  We report the nanoscale electrical imaging results in hexagonal
$Lu_{0.6}Sc_{0.4}FeO_3$ single crystals using conductive atomic force
microscopy (C-AFM) and scanning microwave impedance microscopy (MIM). While the
dc and ac response of the ferroelectric domains can be explained by the surface
band bending, the drastic enhancement of domain wall (DW) ac conductivity is
clearly dominated by the dielectric loss due to DW vibration rather than
mobile-carrier conduction. Our work provides a unified physical picture to
describe the local conductivity of ferroelectric domains and domain walls,
which will be important for future incorporation of electrical conduction,
structural dynamics, and multiferroicity into high-frequency nano-devices.
