3D atomic-scale metrology of strain relaxation and roughness in Gate-All-Around transistors via electron ptychography

Domain: chemistry
Published: 2026-02-23
Editorial outcome: accept

Abstract
Next-generation semiconductor devices are adopting three-dimensional (3D) architectures with feature sizes in the few-nanometer regime, creating a need for atomic-scale metrology to identify and resolve performance-limiting fabrication challenges. X-ray methods provide 3D information but lack atomic resolution, while conventional electron microscopy offers limited depth sensitivity. Here we show how multislice electron ptychography, a computational microscopy technique with sub-Ångström lateral and nanometer-scale depth resolution, enables 3D imaging of buried device structures. We image prototype gate-all-around transistors and directly quantify roughness, strain, and defects at the interface of the 3D gate oxide wrapped around the channel. We find that silicon in the 5-nm-thick channel relaxes away from the interfaces, leaving only ~60% of atoms in a bulk-like structure. From a single dataset, ptychography provides quantitative metrology of atomic-scale interface roughness in 3D, previously accessible only through indirect inference, along with strain and other structural parameters needed for device modeling and process development.

Introduction
The increasing demands of modern computing and the drive to sustain Moore’s Law have led the semiconductor industry to transition from planar to three-dimensional transistor architectures, such as gate-all-around (GAA) structures. This shift enables improved transistor density, performance, and power efficiency, critical for advanced integrated circuits1. GAA transistors, with their nanosheet-based design (Figure 1a-d), provide enhanced electrostatic control by completely surrounding the channel with the gate electrode, unlike planar field-effect transistors (FETs) or FinFETs, which lack full gate-channel coupling1–5. This design is ideal for ultrascaled nodes but, as shown in Figure 1d, the intricate atomic-scale features—including interfaces, potential defects, and combinations of crystalline and amorphous materials of various composition—combined at such small length scales present significant challenges for structural and materials characterization.

Various imaging techniques have been employed to characterize semiconductor devices over the decades, most of which provide only two-dimensional information, whereas 3D metrology tools are necessary to fully characterize the atomic-scale structure and complexity of modern three-dimensional device architectures. Scanning transmission electron microscopy (STEM) with electron energy loss spectroscopy, for instance, was used to establish the critical thickness of amorphous silicon oxide required for bulk-like electronic properties6. X-ray ptychographic tomography provides 3D imaging at deep sub-micron scales, offering insights into larger features like interconnects and the very intricate back-end elements7. Atom probe tomography (APT) enables nm-scale characterization with chemical specificity, though challenges exist due to local charging effects near insulators8. Atomic electron tomography (AET) has reported atomic-scale 3D imaging in small nanoparticles, particularly for heavier elements9. However, its reliance on annular dark field (ADF) STEM limits sensitivity to light elements and introduces channeling artifacts in thicker samples, making it challenging to scale to larger, device-relevant fields of view.

Conventional atomic-resolution imaging techniques, while capable of sub-Ångström lateral resolution, often fail to provide depth-resolved structural information and suffer from well-documented violations of the projection approximation from multiple scattering artifacts10–12.

As the critical dimensions of the GAA transistors approach the sub-10 nm regime, the limitations of existing metrologies become increasingly pronounced. Even a single defect or pinhole (a small void or discontinuity at the dielectric-semiconductor interface) can significantly impact device performance, yet such features remain difficult to detect with current techniques1,13–15. Interface roughness and strain are particularly important to quantify, as they can strongly degrade carrier mobility and shift threshold voltage14–21. Indeed, recent quantum transport simulations confirm surface-roughness scattering as the predominant factor limiting electron mobility in ultrascaled silicon devices21. However, there is a large variance in previously reported measurements of interface roughness21–24, indicating both sensitivity to processing conditions and the limitations of conventional projective imaging methods, which tend to underestimate the true three-dimensional nature of interface roughness. These challenges underscore the need for direct, atomic-resolution 3D measurements to more accurately characterize buried interfaces and enable predictive models of device behavior. This need has been explicitly identified in recent semiconductor roadmaps and national initiatives, including the CHIPS for America Act’s Grand Challenges and review articles1,25.

Here, we show how multislice electron ptychography (MEP) addresses the critical metrology gap in device characterization highlighted above, namely, the need for Ångström-scale resolution, three-dimensional imaging capabilities, and simultaneous sensitivity to both light and heavy atoms26–28 (in the context of GAA devices, light elements refer to Si and O, while heavy refers to Hf), thereby enabling direct measurement of atomic structure, strain, and interface roughness in 3D. By scanning a convergent electron probe across the sample and collecting diffraction patterns from overlapping illuminated regions, MEP retrieves the phase shift introduced by the object’s atomically-resolved electrostatic potential, providing nanometer-scale depth information through post-processing of scanning diffraction data29. We demonstrate that MEP enables high-fidelity three-dimensional imaging of GAA transistors, validate its accuracy on simulated structures and show experimentally that it outperforms conventional through-focal STEM in resolution and dose efficiency. Using MEP on early-stage GAA test structures from Interuniversity Microelectronics Centre (imec), we directly image stacking faults, pinholes, interface roughness, and strain relaxation in the crystalline-Si (c-Si) channel, capturing buried detrimental structural features well before electrical testing is possible. In these GAA devices, in the c-Si channel (labelled c-Si in Figure 1d), we measure the structural transition from strained interfacial silicon to bulk-like silicon to span over 2 nanometers of the 5-nanometer-wide channel (over 40%). Comparing these findings to a conformal amorphous-SiO (a-SiO ) layer on epitaxial c-Si, we find that the strain relaxation is process-dependent and can serve as a metric for interface quality. By tracking interfacial atoms, we extract 3D maps of interface roughness amplitude and correlation length, uncovering asymmetries tied to the processing history of the interfaces. Formation of the critical GAA structure occurs early in the overall CMOS fabrication process, which comprises roughly a thousand steps and typically takes 3-4 months to complete from start to finish. Early structural feedback from MEP performed on dedicated witness wafers (test wafers fabricated alongside production wafers for process monitoring, including by destructive analysis techniques) can provide rapid insight into structural defects and interface quality early in the fabrication flow. This will accelerate the learning cycle during process development by enabling direct correlation between process changes, structural outcomes, and complementary short-loop electrical testing, reducing costly iterations. Moreover, since MEP captures atomic-scale interface roughness and strain, which are both key determinants of carrier mobility, it offers a direct structural metric for building predictive models of device performance and yield, enabling earlier-stage screening and optimization.

Results
Three-Dimensional Imaging by Depth Sectioning

Accurate atomic-scale measurements of interface roughness and structural transitions in 3D are essential for understanding and modeling the electronic behavior of advanced semiconductor devices16–20. While (S)TEM analysis is inherently destructive, requiring samples to be prepared from a chip portion (Figure 1a, b), it is standard practice in industry to dedicate test structures on a witness wafer for this analysis. Figure 2 provides an overview of the microscope configuration (Figure 2a), the depth-information acquisition principles for MEP (Figure 2b, c) and through-focal imaging approaches (Figure 2c, d), and a direct experimental comparison of these methods (Figure 2e-g).

Through-focal annular dark field (tf-ADF) and through-focal integrated differential phase contrast (tf-iDPC) are conventional STEM imaging methods that can potentially obtain limited 3D information by acquiring a series of 2D images, each with a small depth of field, at different defocus values (Figure 2c-d)10,12,30–35. tf-ADF uses high-angle scattering (Figure 2a, blue), which, in very thin samples, exhibits approximately Z2-dependent contrast (Figure 2f). tf-iDPC uses low-angle scattering (Figure 2a, purple) to approximate the projected potential, yielding, in very thin samples, linear or sublinear atomic number contrast and improved sensitivity to light elements (Figure 2g)36.

Both methods rely on a linear, weak-phase imaging approximation—i.e., that the probe shape remains unchanged from its free-space solution as it propagates through the specimen—an assumption that fails for device-relevant thicknesses (tens of nanometers) due to multiple scattering. At these thicknesses, the probe wavefront is repeatedly altered as it interacts with successive atomic planes (multiple scattering) and can be guided along the positively-charged columns of atomic nuclei (channeling)37, breaking the direct relationship between image intensity and the underlying atomic potential. These nonlinear probe-sample interactions produce well-documented artifacts10,11,30,36, including axial elongation, contrast reversals, and positional inaccuracies, which lead to mis-localized interfaces and spurious depth-dependent features (Figure 2e–g).

