Harnessing atomic-scale order at grain boundaries for giant flexoelectricity

Abstract

Flexoelectricity is a ubiquitous electromechanical coupling mechanism that produces a polarization response to strain gradients and requires no material symmetry constraints. Here, we investigated flexoelectricity in the La_0.24Sr_0.76Al_0.62Ta_0.38O₃ (LSAT) grain boundaries using atomic-resolution scanning transmission electron microscopy, including high-angle annular dark-field (HAADF) imaging, energy-dispersive x-ray spectroscopy (EDX), and electron energy-loss spectroscopy (EELS). Our results reveal that tantalum (Ta) segregates in the grain boundaries, forming unique chemically ordered structures. EELS uncovers pronounced distortions of the aluminum (Al)/Ta─oxygen (O) octahedra in the grain boundaries. We exploited the pronounced structural inhomogeneity of the 36.8° grain boundary to achieve a large strain gradient (∼2.0 per nanometer) within two to three unit cells, resulting in an atomic-scale flexoelectric displacement of up to ∼114.8 picometers. Quantitative analysis indicates that the flexoelectric displacement correlates with local nonstoichiometry induced by Ta segregation. We further demonstrated that the segregation-enhanced strain gradient exists generally in both symmetric and asymmetric grain boundaries. Our atomic-scale findings provide insight into tunable giant flexoelectricity in electroceramic grain boundaries.

Ta segregation at LSAT grain boundaries enhances large strain gradients and enables atomic-scale tunable giant flexoelectricity.

INTRODUCTION

Flexoelectricity is a universal, strongly scale-dependent electromechanical coupling mechanism (1). The flexoelectric effect exhibits a pronounced effect at the nanoscale and can mediate magnetic order modulation (2), polar switching (3, 4), and optoelectronic characteristics (5, 6). In certain nanoscale systems, the strain gradient-induced flexoelectric effect can exceed piezoelectricity and thus become the dominant mechanism for polarization (7, 8). Microscopic defects, such as dislocations (9, 10), vacancies (11), and interfaces (12), can lead to intense local lattice distortions. Such an inhomogeneous structural environment can induce local symmetry breaking and charge separation, thereby providing conditions for a flexoelectric response (13). Studies have demonstrated that localized polarization induced by flexoelectricity can emerge around dislocations even at room temperature (14, 15). The discontinuity in the dipole moments can result in the accumulation of bound charges at defect cores (16, 17). The resulting charge imbalance requires atomic-scale reconstruction or charge compensation mechanisms, such as surface adsorption (18), lattice deformation (17, 19), redistribution of free charge carriers (20), and interface charges (21). Dielectric oxides have a large flexoelectric coefficient and thus hold broad promise for application in future nanosized devices (4). Nevertheless, their inherent brittleness makes it challenging to achieve tunable strain gradients through precise microstructure design, limiting the achievement of high flexoelectric response at the nanoscale.

Symmetric grain boundaries (GBs) in bicrystals characterized by specific orientation relationships usually display periodic structural units along the GB plane (22, 23), and atomic-scale defect ordering can further engineer such structural periodicity (24). The breaking of translational symmetry endows GBs with properties not found in intrinsic bulk crystals (9), including flexoelectricity (25), antiferroelectricity (26), and spin-valve magnetoresistance (27). In a bonded symmetric GB, the atomic arrangements exhibit local configurations with substantial geometric asymmetry, which may result in the formation of dipole moments (28). Modulating the geometry of the GB enables effective tuning of the local strain gradient. In addition to the artificial design of GB geometries, segregation-induced GB phase transitions can occur with tunable local chemistry and configuration via appropriate dopant selection (29), creating promising phases with properties that cannot be achieved by current processing techniques (30, 31). In recent years, research has explored interface-induced strain (32), vacancy formation energy (21), crystallographic rotation axis (33), and free-surface analogies (22) to understand the random and ordered segregation mechanisms (34). However, conventional macroscopic characterizations and measurements only provide collective information about the bulk material and can be influenced by irrelevant factors, thereby limiting experimental investigations at the subunit cell level (35, 36). Within the complex structure of GBs, elemental segregation-induced local nonstoichiometry and strain gradients may coexist. Although a previous study has demonstrated that strain gradient and elemental segregation both govern flexoelectric displacement (37), these two factors have only been investigated separately, whereas their atomic-scale coexistence remains unexplored.

