Building a free-energy functional from atomically resolved imaging: Atomic-scale phenomena in La-doped $\mathrm{BiFe}{\mathrm{O}}_3$

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 $AB{\mathrm{O}}_3$ 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 $AB{\mathrm{O}}_3$. We show that FSM explains the coexistence of rhombohedral, orthorhombic, and spatially modulated phases, observed by atomic-resolution STEM in La-doped $\mathrm{BiFe}{\mathrm{O}}_3$. 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 $\mathrm{L}{\mathrm{a}}_x\mathrm{B}{\mathrm{i}}_{1-x}\mathrm{Fe}{\mathrm{O}}_3$ polymorphs. This approach allows the exploration of ferroic behaviors from experimental data only, without additional assumptions on the nature of the order parameter.

Figure 1

STEM of the $\mathrm{L}{\mathrm{a}}_{0.17}\mathrm{B}{\mathrm{i}}_{0.83}\mathrm{Fe}{\mathrm{O}}_3$. The interface between the orthorhombic O (left) and rhombohedral R (right) phases is shown. (a) Atomic model of the O-Pbam antiferroelectric type structure + +− − [2]. (b) Atomic model of the rhombohedral ferroelectric $R3c$ structure + + + + [17]. (c) Atomic-resolution HAADF image. (d) Atomic resolution ABF image. (e)A-site displacements. (f). Box and whisker plot of the [101] displacement component per layer (∼35 datapoints per column).

Figure 2

Distribution of order parameters $b_1$ and $b_2$ near the interface between R phase (left) and O phase (right) (a) for $c=0$ and different values of parameter $\chi =1.05$, 1.15, 1.3, and 1.7 (black, blue, magenta, and red curves, respectively); (b) for $\chi =1.15$ and different values of parameter $c=0$, 0.1, 0.2, and 0.3 (black, blue, magenta, and red curves, respectively).

Figure 3

Distribution of the normalized order parameters $A_i$ (in cyclic order) near the boundary between R domain (left) and O domain (right).

Figure 4

Cations A-displacement map according to theoretical calculations near the boundary between R domain (left) and O domain (right).