  Ferroelectric domain walls are quasi-2D systems that show great promise for
the development of non-volatile memory, memristor technology and electronic
components with ultra-small feature size. Electric fields, for example, can
change the domain wall orientation relative to the spontaneous polarization and
switch between resistive and conductive states, controlling the electrical
current. Being embedded in a 3D material, however, the domain walls are not
perfectly flat and can form networks, which leads to complex physical
structures. We demonstrate the importance of the nanoscale structure for the
emergent transport properties, studying electronic conduction in the 3D network
of neutral and charged domain walls in ErMnO$_3$. By combining tomographic
microscopy techniques and finite element modelling, we clarify the contribution
of domain walls within the bulk and show the significance of curvature effects
for the local conduction down to the nanoscale. The findings provide insights
into the propagation of electrical currents in domain wall networks, reveal
additional degrees of freedom for their control, and provide quantitative
guidelines for the design of domain wall based technology.
