Formation of hierarchical domain patterns is one of the most fundamental properties of ferroelectric materials. Domains appear during symmetry-lowering structural phase transitions, with the number of energetically equivalent variants determined by the corresponding symmetry reduction. The properties of domain patterns are particularly intriguing when domains host two order parameters such as polarization and strain. The switching between domains underpins the giant piezoelectric effect, enhanced dielectric permittivity, and shape-memory.

Investigation of complex three-dimensional topologies of domain patterns remains challenging. Most microscopy techniques (TEM, PFM, etc.) are able to probe only the surface and have limited access to the bulk. In this context, X-ray crystallography is more promising as a bulk-penetrating technique, sensitive to lattice parameters, and, crucially, to the orientational relationship between domains. Yet, also this technique is not straightforward due to the need to interpret reciprocal rather than real-space fingerprints of domains.

The goal of this work is to enhance the capability of high-resolution single-crystal X-ray diffractometry for validating the laws governing the formation of ferroelectric and ferroelastic domain patterns. We employ the concept of permissible domain walls (PDW), where PDW [1] denotes a planar interface that connects two "ferroelastic" domains without any 2D lattice mismatch. We adapt the existing PDW-formalism to enable predicting the splitting of Bragg peaks in reciprocal space, considering that such splitting is directly accessible from X-ray diffraction experiments. This provides a straightforward tool to interpret X-ray diffraction data and assign individual sub-Bragg peaks to domains [2]. We have developed both a theoretical framework and a computer program for predicting permissible domain walls between domains of arbitrary symmetry, with a particular focus on the connection between domains of monoclinic symmetry [3], separating them into fixed (W-) and rotatable (S-) walls. We demonstrate how the results of our work are useful for investigating domain formation during phase transitions and understanding the domain pattern's response to external electric fields.

[1] J. Fousek, V. Janovec. (1969). J. Appl. Phys. 40, 135–142.
[2] S. Gorfman, D. Spirito, G. Zhang, C. Detlefs, N. Zhang. (2022). Acta Cryst. A78, 158–171.
[3] I. Biran, S. Gorfman. (2024). Acta Cryst A80, 112 - 128.

The cubic perovskite BaTiO₃ comprises one of the most renowned ferroelectric materials. Long-range ordering of Ti displacements along the cubic <111> directions is thought to trigger a rich sequence of phase transitions away from cubic symmetry, producing the polarisation observed on a macroscopic scale. While cubic BaTiO₃ has been scrutinised for its appealing ferroelectric properties, its six-layer hexagonal (6H) counterpart has received far less scrutiny. 6H-BaTiO₃ has been reported to exhibit ferroelectricity below 74 K, but the origins of its low-temperature polarisation remain controversial owing to insufficient characterisation of the ferroelectric phase. Prior heat capacity and Raman scattering measurements have contested the possibility of any symmetry breaking at Tc, further shrouding the origins of ferroelectricity in this material. Here, we resolve this prior controversy through detailed synchrotron X-ray/neutron diffraction measurements combined with symmetry-mode Rietveld refinements. Notably, we identify a previously undetected phase separation process below ~220 K that reconciles several experimental discrepancies surrounding the ferroelectric state in 6H-BaTiO₃. Our subsequent symmetry analysis supports the emergence of a rich domain structure in which an improper ferroelectric phase manifests through the rotation of a primary piezoelectric order parameter. Our results highlight 6H-BaTiO₃ as a highly promising system in which to study the coupling between piezoelectricity, ferroelasticity, and ferroelectricity, and how this coupling can be exploited to yield functionality from complex domain structures.

Our ability to study and advance the science of solid state reactions relies on the tools and techniques with which we can monitor them. However, some basic techniques such as Differential Thermal Analysis and Thermogravimetric Analysis monitor macroscopic phenomena and are insensitive to the microscopics and phase information. In situ XRD during the solid-state reaction is an established method to study the formation mechanisms of perovskites from starting oxides and carbonates. Here, we highlight several examples of fundamental insight garnered from such measurements in important and emerging materials including Na0.5Bi0.5TiO3-BaTiO3, K0.5Na0.5NbO3, (Ba,Sr)TiO3, and BiFeO3. Our most recent work on the synthesis of BiFeO3 from starting oxides of Bi2O3 and Fe2O3, recently published, will be a highlight of this presentation. In the synthesis of BiFeO3, secondary phases such as Bi25FeO39 (sillenite) and Bi2Fe4O9 (mullite) are often observed and thought to be a of the interdiffusion of the precursors. In this recent work, we demonstrate that Bi2O3 first reacts with Fe2O3 to form sillenite Bi25FeO39, which then reacts with the remaining Fe2O3 to form BiFeO3. Therefore, the synthesis of perovskite BiFeO3 is shown to occur via a two-step reaction sequence with Bi25FeO39 as an intermediate compound. Because Bi25FeO39 and the γ-Bi2O3 phase are isostructural, it is difficult to discriminate them solely from X-ray diffraction. Evidence is presented for the existence of the intermediate sillenite Bi25FeO39 using quenching experiments, comparisons between Bi2O3 behavior by itself and in the presence of Fe2O3, and crystal structure examination. With this new information, a proposed reaction pathway from the starting oxides to the product is presented. In summary, we reveal important microstructural interactions during solid state processing of perovskite compounds that is used to inform process improvements and synthesis consistency.