Microscale Mapping of Structure and Stress in Barium Titanate

Cross-correlation of electron backscatter diffraction (EBSD) patterns was used to generate rotation, strain, and stress maps of single-crystal tetragonal barium titanate (BaTiO3) containing isolated, small, sub-micrometer a domains separated from a c-domain matrix by 90° domain boundaries. Spatial resolution of about 30 nm was demonstrated over 5 μm maps, with rotation and strain resolutions of approximately 10−4. The magnitudes of surface strains and, especially, rotations peaked within and adjacent to isolated domains at values of approximately 10−2, i.e., the tetragonal distortion of BaTiO3. The conjugate stresses between a domains peaked at about 1 GPa, and principal stress analysis suggested that stable microcrack formation in the c domain was possible. The results clearly demonstrate the applicability of EBSD to advanced multilayer ceramic capacitor reliability and strongly support the coupling between the electrical performance and underlying mechanical state of BaTiO3-containing devices.


Introduction
The multilayer ceramic capacitor (MLCC) [1,2] is the "workhorse of the electronic components industry" [3] and a pervasive and critical element for electrical decoupling, filtering, and many other functions in advanced devices. A modern cell phone incorporates approximately 1000 MLCCs, and an electric vehicle incorporates about 10,000 MLCCs [3,4]; about three trillion MLCCs were manufactured in 2018 [4]. The external form factors of MLCCs are extremely small, ranging from millimeters in scale to smaller than 500 µm × 250 µm [4,5]. Internally, an MLCC consists of stacked layers of dielectric polycrystalline ceramic material, usually barium titanate (BaTiO 3 )-the subject of this work, interdigitated with layers of conducting metal electrodes (usually nickel) [1,2]. The polycrystalline ceramic grain size is typically hundreds of nanometers [6], and the ceramic layers are typically tens of micrometers thick separated by micrometer-scale electrodes [6]. Up to 1000 layers may comprise a single MLCC [5], depending on capacitance requirements.
The manufacturing yield and operational reliability of an MLCC are predominantly determined by three phenomena effective at three different length scales: (1) At the millimeter scale of the MLCC, primarily affecting device yield and primarily mechanical in origin, MLCCs may be stressed and fracture through attachment to, or flexing of, the host printed circuit board [7]. (2) At the 10 µm scale of the dielectric layers, primarily affecting MLCC reliability in early use and primarily electrical in origin, the 2 https://doi.org/10.6028/jres.125.013 ceramic may break down at an electrical or thermal "hot spot" or defect, leading to an electrical short [8,9]. (3) Finally, at the sub-micrometer scale of the grains, primarily affecting MLCC reliability in extended use and originating in mechanical and electrical coupling, the ceramic may lose its dielectric response, leading to a loss of component capacitance and thus device performance [1,10]. Of these three phenomena, the first two are extrinsic to BaTiO 3 , and their prevalence can be reduced through improved manufacturing to reduce circuit board flexure and the number of breakdown-related defects. However, the third phenomenon is intrinsic to BaTiO 3 and related dielectrics. Specifically, on cooling a polycrystalline dielectric during MLCC fabrication, the crystallographic structure of BaTiO 3 leads to "locked-in" grain-scale residual stresses. These stresses influence subsequent microstructural development, including the initiation of microcracks, and, often more practically, impede the ability of the material to alter polarization in response to a changing electric field. Thus, improving the short-term yield and long-term electrical capacitance and reliability of MLCCs requires knowledge of the internal residual stress state and how that stress depends on underlying structure. Mapping of stress in BaTiO 3 at the microscale is the subject of the current work.
