Bragg coherent diffraction imaging allowing simultaneous retrieval of three-dimensional shape and strain distribution for 40–500 nm particles

Ferroelectrics show particle-size-dependent dielectric properties, notably in the range of ∼1 μm or less, called the ferroelectric size effect. ^1–9) The size effect appears in the case of tetragonal barium titanate (BaTiO₃, BTO) as a decrease in its tetragonality (c/a ratio) ^4–6,10) and its dielectric permittivity with a reduction in grain size (<1 μm) in ceramics ^4,5) or particle size (<100 nm). ^7,9,11) The decrease in the c/a ratio and the permittivity corresponds to a decrease in its spontaneous electric polarization. The polarization disappears at a critical size, of which various values have been reported, such as 10–30, ⁵⁾ 40, ⁶⁾ 26, ¹²⁾ and 8 nm. ¹³⁾ The critical size variations are attributed to the shape of the crystals (such as facets), which affects the polarization orientation on the nanoscale. ¹³⁾ The ferroelectric properties in the submicrometer to nanometer scale (called mesoscale hereafter) are influenced greatly by the size and shape of the crystals.

Another aspect of the size effect is that the dielectric permittivity of fine ferroelectric BTO particles has a maximum value at a grain size of ∼1 μm for ceramics or a particle size of ∼100 nm. ^4,9) This behavior is attributed to the increase in the number of 90° domain walls per unit volume of a particle or grain; ^4,9,14) meanwhile, the size of one 90° domain is reduced. The domain sizes can be determined by considering the domain widths and the number of domain walls under the assumption that the ferroelectric crystal is square and has a perfectly striped domain structure with a negligible domain wall thickness. ^15,16) Actual fine ferroelectric particles have various outer shapes; consequently, their domain arrangement is quite complicated. ^13,17,18) The ferroelectricity in a fine particle can be understood by simultaneously knowing its outer shape, size, and domain arrangement.

The 90° domain orientation of tetragonal BTO, which is a typical room-temperature ferroelectric material, is distinguishable according to Bragg reflections. ¹⁹⁾ The decrease in tetragonality reduces the reciprocal space distance between a pair of reflections from the 90° domains (e.g. 200 and 002; 110 and 101). ^8,19) The 90° domain boundary is strongly distorted due to the lattice connection between adjacent domains; ^20–22) therefore, the shape of these domains appears as a strain contrast. Bragg coherent diffractive imaging (Bragg-CDI) ^23–25) is one of the most promising methods to simultaneously investigate the outer shape and strain distribution at the mesoscale in a fine crystalline particle. Third-generation X-ray synchrotron light sources, such as SPring-8, have a spatial coherence length of several micrometers, called coherent X-rays. Irradiation by coherent X-rays upon single-crystalline particles results in a Bragg reflection with speckles, including the structural information of the outer shapes, sizes, and mosaicity of the crystal (we refer to the coherent X-ray diffraction or scattering patterns ²⁶⁾ as a speckle). Phase retrieval of the appropriately sampled speckle pattern can simultaneously reconstruct the outer shape, size, and strain distribution. Since Robinson et al. developed Bragg-CDI, ^23–25) the structural heterogeneity of crystalline materials at the mesoscale has been actively investigated, as seen in many reports (e.g. dislocations inside Pd fine crystal and polycrystalline film due to hydrogen absorption, ^27,28) change in strain distribution inside a BTO fine particle induced by an external electric field ²⁹⁾). The strain distribution for ferroelectric BTO, visualized with Bragg-CDI, reflects the 90° domain structure, including the domain walls. In addition, Bragg-CDI nondestructively obtains these data, which is an essential ability for detailed domain observation.

Single-particle measurements for a wide range of particle sizes from several tens of nanometers (expected to lose ferroelectricity) to over 300 nm (regarded as bulk crystals) using a Bragg-CDI are expected to further elucidate BTO's mesoscale ferroelectricity. We have provided a visualization of the three-dimensional outer shape and internal strain distribution for 100–300 nm class crystals using the Bragg-CDI method. ³⁰⁾ More recently, some researchers proposed visualizing the domain structure of ferroelectric BTO with particle sizes of 200–300 nm using Bragg-CDI. ^31,32) Diao et al. demonstrated the 90° domains undergoing ferroelectric phase transition with increasing temperature, ³¹⁾ and Liu et al. showed that the domains transform with applied uniaxial pressure. ³²⁾ A better understanding of size effects in BTO requires improvements upon the Bragg-CDI technique to expand the range of measurable particle sizes. In this study, we expanded the observable particle size to 40–500 nm by advancing Bragg-CDI development to understand the size effect via the simultaneous visualization of the outer shape and strain distribution for ferroelectric BTO nanocrystals.