Here, we investigated flexoelectricity in three types of La_0.24Sr_0.76Al_0.62Ta_0.38O₃ (LSAT) GBs by using atomic-resolution scanning transmission electron microscopy (STEM) with high-angle annular dark-field (HAADF) imaging, energy-dispersive x-ray spectroscopy (EDX), and electron energy-loss spectroscopy (EELS). We directly visualized the atomic arrangements by HAADF imaging. EDX mapping exhibits a general tantalum (Ta) segregation phenomenon, forming unique chemically ordered structures across the GBs. EELS uncovers more pronounced symmetry breaking of the aluminum (Al)/Ta─oxygen (O) octahedra and more oxygen vacancies in the 24° GB than in the 36.8° GB. Notably, a pronounced strain gradient within the 36.8° GB cores induces substantial atomic displacements, with the maximum displacement reaching 114.8 pm. Quantitative measurement in a general GB demonstrates the universality of the flexoelectric effect in LSAT GBs. Further analysis reveals that substantial flexoelectric displacement closely correlates with Ta segregation. Our atomic-scale findings provide insight into the emergent functionalities that act as independent functional units at oxide interfaces (38). The tunable giant flexoelectricity in electroceramic GBs promises to guide GB engineering in nanoelectronics and nanoelectromechanical systems.

RESULTS

Atomic arrangements and element distribution

To precisely determine the atomic arrangements and chemical compositions, advanced STEM techniques were used to accurately determine the atomic arrangements and chemical compositions. Figure S1 shows the schematic model of the bicrystal. Atomic-resolution EDX mapping (fig. S2) determines the bulk composition to be LSAT. Figure 1A depicts the HAADF image of 24° LSAT GB. The magnified view reveals a periodic structural unit at the GB core. Atomic-resolution EDX mapping in Fig. 1B uncovers distinct net counts of lanthanum (La), strontium (Sr), and O (Fig. 1B, a to c) in the GB compared to those in the bulk, indicating local nonstoichiometry and oxygen deficiency. In contrast to Al, Ta shows increased net counts in the GB (Fig. 1B, d and e). The composite map (Fig. 1B, f) also displays the preferential segregation as enhanced green contrast. As shown in Fig. 1C, Ta reaches a maximum relative occupancy of ∼79% within the Al/Ta─O columns, with a minimum value exceeding 50%. Figure 1D further verifies the segregation phenomenon of 36.8° LSAT GB. Corresponding EDX mapping (Fig. 1E) reveals smaller compositional variation between the bulk and GB compared to that in the 24° GB. The overall net counts of La and Sr in the interface relative to the bulk (Fig. 1E, a and b) follow a trend similar to that observed in the 24° GB. Compared to the 24° GB, the O map (Fig. 1E, c) of the 36.8° GB suggests few oxygen vacancies in the interface because of a slight contrast. Further analysis (Fig. 1E, d to f) reveals higher Ta net counts in the 36.8° GB, consistent with the Ta-rich distribution in the 24° GB. For the Al/Ta─O columns in the 24° GB (Fig. 1F), Ta reaches a maximum relative occupancy of ∼73%, with only minor fluctuation near the maximum compared to that in the 36.8° GB.

Fig. 1. Atomic arrangements and element distributions in the 24° and 36.8° GBs of LSAT.