In a recent series of studies [11][12][13], spatial variations of deformation and stress in single-crystalbased structures of BaTiO 3 were mapped using electron backscatter diffraction (EBSD) techniques in a scanning electron microscope (SEM). At room temperature, BaTiO 3 has a tetragonal unit cell [14,15] with a and c axes. The unit-cell [001] c dimension is about 1 % relatively longer than the [100] and [010] a dimensions, and the c axis is the direction of an electrical dipole moment. As a consequence, to minimize elastostatic and electrostatic energy [16], crystals of BaTiO 3 usually divide into regions-known as domains-of differing unit cell orientation and polarization. The domain orientations are often related by symmetry, and particularly important domain boundaries are those at which the unit cells change orientation by nearly 90° and 180°. The early work of Bradt and Ansell [17][18][19][20] showed that electrical aging in BaTiO 3 was due to the gradual formation of 90° domains as materials, particularly polycrystals, minimized internal stress in the tetragonal state. Later, the work of Arlt et al. [16,21] showed that polycrystal grain-size effects in BaTiO 3 were also due to 90° domain formation to minimize internal stress. Three-dimensional modeling of single crystals also showed the critical importance of electromechanical coupling in establishing domain structures [22,23]. More recent works [10,24,25] have shown the direct effects of externally applied stress on both electrical aging and 90° domain formation and alignment, further supporting the view that device electrical behavior reflects the material mechanical stress state (and showing that aging is probably not a chemical effect, as suggested very early [26]). The mechanical variations noted in the studies above [11][12][13] arose in multidomain structures in which the a and c domain axes alternated in orientation perpendicular to the crystal sample surface, termed the a and c domains, respectively. Two structures were studied: The first structure consisted of large lamellar domains up to 10 µm wide extending through a millimeter-scale crystal [11,13]. The second structure consisted of smaller domains about 1 µm wide in bundles terminating in a similarly sized crystal [12,13]. As well as providing fundamental insights into the microstructure of BaTiO 3 , these studies demonstrated the power of EBSD to generate two-dimensional (2-D) maps of near-surface stress, strain, and rotation with sufficient resolution for application to MLCC structures, i.e., about 10 −4 resolution in strain and rotation with conjugate stress resolution of about 10 MPa, and lateral spatial resolution of about 50 nm. However, the studies were restricted to pseudo-periodic structures large relative to MLCC grain sizes.
Here, extending the previous studies, isolated domains, comparable in size to MLCC grains, were examined by EBSD in the SEM. This article begins with a brief overview of the materials and methods, noting the similarities to those used previously and two major differences: (1) The method was applied at a much smaller scale, and (2) individual isolated microdomains were examined. Strain results are then presented as previously (also in the Appendix), but the emphasis here is on stress, leading to presentation in principal coordinates and enabling comparison with the similarly measured stress field associated with an isolated micro-indentation [27,28] and insight into possible fracture paths. A discussion follows the previous philosophy of inferring subsurface domain structure from the experimental constraint of surfacebased EBSD rotation measurements. The importance of dislocations at domain terminations is emphasized. To provide context, the work here forms part of an extensive set of techniques developed at the National Institute of Standards and Technology (NIST) to map stress: Neutron diffraction at centimeter length scales is used to map stress in large metal structures such as railway tracks [29]; X-ray diffraction at millimeter length scales is used to map stress in metal sheets for automobile applications [30] and at micrometer length scales to map stress in metal lines for microelectronic applications [31]; optical fluorescence techniques at sub-micrometer length scales are used to map stress distributions in polycrystalline ceramics to improve fracture resistance [32]; optical Raman spectroscopy techniques, also at sub-micrometer length scales, are used to map stress and strain distributions in single-crystal [27,28] and polycrystalline silicon (Si) [33] for microelectromechanical systems (MEMS) applications; and electron diffraction (EBSD) techniques are used at the near-nanometer scale to map stress, stain, and rotation in Si [34] and SiGe alloys for microelectronic applications [35].

2.
Materials and Methods

Materials