This study aims to achieve two primary goals. The first is to reduce background noise due to X-ray scattering by air. To this end, we newly prepared a vacuum chamber for samples, enabling us to obtain high-contrast X-ray diffraction pattern for a shorter time. The second is to optimize a real-space constraint; our modified phase-retrieval algorithm can use appropriate real-space constraints with shrinking support to refine the phase distribution. We also introduced sample-handling procedures, which are commonly used in transmission electron microscopy (TEM) techniques, which helped us efficiently detect and trace a single fine particle. Further, the observable scale range of 40–500 nm covers almost the entire inapplicable scale range for other microscopic probing techniques, such as TEM and optical microscopy. The Bragg-CDI technique covered a wide measurable range to help unveil the mesoscale phenomenon of the ferroelectric size effect.

Bragg-CDI was carried out on the coherent X-ray scattering apparatus ³⁰⁾ installed at BL22XU of SPring-8. ³³⁾

Three types of BTO and Pd particles were prepared: Pd with an average size a = 40 nm and a cube-like shape (Kyoto Univ. and Kyushu Univ.); ³⁴⁾ BT-Y2c with an average particle size a = 200 nm and a cube-like shape (Yamanashi Univ.); ³⁵⁾ BT-S3 and BT-S5 with an average particle size a = 300 and 500 nm, respectively, and a rich-curved shape (Sakai Chemical Industry Co., Ltd.). These particle sizes were determined from TEM or a scanning electron microscope.

Figure 1 shows a schematic diagram of the present experimental setup; Table I shows the experimental parameters optimized for the samples (described in detail below). X-rays from an in-vacuum undulator were monochromatized by a liquid-nitrogen-cooled Si(111) double-crystal monochromator. The photon energy was tuned to 8.0 keV (λ = 1.425 Å) or 8.7 keV (λ = 1.55 Å) depending upon the experimental hutch. The samples were supported on support materials using the following procedure: (1) the fine particles were homogeneously dispersed in ethanol using an ultrasonic cleaner, (2) drop of the solution was placed on the commercially available support membranes, and then (3) the membranes were dried in a low vacuum for several hours. In addition, the Pd and BT-Y2c on the membranes were coated with Au film with a thickness of several tens of nanometers using a sputter coater (SC-701MkⅡ, SANYU ELECTRON Co., Ltd.) to prevent thermal drift. Here we used two types of membranes: a scanning transmission electron microscopy (STEM) grid (ELS-C10, Okenshoji) with a laminated structure consisting of 15–20 nm thick polymer, 20–25 nm thick carbon, 25 μm thick copper, and a 200 nm thick SiN membrane window (X05-A200Q25, Filgen Inc.). The STEM grid is commonly used in TEM; therefore, the copper base of the STEM grid has an array of 100 μm diameter holes that allow electrons to pass (see the image in the middle left of Fig. 1). We used this grid array to trace the position of the sample. An optical microscope (VHX-7000, Keyence Corp.) image of the grid on the STEM grid (upper left of Fig. 1) shows that fine particles (BT-S3 in this case) were homogeneously dispersed and supported on the STEM grid; we also confirmed the same condition for fine particles on the SiN membrane window. After the membrane was mounted on a diffractometer via a membrane holder, fine particles on the membrane were observed with a high-resolution lens-coupled X-ray imaging detector, XRDCH-S0-C1 (SIGMAKOKI Co., Ltd.). ³⁷⁾ As shown in the lower right of Fig. 1, the X-ray imaging detector observed the pattern of the grid array of the STEM grid; further, the detector can image fine particles via a high-magnification lens. ³⁰⁾ The X-ray coherency was not disturbed by passing through the membranes. For this direct imaging, an X-ray beam with an area greater than 300 × 300 μm² was used without focusing.

