Building a free-energy functional from atomically resolved imaging: Atomic-scale phenomena in La-doped BiFe O3

Scanning transmission electron microscopy (STEM) has enabled mapping of atomic structures of solids with sub-picometer precision, providing insight to the physics of ferroic phenomena and chemical expansion. However, only a subset of information is available, due to projective nature of imaging in the beam direction. Correspondingly, the analysis often relies on the postulated form of macroscopic Landau-Ginzburg energy for the ferroic long-range order parameter, and some predefined relationship between experimentally determined atomic coordinates and the order-parameter field. Here, we propose an approach for exploring the structure of ferroics using reduced order-parameter models constructed based on experimental data only. We develop a four-sublattice model (FSM) for the analytical description of A-cation displacement in (anti)ferroelectric-antiferrodistortive perovskites of ABO3 type. The model describes the displacements of cation A in four neighboring unit cells and determines the conditions of different structural phases' appearance and stability in ABO3. We show that FSM explains the coexistence of rhombohedral, orthorhombic, and spatially modulated phases, observed by atomic-resolution STEM in La-doped BiFeO3. Using this approach, we atomically resolve and theoretically model the sublattice asymmetry inherent to the case of the A-site La/Bi cation sublattice in LaxBi1-xFeO3 polymorphs. This approach allows the exploration of ferroic behaviors from experimental data only, without additional assumptions on the nature of the order parameter.