Annealing dependence of ferroelectric domain patterns in h-\textit{RE}MnO$_{\mathrm{3}}$

Topological vortices with complex ferroelectric domains and domain walls exist in hexagonal rare-earth manganites(h-\textit{RE}MnO$_{\mathrm{3}})$. When h-\textit{RE}MnO$_{\mathrm{3}}$ crystals are grown below the ferroelectric-trimerization transition temperature (T$_{\mathrm{c}})$, they exhibit stripe domains. However, when a h-\textit{RE}MnO$_{\mathrm{3}}$ crystal with stripe domains is heated above and cooled down across T$_{\mathrm{c}}$, vortex domains emerge in the crystal. The networks of vortices are found to be in two different types : type-I domains with roughly equal fractions of upward and downward polarization domains and type-II domains with one dominant polarization. In this presentation, we report the post annealing effect on the ferroelectric topology under varying the ambient condition with N$_{\mathrm{2}}$ gas. We observed two different types (type-I and type-II) of vortex domains using piezoresponse force microscopy and compared the topology change with Monte Carlo simulation with varying internal electric field. We observed domains of YMnO$_{\mathrm{3}}$ start to increase upward polarization domain area fraction from 10{\%} to 65{\%} through the post annealing processes. We analyzed the correlation between internal electric field and off-stoichiometry condition.