MEP uses the same optical configuration as tf-ADF and tf-iDPC (Figure 2a) but overcomes their limitations—probe–sample coupling and the lack of any correction for multiple scattering in thicker specimens—by recording a momentum-resolved scattering distribution and reconstructing both the probe and the sample potential38,39. The multiple scattering is accounted for using a multislice forward model27,40,41 (Figure 2b). In practice, this is achieved by acquiring a four-dimensional scanning transmission electron microscopy (4D-STEM) dataset: a defocused probe is raster-scanned over the sample, ensuring overlap between adjacent illuminated regions, and recording a two-dimensional diffraction pattern at each probe position in real space. The MEP reconstruction from these datasets explicitly accounts for the probe shape and its evolution—including multiple scattering, channeling, partial coherence, and sample tilts—by iteratively propagating the recovered probe though successive slices of the specimen (Figure 2c), enabling quantitatively accurate 3D reconstructions even for ~40 nm-thick specimens.

In MEP, the depth information is encoded via parallax: each Ronchigram acts as a shadow image of the sample with the observer effectively located at the probe crossover point. With the probe crossover ~10 nm above the sample, atoms at different depths shift across the defocused diffraction patterns at different rates as the probe scans (by parallax), allowing depth resolution from a single 4D dataset42. Unlike through-focal methods (or tomography approaches), which require multiple separate x-y raster scans, each representing a separate exposure of the sample to the electron beam, only a single x-y scan and its resulting 4D-STEM dataset is needed for MEP, from which 3D structural information is recovered (Figure 2b-d). Thus, depth information is encoded within the same x-y raster scan (Figure 2e), without requiring additional scans, thereby reducing the total electron dose.

The resolution of tf-ADF and tf-iDPC is bounded by the diffraction limit set by the numerical aperture of the lenses. The lateral Rayleigh resolution and depth resolution are given by

d = 0.61λ / α (1)

d = 2λ / α^2 (2)

where λ is the electron wavelength and α is the convergence semi-angle43. For a modern aberration-corrected instrument operated at typical settings of 300 kV (λ = 1.97 pm) with α = 30 mrad, these correspond to lateral and depth resolutions of 0.40 Å and 44 Å, respectively. Increasing α further is limited by chromatic aberrations and longitudinal source coherence which degrade image quality43,44. These Rayleigh limits are upper bounds; multiple scattering in real samples degrades resolution beyond the ideal values.

Recent high-speed, high-dynamical-range hybrid-pixel detectors45,46 now enable routine collection of high-quality 4D-STEM datasets suitable for MEP. By capturing scattering to high angles (β, Figure 2a), these detectors extend the accessible spatial frequencies, enabling lateral and depth resolution beyond the diffraction limit arising from the limited probe-forming-aperture semi-angle, α. Under high-dose conditions, the ultimate resolution in MEP is therefore bounded by the detector’s maximum collection angle β (> α) rather than α alone (Figure 2a)27,47–49.

This combination of dose efficiency, resolution, and depth sectioning has enabled 3D imaging across a wide range of materials using MEP28,42,50–56, including our initial conference abstracts on the crystalline-Si/amorphous-SiO interface57–59. The acquisition process is no more complex than that of a standard STEM image, and with recent algorithmic advances60, initial reconstructions that once took a day can now be completed in under an hour (see Typical MEP Workflow Duration in Methods).

Benchmarking of the Imaging Techniques

To systematically assess imaging accuracy, we benchmarked MEP against conventional through-focal methods—tf-iDPC and tf-ADF—using both simulations and experimental reconstructions. Figure 3 summarizes the simulated 3D benchmark of a rough crystalline-silicon (c-Si)/amorphous-silicon oxide (a-SiO )/amorphous-hafnium oxide (a-HfO ) interface, evaluating how each method recovers surface boundaries, depth positions, and buried amorphous features under realistic imaging conditions. These multislice simulations (calculated using abTEM61, see Methods) show that MEP consistently resolves crystalline boundaries, interface roughness, and buried amorphous speckle (Figure 3a-b and Supplementary Figure 1a-b). In contrast, tf-iDPC and tf-ADF display elongation, noise, and depth misattributions that obscure these features (Figure 3c-d, Supplementary Figure 1c-d), consistent with their inability to correct for probe modification from multiple scattering and channeling10,12. This difference is quantified in Figure 3e: for a ground-truth 66-Å-deep Si column, MEP retrieves the correct thickness, whereas tf-ADF and tf-iDPC overestimate it (94 Å and 93 Å) due to axial elongation and apparent surface shifts—artifacts that worsen with increased sample thickness or slight tilt.

To estimate depth blurring, we fitted error functions to the entry and exit edges of the axial line profiles (see Figure 2e and Supplementary Figure 2). MEP shows an edge blur of only 22 Å, while infinite dose tf-ADF shows 40 Å. The finite-dose tf-ADF profile is too noisy to fit, mirroring the experimental situation. tf-iDPC shows a 57 Å edge blur, comparable to the full sample thickness, indicating that the column boundaries cannot be meaningfully localized. Supplementary Figure 2 shows that MEP’s depth-sectioning performance remains stable across depth-slice spacings and detector collection angles, provided Nyquist sampling is met, and sufficient intensity is captured at the highest collected scattering angles.

The improved depth fidelity of MEP translates directly to the ability to recover buried structures: using a simulation study of a pMOS device model62, Supplementary Figure 3 shows that MEP resolves an amorphous Ta-filled pinhole at 14 nm at the correct depth, whereas the feature is indistinct in tf-iDPC and barely visible in tf-ADF (with the silicon channel completely obscured), despite the latter using more than twice the dose.

We next applied the same evaluation framework to experimental datasets acquired on a prototype imec GAA device (Figure 1), using all three methods under identical imaging conditions (described in Methods). A direct comparison is shown in Figure 2e-g. Supplementary Figure 4 highlights the improved resolution and the higher signal-to-noise ratio (SNR) achieved by MEP. MEP achieves a lateral information limit of 0.49 Å, compared to 0.66 Å for tf-iDPC and 0.83 Å for tf-ADF (Supplementary Figure 4), using only half the electron dose (0.5 × 10^5 e- Å^-2 for the single MEP dataset versus 1 × 10^5 e- Å^-2 for either the full tf-iDPC or tf-ADF image series).

From isolated single-atom features in the experimental MEP reconstructions (Figure 4a,b), we measured an axial blur of 40 ± 7 Å (Figure 4c and Supplementary Figure 5). Equivalent estimates were not possible for tf-ADF or tf-iDPC because finite-dose images are too noisy (consistent with limitations seen in simulations) and individual atoms likely move between defocus frames. Thus, MEP provides sufficient detail for quantitative 3D analysis, while tf-iDPC and tf-ADF could not reliably recover depth information under comparable conditions (Supplementary Table 1).

Two factors determine the practical resolution. First, the geometric limit is set by α for tf-ADF and tf-iDPC, and by the detector’s outer collection angle β for MEP. In our experiments, our choice of β was constrained by the ~40 nm specimen thickness and the detector’s finite pixel count, which limited the attainable collection angle without undersampling. Second, dose can also limit the achievable resolution: because each method accesses a different fraction of the scattered electrons (Figure 2a), not all have enough signal to reach their geometric limits. MEP collects nearly all scattered electrons (~99%), whereas tf-ADF uses <1%, and tf-iDPC—despite the high SNR in the bright-field disk—cannot convert this advantage into reliable depth localization due to uncorrected multiple scattering. As shown in Figure 2g, tf-iDPC’s axial profile is broadened to nearly the full sample thickness, reflecting its practical limitation. Additionally, the experimental MEP depth resolution is broader because a stronger regularization (0.5 vs. 0.1 in simulation) was used for stable convergence, which inherently suppresses high-frequency detail.

The limited depth resolution of the conventional methods also leads to ambiguity between true device features and surface damage from sample preparation. For example, a single ADF projection suggests a hafnium oxide intrusion into the Si channel (Figure 1d) but cannot determine its depth. MEP resolves this ambiguity: its 3D reconstruction (Figure 1e-g) shows the intrusion 10 nm beneath the lamella surface, confirming it occurred during device fabrication rather than as a TEM-preparation-induced redeposition. Additional depth sections from a planar c-Si/a-SiO /a-HfO stack (Figure 4a-b; Supplementary Figure 5) likewise demonstrate that MEP distinguishes intrinsic amorphous/crystalline structure from Focused Ion Beam (FIB) preparation-related surface artifacts. The triangular wedge geometry of the lamella is clearly visible in MEP depth slices (Figure 4b), underscoring MEP’s ability to map both sample features and preparation-induced morphology.

Through simulations and experiments, we found that MEP provides quantitatively reliable 3D reconstructions with minimal depth broadening, while conventional through-focal methods are fundamentally limited by probe-evolution artifacts. Leveraging MEP’s robust depth fidelity, we next quantify atomic-scale roughness and strain in GAA structures, capturing buried irregularities that directly impact device performance.

