  The superconducting phase transition in
YBa$_{2}$Cu$_{3}$O$_{7-\textit{$\delta $}}$(YBCO) thin film samples doped with
non-superconducting nanodot impurities of CeO$_{2}$ are the focus of recent
high-temperature superconductor studies. Non-superconducting holes of the
superconducting lattice induce a bound-state of circulating paired electrons.
This creates a magnetic flux vortex state. Examining the flow of free-electrons
shows that these quantized magnetic flux vortices arrange themselves in a
self-assembled lattice. The nanodots serve to present structural properties to
constrict the "creep" of these flux vorticies under a field response in the
form of a pinning-force enhancing the critical current density after phase
transition. In this work, a model for characterizing the superconducting phase
by the work done on electron pairs and chemical potential, following the
well-known theories of Superconductivity (Bardeen-Cooper-Scheifer \&
Ginzburg-Landau), is formulated and tested.A solution to the expression for the
magnetic flux, zero net force and pair velocity will generate a setting for the
optimal deposition parameters of number density, growth geometry and mass
density of these nanodot structures.

Superconductivity