Ferroelectric Size-effect on BiFeO$_{3}$ Films in Ultra-high Vacuum

The ferroelectric size effect is a highly pursued and controversial topic encompassing the scaling of polar distortion, domain structure and switchable polarization. We have studied epitaxial BiFeO$_{3}$ films using ultrahigh vacuum piezoresponse force microscopy. The films in vacuum develop out-of-plane polarization domains which, based on their lateral dimensions, drastically violate the Kittel law. From the analysis of the piezoresponse amplitude, we have established that the presence of a topographic island induces in-plane rotation of the polarization vector by 90$^{o}$. These findings are analyzed using a first-principles based model Hamiltonian approach. We have also obtained stable and reproducible piezoresponse hysteresis loops on the 2 nm films. Experiments conducted at the Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, sponsored by the Division of Scientific User Facilities, U.S. Department of Energy.