HAADF images of the 24° (A) and 36.8° (D) LSAT GBs (colored for clarity). The periodic structural units are enlarged in the insets, with atoms highlighted by polygons. Scale bars, 2 nm. Atomically resolved EDX mappings showing the distribution of elements across the 24° (B) and 36.8° (E) LSAT GBs: individual distributions of La (a), Sr (b), O (c), Al (d), and Ta (e) and the composite map of La + Sr + Al + Ta (f). Scale bars, 1 nm. Schematics of the atomic structure illustrating element segregation at the 24° (C) and 36.8° (F) GB cores. The relative occupancy is calculated from a combination of the HAADF images [(A) and (D)] and EDX mappings [(B) and (E)]. Element net counts at the equivalent lattice sites are normalized. Colors in schematics correspond to the colors in the EDX mappings.

EELS characterization of the 24° and 36.8° LSAT GBs

EELS characterization was performed at the O K-edge and Al L-edge. The O K-edge spectra (Fig. 2, A and B) display a characteristic peak at ∼538 eV in both GB cores, corresponding to La/Ta 5d─O 2p hybridized states (39). Notably, the peak intensity decreases at the 24° GB core compared to that in the bulk. In contrast, the O K-edge fine structure of the 36.8° GB core closely resembles the bulk, exhibiting only a small decrease in the high-energy shoulder (peak B) (40). The difference indicates more oxygen vacancies and greater lattice distortion at the 24° GB core. Average spectra from the GB core and adjacent grains reveal that the O K-edge prepeak shifts to a higher energy loss at the 24° GB core, which further supports the presence of oxygen vacancies (41). At the 24° GB core, peak A exhibits a higher relative intensity compared to peak B, indicating stronger La─O hybridization interactions (42, 43). Spectral region C corresponds to transitions to O 2p states hybridized with Al 3p states (44). The altered fine structure of region C in the GB spectrum reflects modified hybridization between Al and O states. Stronger metal-oxygen hybridization promotes ferroelectricity in perovskites such as PbTiO₃ and BaTiO₃ (42). Therefore, the enhanced La─O hybridization at the GB core aligns with the presence of net polarization. The Al L-edge spectrum serves as a fingerprint for changes in Al coordination symmetry. According to Fig. 2C, the Al L₃-edge threshold energy in the bulk is ∼74.9 eV, whereas in the 24° GB core, it shifts to ∼74.0 eV. The threshold energy at the Al L₃-edge decreases by ∼0.9 eV for the 24° GB, indicating distorted octahedral symmetry (45). A decrease in the L₃/L₂ intensity ratio from ∼1.24 in the bulk to ∼1.22 at the 24° GB core also supports this conclusion (46). The shift of the Al-L_2,3 excitonic peak to lower energy loss in the 24° GB further confirms the distortion of the local Al coordination environment. In contrast, the Al L-edge line shape and threshold energy of the 36.8° GB perturb only weakly (Fig. 2D).

Fig. 2. Electronic structures in the 24° and 36.8° GBs of LSAT.

O K-edge spectra for the 24° (A) and 36.8° (B) GBs. A main peak (∼538 eV) arising from La/Ta─O hybridized states is preserved in the 24° GB. Bulk (red and orange) and GB cores (blue) are clarified by different colors. a.u., arbitrary units. Al-L_2,3 edges and La N-edges spectra for the 24° (C) and 36.8° (D) GBs. The corresponding Al L₃ (green) and L₂ (yellow) edges are marked for clarity.