The sample for all experiments here was a BaTiO 3 single crystal formed by the top seeded solution growth method (MSE Supplies, Tucson, AZ) and used in previous strain and rotation studies of bundled domains [12,13]. 1 The as-received sample was a plate, 5 mm × 5 mm × 1 mm, and nominally [100] × [010] × [001], electrically poled such that the majority of the sample was a single c domain, i.e., a domain with c axis perpendicular to the large faces. A single large face of the sample as received was prepared by chemical-mechanical polishing. An additional polishing step using colloidal silica for 5 min was performed on this face. Optical micrographs of two regions of the polished surface are shown in Fig. 1. Figure 1(a) shows a set of features that appeared over the entire sample surface. These features were lenticular, ranging in length from (1 to 10) µm and width from (0.1 to 2) µm, and they exhibited a slightly raised topography with long dimensions parallel to the sample (and micrograph) edge. As will be shown below, these features are isolated 90° domains with the c axis nearly parallel to the surface and an a axis perpendicular to the surface and will be referred to as isolated a domains. Such surface features are not often reported, although isolated bulk a-c domain structures are common, and surface steps associated with bulk 90° domain boundaries (and their removal by stress) are apparently usual in crystal growth studies [36,37]. Similar features have recently been observed on lead zirconate titanate [38]. Figure 1(b) shows a second set of features that appeared infrequently over the sample surface. These features were in the form of blocky, almost-square, mesas with somewhat irregular (10 to 50) µm edges parallel to the sample (and micrograph) edges, and they exhibited significantly raised topography relative to the surrounding matrix. The size, shape, and topography suggest these features are domains of reversed c-axis polarity, separated from the matrix by 180° domain boundaries. These types of domains have been reported often since the earliest BaTiO 3 studies, especially the irregularly shaped boundaries [39][40][41][42][43][44][45][46][47][48][49], which are sometimes described as exhibiting a "watermark" geometry [42,[45][46][47][48][49][50]. These features will be referred to as isolated c domains. For both isolated a and c domains, the chemical-mechanical and colloidal silica polishing steps described above caused differential etching of the different domain polarization orientations on the surface, giving rise to the observed topography. The clearly raised nature in Fig. 1(b) suggests the features are negative polarization c domains in a positive c matrix [39,40,43,47,[50][51][52]. The slightly raised topography of the isolated a domains in Fig. 1     Schematic diagrams of the isolated domains and the X 1 -X 2 -X 3 sample coordinate system are shown in Fig. 2, such that the X 1 -X 2 planes are the image planes of Fig. 1. The arrows indicate the orientations of crystallographic [001] c axes. Figure 2(a) shows isolated a domains with extended surface dimensions aligned parallel to X 2 in the X 1 -X 2 plane. The perpendicular plane, X 1 -X 3 , has frequently been reported as the image plane in many optical and transmission electron microscope (TEM) studies of isolated a domains. The X 1 -X 3 subsurface cross section of Fig. 2(a) shows the commonly observed crystallographic [101] and [1 � 01] domain boundary orientations bisecting X 1 and X 3 , with the domains terminating in the subsurface in wedge-shaped formations [17-19, 42, 45, 46, 48, 53, 54]. Figure 2(b) shows an isolated c − domain in a c + matrix with edges approximately aligned with the sample X 1 and X 2 edges. The subsurface extensions in the X 3 direction of reversed c domains through the thickness of samples have been observed many times, including the earliest studies [31,33,46] and more recent digital reconstructions [55,56]. A combination of EBSD orientation analysis and simulation has recently demonstrated images of reversed c domains in lithium niobate [57].
Many isolated domains were examined using the orientation and strain analyses described below, and representative results from a single isolated a domain and a single isolated c domain are presented herein.

Orientation and Deformation Analysis
The orientation and deformation of the sample matrix and domains were determined from electron backscatter patterns (EBSPs) obtained from the polished sample surface. The surface was not coated prior to loading into a field emission SEM (Hitachi S4700 FESEM, Hitachi High Tech, Tokyo, Japan) for analysis. High-resolution EBSPs were recorded using an accelerating voltage of 20 kV and a beam current of ≈ 2 nA with the sample normal tilted 70° about X 1 relative to the SEM electron beam. Asymmetric 2-D grids of EBSPs were collected from the X 1 -X 2 plane. For the a domain, the grid was 175 × 20 steps in the X 1 × X 2 directions, respectively, generating 3500 EBSPs from points separated by 30 nm in the X 1 direction and 250 nm in the X 2 direction, producing high-resolution maps approximately 5.25 µm × 4.75 µm. For the c domains, the grid was 150 × 17, generating 2550 EBSPs separated in the X 1 direction by 400 nm and in the X 2 direction by 2.8 µm, and creating 60 µm × 48 µm maps. Each EBSP consisted of an image of 1344 × 1024 pixels; no binning was applied to the EBSPs, which were recorded at high gain with automatic and static background correction. Each EBSP was collected in approximately 1 s and was of sufficiently high quality such that each EBSP could be indexed to obtain crystal orientation at a scan point using Oxford HKL Flamenco software (version 5.0.9.1, Oxford Instruments, Abingdon, U.K.). Detection of the edges of Kikuchi bands in a circular region (radius = 511 pixels) centered on the middle of an EBSP was performed using a Hough transform method operating at maximum achievable resolution. Indexing was determined from the automatic detection of five to six bands and provided the local orientation of the tetragonal (a, a, c) crystal axes relative to the X 1 -X 2 -X 3 sample axes in terms of Euler angles. Two domain types were observed. A local crystal orientation in the sample was regarded as part of a c domain if the (c, X 3 ) angle was close to 0, and as part of an a domain if an (a, X 3 ) angle was close to 0. In practice, (a, X 2 ) angles were always close to 0, such that the transformation from c domain to a domain was accomplished by ± 90° rotation about X 2 .
