Although enhanced conductivity at ferroelectric domain boundaries has been found in BiFeO$_3$ films, Pb(Zr,Ti)O$_3$ films, and hexagonal rare-earth manganite single crystals, the mechanism of the domain wall conductivity is still under debate. Using conductive atomic force microscopy, we observe enhanced conductance at the electrically-neutral domain walls in semiconducting hexagonal ferroelectric TbMnO$_3$ thin films where the structure and polarization direction are strongly constrained along the c-axis. This result indicates that domain wall conductivity in ferroelectric rare-earth manganites is not limited to charged domain walls. We show that the observed conductivity in the TbMnO$_3$ films is governed by a single conduction mechanism, namely, the back-to-back Schottky diodes model tuned by the segregation of defects.
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