  Focused ion beam (FIB) and scanning electron microscopy (SEM) are used to
reversibly switch improper ferroelectric domains in the hexagonal manganite
ErMnO$_3$. Surface charging is achieved by local ion (positive charging) and
electron (positive and negative charging) irradiation, which allows controlled
polarization switching without the need for electrical contacts. Polarization
cycling reveals that the domain walls tend to return to the equilibrium
configuration obtained in the as-grown state. The electric field response of
sub-surface domains is studied by FIB cross-sectioning, revealing the 3D
switching behavior. The results clarify how the polarization reversal in
hexagonal manganites progresses at the level of domains, resolving both domain
wall movements and the nucleation and growth of new domains. Our FIB-SEM based
switching approach is applicable to all ferroelectrics where a sufficiently
large electric field can be built up via surface charging, facilitating
contact-free high-resolution studies of the domain and domain wall response to
electric fields in 3D.
