Using aberration corrected scanning transmission electron microscopy combined with advanced imaging methods, we directly observe atom column specific, picometer-scale displacements induced by local chemistry in a complex oxide solid solution. Displacements predicted from density functional theory were found to correlate with the observed experimental trends. Further analysis of bonding and charge distribution was used to clarify the mechanisms responsible for the detected structural behavior. By extending the experimental electron microscopy measurements to previously inaccessible length scales, we identified correlated atomic displacements linked to bond differences within the complex oxide structure.

Complex oxide solid solutions exhibit a range of properties that are intimately linked to picometer-scale atomic shifts within the crystal structure.1–7 To date, however, these displacements are often difficult to unambiguously detect by conventional means.8 Rather, direct real-space measurements with picometer precision in combination with theoretical investigations can provide insights into the origin of fine scale atomic displacements.

Atomic resolution electron microscopy has served as an indispensable tool for the real-space structural analysis of materials.9,10 High-angle annular dark field scanning transmission electron microscopy (HAADF STEM) has proven pivotal for structural analysis as the image intensities scale approximately as the square of the atomic numbers (Z) present.11 This enables direct identification of light and heavy elements within a structure.12–14 Crystallographic analysis using STEM, however, has been stymied by sample drift and noise that traditionally limit measurement precision to about 5–10 pm.13,15,16 This challenge hindered the possibility to connect complex structural responses to local chemistry. The recent introduction of revolving STEM17 and non-rigid image registration18 has now overcome these challenges. The picometer precise measurements enabled by these techniques are now opening an avenue to study crystallography with the combination of interpretability, chemical sensitivity, and directly in real-space.

In this Letter, we select (La0.18Sr0.82)(Al0.59Ta0.41)O3 (LSAT) as a model oxide solid solution to study atomic displacements in a complex chemical environment. The LSAT crystal poses a particularly challenging test case as atomic forces resulting from the intricate local chemical environment lead to structural distortion. LSAT adopts the cubic perovskite ABO3 structure (space group Pm3¯m) with the corners occupied by (La, Sr), (Al, Ta) located at the cube center, and the oxygen anions positioned on the faces (Figure 1(a)). When the LSAT structure is projected along 100, chemically distinct atom columns are observed that contain either La/Sr, Al/Ta/O or exclusively O. For convenience in referencing the 100 STEM images, we define two distinct cation containing sub-lattices: the A sub-lattice containing La and Sr, and the B sub-lattice consisting of Al and Ta with overlapping oxygen anions.

FIG. 1.

(a) A 3 × 3 × 3 supercell of the LSAT structure superimposed with charge density for the La/Sr and Al/ Ta containing planes, vertical and horizontal slices, respectively. (b) Atomic resolution EDS where the diamond indicates an Al rich column. (c) Sub-section of a RevSTEM image along 100 with the labels “A” and “B” denoting the corresponding sub-lattices. Contrast and brightness have been adjusted to highlight the atom column intensity differences. Strong and weak B sub-lattice atom columns are indicated by circles and diamonds, respectively. Supplementary material Figure S1 provides the original full-size image.30 Average A-A (d) and B-B (e) NLN distance around each sub-lattice atom column. The indicated scale bars represent 1 nm.

FIG. 1.

(a) A 3 × 3 × 3 supercell of the LSAT structure superimposed with charge density for the La/Sr and Al/ Ta containing planes, vertical and horizontal slices, respectively. (b) Atomic resolution EDS where the diamond indicates an Al rich column. (c) Sub-section of a RevSTEM image along 100 with the labels “A” and “B” denoting the corresponding sub-lattices. Contrast and brightness have been adjusted to highlight the atom column intensity differences. Strong and weak B sub-lattice atom columns are indicated by circles and diamonds, respectively. Supplementary material Figure S1 provides the original full-size image.30 Average A-A (d) and B-B (e) NLN distance around each sub-lattice atom column. The indicated scale bars represent 1 nm.

