Permissible domain walls in monoclinic ferroelectrics. Part II. The case of MC phases.

Monoclinic ferroelectric phases are prevalent in various functional materials, most notably mixed-ion perovskite oxides. These phases can manifest as regularly ordered long-range crystallographic structures or as macroscopic averages of the self-assembled tetragonal/rhombohedral nanodomains. The structural and physical properties of monoclinic ferroelectric phases play a pivotal role when exploring the interplay between ferroelectricity, ferroelasticity, giant piezoelectricity and multiferroicity in crystals, ceramics and epitaxial thin films. However, the complex nature of this subject presents challenges, particularly in deciphering the microstructures of monoclinic domains. In Paper I [Biran & Gorfman (2024). Acta Cryst. A80, 112-128] the geometrical principles governing the connection of domain microstructures formed by pairing MAB type monoclinic domains were elucidated. Specifically, a catalog was established of `permissible domain walls', where `permissible', as originally introduced by Fousek & Janovec [J. Appl. Phys. (1969), 40, 135-142], denotes a mismatch-free connection between two monoclinic domains along the corresponding domain wall. The present article continues the prior work by elaborating on the formalisms of permissible domain walls to describe domain microstructures formed by pairing the MC type monoclinic domains. Similarly to Paper I, 84 permissible domain walls are presented for MC type domains. Each permissible domain wall is characterized by Miller indices, the transformation matrix between the crystallographic basis vectors of the domains and, crucially, the expected separation of Bragg peaks diffracted from the matched pair of domains. All these parameters are provided in an analytical form for easy and intuitive interpretation of the results. Additionally, 2D illustrations are provided for selected instances of permissible domain walls. The findings can prove valuable for various domain-related calculations, investigations involving X-ray diffraction for domain analysis and the description of domain-related physical properties.

Permissible domain walls in monoclinic MAB ferroelectric phases.

The concept of monoclinic ferroelectric phases has been extensively used over recent decades for the understanding of crystallographic structures of ferroelectric materials. Monoclinic phases have been actively invoked to describe the phase boundaries such as the so-called morphotropic phase boundary in functional perovskite oxides. These phases are believed to play a major role in the enhancement of such functional properties as dielectricity and electromechanical coupling through rotation of spontaneous polarization and/or modification of the rich domain microstructures. Unfortunately, such microstructures remain poorly understood due to the complexity of the subject. The goal of this work is to formulate the geometrical laws behind the monoclinic domain microstructures. Specifically, the result of previous work [Gorfman et al. (2022). Acta Cryst. A78, 158-171] is implemented to catalog and outline some properties of permissible domain walls that connect `strain' domains with monoclinic (MA/MB type) symmetry, occurring in ferroelectric perovskite oxides. The term `permissible' [Fousek & Janovec (1969). J. Appl. Phys. 40, 135-142] pertains to the domain walls connecting a pair of `strain' domains without a lattice mismatch. It was found that 12 monoclinic domains may form pairs connected along 84 types of permissible domain walls. These contain 48 domain walls with fixed Miller indices (known as W-walls) and 36 domain walls whose Miller indices may change when free lattice parameters change as well (known as S-walls). Simple and intuitive analytical expressions are provided that describe the orientation of these domain walls, the matrices of transformation between crystallographic basis vectors and, most importantly, the separation between Bragg peaks, diffracted from each of the 84 pairs of domains, connected along a permissible domain wall. It is shown that the orientation of a domain wall may be described by the specific combination of the monoclinic distortion parameters r = [2/(γ - α)][(c/a) - 1], f = (π - 2γ)/(π - 2α) and p = [2/(π - α - γ)] [(c/a) - 1]. The results of this work will enhance understanding and facilitate investigation (e.g. using single-crystal X-ray diffraction) of complex monoclinic domain microstructures in both crystals and thin films.

Identification of a coherent twin relationship from high-resolution reciprocal-space maps.