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This paragraph defines the experimental parameters for the Bragg-CDI (the specific parameters for each sample are provided in Table I). The X-rays were focused with a refractive lens (Karlsruhe Institute of Technology) ³⁸⁾ to dimensions of vertical (w_V) × horizontal (w_H) μm². The focused X-rays illuminated the fine particles on the membrane; a two-dimensional photon-counting detector EIGER 500 K or PILATUS 300 K (both DECTRIS Ltd.) collected the diffracted X-rays in three dimensions. The pixel size of both the EIGER 500 K and PILATUS 300 K were 75 × 75 and 172 × 172 μm² (P²), respectively. The detector was positioned at R (m) downstream from the samples. We chose Bragg reflections 110 for BTO and 200 for Pd, respectively. The Pd, BT-Y2c, and BT-S3 samples have small particle sizes and therefore were placed in a vacuum chamber to ensure the contrast of the diffraction patterns. The vacuum chamber has an aperture made from polyimide, which did not disturb the X-ray coherency. The upper right of Fig. 1 shows a typical coherent X-ray diffraction pattern of BT-Y2c. Speckles were observed around the center with good contrast. The vacuum ambient also shortened the scan time, which helped prevent sample drift; the scan time of the speckle pattern in total (t_tot) was 12.5 min for BT-Y2c, which is over ten times less than in the previous report. ³⁰⁾ We collected three-dimensional speckle patterns around the Bragg position by changing the Bragg angle (ω scan) by, at most, Δω ∼ 3°. Note that these experimental settings met the following requirements for phase retrieval of the observed diffraction pattern: the particle size a completely within the transverse and longitudinal coherence lengths, far-field diffraction, and oversampled diffraction patterns.

We reconstructed three-dimensional images of the fine-particle samples using the three-dimensional Bragg diffraction patterns. A series of hybrid input–output (HIO) and error-reduction (ER) algorithms, ³⁹⁾ which are frequently used in CDI, generate the phase retrieval of the diffraction patterns. We carried out these algorithms using the following procedure: (1) perform the HIO-ER algorithm for approximately 1000 cycles, (2) shrink the real-space constraint (support), and then (3) perform the ER algorithm for approximately 200–1000 cycles. The shrunk support was produced by Gaussian blur filtering of the three-dimensional reconstructed image, and the threshold for the contour of its electron density was set such that the support completely wrapped the image. A real-space constraint commonly used in CDI, called shrink wrap, is updated during successive cycles. ⁴⁰⁾ In contrast, almost all iterative operations in this research converged within the procedure described above. We set the initial phase randomly, tried various initial phases, and confirmed that each reconstructed image was similar.

A coherent X-ray diffraction pattern consists of an intense diffraction spot containing information on average crystal structure and anisotropic weak speckles around this intense spot which contain information on the atomic displacement from the average position. The atomic displacement appears as an optical path difference between a diffracted wave from the average structure and a diffracted wave from the displaced one. The optical path difference normalized by the lattice spacing corresponds to a phase. Herein, the phase reconstructed by Bragg-CDI is represented by φ = u/d_hkl using the local displacement u and the lattice spacing d_hkl (h, k, l are diffraction indices). ²⁴⁾ The phase origin is basically undetermined; however, in this paper, φ = 0 can be treated as the phase origin for the following reason. The center of the global phase distribution in real space is almost determined using analytical procedure (1) with the HIO-ER algorithms. In this procedure, we used the phase limitation centered on 0 (−π/2 ≤ φ ≤ π/2); therefore, the phase φ = 0 behaved as the origin.