Close modal

(La0.18Sr0.82)(Al0.59Ta0.41)O3 (MTI Corporation Richmond, CA) single crystals were prepared for electron microscopy by mechanical wedge polishing with an Allied Multiprep.19 Imaging was performed using a probe-corrected FEI Titan G2 60–300 kV S/TEM equipped with an X-FEG source operated at 200 kV. A custom program was used to acquire the RevSTEM image series by interfacing with the microscope through TEM Imaging and Analysis (TIA).17 The sample thickness for (Figure 1(c)) was determined to be 14 ± 1 nm.20 Density functional theory (DFT) calculations were performed using the Vienna Ab initio Software Package (VASP) version 5.3.3.21–24 To approximate the solid, special quasi-random structures (SQS) were generated25 using the Monte Carlo algorithm26 within the Alloy Theoretic Automated Toolkit (ATAT).27–29 Further details of DFT and SQS calculations are provided in the supplementary material.30 

As a critical first step to identify the underlying structure of the solid solution, measurement of the elemental distribution within LSAT is required. This is accomplished using atomic resolution energy dispersive X-ray spectroscopy (EDS) (Figure 1(b)).31–33 The stronger Al X-ray signal at the darkest atom columns in the HAADF image (Figure 1(c)) confirms that these positions are Al-rich compared to other B sub-lattice columns. Shown in the RevSTEM dataset (Figure 1(c)), the compositional fluctuation is directly observed. To aid visual inspection, particularly strong and weak B sub-lattice atom columns are identified in Figure 1(c) for those within ±1.5σ of the average B column intensity. This strong contrast results from the large atomic number difference between Al (Z = 13) and Ta (Z = 73), compared to Sr (Z = 38) and La (Z = 57).

In Figures 1(d) and 1(e), each atom column is outlined by the average nearest like-neighbor (NLN) distance with the extreme contraction and expansion colored as blue or red, respectively, while minor distortions are colored aqua/green/yellow. Inspection of these maps reveals a wide distribution for the A-A NLN distances (Figure 1(d)), while a much narrower distribution is found for the B-B NLN distances (Figure 1(e)). Analysis of the average A-A distances uncovers that significant deviations are correlated with the B site intensity (Figure 1(d)). Specifically, there is a tendency for significant contraction around the dark Al-rich atom columns while the bright Ta-rich columns correspond to expansion. Importantly, this trend has been observed across RevSTEM images acquired at different thicknesses.

Emphasizing these trends, the average NLN distances around each B and A atom column are plotted against the opposing sub-lattice intensity ratio as shown in Figures 2(a) and 2(b), respectively. The intensity ratio is determined by measuring the intensity for a particular atom column and dividing by the average sub-lattice intensity. The average A-A distance shows moderate linear correlation with the corresponding B atom column intensity (R0.5), while the B sub-lattice distances show little to no correlation to A atom column intensity (R0.2). These trends point to the strong influence of the atomic displacements mediated by fluctuations in local chemistry. Furthermore, the scatter in these plots is a direct consequence of the statistical nature of the solid solution alloy and each measurement point represents the average local distortion.

FIG. 2.

(a) Average A-A and (b) B-B NLN distance as a function of the opposing sub-lattice intensity ratio and compared to the 95% confidence intervals determined from the simulated dataset. (c) Average NLN distances mapped across one of the 100 tiled 3 × 3 × 9 supercells.

FIG. 2.

(a) Average A-A and (b) B-B NLN distance as a function of the opposing sub-lattice intensity ratio and compared to the 95% confidence intervals determined from the simulated dataset. (c) Average NLN distances mapped across one of the 100 tiled 3 × 3 × 9 supercells.

Close modal

The LSAT solid solution creates a complex local charge environment on both cation sub-lattices. The A sub-lattice contains La3+ and Sr2+, the B sub-lattice contains Al3+ and Ta5+, and O2– neutralizes the total charge. To understand the nature of bonding and the origin of forces in this complex system that drives the aforementioned distortions, we turn to DFT applied to a 3 × 3 × 3 SQS cell. DFT calculations reveal that the charge of cations in the unrelaxed structure as measured by Bader's method34,35 do not reside in their full ionization states. Rather, La, Sr, Al, and Ta cations are determined to have positive charges of 2.04 ± 0.02, 1.56 ± 0.01, 2.43 ± 0.01, and 2.54 ± 0.01 |e|, respectively. The small standard deviations indicate that all cations are insensitive to their local chemical environment. Ta atoms have the most significant deviation from their formal charge. This is due to Ta-O bonds having the most significant covalent character as seen in the higher concentration of electrons between Ta-O atoms in Figure 1(a).

