  We study the influence of oxygen vacancies on the formation of charged
180$^\circ$ domain walls in ferroelectric BaTiO$_3$ using first principles
calculations. We show that it is favorable for vacancies to assemble in
crystallographic planes, and that such clustering is accompanied by the
formation of a charged domain wall. The domain wall has negative bound charge,
which compensates the nominal positive charge of the vacancies and leads to a
vanishing density of free charge at the wall. This is in contrast to the
positively charged domain walls, which are nearly completely compensated by
free charge from the bulk. The results thus explain the experimentally observed
difference in electronic conductivity of the two types of domain walls, as well
as the generic prevalence of charged domain walls in ferroelectrics. Moreover,
the explicit demonstration of vacancy driven domain wall formation implies that
specific charged domain wall configurations may be realized by bottom-up design
for use in domain wall based information processing.