Atomic-scale characterization in the 24° and 36.8° GBs of LSAT

Quantitative measurements of atomic displacements, strains, and strain gradients were performed in the 24° and 36.8° GBs of LSAT to analyze the flexoelectric response. The atomic displacement map derives from the HAADF image (Fig. 3A), in which overlaid vectors represent the direction and magnitude of the Al/Ta─O column offsets compared to the centers of the surrounding La/Sr columns. Substantial polar displacements occur within two to three unit cells of the GB core. As the schematic in Fig. 3A, the average displacement within the highlighted trapezoidal region around the GB cores reaches ∼48.3 ± 8.2 pm, with a maximum value of 62.8 pm. On both sides of the minimum repeating units, displacement vectors (>10 pm) point toward the GB, forming a “head-to-head” polarization configuration. For the 24° GB center, the polarization direction overall points along the positive x direction (Fig. 3A), and the displacement magnitude is 53.7 ± 3.5 pm (fig. S3B). Strain measurement reveals a pronounced strain component perpendicular to the GB plane (e_zz), whereas the in-plane strain is small (Fig. 3, B and C). Geometric phase analysis (GPA) in fig. S5 verifies the results. Figure 3 (D and E) presents the strain gradients along the direction perpendicular [d(e_zz)/dz] and parallel [d(e_zz)/dx] to the GB, which are noticeable within three to four unit cells. The strain gradient [d(e_zz)/dx] generated within the trapezoidal region is as high as 2.1 nm⁻¹. In the 24° GB, Al/Ta─O planes terminate the GB core between the two periodic structural units (Fig. 3A), whereas La/Sr─O planes terminate those in the 36.8° GB (Fig. 3F) (11). Both terminations can be inferred from Fig. 1 (C and F). The polarization pattern in the GB shows that the displacement vectors (>10 pm) on both sides of the minimum repeating unit point away from the GB, whereas the displacement vector at the bottom runs parallel to the GB. To further evaluate the flexoelectric performance of the LSAT GB, strain and strain gradient distributions were quantitatively mapped (Fig. 3, G to J). Analysis reveals that, although the in-plane strain (e_xx) parallel to the GB is minimal (max ∼7.3%; Fig. 3H), pronounced strain gradients exist along the horizontal [d(e_zz)/dz] and vertical [d(e_zz)/dx] directions, confined to two to three unit cells (Fig. 3, I and J). GPA in fig. S6 also verifies the nonuniform strain distribution. Notably, the trapezoidal core region exhibits a strain gradient [d(e_zz)/dz] up to ∼2.0 nm⁻¹, producing a polar displacement of ∼100.8 ± 8.4 pm with a maximum value of 114.8 pm. The average polar displacement on both sides of the minimum repeating unit is ∼16.6 pm. For the shear strain, its contribution to the overall flexoelectric effect should be very small according to previous theoretical studies (47, 48).

Fig. 3. Atomic-scale displacements and strain gradients in the 24° and 36.8° GBs of LSAT.

Off-center displacement vector maps between the La/Sr and Al/Ta─O columns from the HAADF images of 24° (A) and 36.8° (F) GBs. Scale bars, 1 nm. Arrows indicate the magnitude and direction of polar displacements measured on the basis of the offsets of the Al/Ta─O column with respect to the center of the surrounding four La/Sr columns. Blue vectors, <10 pm; orange vectors, 10 to 50 pm for the 24° GB and 10 to 100 pm for the 36.8° GB; yellow vectors, >50 pm for the 24° GB and >100 pm for the 36.8° GB. The schematics show inhomogeneous deformation around the GB cores, generating net dipole moments. Subunit cell–level mappings of e_zz corresponding to the strain perpendicular to the 24° (B) and 36.8° (G) GB planes, respectively. Scale bars, 1 nm. Subunit cell–level scale mappings of e_xx corresponding to the strain parallel to the 24° (C) and 36.8° (H) GB planes, respectively. Scale bars, 1 nm. Corresponding horizontal [d(e_zz)/dz] strain gradients from e_zz in the 24° (D) and 36.8° (I) GBs. Scale bars, 1 nm. Corresponding vertical [d(e_zz)/dx] strain gradients from e_zz in the 24° (E) and 36.8° (J) GBs. Scale bars, 1 nm. The periodic structural units of the GBs are highlighted by the dashed lines in the mappings.