The oxygen anions exhibit an average charge of 1.37±0.11|e| but have a much larger standard deviation, which indicates a higher sensitivity to the local chemical environment. The identity of the neighboring B sub-lattice cations is found to determine the charge of oxygen. Separating the oxygen charge into groups based on the possible combinations, the three distinct local oxygen charges are −1.51 ± 0.01 |e| for Al-O-Al, −1.33 ± 0.02 |e| for Al-O-Ta, and −1.23 ± 0.02 |e| for Ta-O-Ta. The small standard deviations indicate that the oxygen anions are uniquely identified within these three groups.

The non-uniform charge distribution in the unrelaxed lattice leads to sizable forces on the atoms of up to ∼4 eV/Å. These forces are ultimately minimized through static displacements in the relaxed structure. In an effort to further confirm the origin of the forces as arising from the local charge distribution, we compare the Hellmann-Feynman forces from DFT to those from Coulomb's law using the Bader charges and an Ewald summation method. Forces on atoms in the A sub-lattice derived from the Ewald sum exhibit a minimum absolute percent error (MAPE) of 7% relative to the DFT forces. In contrast, forces on oxygen and B sub-lattice cations have higher MAPE of 102% and 53%, respectively, due to mixed ionic/covalent bonding.

The similarity of the Hellmann-Feynman forces and those from the ideal point charge Coulombic picture for the A sub-lattice cations indicate a predominant ionic interaction between these atoms and their neighbors. La and Sr are thus most sensitive to the local charge distribution, lack bond directionality, and are free to arbitrarily displace to minimize internal energy and reduce the large initial forces. Ultimately, attractive interactions arise between the A sub-lattice cations and the oxygen anions, which have variable charge based on local chemistry. The DFT charge analysis indicates that the most favorable attraction is between A sub-lattice cations and oxygen anions interacting with Al (i.e., Al-O-Al). Further, relaxation of the atomic coordinates shows a tendency for contraction of A sub-lattice cations towards Al rich B columns, consistent with the proposed mechanism and experimental measurements.

To compare the experiment with DFT, STEM images were simulated using the atomic coordinates from the relaxed SQS 3 × 3 × 3 supercell. Two factors are included for comparison with experiment. First, the coordinates are scaled from the DFT lattice parameter of 3.93 Å to the experimental parameter of 3.87 Å. Second, instead of using the 3 × 3 × 3 SQS supercell, the STEM images were simulated using 100 different 3 × 3 × 9 supercells randomly generated by stacking the SQS supercell with different 90° rotations and unit cell translations to better approximate chemical fluctuations seen in the thicker experimental samples. This approximation is justified by the small displacements in the B sub-lattice.

We then apply the same analysis used to determine the NLN distances from experiment to the 100 simulated STEM images (see Figure S2 in the supplementary material for the measurements30). Inspection of one simulated tiled structure (Figure 2(c)) shows the same behavior as the experimental dataset (Figs. 1(d) and 1(e)) with large and small variations observed for the A and B sub-lattices, respectively. Compared with the 95% confidence intervals derived from the simulated dataset, the experiment trends are in near perfect agreement with those predicted for each sub-lattice (Figures 2(a) and 2(b)). Specifically, the confidence intervals contain >95% of the experiment A sub-lattice points and 92.1% for the B sub-lattice. Moreover, the simulated A-A NLN standard deviation is ∼6.7 pm, in agreement with the experiment measurement of ∼7 pm. Similarly, the predicted B-B average NLN has a small standard deviation of ∼1.7 pm compared with ∼3 pm from experiment. This excellent agreement reflects the ability for the relaxed SQS structure to appropriately capture the relatively complex chemical environment of the LSAT structure and provides critical supporting evidence connecting local chemistry and distortion discussed above.