The grid of EBSPs was analyzed to generate stress, strain, and rotation maps using the method of cross-correlation (CrossCourt 3.0, BLG Productions, Bristol, U.K.) [58][59][60]. Reference patterns from points in each data set were assigned (by definition) zero strain and rotation, and all deformation is thus relative to these points. Two reference patterns were used, one in the center of a c domain and one in the center of an a domain (domains were determined from the orientation analysis); all c domains were analyzed with the cdomain reference, and all a domains were analyzed with the a-domain reference. The details of deformation analyses by cross-correlation of EBSPs and extensive application to BaTiO 3 are considered elsewhere [11][12][13][58][59][60]. Briefly, the eight independent components of the traceless distortion tensor in crystal coordinates were determined at each grid location from analysis of 20 regions of interest (256 pixels 7 https://doi.org/10.6028/jres.125.013 × 256 pixels) from each EBSP. To obtain the full nine-component distortion tensor, A ij , the X 1 -X 2 surface was considered to be in mechanical equilibrium and normal traction-free, such that the closure condition 33 = 0 was imposed at all locations. No additional flattening or background subtraction of A ij was required or imposed [27,28]. The local infinitesimal strain tensor, ε ij , was obtained as the symmetric component of the full distortion tensor, = ( + )/2. The local infinitesimal rotation tensor, ω ij , was obtained as the antisymmetric component of the full distortion tensor, = ( − )/2, noting that the closure relation is not required in this case. The polar rotation tensor, , using a single c-domain reference [13] was not used ( = , except for a domains 13 = − 31 = 13 − /2). The stress, σ ij , was obtained from strain using σ ij = c ijkl ε kl and the appropriately rotated domain-dependent elastic constants c ijkl [11,12]. Global mechanical equilibrium was imposed by adjusting individual stress components such that the resultants had mean values of zero, noting that the condition 33 = 0 was automatically fulfilled [12]. The values and orientations of the principal stresses were determined using conventional methods [61].

Results
Optical microscopy, SEM, and EBSD revealed the domain structure, surface strains, and surface stresses. The domains observed here differed from earlier work in that they were relatively isolated, although they exhibited similar patterns of crystal rotation. Stress and strain maps exhibited large spatial variations and correlated with the measured domain structure. Each of these aspects is examined in turn.  Fig. 1(a), and the topography is not very evident. Nevertheless, a dark elongated feature is visible on the left of the image, and a less distinct feature is visible on the right. Much greater contrast is visible in the EBSD index map of the same region in Fig.  3(b), which shows the pixels indexed as "a" in red on a blue "c" field. There is indeed a large elongated a domain in the center left and smaller a domains to the center right and perhaps upper left. Figure 3(c) is a cross-sectional schematic diagram of the c-domain field with two isolated a domains, complementing the plan-view observations of Fig. 3, parts (a) and (b). The schematic cross section is consistent with the orientation variation measurement of Fig. 3(b) and a rigid rotation description of domain boundary structure [13]. In this description, domain boundary unit cells are deformed and exhibit diagonal mirror symmetry, and cells in domains on either side of a boundary are undeformed and related by rotation. The domain boundaries are approximately parallel to c{101} planes and are thus here inclined to both X 3 and X 1 ; see Fig. 2(a). The unit cell c/a ratio of tetragonal BaTiO 3 requires the scalar rigid rotation angle, θ r , to be 0.011 rad (or about 0.63°); see Fig. 3(c) [14,15]. In addition, charge neutrality requires the unit-cell ferroelectric dipoles to maintain relative "head-to-tail" orientations across domain boundaries. Arrows indicate dipole orientations, and dashed lines indicate domain boundary cells with indefinite dipole character. For visualization purposes, the BaTiO 3 tetragonal distortion and domain rotation angle are exaggerated by a factor of 20.