Quantitative correlation between strain gradient, element segregation, and flexoelectric displacement

Such a phenomenon and subunit cell–level measurements motivate further clarification of the correlations among flexoelectricity, elemental segregation, and strain gradient. Figure 4 (A and B) correlates the Al/Ta cationic displacements around the GB cores with the relative Ta net counts for both bicrystals in the GBs and bulk. The results reveal that displacements in both the 24° and 36.8° bicrystals are negligible in the bulk, confirming the centrosymmetric and nonpolar nature of the bulk of LSAT (21). Notably, with increasing Ta net counts in the GB, the polar displacement exhibits an approximately linear growth, indicating that element segregation plays an important role in enhancing local polarization. Moreover, displacements and strain gradients are closely linked (Fig. 4, C and D). Specifically, the 24° and 36.8° GBs exhibit an approximately proportional relationship between displacement and strain gradient from the bulk to the GB. The findings indicate that the flexoelectric response depends on both the strain gradients and the specific atomic patterns in GBs. On the basis of 20 HAADF images, the distributions of polar displacements are measured (Fig. 4E). The average displacement is 45.2 ± 8.4 pm for the 24° GB core, whereas the 36.8° GB core exhibits a larger value of 103.1 ± 5.5 pm. The 36.8° GB core exhibits a notable polar displacement that exceeds the other systems (Fig. 4F). Generally, the reported flexoelectric displacements in defect interfaces positively correlate with strain gradients.

Fig. 4. Quantitative investigation of the atomic-scale flexoelectric effect.

Polar displacements in the bulk and GB cores of the 24° (A) and 36.8° (B) GBs of LSAT are plotted as a function of the Ta x-ray net count. Polar displacements in the bulk and GB cores of the 24° (C) and 36.8° (D) GBs of LSAT are plotted as a function of the strain gradient. (E) Distribution of displacements at the 24° and 36.8° GB cores. The statistics are based on 20 HAADF images. (F) Comparison of the displacements as a function of strain gradient with previous results, including LaAlO₃ GB cores, halide perovskites GB cores, SrTiO₃ dislocation cores, and GB cores of 24° and 36.8° LSAT GBs. The 24° and 36.8° LSAT GB cores are labeled in the figures with black dashed lines. Confidence ellipses for different defect structures are shown schematically as distinct colors.

Atomic-scale characterization in a general GB of LSAT

Figure 5A displays a HAADF image of a general GB that has lower structural symmetry than the 36.8° and 24° GBs. EDX mapping reveals lower net counts for La, Sr, and O in the GB compared to the bulk (Fig. 5B, a to c). Figure 5B (d to f) shows an increased Ta net counts in the GB compared to those in the bulk. Figure 5C visualizes the atomic arrangement at the GB core. In the bulk, off-center displacement vector maps between the La/Sr and Al/Ta─O columns are randomly oriented with negligible magnitudes, confirming the nonpolar nature of the bulk (Fig. 5D). Similar to the relative distributions perpendicular and parallel to the symmetric 24° and 36.8° GBs, the strain perpendicular to (e_zz) the interface is larger than that parallel to (e_xx) the interface in the general GB. Figure 5 (G and H) presents the strain gradients along the direction perpendicular [d(e_zz)/dz] and parallel [d(e_zz)/dx] to the GB within two to three unit cells. Quantitative measurements show that the GB core exhibits a huge strain gradient up to ∼2.2 nm⁻¹ [d(e_zz)/dx], producing a polar displacement of ∼48.0 ± 6.2 pm.

Fig. 5. Atomic-scale characterization in a general GB of LSAT.

(A) HAADF image of a general GB of LSAT. Scale bar, 2 nm. (B) Atomic-resolution EDX mapping showing the distribution of elements across the GBs: individual distributions of La (a), Sr (b), O (c), Al (d), and Ta (e) and the composite map of La + Sr + Al + Ta (f). Scale bars, 1 nm. (C) Magnified image of the region within the gray box in (A). Scale bar, 1 nm. (D) Off-center displacement vector maps between the La/Sr and Al/Ta─O columns corresponding to (C). Blue vectors, <10 pm; orange vectors, 10 to 50 pm; deep red vectors, >50 pm. Subunit cell–level mapping of strain perpendicular to (E) and parallel to (F) the GB. Scale bars, 1 nm. Corresponding subunit cell–level mapping of the strain gradient perpendicular to (G) and parallel to (H) the GB, respectively. Scale bars, 1 nm.