For structural correlations at longer length scales, nth like-neighbor distances are extracted across the entire image and used to construct a projected pair distribution function (pPDF). This is schematically represented in Figure 3(a) for 1 ≤ n ≤ 11. Conceptually, these measurements are similar to the PDF extracted from diffraction data.36 The measurements differ, however, in that pPDFs provide insight into the projected average distortion, are determined directly at the atomic scale, and can be separated according to each sub-lattice. While the pPDFs for A and B sub-lattices appear qualitatively similar (Figure 3(b)), the standard deviation (σ) for the A-A nearest and second like-neighbors, as measured using the pPDF, are considerably larger than those for the B-B distances and are reported in Figure 3(c).

FIG. 3.

(a) Schematic of projected pair distribution function (pPDF). (b) pPDFs for A (red) and B (gray) sub-lattices calculated based on nth like-neighbor atom columns. (c) Comparison of first and second nearest like-neighbor peaks. (d) Standard deviation of the atom column pair distances for A (boxes) and B (circles) sub-lattices and Si (line). (e) Correlation coefficient ϕ as a function of nth-neighbor distance for both sub-lattices.

FIG. 3.

(a) Schematic of projected pair distribution function (pPDF). (b) pPDFs for A (red) and B (gray) sub-lattices calculated based on nth like-neighbor atom columns. (c) Comparison of first and second nearest like-neighbor peaks. (d) Standard deviation of the atom column pair distances for A (boxes) and B (circles) sub-lattices and Si (line). (e) Correlation coefficient ϕ as a function of nth-neighbor distance for both sub-lattices.

Close modal

Extending the σ analysis across the image, the A nth like-neighbor also differ dramatically from the B sub-lattice in terms of trend (Figure 3(d)). Identical analysis was performed using a Si single crystal to demonstrate that the measurement precision is consistently ≈2 pm (Figure 3(d), solid line). Inspection of the first few like-neighbors of LSAT reveals that σB increases as the distance between B sub-lattice columns increases, while σA increases initially and then decreases. The stark difference between the LSAT trends, especially relative to the flat Si σ, indicates the presence of correlated atomic distortion within LSAT. To quantify the degree of correlation, we use the coefficient, ϕ=(σ02σ2)/2σ02, where σ0 is the contribution due to uncorrelated displacements at large pair distances. Specifically, σ0 can be extracted at large distances where the uncorrelated distortion dominates, i.e., when ϕ0. Here, the average σ from 2 < r < 5 nm is used to determine σ0 for the A and B sub-lattices, which was 6.5 pm and 3.9 pm respectively. This approach is commonly employed for PDF analysis,37 but without near perfect imaging enabled by RevSTEM, drift and noise would usually preclude extending precise distance measurements beyond a few angstroms (one unit-cell) in STEM.

As shown in Figure 3(e), the first few like-neighbors for the B sub-lattice exhibit strong, positive correlation while those for A show strong, negative correlation. The decreasing correlation for σB can be explained by the rigid cage introduced by B-O covalent bonding: cooperative distortion results in near neighbor distances that are more similar than those at longer range (top schematic, Figure 3(e)). Intriguingly, atypical negative correlation is observed for the A sub-lattice distortion. This behavior can be understood in the context of the local chemical fluctuation of the B sub-lattice. For Al- or Ta-rich atom columns, the surrounding A sub-lattice atoms contract inward or outward in opposite directions due to the local charge distribution (bottom schematic, Figure 3(e)). Thus, A sub-lattice distortion is anti-correlated. Furthermore, the second like-neighbor σA increases due to compounding distortion contributions from both x and y directions (see further details of the correlation coefficient in the supplementary material30).

Through picometer precise measurements in HAADF RevSTEM, we have demonstrated that relationships between chemistry and local distortion can be interrogated with atomic scale spatial resolution. When combined with DFT, these results offer additional insights to provide a mechanistic picture of unit-cell level structural displacements, which can be particularly important to the functionality of complex oxide solid solutions. The real-space measurement precision rivals that of diffraction determined PDFs, with the distinct advantage of exposing a rich set of real space structural and chemical information. This approach opens the doorway to physical observations, as demonstrated here, and enables opportunities to investigate property defining atomic structure.

The authors acknowledge the use of the Analytical Instrumentation Facility (AIF) at North Carolina State University, which is supported by the State of North Carolina and the National Science Foundation. D.L.I. acknowledges support from NSF Grant DMR-1151568.

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Supplementary Material