Twinning is a common crystallographic phenomenon which is related to the formation and coexistence of several orientation variants of the same crystal structure. It may occur during symmetry-lowering phase transitions or during the crystal growth itself. Once formed, twin domains play an important role in defining physical properties: for example, they underpin the giant piezoelectric effect in ferroelectrics, superelasticity in ferroelastics and the shape-memory effect in martensitic alloys. Regrettably, there is still a lack of experimental methods for the characterization of twin domain patterns. Here, a theoretical framework and algorithm are presented for the recognition of ferroelastic domains, as well as the identification of the coherent twin relationship using high-resolution reciprocal-space mapping of X-ray diffraction intensity around split Bragg peaks. Specifically, the geometrical theory of twinned ferroelastic crystals [Fousek & Janovec (1969). J. Appl. Phys. 40, 135-142] is adapted for the analysis of the X-ray diffraction patterns. The necessary equations are derived and an algorithm is outlined for the calculation of the separation between the Bragg peaks, diffracted from possible coherent twin domains, connected to one another via a mismatch-free interface. It is demonstrated that such separation is always perpendicular to the planar interface between mechanically matched domains. For illustration purposes, the analysis is presented of the separation between the peaks diffracted from tetragonal and rhombohedral domains in the high-resolution reciprocal-space maps of BaTiO3 and PbZr1-xTixO3 crystals. The demonstrated method can be used to analyse the response of multi-domain patterns to external perturbations such as electric field, change of temperature or pressure.

Geometrical prediction of cleavage planes in crystal structures.

Cleavage is the ability of single crystals to split easily along specifically oriented planes. This phenomenon is of great interest for materials' scientists. Acquiring the data regarding cleavage is essential for the understanding of brittle fracture, plasticity and strength, as well as for the prevention of catastrophic device failures. Unfortunately, theoretical calculations of cleavage energy are demanding and often unsuitable for high-throughput searches of cleavage planes in arbitrary crystal structures. A simplified geometrical approach (GALOCS = gaps locations in crystal structures) is suggested for predicting the most promising cleavage planes. GALOCS enumerates all the possible reticular lattice planes and calculates the plane-average electron density as a function of the position of the planes in the unit cell. The assessment of the cleavage ability of the planes is based on the width and depth of planar gaps in crystal structures, which appear when observing the planes lengthwise. The method is demonstrated on two-dimensional graphene and three-dimensional silicon, quartz and LiNbO3 structures. A summary of planar gaps in a few more inorganic crystal structures is also presented.

Algorithms for target transformations of lattice basis vectors.

Simple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.

Ferroelectric domain wall dynamics characterized with X-ray photon correlation spectroscopy.

Technologically important properties of ferroic materials are determined by their intricate response to external stimuli. This response is driven by distortions of the crystal structure and/or by domain wall motion. Experimental separation of these two mechanisms is a challenging problem which has not been solved so far. Here, we apply X-ray photon correlation spectroscopy (XPCS) to extract the contribution of domain wall dynamics to the overall response. Furthermore, we show how to distinguish the dynamics related to the passing of domain walls through the periodic (Peierls) potential of the crystal lattice and through the random potential caused by lattice defects (pinning centers). The approach involves the statistical analysis of correlations between X-ray speckle patterns produced by the interference of coherent synchrotron X-rays scattered from different nanosize volumes of the crystal and identification of Poisson-type contribution to the statistics. We find such a contribution in the thermally driven response of the monoclinic phase of a ferroelectric PbZr0.55Ti0.45O3 crystal and calculate the number of domain wall jumps in the studied microvolume.

Monoclinic distortion, polarization rotation and piezoelectricity in the ferroelectric Na0.5Bi0.5TiO3.

The relationship between crystal structure and physical properties in the ferroelectric Na0.5Bi0.5TiO3 (NBT) has been of interest for the last two decades. Originally, the average structure was held to be of rhombohedral (R3c) symmetry with a fixed polarization direction. This has undergone a series of revisions, however, based on high-resolution X-ray diffraction, total neutron scattering, and optical and electron microscopy. The recent experimental findings suggest that the true average symmetry is monoclinic (space group Cc), which allows for a rotatable spontaneous polarization. Neither polarization rotation nor its potentially important real role in enhanced piezoelectricity is well understood. The present work describes an in situ investigation of the average monoclinic distortion in NBT by time-resolved single-crystal X-ray diffraction under external electric fields. The study presents a high-resolution inspection of the characteristic diffraction features of the monoclinic distortion - splitting of specific Bragg reflections - and their changes under a cyclic electric field. The results favour a model in which there is direct coupling between the shear monoclinic strain and the polarization rotation. This suggests that the angle of polarization rotation under a sub-coercive electric field could be 30° or more.

