  Conductive ferroelectric domain walls--ultra-narrow and configurable
conduction paths, have been considered as essential building blocks for future
programmable domain wall electronics. For applications in high density devices,
it is imperative to explore the conductive domain walls in small confined
systems while earlier investigations have hitherto focused on thin films or
bulk single crystals, noting that the size-confined effects will certainly
modulate seriously the domain structure and wall transport. Here, we
demonstrate an observation and manipulation of conductive domain walls confined
within small BiFeO3 nano-islands aligned in high density arrays. Using
conductive atomic force microscopy (CAFM), we are able to distinctly visualize
various types of conductive domain walls, including the head-to-head charged
walls (CDWs), zigzag walls (zigzag-DWs), and typical 71{\deg} head-to-tail
neutral walls (NDWs). The CDWs exhibit remarkably enhanced metallic
conductivity with current of ~ nA order in magnitude and 104 times larger than
that inside domains (0.01 ~ 0.1 pA), while the semiconducting NDWs allow also
much smaller current ~ 10 pA than the CDWs. The substantially difference in
conductivity for dissimilar walls enables additional manipulations of various
wall conduction states for individual addressable nano-islands via electrically
tuning of their domain structures. A controllable writing of four distinctive
states by applying various scanning bias voltages is achieved, offering
opportunities for developing multilevel high density memories.