Structure
The domain and domain boundary sequence from left to right in Fig. 3(c), using previous notation [13], is c + //a − //c + //a + //c + , in which a letter identifies the domain, a superscript represents the polarization relative to X 3 or X 1 as appropriate, and // represents a domain boundary. Boundary orientation is constrained by the specified surface orientation and polarization. The right-hand sequence in Fig. 3(c), c + //a + //c + , positive θ r , is often depicted (see [13] for a list), and the domain boundaries in this case proceed downward to the right. The left-hand sequence, c + //a − //c + , negative θ r , is less often depicted (e.g., [15]), and, in this case, the domain boundaries proceed downward to the left. In both cases, a key assumption is    Figure 4 shows the region of Fig. 3(b) as a "visual tensor" representation of deformation in the form of color-filled contour maps of the six independent components of ε ij (symmetric) and three independent components of ω ij (antisymmetric). Such representation was used previously for indentations and MEMS structures in silicon [28,34,59,60]. The contour range for all maps is ± [(c/a) −1], ≈ ± 0.01, which is the relative tetragonal distortion of BaTiO 3 and the rigid-domain rotation angle (in radians, see above); the scale and axis orientation are indicated. Perhaps the most striking feature of Fig. 4 is that the deformation is

Stress
As demonstrated previously for lamellar and bundled domains [11,12], stress component maps using the laboratory Cartesian axes were generated from strain data using the elastic constants of BaTiO 3 . For the isolated domains here, a visual representation of the stress tensor in Cartesian axes was generated, shown in Fig. 5, noting that σ 33 = 0 is a boundary condition and is absent. The Cartesian stress components of Fig. 5 strongly resemble the conjugate Cartesian strain components of Fig. 4. Stress information that is unbiassed by the axes selected and that is more directly useful for interpreting potential fracture behavior can be obtained using principal coordinates [28]. Based on the Cartesian stress tensor data, the dominant in-plane principal stresses were determined, and the values are shown as color-filled contour maps in Fig. 6. Two features are quite apparent in these maps: The maximum principal stress, max σ p , shown in Information regarding the orientation of the principal stresses is shown on Fig. 7, which consists of several layers. First, the local values of max σ p are shown as a color-filled contour map, as in Fig. 6(a). Second, the local orientations of min σ p are shown as the short fine lines (for clarity, every fifth orientation is shown). Third, the minimum principal stress trajectories are shown as the continuous bold lines. These trajectories were obtained by (numerically) integrating the differential equation implicit in the determination of the principal stresses. The minimum principal stress trajectories in Fig. 7 are perpendicular to the local maximum principal stresses and as such provide estimates of potential fracture paths [62]. A feature apparent in Fig. 7 is that the trajectories tend to converge in two areas: in the region between the domains and along the edge of the large domain. The implication, given that these are also regions of tensile stress, is that these are areas of greater fracture probability and that cracks would preferentially propagate along these trajectories, somewhat independent of initiation location, leading to cracking in the c domain and delamination of the a domain from the c domain. In more detail, fracture of single-crystal BaTiO 3 is characterized by a toughness of 0.7 MPa m 1/2 [12], representing the equilibrium resistance to crack propagation. The relevant driving force for fracture is the stress-intensity factor, for which the value and variation can be obtained explicitly by integrating a known stress field over an existing or candidate crack path, as demonstrated previously for a BaTiO 3 microstructure [12]. Such an integration was not performed here, but estimates from Fig. 7 suggest a tensile stress zone of about 1 GPa characteristic magnitude and about 1 µm extent, providing a characteristic crack driving force expressed of about 1 MPa m 1/2 . When compared with the toughness of BaTiO 3 , this value suggests that cracks 1 µm in length would be in equilibrium in the BaTiO 3 microstructure of Fig. 1, in the c domain between the a domains. The localized nature of the tensile stress fields along the trajectories in Fig.  7 suggests that such cracks would be in stable equilibrium as "microcracks" and, although perhaps degrading the electrical and stiffness properties of an MLCC, would not lead to catastrophic device failure. The subsequent superposition of a uniform stress by circuit board flexure after MLCC fabrication could propagate such cracks and lead to device fracture and failure [7]. These considerations depend on the microstructural stress state remaining relatively unaltered by the initiation of cracks or delamination. If such effects do alter the stress field, domain wall motion or other relaxation effects are then also possible. The large stresses observed here suggest that such sequential electromechanical effects could be important, and that other accommodation mechanisms, e.g., formation of point defects or finite-width charged domain boundaries, which do not greatly alter the electrical response or mechanical compliance, may also operate.
An important final result is the strain field observed surrounding the very common c domains shown in Figs. 1(b) and 2(b). Consistent with the many previous observations [18,31,32,37,38,42,47,49], no strain of any significance with any distinguishing features or correlation with optical or SEM images was observed at any c domain or, by implication, associated with any 180° domain boundary. (Strictly, this was observed for 180° boundaries parallel to the c polarizations in the separated domains. Strain has been observed at 180° boundaries tilted relative to c [63].) This important null result places the above a-domain observations in context and is supported by the strain tensor with domain overlay shown in the Appendix.