Atomic-Scale Strain and Roughness in Gate-All-Around Devices

Building on MEP’s depth-resolving capabilities, we now apply it to visualize and measure strain and atomic-scale roughness in gate-all-around (GAA) transistors—features that are critical for device performance13,16,17,21 yet poorly captured by conventional methods1,25. Figure 5a shows reconstructed slices through a prototype GAA device, revealing sharp variations in channel structure at different depths. At 18 nm, the crystalline silicon channel appears largely intact, but at 8 nm and 28 nm, large sections are absent—visible as missing-channel regions (‘mouse-bites’)—with associated pinholes and hafnium-rich regions. The full 3D volume (Figure 5b) places these depth slices in context, while a montage of a selected xy section in depth (Figure 5c) shows a stacking fault defect, evidenced by a half-unit-cell lattice shift and bending of adjacent atomic columns. Additional views are shown in Supplementary Figure 6. Green arrows highlight step edges at the c-Si/a-SiO interface, corresponding to interface roughness captured in 3D. These early-stage GAA structures were still undergoing process development, so the presence of such structural variability was expected.

To quantitatively map interface morphology and strain, we tracked the atoms in the silicon channel across depth using Atomap63 (see Methods, Supplementary Figure 7 and Movies 1-3). By grouping atoms into bilayers from the c-Si/a-SiO interface inwards (Figure 6a), we extracted Si–Si spacings per bilayer (Figure 6b, c). We find that in these GAA structures (blue and gray), on average, Si–Si spacing relaxes to its bulk value over four bilayers or 11 Å from each interface. To put this into context, we compare this strain relaxation length to that of a planar and conformal a-SiO layer on epitaxial c-Si (red), where the convergence occurs more abruptly—within just three atomic bilayers or 8 Å. Larger standard deviations in the Si-Si distances per bilayer in GAA structures (blue and gray shaded regions in Figure 6c) compared to the planar interface (red) reflect a more strained and disordered interface, highlighting greater structural disorder compared to the planar interface. Given that in modern devices, including the ones studied here, the shortest direction in the Si channel is around 50 Å, with top and bottom interfaces, this means that over 40% of the silicon in the channel is under strain (Figure 6b, c)—an important consideration for mobility and performance.

Conventional (S)TEM approaches have long struggled to resolve buried interface roughness in 3D, leading to conflicting models: some suggesting exponential height distributions24, while others favor Gaussian fits22,64 even though neither form had strong experimental evidence or a clear physical basis15,21. In both roughness models two fitting parameters are used: RMS amplitude Δ and correlation length Λ, extracted by fitting the autocorrelation function of the measured roughness with exponential or Gaussian models (see Methods and Supplementary Figure 8). Exponential profiles correspond to abrupt, low-correlation roughness, while Gaussian profiles indicate smoother, long-range variations, reflective of the interface's growth history. However, such detail about the actual interface profile has not previously been accessible with conventional (S)TEM techniques, as can be seen in Figure 2e-g and Supplementary Figure 4, where both tf-iDPC and tf-ADF fail to correctly capture the nuanced interface shape. With MEP, we can now directly resolve buried interfaces in 3D (Figures 6d-e, Supplementary Movies 4-5) and determine not only the amplitude Δ and correlation length Λ but also the functional form of the roughness distribution at atomic-scale resolution in nanoscale devices rather than macroscopic planar films.

Importantly, we find that different interfaces can follow different distributions, highlighting that no single functional form universally applies and underscoring the need for direct atomic-scale measurements of roughness. Supplementary Table 2 presents the measured interface roughness parameters for the reference planar interface and the two GAA interfaces, generated by tracking the outermost atoms of each interface. We find that smoother, intact interfaces primarily with step-edges are best described by exponentially decaying roughness (Supplementary Figure 8a-f). In contrast, rougher interfaces, such as the one with a mouse-bite, deviate from exponential model and look more Gaussian, indicating the limits of a simple two-parameter model (Supplementary Figure 8d, g-h). The top and bottom interfaces likely differ due to their different epitaxial histories: the SiGe-on-Si bottom interfaces tend to have mouse-bites and pinholes, while the top Si-on-SiGe interface is smoother, consistent with strain-driven roughening and Si interdiffusion65–67 (Figure 6d-g). Mouse-bites are present in some of the top interfaces too, but less frequently.

These quantitative values of Δ and Λ can be linked to carrier mobility using established scattering formalisms. In the long-wavelength limit, where the carrier wavelength is much larger than the characteristic interface roughness scale Λ, the carriers experience the interface as a smooth averaged potential, and interface roughness scattering rates scale approximately as

τ^-1 ∝ Δ^2Λ^2 (3)

and since mobility (μ) is proportional to the scattering time (τ), mobility scales as

μ ∝ 1 / (Δ^2Λ^2) (4)

leading to reduced mobility for rougher interfaces68. At shorter wavelengths, the dependence is more complex and requires full evaluation of the roughness power spectrum. Recent master-equation simulations have shown that surface-roughness scattering is in fact the predominant mobility-limiting mechanism in ultrathin-body FETs21. Using our measured roughness values and adopting the long-wavelength limit of mobility and interface roughness scaling, we estimate that mobility in these early-stage and defective GAA channels is reduced by approximately 7.5 times for the top GAA interface and 37 times for the bottom GAA interface, compared to the planar reference (Supplementary Table 2). Crucially, our ability to measure interface roughness distribution (Δ and Λ) directly at the buried interface now provides a pathway to testing and refining these scattering models in future studies.

Spatially-resolved roughness analysis also complements our simultaneous strain mapping within the same volumetric dataset, enabling the decoupling of strain and morphology. The variations we found show geometry and process-dependent differences in c-Si/a-SiO interface quality, which we can now directly measure using MEP. This combined metrology is especially crucial at extreme scaling, where roughness and strain strongly influence channel carrier mobility, threshold voltage, and device reliability. Indeed, recent studies on sub-5 nm silicon nanoribbons attribute performance degradation to precisely these effects4, which we can now measure directly from a single dataset, opening a path towards more studies to understand these parameters better for more predictive and realistic modeling. Interface roughness, and the need to measure it directly in buried interfaces in 3D is also relevant for quantum devices, where interface disorder leads to charge noise and decoherence, as recently demonstrated for Si/SiGe spin qubits64.

Methods
Device Fabrication

The GAA devices were fabricated following a customized version of the full electrical device flow developed by imec, with simplifications to enable direct access to critical structural features for characterization. Specifically, the process follows the basic CFET flow69, modified to produce 3× nanosheet GAA structures using a different starting epitaxial stack. The middle dielectric isolation and Epi source/drain modules were skipped to allow early access to the nanosheet release and gate stack deposition steps. The devices were formed by first growing a superlattice of alternating Si/SiGe single-crystal epitaxial layers. The SiGe layers serve as sacrificial materials that are selectively etched to release the Si nanosheets, which ultimately act as the transistor channels. The resulting gaps are then used to deposit the gate stack (typically a-SiO /a-HfO /metal gate). An interfacial a-SiO₂ layer is first formed by oxidizing the Si nanosheets, followed by high-k and metal gate deposition via atomic layer deposition (ALD), which offers the precise thickness control and conformality needed to uniformly cover the nanosheets on all sides. This stacked geometry enables each ALD step to coat multiple Si channels simultaneously3,69–71.

The planar interface (c-Si/a-SiO /a-HfO ) blanket film sample used for comparison was fabricated separately. The Si portion of this sample corresponds to the starting Si wafer, while the SiO layer is the native oxide present on the wafer surface, which serves as an effective nucleation layer for subsequent ALD deposition. The HfO film was deposited at 300 °C using ALD with alternating HfCl 4 and H 2 O pulses and N 2 purge steps between pulses.

Electron Microscopy Sample Preparation

The c-Si/a-SiO /a-HfO cross-sectional lamella was prepared using a standard cross-section Focused Ion Beam (FIB) lift-out procedure on Thermo Fisher Scientific Helios G4 UX FIB. The lamella is 10-20 nm thick in the region of interest. The GAA sample was prepared by Eurofins Nanolab Technologies with thickness of ~30-40 nm in the region of interest.

Data Acquisition

All the experimental datasets were acquired using a Cs-corrected Thermo Fisher Scientific Spectra 300 X-CFEG at 300 keV with a probe convergence semi-angle of 30 mrad. The MEP datasets were acquired at 10-15 nm nominal overfocus values, using an EMPAD-G2 detector, with scan step sizes in real space of 0.43 Å or 0.81 Å, scan dwell time of 100 s, and diffraction space sampling of 0.65 mrad per pixel or 0.82 mrad per pixel. The scan fields of view were ~10-20 nm with 256x256 pixel scans. A total dose of 0.5 - 2.5 × 10^5 e- Å^-2 was used for the 4D-STEM datasets.

