Size-dependent infrared phonon modes and ferroelectric phase transition in BiFeO$_3$ nanoparticles

One emergent property of ferroelectric nanoparticles is the sized-induced structural distortion to a high-symmetry paraelectric phase at small particle sizes. Finite length scale effects can thus be advantageously employed to elucidate ferroelectric transition mechanisms. In this work, we combine infrared spectroscopy with group theory and lattice dynamics calculations to reveal the displacive nature of the ferroelectric transition in BiFeO$_3$, a room temperature multiferroic. Systematic intensity and frequency trends in selected vibrational modes show that the paraelectric phase is \emph{Pm}\=3\emph{m} and the lowest frequency A$_1$ feature is the soft mode that drives the first order transition. Finite length scale effects are also evident in the electronic structure with a red shifted band gap in nanoscale BiFeO$_3$ compared with that of the rhombohedral film, a result that can impact the development of ferroelectric photovoltaics and oxide- based electronics. Taken together, these findings demonstrate the foundational importance of size effects for enhancing the rich functionality and broad utility of transition metal oxides.