Discussion
The observations here have extended the application of EBSD to small-scale deformation measurements in BaTiO 3 . In a materials engineering sense, the demonstration of strain and rotation resolutions of approximately 10 −4 with image pixel spacing of approximately 30 nm over maps about 5 µm square (Fig. 4) brings EBSD into the realm of commercial MLCC device dimensions. The use of such information to generate similarly sized maps of principal stresses and principal stress trajectories with resolutions of about 10 MPa (Figs. 5, 6, and 7) is of great practical importance for optimizing MLCC yield and reliability, which are currently limited by domain locking and cracking and fracture. In a materials science sense, study of isolated domains expands the number of different features examined in BaTiO 3 , enabling similarities and differences in underlying domain microstructure to be identified. Here, the similarities were noted in the strain variations within a single isolated domain that were comparable to those observed in lamellar [11] and bundled [12] domains. In particular, the strains suggested that unit cells within domains were more "cubic" at domain edges; e.g., a-domain tetragonal cells were compressed in plane and expanded out of plane to accommodate inclusion in a surrounding c-domain matrix (Fig. 4). Differences were noted in the rotation variation across multiple domains. In the lamellar and bundled structures, the rotation variations were all of one sign, suggesting that multiple lamellae and bundles generated macroscopic curvature of the single-crystal surface, an idea suggested in the earliest work [53]. Here, the rotation variations were of opposite signs, e.g., Fig. 4, suggesting that the isolated domains did not lead to surface curvature. The fundamental structure of BaTiO 3 , constrained by the isolated nature of the domains (Fig. 3), the localized rotation (Fig. 4), and the previous observations of subsurface termination [17-19, 42, 45, 46, 48, 53], are now considered. The individual rotation and domain sequences in Fig. 3(c), 0//−θ r //0 and c + //a − //c + (left) and 0//θ r //0 and c + //a + //c + (right), are similar to those observed in previous investigations of large (> 10 µm) lamellar domains [11,13]. Such lamellae were observed to extend completely through a millimeter-scale sample, and thus the sequences of Fig. 3(c) were considered to be adequate descriptions of the complete domain structure. The rotation and domain sequences of Fig. 3(c) are also similar to those that were most frequently observed in investigations of small (1 µm) domains in bundles [12,13]. The bundles were not observed to extend through the sample, and hence it was inferred that the domains terminated in the subsurface. The inference was strengthened by the occasional observation of simultaneous 0//θ r //2θ r and c + //a + //c + sequences. These sequences provided a clear indication of subsurface intersecting domain boundaries and a-domain termination. The consequent implication was that the surrounding c-domain material was connected beneath the surface by a dislocation-mediated low-angle grain boundary [13]. In the case of the bundles, Fig. 3(c) was thus considered to be only a description of the domain surface structure. In the case here of ultrasmall (0.1 µm) isolated a domains remnant from the c-domain poling process, the rotation and domain sequences of Fig. 3(c) are also observed, similar to the lamellae and the bundles. However, as for the bundles but different from the lamellae, the isolated domains were not observed to extend through the sample. This observation and the small size again suggest subsurface termination, although with a different surface constraint and thus different termination structure from that considered earlier [13]. The probable structure is considered below and has implications not just for isolated domains but for the majority of bundled domains. Perhaps the greatest constraint on inferred subsurface structure is surface rotation. Figure 8 shows line scans of rotations taken from the center of the maps in the X 1 (horizontal) direction in Fig. 4. In contrast to Fig. 4, the rotations are expressed in degrees, noting that the rigid rotation angle as shown in Fig. 3(c) is about 0.63°. Figure 8 makes clear the greater magnitude of the ω 13 rotation relative to the ω 12 and ω 23 rotations, the spatial localization of this rotation to the a domains, the reversal of sign from one a domain to the next, and the approximate agreement between the magnitudes of the peak values within the a domains with the rigid rotation value. The negative-zero-positive rotation sequence of ω 13 in Fig. 8 also agrees very well with that shown in the schematic a − //c + //a + central domain sequence in Fig. 3(c). Building on these agreements, and consistent with charge neutrality and domain indexing, Fig. 9(a) is a schematic crosssectional diagram of an isolated a domain terminated in a c-domain field. The subsurface structure of the a domain in Fig. 9(a) was constructed as follows, as the simplest of many alternatives: A single undeformed c domain with a surface step of height one unit cell was considered as an initial configuration. An area of material on the surface of the domain at the step was removed as a discrete number of cells. The initial cdomain cells are indicated by blue arrows parallel to X 3 . A single a domain of 4 × 4 cells was rotated by −θ r and inserted into the excised c domain. The initial a-domain cells are indicated by red arrows slightly inclined to X 1 . An ideal rigid rotation domain boundary was formed along the left edge of the a domain. A near-ideal boundary was formed along the right edge of the a domain by weakly deforming a-domain and boundary cells. Domain boundary walls are indicated by dashed lines.
The finite domain structure of Fig. 9(a) is perturbed from the semi-infinite structure of Fig. 3(c) due to the connected nature of the single c-domain matrix. The initial excised c domain included a step height that was filled by the inserted a domain: In the semi-infinite case, in which the c domains are unconnected, step height is determined solely by the assumed width of the intervening a domain. In the finite case here, step height was determined by a discrete number of c-domain cells, and weak deformation at the right boundary was required to accommodate the imposed c-domain geometry. A surface step is required by the a-domain rotation, and, as a consequence, deformation is required to maintain domain compatibility at the top surface. In contrast to the weak subsurface perturbation in Fig. 3(c), the subsurface structure of Fig. 9(a) is strongly perturbed by the imposition of compatibility at the a-domain termination. Cells in both the a domain and c domain were strongly deformed in Fig. 9(a) so as to form a continuous lattice at the subsurface domain boundary. In one area, lattice continuity could not be maintained, and this area is shown shaded in Fig. 9(a).
