A Deep Convolutional Neural Network to Analyze Position Averaged Convergent Beam Electron Diffraction Patterns

PACBED analysis

• PACBED patterns depend strongly on sample thickness and tilt and can reveal polarity, composition changes, and octahedral distortions.Their thickness dependence is non-intuitive because of dynamical diffraction, although visual matching can often identify experimental patterns.Source: §1 · pp. 1–2

Existing methods

• Least-squares fitting searches simulated-pattern libraries but can be time-consuming across broad parameter ranges and vulnerable to local minima.The approach has been widely used for thickness, tilt, and related measurements.Source: §1 · p. 2
• Pattern alignment is difficult because users generally must locate the center and calibrate pixel scale, while tilt displaces the intensity center and overlapping disks defeat Hough-transform approaches.These constraints complicate automated analysis before parameter fitting.Source: §1 · p. 2

Contribution

• Deep CNNs automatically extract PACBED pattern size, center, rotation, specimen thickness, and tilt.The networks are designed to analyze PACBED patterns without requiring human supervision.Source: §1 · p. 2

Results

• After training, the CNN approach is orders of magnitude faster than LSF while maintaining accuracy and supports automated processing of large 4D STEM datasets.The paper also examines generalization to other materials and a hybrid CNN–LSF approach for fast, robust analysis.Source: §1 · p. 2

Experimental data

• Experimental PACBED datasets used SrTiO3 along [100] and PbMg1/3Nb2/3O3 along [110], with random tilts up to approximately 4 mrad and varied acquisition noise.Measurements used a 200 kV probe-corrected STEM with 13.6 or 19.1 mrad convergence semi-angle.Source: §2.1 · pp. 2–3

Simulated data

• Bloch-wave simulations generated PACBED libraries spanning thickness, tilt, and probe conditions for CNN training.Thicknesses were sampled in 1 nm increments, with tilts up to 4 mrad along [100] and [010].Source: §2.2 · p. 3

Network training

• The CNNs were based on AlexNet and trained by stochastic gradient descent with dropout and converted 227×227 RGB inputs.The first two fully connected layers used dropout at a ratio of 0.5 to reduce overfitting.Source: §2.3 · p. 3

Benchmarking

• Least-squares fitting benchmarked CNN thickness and tilt estimates by minimizing intensity differences after scaling, rotation, and shifting simulated patterns.For specimens above approximately 70 nm, LSF tended to converge to thinner values because of excess inelastic-scattering background, which was corrected empirically.Source: §2.4 · p. 4

Pipeline

• The pipeline first measures zero-order disk size, disk center, and pattern rotation before CNNs estimate thickness and tilt.The procedure is organized as multiple CNNs with automated initialization and iterative refinement.Source: §3 · p. 4

Alignment

• Gaussian fits provide rough center and size estimates, after which CNNs refine shifts and recenter the cropped PACBED patterns.Precision of the initial estimates is not essential because the variables are iterated.Source: §3 · p. 4

Thickness and tilt

• Background subtraction before thickness and tilt estimation improves performance, particularly for sample thicknesses above approximately 70 nm.The background value is converged during thickness and tilt determination.Source: §3 · p. 4
• Only positive tilts from 0–4 mrad along [100] and [010] are measured directly, using four-fold symmetry to recover tilt signs from quadrant intensities.This reduces the complexity of CNN-based tilt measurement.Source: §3 · p. 4

Network design

• AlexNet’s initial convolutional kernels speed training and improve PACBED pattern identification before PACBED-specific classification layers are substituted.The network retains transferred AlexNet weights while adapting the output categories to PACBED tasks.Source: §3.1 · p. 5

Robustness

• Including tilt variation and experimental augmentations improves network generalization to distortions, noise, blur, and lens aberrations.Sub-mrad tilts are difficult to avoid experimentally, so their inclusion improves measurement accuracy.Source: §3.1 · p. 5