The DPC images were acquired on the Thermo Fisher Scientific Panther detector (8 sections) and the ADF images were acquired using the Thermo Fisher Scientific ADF detector, with total through focal series dose of 1 × 10^5 e- Å^-2 using an automated acquisition method outlined here72. Nominal defocus values were corrected by a factor of two based on calibration. For the through-focal datasets, 20 focal planes were acquired with a defocus step-size of 20 Å. High-pass filtering was applied to the tf-iDPC images to suppress low-frequency artifacts, ensuring consistency with the ptychographic reconstructions. All experimental parameters for both MEP and through-focal imaging are summarized in Supplementary Table 3.

Ptychography Reconstruction

Multislice electron ptychography (MEP) was employed to reconstruct the 3D atomic potential of the sample with sub-Ångström lateral and 3-4 nm depth resolution. MEP solves the inverse problem of structure retrieval using 4D-STEM datasets, where a coherent electron beam is raster-scanned across the sample, recording diffraction patterns at each scan position. These diffraction patterns encode both phase and amplitude information of the transmitted electron waves due to overlap in the diffraction spots, and encode depth information due to the parallax effect, allowing for depth-resolved structural reconstruction.

To model the sample, it is conceptually divided into thin slices perpendicular to the beam propagation direction, and phase retrieval is performed using the maximum-likelihood multislice ptychography algorithm implemented in the fold_slice package. The version used in this work includes a tilt propagator addition and is available on Zenodo (DOI:10.5281/zenodo.15882443). This algorithm iteratively refines both the probe and sample structure, using a gradient-based approach and multiple probe modes to account for the partial coherence of the electron beam. The optimal reconstruction parameters – convergence semi-angle, defocus, and sample thickness – were determined using a Bayesian optimization model73, minimizing data error as the objective function.

The reconstruction used 4 probe modes and >1000 iterations, with a slice thickness of 1 nm and depth regularization of 0.4–0.7 (reconstruction summaries in Supplementary Figure 9). Since each slice in the reconstruction represents a phase object, the amplitude of each slice should be close to one, with deviations primarily due to scattering beyond the outer collection angle of the detector (thus hafnium oxide looks darker), as is shown in Supplementary Figure 9. High-pass filtering was applied to the final phase reconstructions to eliminate low-frequency artifacts. Supplementary Figure 9 provides a visual summary of the object phase and amplitude, with intensity histograms, as well as the different probe modes at the entrance surface for all the experimental datasets considered in this work. The fully-converged ptychographic reconstructions were computed using an in-house cluster, requiring approximately one day for completion.

Typical ptychography workflow duration

The total measurement time for MEP experiments is comparable to standard high-quality TEM workflows. Focused ion beam (FIB) sample preparation for a <50 nm lamella typically requires half a day for an experienced user, followed by approximately half a day of microscope time (including alignment and tuning). Each 4D-STEM dataset (256 x 256 scan positions) is acquired in about 7 seconds on the EMPAD-G2 detector. Reconstruction typically takes from an hour to a day, depending on user familiarity, computational resources, degree of convergence needed for imaging vs quantification, and the algorithm used. These times are representative of our setup and will vary between laboratories.

Reconstruction time scales roughly linearly (with a small offset for setup) with the number of probe positions (i.e. area) and with the number of reconstructed depth slices60; however, the iterative reconstruction is efficiently accelerated on GPUs and can be distributed across compute nodes. Thus, reconstruction a large area can be done in a similar time, provided more compute resources are available, enabling practical times for larger volumes and multi-device studies.

Simulation of Electron Microscopy Data

To evaluate the imaging performance of multislice electron ptychography (MEP) and compare it with other imaging modes, we conducted multislice simulations of scanning transmission electron microscopy (STEM) images and 4D-STEM datasets. To validate the accuracy of MEP, we benchmarked its performance in reconstructing a technologically relevant c-Si/a-SiO /a-HfO interface structure and a pMOS structure12,62. The simulations were performed using the abTEM multislice simulation package61. The simulation approach provides a controlled environment for assessing resolution, sensitivity, and depth reconstruction accuracy while incorporating experimental conditions such as realistic aberrations and electron dose (through detector noise).

The simulations were conducted under conditions mimicking experimental setups, using a 300 keV acceleration voltage and an aberrated electron probe with a convergence semi-angle of 30 mrad. The simulated probe was subjected to spherical (Cs = 1 μm) and chromatic (Cc = 2.46 mm) aberrations with an energy spread of 0.4 eV. The chromatic focal blur can be estimated as

Δz ≈ Cc * (dE / E0) ≈ 3.3nm (5)

which is of the order of the expected depth resolution, 4.4 nm, and does not affect the imaging modes drastically. On the other hand, spherical aberration shifts the effective entrance surface location by approximately

Δz ≈ sqrt(Cs * λ) (6)

thus shifting the depth features primarily for the tf-ADF and tf-iDPC images. The total electron dose was kept constant at 2.5 × 10^5 e- Å^-2 across all imaging methods to ensure a fair comparison and Poisson noise was applied to the diffraction patterns to match experimental conditions. Further simulation details can be found in Supplementary Table 4.

Simulated (4D-)STEM datasets were generated by real space scan step size of 0.14 Å for tf-iDPC and tf-ADF images and 0.44 Å for c-Si/a-Si O /a-HfO MEP and 0.75 Å for GAA MEP. The extent of the pixelated detector was like experimental conditions with maximum collection angle of around 60 mrad and 128x128 diffraction patterns. Using these datasets, various imaging modes were calculated, including MEP, through focal integrated differential phase contrast (iDPC), and through focal annular dark field (tf-ADF) imaging, results of which are presented in Figure 3 and Supplementary Figures 1-3. Chromatic aberration was added by averaging simulations at different defocus values according to a gaussian focal spread. Ptychographic reconstructions of the 4D-STEM datasets were performed using the fold_slice package; iDPC images were generated from 4D-STEM datasets by computing DPCx and DPCy images, followed by Fourier integration. ADF images were simulated by integrating intensities in annular detectors. For iDPC, only the bright disk was used (30 mrad) while for ADF the inner and outer angles for integration were 60 – 150 mrad. Unlike ptychographic reconstruction, iDPC and ADF imaging require multiple through-focal datasets to approximate 3D structural information, increasing the total dose and complexity.

The reconstructed images were compared to the ground truth atomic potential, demonstrating that MEP successfully recovered key structural features, including interface roughness, amorphous intrusions, and Si atomic column extend (see Figure 3 and Supplementary Figures 1-3). These features are critical for evaluating the performance and reliability of GAA transistors. Compared to iDPC and ADF, MEP provided improved 3D reconstruction accuracy and dose efficiency, mitigating artifacts associated with multiple scattering and channeling.

Atom Tracking and Strain Relaxation Analysis

Atom positions were identified using the Atomap package63 (see Supplementary Figure 7 and Supplementary Movies 1,2 and 4). Custom-written Python scripts were used to segment tracked atoms into layers and analyze interfacial structures (see Supplementary Figure 7 and Supplementary Movies 1,2 and 4). Si-Si projected nearest neighbor distances were calculated for each layer for all the depths for the datasets discussed in this work (see Figure 6c). Distance mappings were generated to visualize variations in bonding configurations across the layers (for an example see Figure 6b). Jupyter Notebooks for this analysis are available on Zenodo (DOI:10.5281/zenodo.15882443).

Three-Dimensional Visualization

Tomviz was used for volumetric visualization of atomic cuboids shown in Figure 1e, Figure 5b and Supplementary Figure 6a74. The c-Si/a-SiO interface shape was constructed using the outer-most tracked Si atoms per layer and a 3D surface was constructed using Open3D python package, results of which are shown in Figure 6e-f and Supplementary Movies 3 and 575. Jupyter Notebook for this analysis is available on Zenodo (DOI:10.5281/zenodo.15882443). Supplementary Movie 6 was rendered in ParaView using synchronized views of the full 3D volume and Z-slices from the GAA-1 reconstructed dataset76.

Interface Roughness Quantification and Mobility Estimation

To quantify interface morphology, we compute standard roughness metrics: the root-mean-square (RMS) height Δ and in-plane correlation length Λ, directly from the tracked 3D atomic positions. For each dataset, the outermost interfacial Si atoms were segmented into facet-specific groups: planar, top-GAA, and bottom-GAA interfaces. A reference plane was first fitted to the local atomic surface for each facet using a least-squares approach. Height residuals were then computed as the perpendicular distances of each atom from this fitted plane.

