First-principles study of two-dimensional electron and hole gases at the head-to-head and tail-to-tail 180$^circ$ domain walls in PbTiO$_{3}$ ferroelectric thin films

We study from first-principles the structural and electronic properties of head-to-head (HH) and tail-to-tail (TT) 180$^circ$ domain walls in isolated free-standing PbTiO$_{3}$ slabs. For sufficiently thick domains ($n$ = 16 unit cells of PbTiO$_{3}$), a transfer of charge from the free surfaces to the domain walls to form localized electron (in the HH) and hole (in the TT) gases in order to screen the bound polarization charges is observed. The electrostatic driving force behind this electronic reconstruction is clearly visible from the perfect match between the smoothed free charge densities and the bound charge distribution, computed from a finite difference of the polarization profile obtained after the relaxation of the lattice degrees of freedom. The domain wall widths, of around six unit cells, are larger than in the conventional 180$^circ$ neutral configurations. Since no oxygen vacancies, defects or dopant atoms are introduced in our simulations, all the previous physical quantities are the intrinsic limits of the system. Our results support the existence of an extra source of charge at the domains walls to explain the enhancement of the conductivity observed in some domains walls of prototypical, insulating in bulk, perovskite oxides.