Empirical approach to measuring interface energies in mixed-phase bismuth ferrite

In complex oxide heteroepitaxy, strain engineering is a powerful tool to obtain phases in thin films that may be otherwise unstable in bulk. A successful example of this approach is mixed-phase bismuth ferrite ($\mathrm{BiFe}{\mathrm{O}}_{3}$) epitaxial thin films. The coexistence of a tetragonal-like ($T$-like) matrix and rhombohedral-like ($R$-like) striations provides an enhanced electromechanical response, along with other attractive functional behaviors. In this paper, we compare the energetics associated with two thickness-dependent strain relaxation mechanisms in this system: domain walls arising from monoclinic distortion in the $T$-like phase, and the interphase boundary between the host $T$-like matrix and tilted $R$-like phases. Combining x-ray-diffraction measurements with scanning probe microscopy, we extract quantitative values using an empirical energy balance approach. The domain wall and phase-boundary energies are found to be 113 \ifmmode\pm\else\textpm\fi{} 21 and $426\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0.16em}{0ex}}23\phantom{\rule{0.16em}{0ex}}\mathrm{mJ}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{--2}$, respectively. These numerical estimates will help us realize designer phase boundaries in multiferroics, which possess colossal responses to external stimuli, attractive for a diverse range of functional applications.