Inspection of Fig. 9(a) shows that the shaded area is the core of a dislocation with Burgers vector b = a[100] + c[001] and line direction perpendicular to the page (X 2 direction). Figure 9(b) shows the same unit cells as in Fig. 9(a) but as light-gray lines. Diagonal dark lines indicate the traces of (101) planes and halfplanes of unit cells. The dislocation consists of two terminating half-planes of unit cells (as opposed to halfplanes of atoms as in metals: a consequence of charge neutrality). The Burgers vector is also demonstrated by considering the Burgers circuit, shown using the finish-start convention (dashed-line with arrows). The shown dislocation can also be decomposed into two orthogonal dislocations, b = b 1 + b 2 , where b 1 = a[100], and b 2 = c[001], with one set of extra half-planes of unit cells below the dislocation and normal to the X 1 direction, and a second set to the right of the dislocation and normal to the X 3 direction. Figure 9 is easily extended to isolated domains much larger in terms of the number of cells involved and in which the surface steps are larger, perhaps visible as in crystal growth studies [36,37]. In those cases, although the left and right domain boundaries remain unaltered, the subsurface termination takes on a serrated form consisting of repeated instances of Fig. 9 and a periodic array of dislocations. The dislocation array spacing is entirely set by the angular deviation of the a domain from the c-domain matrix. The number of dislocations, N, is set by the size of the isolated domain and can be approximated by N ≈ θ r l/c, where l is the domain width. For the left domain in Fig. 3(b), l ≈ 0.7 µm, and for BaTiO 3 [14], c ≈ 0.4 nm, and θ r ≈ 10 mrad, such that N ≈ 20, and the total Burgers vector for the isolated domain is Nb ≈ c [20 0 20]. This surface inclusion domain structure, delimited by parallel rotation domain boundaries, should be compared with a similar structure delimited by intersecting domain boundaries studied earlier [13]. In the 16 https://doi.org/10.6028/jres.125.013 earlier case, the terminated a domain was accommodated by tilt in the c-domain matrix, modelled as a lowangle grain boundary and periodic array of edge dislocations. Once again, dislocation spacing was set by angular deviation of the a domain from the c-domain matrix, but the Burgers vector of these dislocations was approximately b = a [100], and the number of dislocations was set by the thickness of the sample. It is clear that subsurface termination of domains within differently oriented matrices requires accommodating dislocations to maintain structural compatibility. Recent observations [64] using dark-field X-ray microscopy of a lamellar domain array in a c-domain crystal indicated 〈110〉 trace orientations, subsurface terminations, and curved domain boundaries, probably accommodated by dislocations, with extended strain fields, all as inferred here from EBSD. For bundled domains, many dislocations would be required to accommodate the multiple changes in orientation, and hence a prediction from the above considerations is that the surfaces of bundles should be rumpled or rough, and this is borne out by experimental observation [12,64]. In addition, for both bundled and isolated domains, the stress and strain fields associated with dislocations should exist within domains and extend well beyond domain boundaries, and this is also borne out by observation. An interesting engineering question is whether, during an electrical poling operation, isolated and bundled domains form as a consequence of dislocation-based domain "pinning" or dislocations form as a consequence of incomplete poling, leaving isolated or bundled domains. Termination of domain boundaries may also be accommodated by charged domain boundaries as reviewed extensively elsewhere [65,66]. Such boundaries remove the strict "head-to-tail" domain boundary polarization requirement and allow the formation of local non-dislocation structures to accommodate domain terminations and rotations, but they do not remove all surface "rumpling" [66]. Electrical boundary conditions are likely to be critical to the formation of such boundaries, and thus the removal of dislocations, as free-charge carriers are required for screening charged features, providing even greater electromechanical coupling.