The RMS roughness was defined as the average amplitude of deviation of these residual heights across each facet. To assess the lateral spatial coherence of the roughness, we computed the 2D autocorrelation function (ACF) of the roughness map. The in-plane correlation length Λ was extracted by fitting the ACF envelope with both exponential and Gaussian models:

Exponential decay: C(r) ∝ e^(-r/Λ) (7)

Gaussian decay: C(r) ∝ e^(-r^2/Λ^2) (8)

Both fitting models are presented (Supplementary Figure 9h–i), as there is longstanding discussion in the literature about which model more accurately represents interfacial roughness. Since our MEP reconstructions yield fully 3D atom-resolved data, we can compute the autocorrelation either radially or along specific in-plane directions (x and z), shown in Supplementary Figure 8. This enables an analysis of anisotropy in the roughness correlation. Axis-resolved correlation lengths are reported alongside radially averaged values to capture the directional dependence of morphological fluctuations.

These quantitative roughness metrics map directly onto mobility via standard interface-roughness (IR) scattering formalism14. In planar channels, a compact and widely used scaling for long-wavelength limit follows the relation given in Equation (4). A full treatment requires evaluation of the roughness power spectrum and numerical calculation of scattering rates21. Thus, smoother (smaller Δ) and less correlated (smaller Λ) interfaces yield higher mobility, all else equal. Using this relation and assuming all other relevant parameters to be the same for all interfaces (i.e. carrier density, effective fields), we estimate relative mobility reductions for the measured planar and GAA interfaces, as detailed in Supplementary Table 2.

