Emergence of Long-Range Order in ${\mathrm{BaTiO}}_3$ from Local Symmetry-Breaking Distortions

By using a symmetry motivated basis to evaluate local distortions against pair distribution function data, we show without prior bias, that the off-center Ti displacements in the archetypal ferroelectric ${\mathrm{BaTiO}}_3$ are zone centered and rhombohedral-like across its known ferroelectric and paraelectric phases. We construct a simple Monte Carlo model that captures our main experimental findings and demonstrate how the rich crystallographic phase diagram of ${\mathrm{BaTiO}}_3$ emerges from correlations of local symmetry-breaking distortions alone. Our results strongly support the order-disorder picture for these phase transitions, but can also be reconciled with the soft-mode theory of ${\mathrm{BaTiO}}_3$ that is supported by some spectroscopic techniques.

Figure 1

Fits to $G(r)$ at the various temperatures indicated using symmetry-adapted displacements belonging to the ${\mathrm{\Gamma }}_4^{-}$ irrep only. The enlarged region around 1.8–2.2 Å shows the Ti-O bonds, which change most visibly as a function of temperature. The fitting procedure is described in the text.

Figure 2

(a) BWMAs from PDF refinements for all modes plotted against temperature. Modes belonging to ${\mathrm{\Gamma }}_4^{-}$, $X_5^{+}$, $M_2^{-}$, and all other irreps are colored blue, magenta, cyan, and red respectively. Insets show symmetry-adapted displacements belonging to the ${\mathrm{\Gamma }}_4^{-}$ irrep, which correspond to the three modes with the highest BWMA values and (b) shows their combined $+--$ coupling [$\mathrm{Ti}(T_{1u})$, $\mathrm{O}(A_{2u})$, and $\mathrm{O}(E_u)$, respectively] along the rhombohedral order parameter direction as observed locally in this study at all temperatures.

Figure 3

(a) MC simulation reproducing the tetragonal, orthorhombic, and rhombohedral phase transitions, with the square of the average polarization serving as an order parameter. Insets below the polarization line identify the relevant interactions penalized by our Hamiltonian, placed at temperatures where their energy scales become important in the simulation; above the line is a representation of the chainlike correlations present in the cubic phase ($J_0$, hardwired into our simulations) showing how the projection of the polarization along the direction of a chain is always preserved (black arrow). (b)–(e) A portion of a MC configuration for each phase, with the polar vector projections along the $⟨100⟩$ directions, and the calculated diffuse scattering in the planes indicated. It is the interchain disorder between the polar vector projections (white and black arrows) that gives rise to diffuse planes of scattering in reciprocal space. In the cubic phase it is in all three $⟨100⟩$ directions, giving three sets of intersecting orthogonal diffuse scattering planes in reciprocal space, and each successive phase transition represents interchain orderings along successive $⟨100⟩$ directions. An enlarged version of this figure is given as part of the Supplemental Material [33].