  The suppression of ferroelectricity in ultrathin films of improper
ferroelectric hexagonal ferrites or manganites has been attributed to the
effect of interfacial clamping, however, the quantitative understanding and
related phenomenological model are still lacking. In this work, we report the
paraelectric-to-ferroelectric phase transition of epitaxial h-ScFeO3 films with
different thickness through in-situ reflection high-energy electron diffraction
(RHEED). Based on the interfacial clamping model and the Landau theory, we show
that the thickness-dependence of the ferroelectric Curie temperature can be
understood in terms of the characteristic length of interfacial clamping layer
and the bulk Curie temperature. Furthermore, we found that the critical
thickness of improper ferroelectricity is proportional to the characteristic
length of interfacial clamping layer. These results reveal the essential role
of mechanical clamping from interface on the improper ferroelectricity of
hexagonal ferrites or manganites, and could serve as the guidance to achieve
robust improper ferroelectricity in ultrathin films.