Local-scale structures across the morphotropic phase boundary in PbZr1-x Ti x O3.

Lead zirconate titanate (PZT) is one of the most widely studied piezoelectric materials, mainly because of its 'mysterious' relationship between the so-called morphotropic phase boundary (MPB) and its strong piezoelectric coupling factor. Using results from a pair distribution function analysis, this paper examines how the complex local structure in PZT affects the long-range average structure across the MPB. A monoclinic M C type structure is discovered in PZT. A first-order transformation between the monoclinic M A and M C components in both the average and local structures explains the sudden change in piezoelectric effect around these compositions. The role of polarization rotation in the enhancement of the piezoelectric properties is discussed with respect to the composition of PZT. The structure-property relationship that is revealed by this study explains the unique properties of PZT, and may be applicable in the design of new MPB-type functional materials.

Picometer polar atomic displacements in strontium titanate determined by resonant X-ray diffraction.

Physical properties of crystalline materials often manifest themselves as atomic displacements either away from symmetry positions or driven by external fields. Especially the origin of multiferroic or magnetoelectric effects may be hard to ascertain as the related displacements can reach the detection limit. Here we present a resonant X-ray crystal structure analysis technique that shows enhanced sensitivity to minute atomic displacements. It is applied to a recently found crystalline modification of strontium titanate that forms in single crystals under electric field due to oxygen vacancy migration. The phase has demonstrated unexpected properties, including piezoelectricity and pyroelectricity, which can only exist in non-centrosymmetric crystals. Apart from that, the atomic structure has remained elusive and could not be obtained by standard methods. Using resonant X-ray diffraction, we determine atomic displacements with sub-picometer precision and show that the modified structure of strontium titanate corresponds to that of well-known ferroelectrics such as lead titanate.

A rapid two-dimensional data collection system for the study of ferroelectric materials under external applied electric fields.

Synchrotron X-rays on the Swiss Norwegian Beamline and BM28 (XMaS) at the ESRF have been used to record the diffraction response of the PMN-PT relaxor piezoelectric 67% Pb(Mg1/3Nb2/3)O3-33% PbTiO3 as a function of externally applied electric field. A DC field in the range 0-18 kV cm-1 was applied along the [001] pseudo-cubic direction using a specially designed sample cell for in situ single-crystal diffraction experiments. The cell allowed data to be collected on a Pilatus 2M area detector in a large volume of reciprocal space using transmission geometry. The data showed good agreement with a twinned single-phase monoclinic structure model. The results from the area detector were compared with previous Bragg peak mapping using variable electric fields and a single detector where the structural model was ambiguous. The coverage of a significantly larger section of reciprocal space facilitated by the area detector allowed precise phase analysis.

Time-resolved X-ray diffraction reveals the hidden mechanism of high piezoelectric activity in a uniaxial ferroelectric.

High piezoelectric activity of many ferroelectrics has been the focus of numerous recent studies. The structural origin of this activity remains poorly understood due to a lack of appropriate experimental techniques and mixing of different mechanisms related to ferroelectricity and ferroelasticity. Our work reports on the study of a uniaxial Sr_{0.5}Ba_{0.5}Nb_{2}O_{6} ferroelectric where the formation of regions with different spontaneous strains is ruled out by the symmetry and where the interrelation between piezoelectricity and ferroelectricity can be inspected in an isolated fashion. We performed x-ray diffraction experiments on a single crystalline sample under alternating electric field and observed an unknown hidden-in-the-bulk mechanism, which suggests that the highest piezoelectric activity is realized in the volumes where nucleation of small ferroelectric domains takes place. This new mechanism creates a novel roadmap for designing materials with enhanced piezoelectric properties.

