Modelling reduced field dependence of $j_c$ in YBCO films with nanorods

Flux pinning is still the main limiting factor for the critical current of the high-temperature superconductors in high fields. In this paper we model the field dependence of the critical current in thin films with columnar defects aligned with the field. The characteristic shape of the critical current of the BaZrO$_3$ doped YBa$_2$Cu$_3$O$_{6+x}$ thin films is reproduced and explained. The model is based on solving the Ginzburg-Landau equations with columnar defects present in the lattice. The size of the columnar defect is found to be of key importance in explaining the rounded shape of the critical current of the BaZrO$_3$ doped YBa$_2$Cu$_3$O$_{6+x}$ thin films. It is also found that the size of the rod changes the long range order of the vortex lattice.