  We report a theoretical investigation of a charged 180$^\circ$ domain wall in
ferroelectric PbTiO$_3$, compensated by randomly distributed immobile charge
defects. For this we utilize atomistic shell-model simulations and continuous
phase-field simulations in the framework of the Ginzburg-Landau-Devonshire
model. We predict that domain walls form a zig-zag pattern and we discuss its
properties in a broad interval of compensation-region widths, ranging from a
couple to over a hundred nanometers.
