  First-principles calculations for pristine neutral ferroelectric domain walls
in BiFeO$_3$ reveal that excess electrons are selectively trapped by the domain
walls, while holes are only weakly attracted. Such trapped excess electrons may
be responsible for the thermally activated electrical conductivity at domain
walls observed in experiments. In the case of a periodic array of domain walls,
the trapped excess electrons create a zigzag potential, whose amplitude depends
on the electron concentration in the material and the domain-wall distance. The
potential is asymmetric for 71{\deg} and 109{\deg} domain walls. This could
modify the open-circuit voltage in a solar cell and hence influence the
photoelectric effect in BiFeO$_3$.