To fully capture mobility degradation from interface roughness, one must evaluate the complete roughness power spectrum S(q) (the Fourier transform of the measured autocorrelation) together with the electronic form factor and screening effects that govern how carriers sample the interface. In modern master-equation and simulations, which treat these ingredients self-consistently, interface roughness scattering has been identified as the dominant mobility-limiting mechanism in ultrathin-body FETs21. By directly measuring both the RMS roughness amplitude (Δ) and correlation length (Λ) from our 3D atomic reconstructions, we now provide realistic inputs for such models, replacing the common practice of treating roughness as an adjustable fitting parameter. This opens the door to quantitative, predictive tests of process-structure-performance relationships in future device simulations.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This study of a GAA device using multislice electron ptychography is precisely the type of experiment needed to further demonstrate and reinforce why MEP is so powerful, and why it will likely largely replace conventional imaging in the near future. This is clearly laid out by the authors, who compare traditional methods (ADF, iDPC) that do not come close to yielding the same insights from a single dataset. Further, all claims are well validated and supported. Given the timeliness, the manuscript should be accepted and published posthaste.\nThere a couple of minor points:\n“By retrieving the object’s phase.” ​ Perhaps better to refer to this as retrieving the phase shifts introduced by the object’s atomically resolved electrostatic potential? In other words, does “object phase” have a broadly understood meaning?\nUltra-scaled has a dash in some places, and no dash in others.", "reviewer_2": "In high resolution imaging, the depth of field limitations put a limit on the sample thicknesses that can be imaged. Conventional STEM depth-resolved approaches can produce images with a lateral resolution of 0.66-0.89 Å and a depth resolution of ~95 Å. By changing the focus, the resulting 2D images can be stacked to recover the 3D sample structure. By employing multi-slice electron ptychography (MEP) the authors were able to demonstrate that the lateral resolution improves down to 0.49 Å and the depth of field can be extended down to 64 Å. In addition, MEP achieved this imaging performance with at least 2-times lower dose. In summary, MEP enables imaging of thicker sample structures at a higher resolution and with less dose.\nWhile MEP has been developed by prior works, it is the first application (to the best of my knowledge) for metrology of modern semiconductor devices such as the GAA transistor. The 3D structural information looks impressive and highly useful for the semiconductor industry. The surface roughness of the measured transistor features was used to estimate the effect on the charge carrier mobility, enabling optimization of semiconductor device electronic properties. Overall, the presented results demonstrate the capabilities of electron microscopy for quantitative 3D metrology of interface roughness, strain relaxation and atomic-scale defects.\nGiven that the semiconductor industry is moving towards 3D transistor architectures such as GAA, the need for such 3D metrology tools will only increase. These measurement techniques are also timely since these new transistors will start being introduced to consumer electronic in the next years. Therefore, I believe such metrology capabilities are relevant to the broader semiconductor industry.\nBased on my understanding, the MEP methodology is sound and the improvements compared to state-of-the-art are consistent. I do find that there quite a lot of details missing. The details that are within the manuscript they are hard to extract. For example, one of the primary claims is that MEP provides superior spatial and axial resolution, yet finding quantitative numbers is difficult. The axial resolutions comparison to state-of-the-art methods can only be found within one of the image panels and are not searchable within the text. I think the manuscript would benefit greatly from improved structure and flow. I found it quite difficult to track what is going on and had to jump around the text to find the basic quantitative numbers. For example, in MEP the sample is split into slices or during through-focal imaging, a certain number of slices are acquired. Yet there is no number of how many slices were reconstructed/acquired for each method, making it difficult to compare.\nAs an imaging method developer I found a lot of details missing, making it difficult to draw conclusions about the imaging performance compared to other methods. But from the point of view of the transistor metrology I found the presented results impressive and convincing. I think the presented results meet the required of Nature Communications given their impact.\nComments\n1. Line 42 - at this point I think it would be good to say that most metrologies provide 2D images whereas 3D metrology tools are necessary for the metrology of 3D structures. This way it will be clearer for the reader moving forward.\n2. Line 71 - \"address the metrology gap highlighted above\". Could the authors please explicitly state the gap i.e., the need for both angstrom resolution and 3D imaging capabilities. Otherwise it can be not entirely clear for the reader.\n3. Line 72 - \"By retrieving the object’s phase from overlapping diffraction patterns, MEP provides depth information through post-processing of scanning diffraction data\". This description is misleading, since the diffraction patterns themselves don't overlap. Instead the overlap is of the neighboring sample illumination patches, which produce these diffraction patterns. Please rephrase by e.g., the sample is scanned across an electron beam and diffraction patterns are collection from overlapping illumination patches.\n4. Line 87 - \"Formation of the critical GAA structure is one of the earliest steps in a CMOS process that can take 3-4 months\". Are the authors claiming that the lithography of GAA transistors takes 3 months during a single wafer production? Please clarify.\n5. Line 88 - \"Early structural feedback from MEP on witness wafers\". What is a witness wafer? The MEP imaging process requires extraction of a thin lamella, which would also be highly disruptive and destructive to the manufacturing process. Could the authors here explain in more detail how MEP could be integrated into the manufacturing process given the constraints of the technique.\n6. Line 99 - \"tf-ADF signal is ... proportional to the square of the atomic number\" and \"tf-iDPC ... yields linear or sublinear atomic number contrast\". Comparing figure 4b and 4c, the contrast seems to be reversed. If the main differences are linear and squared contrast relationships, then why one is inversely proportional while the other is directly? I would expect both images to have the same direct contrast relationship. Please elaborate why.\n7. Line 107 - \"correspond to a lateral resolution of 0.40 Å and a depth resolution of 44 Å\". Could the authors please explain why the depth resolution should be 44 Å, but in Figure 3, the resulting resolutions are significantly worse? Moreover, MEP achieves 64 Å depth resolution based on Figure 3, which is worse than the quoted imaging resolution of 44 Å. Could the authors please dedicate a paragraph to discuss these theoretical and achieved imaging performance? The whole article states claims such as \"better depth resolution\" etc., yet there are no quantitative comparison within the text which would state how much better the imaging performance of MEP is compared to other techniques. At the moment the claims are qualitative and scattered throughout the text. The depth resolutions are only shown in Figure 3e and do not appear in the text, which is strange given the emphasis on how much better the depth sectioning is with MEP.\n8. Line 111 and 114 - \"these issues\", \"these limitations\". Could the authors phrase the issues/limitations more explicitly. e.g., \"the primary issue is multiple scattering... and the with MEP we overcoming multiple scattering through multi-slice ptychography, which models the multiple scattering process.\"\n9. Line 115 - first mention of 4D-STEM dataset, please explain here or earlier. Potentially during the description of ptychography and that it acquires 4D-STEM data.\n10. Line 117 - \"multiple scattering and channeling effects\". I am not familiar what are the channeling effects and how multi- slice ptychography can overcome them. Could the authors explicitly mention these artefacts in the earlier section of the text and explain why multiple-scattering happens and what are channeling effects. Potentially around Line 111. Please be clear on the challenges and how they are solved.\n11. Line 128 - \"Unlike tf-ADF and tf-iDPC (or tomography approaches), which require multiple scans, only a single 4D- STEM dataset is needed for MEP, which can reduce the total electron dose (Figure 2b-d, Extended Data Figure 3).\" Please clarify what the authors mean by \"scan\", since 4D-STEM is also produced by raster-scanning the sample. The authors should clarify that 3D information in MEP is obtained from the same dataset that would be acquired for a 2D sample structure recovery i.e., depth information basically comes \"for free\". With other approaches the 3D information is does not come \"for free\" and instead required acquisition of many 2D images e.g., multiple through-focus slices or rotation angles. Without this information some readers might be confused.\n12. Line 141-148 - I am missing details on the imaging methods. How many slices are the authors reconstructing with MEP? How many through focus images are the authors acquiring? This would give directly the expected axial resolution for a given sample thickness. What are the performance differences between MEP compared to the through-focus methods. Could the authors expand?\n13. Line 166 - \"this enhanced resolution is limited by the small diffraction collection angle required for the ~38 nm thick section, whereas the poorer resolution of ADF and iDPC is primarily dose-limited, due to radiation damage concerns\". This explanation is not clear. The collection angle for MEP is higher, which allows acquisition of a wider scattering leading to higher resolution, as one would expect. Plus the authors have dose limitations for ADF and DPC leading to even lesser resolution. My questions would be: is the resolution difference explained by the detection acceptance angles beta and alpha alone? Or it also requires the additional discussion on radiation dose? Could the authors please discuss the resolution from both aspects.\n14. Moreover, is the dose for a 2D image acquisition in both MEP and DPC/ADF is the same? ADF/DPC requires many through-focus slices which multiplies the dose requirements. Did the authors reduce the dose for each of the through-focus slices in DPC/ADF or kept it the same? It would be good to clearly establish at the beginning what are the fundamental data acquisition differences between DPC/ADF vs MEP, in terms of dose, data acquisition etc.\n15. Line 196 - The following paragraph discusses strain, but it is not explained how they authors obtain the strain information. The resulting MEP reconstruction is simply the 3D structure of the sample. How is the strain recovered from the acquired data?\n16. Line 226 - \"In both cases two fitting parameters are used...\". Could the authors please elaborate what is actually being fitted? I would assume that the authors are fitting the extracted atoms and their positions using Atomap, but it is just a guess on my part.\n17. Line 249 - \"In the long-wavelength limit...\". Could the authors briefly explain what is the wavelength referring to here and how it relates to the carrier mobility mentioned previously.\n18. Line 271 - \"Importantly, for the information gained, MEP is relatively time- and resource-efficient: from sample preparation to a fully reconstructed 3D image takes only about two days, providing rapid atomic-scale structural feed\". Could the authors expand a bit more on the time distribution for each step? E.g., how much slower is the sample preparation compared to imaging itself and the data reconstruction.\n19. Line 271 - The paragraph starting here seems to be related more to the \"general discussion\" and seems unrelated to the findings within the current section. Consider integrating into the discussion.\nMinor comments\n1. Line 37 - explain FET abbreviation (Field effect transistor).\n2. Line 45 - \"X-ray ptychographic tomography\" rather than just \"X-ray ptychography\".\n3. Line 54 - \"projection\" to \"violation of the projection approximation\".