  Statistical thermodynamical and kinetically-limited models are applied to
study the origin and evolution of space charges and band-bending effects at low
angle [001] tilt grain boundaries in YBa$_2$Cu$_3$O$_7$ and the effects of Ca
doping upon them. Atomistic simulations, using shell models of interatomic
forces, are used to calculate the energetics of various relevant point defects.
The intrinsic space charge profiles at ideal surfaces are calculated for two
limits of oxygen contents, i.e. YBa$_2$Cu$_3$O$_6$ and YBa$_2$Cu$_3$O$_7$. At
one limit, O$_6$, the system is an insulator, while at O$_7$, a metal. This is
analogous to the intrinsic and doping cases of semiconductors. The site
selections for doping calcium and creating holes are also investigated by
calculating the heat of solution. In a continuum treatment, the volume of
formation of doping calcium at Y-sites is computed. It is then applied to study
the segregation of calcium ions to grain boundaries in the Y-123 compound. The
influences of the segregation of calcium ions on space charge profiles are
finally studied to provide one guide for understanding the improvement of
transport properties by doping calcium at grain boundaries in Y-123 compound.
