Surface-Adaptive Electron Ptychography (SAEP)

• Surface-Adaptive Electron Ptychography (SAEP) is an advanced imaging technique that adapts probe defocus and slice thickness to recover detailed 3D atomic potential, local surface topography, and thickness variations.
• It employs an iterative multi-slice reconstruction workflow with a least-squares maximum-likelihood cost function to accurately map both atomic-scale and morphological features.
• SAEP outperforms conventional ptychography by enabling sub-ångström resolution and robust analysis of beam-sensitive materials like zeolites and metal–organic frameworks for correlated structure–property studies.

Surface-Adaptive Electron Ptychography (SAEP) is an advanced extension of multi-slice electron ptychography designed to recover the three-dimensional atomic potential, local surface topography, and thickness variation in beam-sensitive, nanostructured materials. SAEP achieves sub-ångström resolution for both bulk and surface features by adaptively optimizing probe defocus and slice thickness on a per-position and per-slice basis during iterative reconstruction. This methodology enables high-fidelity mapping of both local (atomic) and global (morphological) specimen characteristics, directly facilitating correlated structure–property studies, especially in materials where surface morphology and thickness variations critically affect function, such as zeolites and metal–organic frameworks (Zhang et al., 24 Apr 2025).

1. Theoretical Foundations: Forward Model and Cost Function

SAEP builds on the multi-slice formalism in which the specimen is partitioned along the beam direction into $N$ slices, each with a projected potential $V_k(\mathbf{r})$ and an adaptive thickness $t_k$. At every probe scan position $\mathbf{r}_j$, the incident probe $P(\mathbf{r})$ is propagated through (1) a vacuum layer of variable thickness $A_{d,j}$, representing scan-position-specific local defocus, and subsequently (2) each specimen slice with adaptive Fresnel propagation distance $A_{z,k}$. The transmission function for slice $k$ is: $$T_k(\mathbf{r}) = \exp\left[i\sigma V_k(\mathbf{r}) t_k \right],$$ where $\sigma$ is the electron–potential interaction constant.

The exit wave after $N$ slices for position $j$ is: $$\Psi_{N,j}(\mathbf{r}) = T_N\,\mathcal{P}\Bigl\{\,T_{N-1}\,\mathcal{P}\{\cdots T_1\,\mathcal{P}\{\tilde{P}_j;A_{d,j}\}\cdots;A_{z,2}\};A_{z,1}\Bigr\},$$ where $$\tilde{P}_j(\mathbf{r}) = P(\mathbf{r}-\mathbf{r}_j)$$. The measured far-field intensity at each detector pixel $\mathbf{q}$ is then

$$I_j(\mathbf{q}) = |\mathcal{F}[\Psi_{N,j}](\mathbf{q})|^2.$$

SAEP employs a least-squares maximum-likelihood (LSQML) cost function in real space, comparing the updated exit-wave $U_{N,j}$ with a “corrected” exit-wave $X_{N,j}$ after applying the reciprocal-space amplitude constraint: $$E = \sum_{j=1}^M \| X_{N,j}(\mathbf{r}) - U_{N,j}(\mathbf{r}) \|_2^2.$$ Gradients with respect to both $A_{d,j}$ (local defocus) and $t_k$ (slice thicknesses) are computed analytically and updated via least-squares line search, ensuring adaptive convergence for both surface and bulk attributes (Zhang et al., 24 Apr 2025).

2. Iterative SAEP Reconstruction Workflow

The SAEP pipeline consists of the following steps:

1. Data Acquisition:
   • Record 4D-STEM diffraction patterns $I_j(\mathbf{q})$ on a pixelated detector for each probe scan position $\mathbf{r}_j$.
   • Typical scan parameters include $256 \times 257$ points, $96\%$ overlap, and $1.123\ \text{\AA}$ step size.
2. Pre-processing:
   • Calibrate the incident probe in vacuum.
   • Estimate coarse specimen thickness via single-slice or low-resolution multi-slice ptychography.
3. Initialization:
   • Insert a vacuum or “defocus” layer at each scan position ($A_{d,j}=0$) to account for surface topography.
   • Partition the sample into $N$ slices with initial thicknesses $t_k$.
   • Initialize propagation distances and defocus from coarse pre-reconstruction.
4. Iterative Updates:
   • Forward propagate the probe wavefield using the current estimates of $A_{d,j}$ and $t_k$.
   • Enforce amplitude constraints in reciprocal space and back-propagate to obtain $X_{N,j}$.
   • Compute updates for probe, transmission functions $T_k$, defocus $A_{d,j}$, and slice thicknesses $t_k$ via LSQML-based gradients and line search updates:

   $$\Delta A_{d,j} = -\frac{\langle \Delta U_{1,j},\; \frac{\partial U_{1,j}}{\partial A_{d,j}} \rangle}{\|\frac{\partial U_{1,j}}{\partial A_{d,j}}\|^2}, \qquad \Delta t_k = -\frac{\sum_j \langle \Delta U_{N,j},\; \frac{\partial U_{N,j}}{\partial t_k} \rangle}{\sum_j \|\frac{\partial U_{N,j}}{\partial t_k}\|^2}.$$

• Monitor convergence metrics such as cost function $E$ and parameter variation.
• Optionally anneal step-sizes or regularize thresholds, e.g., $T(u, \epsilon)$ truncation.

