Quantifying Jahn-Teller distortion at the nanoscale with picometer accuracy using position averaged convergent beam electron diffraction

Using position averaged convergent beam electron diffraction (PACBED) the Jahn-Teller distortion in $\mathrm{LaMn}{\mathrm{O}}_3$ is quantitatively measured using a straightforward pattern-matching approach. The fit between experimental patterns and PACBED patterns simulated using the quantum excitation of phonons model allows a three-dimensional measure of octahedral distortion and rotation information from the near transmitted disk region with picometer precision. The effects of plasmon and other inelastic components on quantification using this method are investigated and discussed. The results provide an avenue for accurate local studies of the crystal structure origins of emergent physics in parallel with high-resolution annular dark field scanning transmission electron microscopy imaging at interfaces and defects in quantum materials.

Figure 1

Experimental STEM image and PACBED patterns. (a) HAADF-STEM image for $\mathrm{LaMn}{\mathrm{O}}_3$ along the [001] direction with an overlay of the lattice. Nonlinear drift distortion is corrected using image pairs with orthogonal scan directions [20]. PACBED patterns acquired from the area indicated by the red dashed box in (a) with energy filter off (b) and energy filter on (c). Normalized intensity integration along the red and green boxes is shown in (d) to highlight differences between the two images. The asymmetry along the intensity profile is likely caused by a slight deviation of the sample from the exact zone axis. The transmitted disk size is 9.7 mrad.

Figure 2

Intensity variation in PACBED with varying octahedral parameters. (a) Crystal structure of $\mathrm{LaMn}{\mathrm{O}}_3$. PACBED (left) and the squared difference between the perfect structure and the distorted one (right) as a function of (b) δ-type rotation, (c) γ -type rotation, and (d) bonding ratio. The white solid and dashed circles indicate the transmitted disk (1α) and 1.4α, respectively. The intensity in each pattern is normalized to show the contrast detail and its scale is arbitrary.

Figure 3

Comparison between experimental energy filtered PACBED and simulated patterns as a function of the simulated octahedral distortion parameter. (a) Minimum ${\chi }^2$ in the permutation of δ and γ type octahedral distortions ($0^{\circ }-{15}^{\circ }$) with fixed bonding ratio J (0.7–1). The Mn-O bond lengths (b) correspond to the simulated structure of each minimum ${\chi }^2$ in (a). Comparison between the down-sampled pattern and simulated pattern that determined the best fitting parameters in (c). The measurement steps are controlled by following a rough-to-fine strategy. The gray dashed line in these figures shows the position of the best fit condition.

Figure 4

The effect of inelastic scattering on the measurement of the octahedral distortion parameter and its amelioration by removing a background. Black squares are the measurement using the original unfiltered PACBED pattern. Then different background is produced using Gaussian blur to remove from the original pattern 2α Gaussian blur radius for red circle line and 1α Gaussian blur radius for green triangle line. Black arrows indicate the best fit result for each measurement, while the gray dashed line marks the result measured from energy filtered PACBED.