The introduction of the vacuum chamber has enabled the reduction of background levels by a factor of approximately 1/100 compared to our previous measurements; we have succeeded in obtaining coherent X-ray diffraction data for 40 nm fine crystals with high contrast. Figure 2 shows the results obtained by Bragg-CDI for a particle from the Pd fine crystal with a cube-like shape. The scattering pattern shown in Fig. 2(a) consists of an intense diffraction spot (yellow) and a crisscrossed fringe around the spot (light blue). The crisscrossed scattering pattern indicates the Pd particle's cube-like shape. The reconstructed image actually showed an approximately cubic shape, as shown in Fig. 2(b). A typical particle size of the Pd sample is already known to be 40 × 40 × 40 nm³, as shown in the lower portion of Fig. 2(b). The reconstructed image has approximately the same size as this box. The resolution in real space was approximately 9 nm. Figure 2(c) shows the phase distribution (corresponding to the strain) cut by the blue plane in Fig. 2(b). Except for a part of the edge (the surface in three-dimensional image), the phase difference inside the cross-sectional image was at most ∼30° (∼0.16 Å, expressed by displacement). This small phase difference indicates low structural heterogeneity inside the Pd fine crystal. Thus, the present Bragg-CDI apparatus enabled the outer shape, size, and strain distribution to be observed in a 40 nm fine crystal.

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It should be noted that the support-shrinking operations clarified the contour of the reconstructed images and quantified the phase contrast. Figure 3 shows a comparison of the reconstructed image of a particle from BT-Y2c before and after the support-shrinking operation. The scattering pattern shown in Fig. 3(a) consists of an intense diffraction spot, a crisscrossed fringe around the spot, and a fish-spine-like fringe extending from the spot. The tail of the fish-spine-like fringe is directed to 101 (or 10-1) reflection, although the 101 (or 10-1) reflection does not appear. Phase retrieval of the diffraction pattern of split peaks often generates a misaligned structure (artifacts) in images [e.g. as seen in Fig. 2(g) in Ref. 31]; therefore, we carried out phase retrieval only around the crisscrossed speckle (inside the area marked by a dashed line). The upper side of Fig. 3(b) shows an image that has been reconstructed using cuboidal support corresponding to an oversampling ratio of 2 × 2 × 1.2. A cubic image corresponds to the outer shape of the BT-Y2c particle; however, messy shapes appear simultaneously around the cubic shape. As shown in Fig. 3(c), the reconstructed phase-distribution image exhibits an unclear cubic shape, as well as noisy patterns across the support. The phase was limited from −π/2 to +π/2 to prevent the divergence of these noisy patterns; however, this limitation spoils the quantitativeness of the phase contrast. The shrunken support enabled phase retrieval with a full-circle range to reproduce the quantitative phase information. The lower side of Fig. 3(b) shows BT-Y2c's reconstructed image, upon which the support-shrinking operation was applied. The cubic outer shape appears clearly without noise. The phase information exists only inside the cubic shape [see Fig. 3(d)]. The phase difference inside the sample is ∼160°, indicating considerable strain along an antiphase boundary (white). The phase-retrieval operation with a support shrinkage provided a clear image for the outer shape and quantitative phase information.

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The development of the Bragg-CDI, as demonstrated above, enabled us to broaden the observable particle size to 40–500 nm, compared to the previous report. ³⁰⁾ Figure 4 shows the internal strain distribution for a series of samples obtained by the present Bragg-CDI method. The images for the Pd and BT-Y2c are the same as shown in Figs. 2(c) and 3(d), respectively. A particle from BT-S3 shows the characteristics of a curved outer shape, a particle size of approximately 300 nm, and a rounded internal antiphase boundary; the boundary also shows a narrower shape than BT-Y2c. The area with positive phases was significantly wider than that with negative phases; the maximum contrast of the inner phase was ∼160°. For BT-S5, the outer shape was curved as in BT-S3, whereas the internal antiphase boundary was quite sharp and linear. The area with negative phases was slightly wider than that of positive; the maximum contrast of the inner phase was ∼170°. As shown in the series of reconstructed images, the inner antiphase boundaries for BTOs tended to be obscured and complicated as the particle size decreased. This trend was the same for the other cross sections of these samples. Note, however, that abnormally sharp phase changes in the 20–30 nm width vicinity of the surface are likely attributed to surface distortions or artifacts because those changes occurred with a significant decrease in electron density. Next, we discuss the phases and their boundaries of the inner samples, excluding those near the surface. All the BTO samples had a phase contrast of nearly 180°. Such antiphase contrast in a ferroelectric material is most likely due to a ferroelastic domain. ³¹⁾ We assume that two regions with different phases represent the ferroelastic domains (90° domains for tetragonal BTO), and the region between them with a phase of ∼0° represents the domain boundary. Considering the particle-size dependence of BTO fine crystal's c/a ratio and dielectric constant, ^7,9,11) the structural properties of 500 nm BTO, including its domain patterns, are regarded as bulk crystals. Those properties for BTO with a particle size of <300 nm are influenced by the size effect, ^7,9,11) making the domain shape finer and more complex. ^13,17,18) Our results are consistent with the hypothesis above. BT-S5 with a particle size of 500 nm had a phase boundary with a straight and sharp shape generally seen in the striped 90° domain boundary of BTO bulk crystals. ^14,20,42) BT-S3 and BT-Y2c showed curved and complicated boundaries. The shape of boundaries of BTOs with a particle size of <300 nm is attributed to a structural refinement of the domains due to the size effect. Although the above assumption that the two regions with antiphases represent ferroelastic domains has no conflict, other possibilities should be considered; for example, these phase boundaries could represent defects or grain boundaries. Defects, such as dislocations, appear as a gradual phase variation around the phase-discontinued point. ^28,31) This phase slope, which depends on the degree of a crystal displacement, could appear to be a sharp phase-boundary. Further, grain boundaries induced by grain growth and stress could form into a single particle. The grain boundaries' phase distribution may resemble that of the domain boundary if two or more crystals join with a very small angle. Thus, it remains controversial whether the phase boundaries for the BTOs represent ferroelastic domain boundaries, defects, or grain boundaries. If the Bragg-CDI observes the changes in obvious phase distributions by some stimulation (e.g. a temperature change involving ferroelectric phase transition or an electric field application inducing a polarization reorientation or both), one could conclude the direct observation of the ferroelectric's internal domain structure.

