Superconductivity in mesoscopic high-$T_{c}$ superconducting particles

Based on the Hubbard model in the framework of non-phonon kinematical mechanism and taking into account the discreetness of an electronic energy spectrum, the superconducting critical temperature of a mesoscopic high-$T_{c}$ sphere is analyzed as a function of doping and as a function of a particle's radius. The critical temperature $T_{c}$ is found to be an oscillating function of the radius of a particle. The size-dependent doping regime is revealed in high-$T_{c}$ nanoparticles. Our analysis shows that each oscillation in $T_{c}$ corresponds to the increase of a number of the energy levels in the sphere by one. The amplitude of oscillations of $T_{c}$ increases with decreasing $R$ and can reach a value of 6 K for nanopartilces with sizes about 25 nm, in a good agreement with experimental studies of YBa$_{2}$Cu$_{3}$O$_{7-\delta}$ nanoparticles.