DISCUSSION

Achieving appreciable flexoelectricity in bulk materials generally requires macroscopic bending. In contrast, topological defects such as the atomic-sized trapezoidal units in GBs in LSAT can confine strain gradients within a few unit cells, deterministically triggering nanoscale flexoelectricity (37). Compared to epitaxial strain or crack engineering, GB engineering exhibits distinct advantages. For instance, although large tensile strains exist at crack tips in SrTiO₃, symmetric stretching often suppresses the net flexoelectric polarization (49). Although strained regions can induce polarization by electrostriction or piezoelectricity, these effects typically require noncentrosymmetric crystal structures. In the 36.8° LSAT bicrystal, the dominant polarization vectors in the GB align parallel to the GB interface (Fig. 3F). Although lattice strain (e_xx) along this direction displays a negligible magnitude relative to LSAT GBs of other configurations, the strain gradient d(e_zz)/dx shows a prominent value. The decoupling of strain and polarization direction demonstrates that the flexoelectric effect, rather than piezoelectricity or electrostriction, mainly drives the polarization. We also statistically analyzed the distribution of displacement components along the x and z directions for 24° and 36.8° GBs, respectively. On sides of the GBs, both 24° (fig. S3, C and D) and 36.8° GBs (fig. S3, G and H) exhibit small average displacements, whereas large net x direction displacements appear at the GB center (∼51.5 pm for the 24° GB and ∼−102.18 pm for the 36.8° GB). Figure S4 shows the estimation of errors, showing that the calculated data exhibit only minor deviations within an acceptable error range.

The differences in polarization magnitude and direction between the 24° and 36.8° GBs can be attributed to their distinct geometric configurations and strain field distributions. Owing to the broken translational symmetry, the minimum repeating structural unit of the 24° GB contains more atoms than that of the 36.8° GB (Fig. 1D). However, the structural distortion is not solely mechanical but also correlates closely with chemical heterogeneity (30), and GB engineering has also shown great potential in modulating mechanical properties through solute segregation (50). The large displacements thus represent the superposition of mechanical flexoelectricity and defect-induced chemical expansion (11, 31). We proposed that atomic radius mismatch constitutes a fundamental mechanism for the segregation-induced local nonstoichiometry. The segregation of larger Ta⁵⁺ ions replacing smaller Al³⁺ ions in the GB induces a notable local strain. To alleviate the lattice instability, strain gradients are generated perpendicular to the GB. Consequently, the chemically driven lattice relaxation creates the conditions for a flexoelectric response (31, 51). The substantial displacement in the 36.8° GB thus provides a scenario for understanding the electrical activity: Ta segregation induces lattice distortions, enhances local strain gradients, and thus drives flexoelectric polarization. EELS characterization demonstrates that oxygen vacancies occur at the GBs, and these phenomena are especially notable in the 24° GB. However, the observed flexoelectric displacement does not exhibit a simple positive correlation with the concentration of such defects. Instead, the dominant contribution arises from the large strain gradient enhanced by Ta segregation–induced lattice mismatch, as evidenced by the substantially larger displacement in the 36.8° GB despite its lower oxygen vacancy concentration. Density functional theory DFT calculations can determine the ground-state atomic structures, Ta incorporation energetics, and atomic-scale strain field distributions of the 24° and 36.8° LSAT GBs, which have been crucial for understanding solute segregation and structural transitions in similar oxides (29, 52). In this work, despite the complexity of the system challenging theoretical modeling, HAADF imaging, EDX mapping, and EELS collectively provide atomic-scale insight into the giant flexoelectric effect at GBs.