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The lamellar contrast of the reconstructed image for BT-S5 is considered to be attributable to the mutual interference of the 110 and 101 (or 10-1) reflections. The BT-S5 sample is illuminated by a coherent plane wave, therefore, the diffracted waves from the two domains interfere with each other. The interference pattern in real space is a hologram of the domains, formed by the diffracted wavefront of one domain superimposed on the diffracted wavefront from another domain as a reference wave. The interference wave has a period corresponding to the reciprocal space distance between the two reflections. For the BT-S5 sample, the reciprocal space distance between the two reflections is 0.0329 Å⁻¹. In real space, this value corresponds to 191 Å, which closely matches the period of the lamellar contrast for BT-S5 in Fig. 4.

We have improved the Bragg-CDI apparatus ³⁰⁾ at BL22XU in SPring-8 to expand the applicable particle size and applied the Bragg-CDI technique for Pd and fine ferroelectric BTO crystals with a particle size of 40–500 nm. The vacuum chamber for the sample enabled us to obtain a high-contrast diffraction pattern of a 40 nm Pd particle. The reconstructed three-dimensional image showed the outer shape, size, and internal phase (strain) for a single particle. The phase-retrieval operation with an updated real-space constraint provided a clear shape and accurate phase (strain) distribution to the reconstructed image. The BTO particle with a size of 500 nm (BT-S5) had a straight and sharp phase boundary, whereas the smaller BTOs (BT-S3 and BT-Y2c) had curved or fine boundaries. Bragg-CDI thus allows the observation of the outer shape, size, and inner phase distribution for a single particle with a size of tens to hundreds of nanometers, which may lead to a simple understanding of mesoscale ferroelectricity.

We would like to thank Prof. Y. Takayama (Univ. Hyogo) for comments about Shrink-wrap processing. We would like to thank Profs. Y. Sato (Kyushu Univ.), K. Tsuda (Tohoku Univ), and S. Mori (Osaka Pref. Univ.) for suggestion about STEM grid handling. We would like to thank Dr. J. R. Harries (QST) for discussions and comments. We thank Japan Synchrotron Radiation Research Institute (JASRI) for providing a chance for use the PILATUS 300 K detector. This work was partly supported by JSPS Grant-in-Aid for Scientific Research (Grant Nos. JP19H02618, JP18H03850, JP18H05518, JP19H05819, JP19H05625, JP18H05517) and The Murata Science Foundation. The synchrotron X-ray scattering experiment was performed at SPring-8 with the approval of JASRI (Proposal Nos. 2017B3761, 2018A3761, 2018B3761, 2019A3761, 2020A3761, 2020A3762).