Longer-range behavior is highlighted in Fig. 8, which shows peaked variations in ω 13 extending over approximately 1 µm on the left and 0.5 µm on the right, somewhat inconsistent with the abrupt, stepped variation implied by Fig. 3(c). Two possible explanations for this difference are that (1) the enclosed a domain is not formed by rigid rotation but by gradual rotation, and Fig. 3(c) is only approximately correct, or (2) the information volume of the EBSD probe convolutes a range of rotations, both laterally and in depth, and Fig. 8 only approximates surface rotations. These are experimental questions to be addressed by further stress mapping observations or possibly modeling or simulations of the long-range elastic distortions and stresses. The idealized domain walls depicted in Fig. 3 and in many prior works give rise to charge-neutral domain walls and stress-free rigid rotations of a domains with respect to c domains, contrary to observation. Similarly, the Ginzburg-Landau-Devonshire (GLD) model, which even takes into account finite domain wall thickness, predicts that stresses and strains are localized to the nanoscale region around the domain wall [67]. However, the boundary conditions in the present work are different. Most notably, the GLD-simulated domain wall was not oblique to a free surface, differing from the present case. Allowing relaxation of the domain wall in the context of the GLD model or even a domain wall topological model [68,69] may reveal the source of the observed long-range elastic fields.
Finally, it should be emphasized that the results presented here do not address MLCC structures directly, but they do provide a measurement tool and demonstration measurements with the appropriate spatial and deformation resolutions to investigate reliability-limiting features that might arise in commercially fabricated MLCCs. Such MLCCs often use the doped BaTiO 3 X7R formulation [1,2], with microstructures significantly affected by the metal electrodes, screen printing of layers, and initial particle size. All these features could influence the formation and motion of domain boundaries and consequent internal stress states in a real MLCC, well beyond the range of structures observed in a single crystal. The work here provides a basis for using EBSD to investigate real MLCC structures.

Conclusions
Measurements of deformation and stress using high-resolution EBSD can be extended to spatial resolutions of 30 nm with strain and rotation resolutions of approximately 10 −4 and conjugate stress resolutions of approximately 10 MPa. The capability was demonstrated here on domain features in tetragonal BaTiO 3 . In particular, EBSPs from isolated, sub-micrometer a domains, surrounded by 90° 18 https://doi.org/10.6028/jres.125.013 domain boundaries within a c-domain field, were indexed for orientation, and detailed 2-D maps of strain, rotation, and stress were generated. The maps indicated that the surface deformation of such isolated domains is dominated by out-of-plane rotation, although the in-plane normal strains are still significant. The strain profiles across isolated domains were very similar to those observed across quasi-periodic lamellar and bundled domains, suggesting a commonality in domain boundary structure and within-domain relaxation for different domain microstructures. Maps of principal stress values were generated, indicating tension in the c-domain region separating isolated a domains. Calculated stress trajectories and assessment of stress values suggested that isolated microcracks in such locations were possible in these BaTiO 3 microstructures. The surface rotation profile across an isolated a domain within a continuous c-domain field constrained the subsurface surface termination of the a domain to include a dislocation with Burgers vector inclined to the surface. The results clearly demonstrate the applicability of EBSD measurements to commercial MLCC structures and reinforce the linkage between electrical performance of MLCCs, and other BaTiO 3 devices, and the underlying mechanical features through the domain structure.

Appendix
This Appendix provides strain information for isolated a and c domains in barium titanate, supporting claims made in the text. Specifically, Fig. A1 shows strain cross sections taken from Fig. 4 in the text, illustrating the similarity of strain variations within an isolated a domain as compared with those previously observed in lamellar and bundled domains and highlighting the resolved strain variations in 30 nm steps. Figure A2 shows strain maps of an area within Fig. 1(b) of the text, illustrating the lack of strain variation within, and adjacent to, an isolated c domain.  Fig. 4. There are no offsets, and zero strain is indicated by a dashed horizontal line. The shaded region indicates the domain boundary edges along the line scans taken from Fig. 3(b). There are many commonalities with the previous observations from multidomain structures [11][12][13]: First, the strain components in the c-domain matrix are little perturbed from the c-domain zero-reference value, as is the ε 22 component within the a domain. Second, the ε 33 strain component increases rapidly adjacent to both domain boundaries within the a domain before decaying to a smaller, but nonzero value. The peak value of the increase is comparable to the tetragonal distortion. Third, the ε 11 component is of opposite sign to the ε 33 component, and mirrors it in magnitude, even in detailed variation. The strain variations in other sections of this a domain and in other a domains, especially the right-most domain from Figs. 3 and 4, were almost identical to Fig. A1: out-of-plane tensile strain, in-plane compressive strain, and little strain along the long axis of the domain. The strain variation is consistent with the a domain accommodated in the c-domain matrix by becoming more "cubic,": shortened c axis and lengthened a axis, especially at domain edges, recalling that the strains are referenced relative to the center of the a domain; see Fig. 3(b). The major difference between Fig. A1 and the previous observations [13] is the spatial resolution: In this scan, the surface dimension of the domain is about 600 nm, and the pixel spacing is 30 nm, demonstrating the nanoscale strain mapping capability of EBSD.