜅 and conductivities 𝜎. The influence of both magnetic field and G-L parameters are considered in the analysis. The Energy of the system is calculated for selected parameters using the formula:

Analysis of the Energy graph one can see when the vortices nucleate.