  Ferroelectric domain walls are attracting broad attention as atomic-scale
switches, diodes and mobile wires for next-generation nanoelectronics. Charged
domain walls in improper ferroelectrics are particularly interesting as they
offer multifunctional properties and an inherent stability not found in proper
ferroelectrics. Here we study the energetics and structure of charged walls in
improper ferroelectric YMnO$_3$, InMnO$_3$ and YGaO$_3$ by first principles
calculations and phenomenological modeling. Positively and negatively charged
walls are asymmetric in terms of local structure and width, reflecting that
polarization is not the driving force for domain formation. The wall width
scales with the amplitude of the primary structural order parameter and the
coupling strength to the polarization. We introduce general rules for how to
engineer $n$- and $p$-type domain wall conductivity based on the domain size,
polarization and electronic band gap. This opens the possibility of fine-tuning
the local transport properties and design $p$-$n$-junctions for domain
wall-based nano-circuitry.