\n4. Line 79 - if crystalline-Si is c-Si, please add the abbreviation in brackets, which appears later in the text.\n5. Line 81 - could the authors please refer here (and int his paragraph in general) to Figure 1 b such that it is clear what the text is referring to. Otherwise for people not familiar with the GAA structure it is unclear.\n6. Line 111 - replace \"distortions in contrast\" to \"contrast distortions\".\n7. Line 132 - \"This enables resolution beyond the diffraction limit imposed by the probe convergence semi-angle α\". The probe convergence does not impose a \"diffraction limit\", only the wave propagation physics imposes a \"diffraction limit\".\n8. Line 144, 223 - \"S/TEM\" and \"(S)TEM\". Please use a consistent naming convention.\n9. Line 149 - would be good to mention that \"c\" stands for crystalline and \"a\" for amorphous.\n10. Line 197 - \"depth-resolving power\" to \"depth-resolving capabilities\".\n11. Line 223 - \"for the first time determining\" to \"determine\"\n12. Line 533 - \"due to linear atomic number contrast, all sections of the structure are visible.\" what does this mean exactly?\n13. Line 623 - \"3D rendering of GAA 2’s surface\", what is the \"2's surface\"?\n14. Figure 1 b - it is unclear what the text is pointing to exactly. Could the authors highlight and assign the layers better by e.g., a contour for each layer. Otherwise it is unclear.\n15. Figure 2 c - \"Visualization of depth-resolved slices obtained by different methods\". I only see three slices, which I assume are from MEP. Where are the other slices \"obtained by different methods\" which here should include also through focal series? Description seems misleading.", "reviewer_3": "Karapetyan et al describe a carefully written study using multislice electron ptychography (MEP) to study the nanoscale local structure of an advanced gate-all-around (GAA) nanosheet transistor. GAA transistor structures are central to next generation microelectronics. Characterizing the nanoscale structure of these nanodevices is a critical metrology challenge for the microelectronics industry. This report describes how MEP can provide nanoscale interface roughness and directly measure the local strain in the silicon channel of a single GAA transistor. They compare MEP to other electron microscopy modes and show how it improves what can be extracted from the microscope. Below I detail a couple of minor improvements that can be made to the manuscript. The manuscript will be of wide interest to the readership of Nature Communications.\nSeeing this level of detail on a single device is amazing and will be of great interest to the semiconductor industry. The authors should note that a chip has billions of transistors and that there is often variation between each individual device. That variation is quite important for product performance. The unaddressed point is statistical sampling. Measuring one out of a population of one billion is not statistically representative of the population. If they measured a second transistor it likely would be different from the first one. This makes the discussion in the manuscript about speeding up the measurement and reconstruction algorithm critical to the utility of the measurement so that it can be applied to many transistors to improve the statistics. I should note that for the purposes of the manuscript it is acceptable that they only report on one transistor device.\nIt would be helpful to non-microscopists to include some additional diagrams in the figures to help one understand where in the device the particular view/slice comes from. The slices sometimes can be non-intuitive, especially with the asymmetric resolution. For example, on figure 1 it would be helpful to add marks on the 3D image in 1C highlighting where in z the slices shown in 1d come from. They should be color coded better. Making 1D the same green as 1C is a little confusing.\nExtended data figure 4 - not clear from the caption what is different between the two images. The figures also appear to have different scalings in the two directions. The vertical dimension is 20 nm according to the arrows and the horizontal scale bar is 1 nm. The difference in scaling is about 3.67x.\nThe methods section does not describe the fabrication of the silicon, silicon oxide, and halfnium oxide blanket film.\nI recommend mentioning in the abstract that the GAA devices are fabricated by imec. With the TSMC coauthors and no imec coauthors it is easy to assume the devices were made by TSMC."}, "author_response": "We thank all three reviewers for their thoughtful and constructive feedback. We\nappreciate the reviewers’ positive assessment of the significance of MEP within the\nsemiconductor metrology toolbox, and we have revised the manuscript accordingly to\nimprove clarity, completeness, and reader accessibility. We have carefully addressed\neach comment in the point-by-point responses below. Our responses are shown in green\nand revised or newly added text in the manuscript is shown in blue, with the relevant\nmodifications in bold. All line numbers referenced in our responses correspond to the\nrevised clean draft (with tracked changes accepted), which is provided separately from\nthe marked-up version.\nBelow is a summary of the key changes:\nTo address the reviewers’ suggestions regarding clarity and structure, we have re-\norganized several sections of the manuscript to better match the typical format of a Nature\nCommunications article. This includes restructuring the imaging-results and analysis\nsections for improved logical flow and adding the previously missing Code Availability\nstatement. We have also expanded the Methods section and added new tables in the\nSupplementary Information that summarize all imaging and reconstruction parameters.\nSeveral figures have been revised for clarity and completeness. In particular, Figure 3\nhas been updated to clearly distinguish the apparent depth extent from the depth blur,\nreducing the potential for misinterpretation. We have moved certain figures earlier in the\nmanuscript where they more naturally support the surrounding discussion, added\nquantitative analyses of axial blur for the different imaging modes, and consolidated all\ndepth-resolution values in Supplementary Table 3. We have also included an explicit\nexperimental estimate of the MEP depth blur. We hope these updates ensure that the\nquantitative performance of MEP relative to tf-ADF and tf-iDPC can be located quickly\nand unambiguously.\nReviewer #1 (Remarks to the Author):\nThis study of a GAA device using multislice electron ptychography is precisely the type\nof experiment needed to further demonstrate and reinforce why MEP is so powerful, and\nwhy it will likely largely replace conventional imaging in the near future. This is clearly laid\nout by the authors, who compare traditional methods (ADF, iDPC) that do not come close\nto yielding the same insights from a single dataset. Further, all claims are well validated\nand supported. Given the timeliness, the manuscript should be accepted and published\nposthaste.\nWe sincerely thank the reviewer for their very positive and encouraging assessment of\nour work, and for recognizing both the significance and timeliness of applying Multislice\nElectron Ptychography (MEP) to advanced GAA transistor metrology. We are grateful for\nthe reviewer’s supportive comments affirming the validity of our results and the potential\nof MEP to transform atomic-scale imaging in semiconductor research.\nBelow, we provide point-by-point responses addressing the reviewer’s comments.\nThere is a couple of minor points:\nComment #1:\n“By retrieving the object’s phase.” ï¿½ Perhaps better to refer to this as retrieving the phase\nshifts introduced by the object’s atomically resolved electrostatic potential? In other\nwords, does “object phase” have a broadly understood meaning?\nResponse:\nWe agree and have revised the text for clarity (lines 79-81):\n“…. MEP retrieves the phase shift introduced by the object’s atomically-resolved\nelectrostatic potential, providing depth information through post-processing of scanning\ndiffraction data.”\nComment #2:\nUltra-scaled has a dash in some places, and no dash in others.\nResponse:\nThank you for catching this, we have changed all to be “ultrascaled” with no dash.\nIn high resolution imaging, the depth of field limitations put a limit on the sample\nthicknesses that can be imaged. Conventional STEM depth-resolved approaches can\nproduce images with a lateral resolution of 0.66-0.89 Å and a depth resolution of ~95 Å.\nBy changing the focus, the resulting 2D images can be stacked to recover the 3D sample\nstructure. By employing multi-slice electron ptychography (MEP) the authors were able\nto demonstrate that the lateral resolution improves down to 0.49 Å and the depth of field\ncan be extended down to 64 Å. In addition, MEP achieved this imaging performance with\nat least 2-times lower dose. In summary, MEP enables imaging of thicker sample\nstructures at a higher resolution and with less dose.\nWhile MEP has been developed by prior works, it is the first application (to the best of my\nknowledge) for metrology of modern semiconductor devices such as the GAA transistor.\nThe 3D structural information looks impressive and highly useful for the semiconductor\nindustry. The surface roughness of the measured transistor features was used to estimate\nthe effect on the charge carrier mobility, enabling optimization of semiconductor device\nelectronic properties. Overall, the presented results demonstrate the capabilities of\nelectron microscopy for quantitative 3D metrology of interface roughness, strain\nrelaxation and atomic-scale defects.\nGiven that the semiconductor industry is moving towards 3D transistor architectures such\nas GAA, the need for such 3D metrology tools will only increase. These measurement\ntechniques are also timely since these new transistors will start being introduced to\nconsumer electronic in the next years. Therefore, I believe such metrology capabilities\nare relevant to the broader semiconductor industry.\nBased on my understanding, the MEP methodology is sound and the improvements\ncompared to state-of-the-art are consistent. I do find that there quite a lot of details\nmissing. The details that are within the manuscript they are hard to extract. For example,\none of the primary claims is that MEP provides superior spatial and axial resolution, yet\nfinding quantitative numbers is difficult. The axial resolutions comparison to state-of-the-\nart methods can only be found within one of the image panels and are not searchable\nwithin the text. I think the manuscript would benefit greatly from improved structure and\nflow. I found it quite difficult to track what is going on and had to jump around the text to\nfind the basic quantitative numbers. For example, in MEP the sample is split into slices or\nduring through-focal imaging, a certain number of slices are acquired. Yet there is no\nnumber of how many slices were reconstructed/acquired for each method, making it\ndifficult to compare.\nconclusions about the imaging performance compared to other methods. But from the\npoint of view of the transistor metrology I found the presented results impressive and\nconvincing. I think the presented results meet the required of Nature Communications\ngiven their impact.\nWe sincerely thank the reviewer for their careful reading of the manuscript, their positive\nassessment of our results, and their recognition of the broader relevance of multislice\nelectron ptychography (MEP) for semiconductor metrology. We greatly appreciate the\nreviewer’s detailed engagement with both the technical aspects of the imaging\nmethodology and the clarity of the manuscript. Their comments have helped us\nsignificantly improve the presentation, accessibility, and completeness of the work.\nIn response to the reviewer’s request for clearer structure and for quantitative imaging\nperformance metrics to be more readily identifiable, we have substantially revised the\nmanuscript for improved flow and readability. These revisions include:\n• Adding measurements of depth resolution, both for simulation and experiment.\n• Adding a paragraph clarifying the theoretical depth limits, the achieved simulated\nand experimental depth resolution, and the distinction between apparent thickness\nand true axial resolution.\n• Introducing Supplementary Table 3, which consolidates all theoretical and\nexperimental depth-resolution values for MEP, tf-ADF, and tf-iDPC in one\nsearchable location.\n• Expanding our explanation of probe–sample interactions (multiple scattering,\nchanneling) and how these degrade conventional through-focal methods but are\nexplicitly modeled and corrected in MEP.\n• Summarizing acquisition and reconstruction parameters in Supplementary Table\n1 (noting that our Zenodo repository had the datasets with the full metadata).