Zig-zag charged domain walls in ferroelectric PbTiO${}_3$

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.