In summary, our findings establish a systematic picture that, in three types of LSAT GBs, segregation generates atomic-scale strain gradients, which induce flexoelectric polarization. By combining advanced STEM techniques and quantitative measurements, we determined the atomic configurations of the 24°, 36.8°, and general GBs and elucidated the electronic origin. Atomic-resolution EDX mapping reveals the generality of Ta segregation. EELS uncovers that, compared to the 36.8° GB, the 24° GB exhibits more pronounced symmetry breaking in the Al/Ta─O octahedra and a higher concentration of oxygen vacancies. Notably, the core region of the 36.8° GB exhibits a strain gradient [d(e_zz)/dx] up to ∼2.0 nm⁻¹, which produces a substantial polar displacement of ∼103.1 ± 5.5 pm (with a maximum of ∼114.8 pm). The general GB also exhibits a huge strain gradient, producing a polar displacement of ∼48.0 ± 6.2 pm. Owing to the strain gradients enhanced by the ionic size mismatch of segregated Ta cations, polar displacements exceeding 100 pm can be achieved at the GB core within a few unit cells. The tunable giant flexoelectricity through GB engineering broadens prospects in nanoelectronics and nanoelectromechanical systems.

MATERIALS AND METHODS

Bicrystal fabrication and TEM cross-sectional sample preparation

An LSAT bicrystal with a mistilt GB was fabricated by using thermal diffusion bonding of two LSAT single crystals at 1800°C for 12 hours in a N₂ atmosphere under a uniaxial stress of 2 MPa. The bicrystal was supplied by Hefei Ke Jing Materials Technology Co. Ltd. The TEM specimens were first thinned by mechanical polishing and then subjected to argon ion milling. The ion milling process was carried out using a PIPS (model 691, Gatan Inc.) with an accelerating voltage of 3.5 kV until a hole was observed. Low-voltage milling was performed with an accelerating voltage of 0.3 kV to remove the surface amorphous layer and minimize the irradiation-damaged layers.

Electron microscopy characterization and image analysis

HAADF images were recorded at 300 kV using an aberration-corrected FEI Titan Themis G2. The convergence semiangle for imaging was 30 mrad, and the collection semiangle snap range was 39 to 240 mrad for HAADF imaging. The EDX experiments were carried out in an aberration-corrected FEI Titan Themis G2 equipped with EDX detectors. The core-loss EELS data were acquired on Titan Themis G2, and the energy dispersion was 0.05 eV per channel. The background of the core-loss EELS data was fitted by power law and then subtracted from the whole space. To obtain quantitative information on the flexoelectricity, the atom positions were determined by simultaneous fitting with two-dimensional Gaussian peaks using a MATLAB code (53). The magnitude of the strain was defined as the difference in the lattice parameters between the strained and ground states. The strain was calculated on the basis of the La (Sr) lattice as $\text{strain}=\frac{a-a_0}{a_0}$, where $a_0$ is the lattice parameter of freestanding LSAT and $a$ is the in-plane lattice parameter perpendicular ($a_z$) or parallel ($a_x$) to the GB plane. The strain gradient was then obtained by taking the differential of the strains along different directions, given as $\mathrm{d}\left(e_{\mathit{zz}}\right)/\mathrm{d}z=2\ast (e_{\mathit{zz}}^n-e_{\mathit{zz}}^{n-1})/(a_z^n+a_z^{n-1})$ and $\mathrm{d}\left(e_{\mathit{zz}}\right)/\mathrm{d}x=(e_{\mathit{zz}}^m-e_{\mathit{zz}}^{m-1})/a_x^m$. GPA was performed using the free FRWRtools plugin for Digital Micrograph (developed by Koch et al. and available at https://www.physics.hu-berlin.de/en/sem) based on an original work (54).

Supplementary Materials

This PDF file includes:

Supplemental Text

Figs. S1 to S6