Topology and temperature dependence of the diffuse X-ray scattering in Na0.5Bi0.5TiO3 ferroelectric single crystals.

The results of high-resolution measurements of the diffuse X-ray scattering produced by a perovskite-based Na0.5Bi0.5TiO3 ferroelectric single crystal between 40 and 620 K are reported. The study was designed as an attempt to resolve numerous controversies regarding the average structure of Na0.5Bi0.5TiO3, such as the mechanism of the phase transitions between the tetragonal, P4bm, and rhombohedral | monoclinic, R3c | Cc, space groups and the correlation between structural changes and macroscopic physical properties. The starting point was to search for any transformations of structural disorder in the temperature range of thermal depoling (420-480 K), where the average structure is known to remain unchanged. The intensity distribution around the {032} pseudocubic reflection was collected using a PILATUS 100K detector at the I16 beamline of the Diamond Light Source (UK). The data revealed previously unknown features of the diffuse scattering, including a system of dual asymmetric L-shaped diffuse scattering streaks. The topology, temperature dependence, and relationship between Bragg and diffuse intensities suggest the presence of complex microstructure in the low-temperature R3c | Cc phase. This microstructure may be formed by the persistence of the higher-temperature P4bm phase, built into a lower-temperature R3c | Cc matrix, accompanied by the related long-range strain fields. Finally, it is shown that a correlation between the temperature dependence of the X-ray scattering features and the temperature regime of thermal depoling is present.

X-ray diffraction by a crystal in a permanent external electric field: electric-field-induced structural response in alpha-GaPO4.

For the first time, site-selective distortion has been investigated for two different structural units in the ternary compound alpha-GaPO(4) under the influence of a permanent external electric field. Based on 54 measured reflection intensities, the electric-field-induced distortion of PO(4) and GaO(4) tetrahedra in alpha-GaPO(4) crystals is evaluated using a model of pseudoatomic displacements introduced recently [Gorfman, Tsirelson & Pietsch (2005). Acta Cryst. A61, 387-396]. A stronger variation of the P-O bond lengths in the PO(4) tetrahedron was found compared to the bonds in the GaO(4) tetrahedron. The different distortions of the tetrahedra owing to the electric field were analysed in terms of the valence charge density of alpha-GaPO(4) and its topological characteristics. The larger charge of the P pseudoatom compared to the Ga atom was recognized as the main reason for the higher sensitivity of the PO(4) tetrahedron to a permanent external electric field.

X-ray diffraction by a crystal in a permanent external electric field: general considerations.

The variations of X-ray diffraction intensities from a crystal in the presence of a permanent external electric field is modeled analytically using a first-order stationary perturbation theory. The change in a crystal, induced by an external electric field, is separated into two contributions. The first one is related to a pure polarization of an electron subsystem, while the second contribution can be reduced to the displacements of the rigid pseudoatoms from their equilibrium positions. It is shown that a change of the X-ray diffraction intensities mainly originates from the second contribution, while the influence of the pure polarization of a crystal electron subsystem is negligibly small. The quantities restored from an X-ray diffraction experiment in the presence of an external electric field were analyzed in detail in terms of a rigid pseudoatomic model of electron density and harmonic approximation for the atomic thermal motion. Explicit relationships are derived that link the properties of phonon spectra with E-field-induced variations of a structure factor, pseudoatomic displacements and piezoelectric strains. The displacements can be numerically estimated using a model of independent atomic motion if the Debye-Waller factors and pseudoatomic charges are known either from a previous single-crystal X-ray diffraction study or from density functional theory calculations. The above estimations can be used to develop an optimum strategy for a data collection that avoids the measurements of reflections insensitive to the electric-field-induced variations.

X-ray scattering amplitude of an atom in a permanent external electric field.

Quantum-mechanical description of the X-ray scattering by the many-electron atom in a permanent external electric field is developed in terms of the perturbation theory. Explicit expression for the electric field induced addition to the atomic scattering factor is derived and calculations for some atoms are performed. It was found that the change of the X-ray structure factor due to an electric field is too small to be detected with existing experimental techniques.