1. Stopping condition:
   • Terminate if relative change in $E$, $A_{d,j}$, or $t_k$ falls below preset tolerances (e.g., $10^{-6}$) (Zhang et al., 24 Apr 2025).

3. Key Advancements over Conventional Multi-Slice Ptychography

Conventional multi-slice electron ptychography (MSEP) assumes uniform defocus and constant slice thickness, recovering only the 3D atomic potential and a global thickness estimate. In contrast, SAEP introduces two principal adaptive features:

• Per-scan-position defocus $A_{d,j}$: Directly models local surface undulation and topography by allowing each probe position to have an individually optimized defocus layer.
• Per-slice thickness $t_k$: Enables accurate description of thickness gradients and morphological variations along the beam direction.

These enhancements allow SAEP to simultaneously resolve atomic-scale structure and longer-range morphological features, including $30$–$40$ nm surface variation and local thickness up to $40$ nm, while maintaining sub-ångström lattice resolution, even at specimen edges or under ultra-low dose imaging conditions (Zhang et al., 24 Apr 2025).

4. Experimental Demonstration and Quantitative Validation

SAEP has been experimentally validated using JEOL GrandARM 300 kV microscopes equipped with MerlinEM pixelated detectors, with typical parameters of $300$ kV accelerating voltage ($\lambda \approx 1.97$ pm), $16$ mrad convergence angle, and probe under-focus near $-100$ nm. Experimental targets include:

• Beta Zeolite: 3D mapping of channel-type correlations with locally measured thickness (regions $50$–$90$ nm), and observation of stacking-fault transformations.
• MOF–MIL-101(Cr) under Ultra-Low Dose: SAEP enabled edge atomic structure recovery at $\sim 50$–$60$ e$^-/\text{\AA}^2$, unattainable with conventional MSEP.
• Large-Field STW Zeolite: Mapping of surface protrusions and collapse features to depths of $\sim 15$ nm across $100 \times 100$ nm$^2$ domains.

Performance metrics from simulation and experiment show mean thickness error near $0.95$ nm, maximum error below $10$ nm, and robust sub-ångström imaging validated via lattice and strain mapping. Surface undulation mapping quantifies up to $30$ nm variation on the upper surface and $20$ nm on the lower surface (Zhang et al., 24 Apr 2025).

5. Comparison with Multi-Focus Ptychography and Other Strategies

Multi-focus electron ptychography employs stacks of 4D-STEM datasets at discrete defocus values, which increases the number of axial intensity constraints and improves interface and surface fidelity (down to FWHM $\sim 3$ nm for interfacial transitions and RMSE below $10^{-3}$ V·Å in homogeneous regions). The central distinction for SAEP is the further introduction of non-uniform defocus spacing near surfaces, adaptive slice grids ($\Delta z_n$ smaller near interfaces), and surface-specific regularizers $\mathcal{R}_{\text{surf}}(V_n)$ to fit physical prior knowledge at the boundaries.

In the context of thick ($\sim 20$–$30$ nm) and beam-sensitive specimens, such as zeolites, SAEP outperforms fixed-defocus MSEP methods, particularly regarding surface and thickness mapping, beam sensitivity, avoidance of reconstruction edge artifacts, and structure–property analysis (Schloz et al., 2024, Zhang et al., 24 Apr 2025).

6. Limitations, Best Practices, and Prospective Developments

Key considerations for SAEP deployment include:

• Data redundancy vs. electron dose: Adaptive methods increase the number of model parameters (per-position defocus and per-slice thickness); dose must be managed to avoid excess specimen damage, particularly at multiple defocuses or fine spatial grids.
• Optimal initialization and convergence: Coarse pre-reconstruction via conventional MSEP is generally required to constrain the solution space for $A_{d,j}$ and $t_k$.
• Handling of partial coherence and probe aberrations: Multiple probe modes may be required for highly defocused data or spatially incoherent sources, increasing computational complexity.
• Application scope: SAEP is particularly suitable for systems where the surface and interface properties dictate function, e.g., catalytic nanoparticles, buried semiconductor junctions, and oxide heterostructures.

A plausible implication is that future SAEP approaches may integrate small-angle tilt series or hybrid tomography for enhanced near-surface 3D fidelity, deploy surface-specific physical regularizers, and implement non-uniform adaptive grids matched to experimental sensitivity requirements (Schloz et al., 2024).

7. Impact and Outlook in Materials Characterization

SAEP enables simultaneous recovery of three-dimensional atomic potential maps, local surface morphology, and quantitative thickness gradients across extended fields of view, with beam sensitivity suitable for fragile porous materials. It directly facilitates correlation of microstructure with local channel topology, stacking faults, and morphological defects in heterogeneous nanomaterials. The integration of surface–thickness adaptivity in iterative multi-slice ptychography represents a significant methodological advance, opening new opportunities for the quantitative study of structure–function relationships in beam-sensitive, morphologically complex systems (Zhang et al., 24 Apr 2025).