• Alignment networks achieve near 100% accuracy within ±2% for disk size, ±2 pixels for center shift, and ±2° for rotation.The networks also align patterns when the zero-order disk is poorly defined in thick or tilted regions.Source: §4 · p. 6
• 0.1 s/pattern is achieved for CNN processing, compared with LSF times exceeding 30 min/pattern.The timing is measured from raw-image input through output.Source: §4 · p. 6
• Near 95.7% and 96.5% of 13.6 mrad CNN thickness measurements match LSF within ±2 nm and ±3 nm, respectively.Source: §4 · p. 8
• For 19.1 mrad patterns, CNN thickness results match LSF 93.6% within ±2 nm and 94.2% within ±3 nm.Source: §4 · p. 8
• CNN tilt measurements match LSF 93.7% and 97.3% within ±0.50 and ±0.71 mrad at 13.6 mrad, and 94.9% and 100% at 19.1 mrad.Source: §4 · p. 8
• LSF is a benchmark rather than an absolute truth and can report incorrect thicknesses in contaminated areas or when pattern quality is low.Source: §4 · pp. 8–9

5.1. Thickness Dependence

• Over 95% of thickness measurements match LSF below 60 nm, while agreement decreases with increasing thickness and is poorest at 100–120 nm.For thick regions, experiment–simulation discrepancies include peripheral background, blurring, and differences in overlapped disks.Source: §5.1 · p. 10
• 0.7/nm and 0.3/nm are optimal uniform background subtractions for SrTiO3 at 13.6 mrad and 19.1 mrad, respectively.Appropriate subtraction optimizes PACBED thickness accuracy by reducing inelastic-scattering background.Source: §5.1 · p. 11

5.2. Pattern Feature Dependence

• Thickness agreement decreases when band-pass masks remove bright, high-contrast regions, indicating reliance on multiple PACBED regions, especially the central-disk periphery.For 13.6 and 19.1 mrad, degradation begins near 18 and 24 mrad mask positions, respectively.Source: §5.1 · pp. 11–12

5.1. Thickness Dependence

• For thicknesses below 10 nm, CNN tilt reliability is limited to about ±1 mrad, while tilt precision is about ±0.5 mrad more generally.LSF also frequently produces incorrect tilt measurements in this thin-specimen range.Source: §5.2 · p. 12

5.4. Dose Dependence

• At doses ≥2.3×10^7 e−/pattern, CNN thickness measurements remain consistent, whereas robust tilt measurements require roughly 4.7×10^7 e−/pattern.Below these thresholds, thickness and tilt errors increase; tilt sensitivity reflects its reliance on peripheral pattern features.Source: §5.4 · p. 14

• 0.1 s/pattern processing makes CNN analysis suitable for quantifying large 4D STEM datasets.GPU-accelerated CNN processing is described as well suited to big PACBED datasets.Source: §6.1 · p. 15
• A 60×60 nm2 region showed a wedge-shaped thickness variation of about 5 nm around an average thickness of 23 nm.The reported thickness variation matters because a 5 nm difference changes STEM image signal by 25%.Source: §6.1 · p. 15
• The same region exhibited tilt up to 4.2 mrad and bowl-shaped curvature despite challenging combined thickness gradients and crystal distortion.The measured tilts can alter electron channeling and signal contrast, while curvature can affect quantitative EDS absorption.Source: §6.1 · pp. 15–16

• CNNs automatically measure PACBED disk size, pattern center, rotation, thickness, and tilt.Source: §7 · p. 17
• Training uses thousands of simulated patterns augmented with geometric distortion, size, shift, noise, and intensity variations.Source: §7 · p. 17
• 0.1 s per pattern processing achieves excellent accuracy relative to brute-force methods while enabling fast 4D STEM analysis.Source: §7 · p. 17
• A response-analysis methodology examines which PACBED features drive CNN behavior, supporting interpretation of the automated measurements.Source: §7 · p. 17
• The approach demonstrates successful neural-network implementation for automated electron diffraction analysis.Source: §7 · p. 17