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Electrical phase diagram of bulk BiFeO$_3$

We study the electrical behavior of multiferroic BiFeO$_3$ by means of first-principles calculations. We do so by constraining a specific component of the electric displacement field along a variety of structural paths, and by monitoring the evolution of the relevant physical properties of the crystal along the way. We find a complex interplay of ferroelectric, antiferroelectric and antiferrodistortive degrees of freedom that leads to an unusually rich electrical phase diagram, which strongly departs from the paradigmatic double-well model of simpler ferroelectric materials. In particular, we show that many of the structural phases that were recently reported in the literature, e.g. those characterized by a giant aspect ratio, can be accessed via application of an external electric field starting from the $R3c$ ground state. Our results also reveal ways in which non-polar distortions (e.g., the antiferrodistortive ones associated with rotations of the oxygen octahedra in the perovskite lattice) can be controlled by means of applied electric fields, as well as the basic features characterizing the switching between the ferroelectric and antiferroelectric phases of BiFeO$_{3}$. We discuss the multi-mode couplings behind this wealth of effects, while highlighting the implications of our work as regards both theoretical and experimental literature on BiFeO$_{3}$.