\n• Improving the flow in the Results sections so that definitions, numerical values,\nand experimental conditions appear in a logical order without requiring the reader\nto jump between sections.\n• Including new figure annotations, improved captions, and a rewritten\nbenchmarking section for clearer comparison of MEP with tf-ADF and tf-iDPC.\nWe are grateful for the reviewer’s careful reading and constructive feedback, which\nsubstantially improved the manuscript. Below we respond point-by-point to all comments.\nComments\n1. Line 42 - at this point I think it would be good to say that most metrologies provide 2D\nimages whereas 3D metrology tools are necessary for the metrology of 3D structures.\nThis way it will be clearer for the reader moving forward.\nWe thank the reviewer for the helpful suggestion. We have added a brief statement\nhighlighting that most conventional metrologies provide 2D information, whereas novel\n3D metrology is required to fully characterize modern 3D atomic-scale semiconductor\nstructures (lines 43-46):\n“Various imaging techniques have been employed to characterize semiconductor devices\nover the decades, most of which provide only two-dimensional information, whereas\n3D metrology tools are necessary to fully characterize the atomic-scale structure\nand complexity of modern three-dimensional device architectures.”\n2. Line 71 - \"address the metrology gap highlighted above\". Could the authors please\nexplicitly state the gap i.e., the need for both angstrom resolution and 3D imaging\ncapabilities. Otherwise it can be not entirely clear for the reader.\nResponse:\nWe have revised the text to explicitly state the metrology gap identified by the roadmaps\nas the need for Ångström-scale resolution, three-dimensional imaging capabilities, and\nthe ability to simultaneously resolve both light and heavy atoms, enabling direct\nmeasurement of atomic structure, strain, and interface roughness in 3D (lines 74-78):\n“Here, we show how multislice electron ptychography (MEP) addresses the critical\nmetrology gap in device characterization highlighted above, namely, the need for\nÅngström-scale resolution, three-dimensional imaging capabilities, and\nsimultaneous sensitivity to both light and heavy atoms, thereby enabling direct\nmeasurement of atomic structure, strain, and interface roughness in 3D.”\n3. Line 72 - \"By retrieving the object’s phase from overlapping diffraction patterns, MEP\nprovides depth information through post-processing of scanning diffraction data\". This\ndescription is misleading, since the diffraction patterns themselves don't overlap.\nInstead the overlap is of the neighboring sample illumination patches, which produce\nthese diffraction patterns. Please rephrase by e.g., the sample is scanned across an\nelectron beam and diffraction patterns are collection from overlapping illumination\npatches.\nResponse:\nWe have revised the sentence to clarify that the overlap occurs in the illuminated regions\nof the sample rather than in the diffraction patterns themselves (lines 78-81):\n“By scanning a convergent electron probe across the sample and collecting\ndiffraction patterns from overlapping illuminated regions, MEP retrieves the phase\nshift introduced by the object’s atomically resolved electrostatic potential, providing\ndepth information through post-processing of scanning diffraction data26.”\n5\n4. Line 87 - \"Formation of the critical GAA structure is one of the earliest steps in a CMOS\nprocess that can take 3-4 months\". Are the authors claiming that the lithography of\nGAA transistors takes 3 months during a single wafer production? Please clarify.\nResponse:\nThe 3-4 months refers to the total duration of the complete CMOS fabrication process,\nwhich involves about a thousand steps. We have revised the text to clarify that GAA\nstructure formation occurs early in this overall process (lines 95-97):\n“Formation of the critical GAA structure occurs early in the overall CMOS fabrication\nprocess, which comprises roughly a thousand steps and typically takes 3-4 months\nto complete from start to finish.”\n5. Line 88 - \"Early structural feedback from MEP on witness wafers\". What is a witness\nwafer? The MEP imaging process requires extraction of a thin lamella, which would\nalso be highly disruptive and destructive to the manufacturing process. Could the\nauthors here explain in more detail how MEP could be integrated into the\nmanufacturing process given the constraints of the technique.\nResponse:\nA witness wafer refers to a dedicated test wafer fabricated in parallel with production\nwafers for process monitoring and metrology for exactly these concerns. We have\nclarified this in the text (lines 97-100):\n“Early structural feedback from MEP performed on dedicated witness wafers (test\nwafers fabricated alongside production wafers for process monitoring, including\nby destructive analysis techniques) can provide rapid insight into structural defects\nand interface quality early in fabrication flow and reduce costly iterations.”\nWhile MEP requires lamella extraction, such destructive analysis is already standard\npractice for structural characterization in semiconductor process development and is\nroutinely performed on witness wafers without affecting production yield.\n6. Line 99 - \"tf-ADF signal is ... proportional to the square of the atomic number\" and \"tf-\niDPC ... yields linear or sublinear atomic number contrast\". Comparing figure 4b and\n4c, the contrast seems to be reversed. If the main differences are linear and squared\ncontrast relationships, then why one is inversely proportional while the other is\ndirectly? I would expect both images to have the same direct contrast relationship.\nPlease elaborate why.\nResponse:\nThis reflects a breakdown of the standard imaging approximations used to describe these\nimaging modes. The statements about tf-ADF and tf-iDPC contrast (~Z2 vs\n~linear/sublinear) refer to their ideal behavior for very thin (a few nm) samples, where the\nelectron probe shape remains essentially unchanged as it propagates through the\n6\nspecimen, so the resulting image can be interpreted directly in terms of the projected\natomic potential of the sample.\nIn our experimental data (revised Figure 2e-g), however, the GAA lamella is tens of\nnanometers thick (~38nm), well beyond the weak-interaction regime, which holds only for\nsamples a few nanometers thick, since electrons interact strongly with matter. At this\nthickness, the electron probe undergoes multiple scattering and electron channeling\n(described in a newly added paragraph, also as per comment 10), which modify the\nwavefront as it traverses the sample. These effects lead to deviations from the ideal\ncontrast relations and can even invert contrast in iDPC. Such behavior is well documented\n[1-3] and is one of our primary motivations for employing MEP, which explicitly models\nthese interactions, avoiding such artifacts.\nTo clarify this, we have added two new paragraphs that explicitly note that the linear or\nquadratic Z-dependence applies only in the thin-sample limit (for iDPC and ADF,\nrespectively) and that deviations arise in realistic, thicker device specimens due to\nmultiple scattering and channeling, which are explained in the following paragraph and\ncause the artifacts in iDPC contrast that the reviewer is pointing out (starting on line 116):\n“…. tf-ADF uses high-angle scattering (Figure 2a, blue), which, in very thin samples,\nexhibits approximately Z2-dependent contrast (Figure 2f). tf-iDPC uses low-angle\nscattering (Figure 2a, purple) to approximate the projected potential, yielding, in very thin\nsamples, linear or sublinear atomic number contrast and improved sensitivity to light\nelements (Figure 2g)36.\nBoth methods rely on a linear, weak-phase imaging approximation—i.e., that the probe\nshape remains unchanged from its free-space solution as it propagates through the\nspecimen—an assumption that fails for device-relevant thicknesses (tens of nanometers)\ndue to multiple scattering. At these thicknesses, the probe wavefront is repeatedly altered\nas it interacts with successive atomic planes (multiple scattering) and can be guided along\nthe positively-charged columns of atomic nuclei (channeling)37, breaking the direct\nrelationship between image intensity and the underlying atomic potential. These nonlinear\nprobe-sample interactions produce well-documented artifacts10,11,30,36, including axial\nelongation, contrast reversals, and positional inaccuracies, which lead to mis-\nlocalized interfaces and spurious depth-dependent features (Figure 2e–g).”\n1. Bosch, E. G. & Lazić, I. Analysis of depth-sectioning STEM for thick samples and 3D\nimaging. Ultramicroscopy 207, 112831 (2019).\n2. Lazić, I., Bosch, E. G. T. & Lazar, S. Phase contrast STEM for thin samples: Integrated\ndifferential phase contrast. Ultramicroscopy 160, 265–280 (2016).\n7\n3. Bürger, J., Riedl, T. & Lindner, J. K. N. Influence of lens aberrations, specimen\nthickness and tilt on differential phase contrast STEM images. Ultramicroscopy 219,\n113118 (2020).\n7. Line 107 - \"correspond to a lateral resolution of 0.40 Å and a depth resolution of 44\nÅ\". Could the authors please explain why the depth resolution should be 44 Å, but in\nFigure 3, the resulting resolutions are significantly worse? Moreover, MEP achieves\n64 Å depth resolution based on Figure 3, which is worse than the quoted imaging\nresolution of 44 Å. Could the authors please dedicate a paragraph to discuss these\ntheoretical and achieved imaging performance? The whole article states claims such\nas \"better depth resolution\" etc., yet there are no quantitative comparison within the\ntext which would state how much better the imaging performance of MEP is compared\nto other techniques. At the moment the claims are qualitative and scattered throughout\nthe text. The depth resolutions are only shown in Figure 3e and do not appear in the\ntext, which is strange given the emphasis on how much better the depth sectioning is\nwith MEP.\nResponse:\nAs discussed below we have now added measurements and discussion of depth\nresolution. What we had in Figure 3e was not meant to represent the depth resolutions\nthemselves. Instead, they show where a known 66 Å-thick Si column appears when\nimaged using each method. The question being evaluated is: given a true axial thickness\nof 66 Å, what thickness and apparent depth location of the column does each method\nrecover? MEP correctly reconstructs 66 Å, whereas tf-ADF and tf-iDPC return ~93–94 Å,\nreflecting axial elongation and surface-shift artifacts as a consequence of the probe–\nsample coupling and non-linear propagation discussed above. MEP has properly\naccounted for these effects. We have changed the text to make this clearer (lines 183-\n186).\nTo estimate the depth resolution (or in this case the axial point-spread width), we fitted\nerror functions to the leading and trailing edges of each axial profile in Figure 3e,\nessentially a measure of the edge-spread function. Below is the updated text (lines 187-\n194):\n“To estimate depth blurring, we fitted error functions to the entry and exit edges of\nthe axial line profiles (see Figure 2e and Supplementary Figure 2). MEP shows an\nedge blur of only 22 Å, while infinite dose tf-ADF shows 40 Å. The finite-dose tf-\nADF profile is too noisy to fit, mirroring the experimental situation. tf-iDPC shows\na 57 Å edge blur, comparable to the full sample thickness, indicating that the\ncolumn boundaries cannot be meaningfully localized. Supplementary Figure 2\nshows that MEP’s depth-sectioning performance remains stable across depth-slice\nspacings and detector collection angles, provided Nyquist sampling is met, and\nsufficient intensity is captured at the highest collected scattering angles.”\nnew gray dashed lines mark the true sample thickness, and the red-shaded regions\nindicate the edge blurring of each method, so that the distinction between ground truth,\napparent thickness, and depth blur is more apparent. These quantitative values are now\nexplicitly included in the main text (lines 188-191) and the revised Figure 3e and its\ncaption (lines 422-431):\n“Figure 3. … e Depth-intensity profiles of the ×-\na Atomic Potential b MEP c tf-ADF dtf-iDPC e\nxy zy zy zy zy 66 Åthick c-Si marked c-Si column compared with the true axial\ne-beam extent of the 66 Å-thick Si segment (top). The grey\nExit\na-HfO\n2 surface dashed lines show the true positions of the entrance\nand exit surfaces, while the solid grey lines show the\nMEP apparent location of these surfaces for the different\nmethods and the corresponding column widths.\n22 Å\na-SiO\n66 Å\nblur (depth broadening) from error-function fits to each\ntf-ADF\nmethod. Because the noisy tf-ADF signal cannot be\n40 Å\nreliably fitted (blue crosses), the tf-ADF blur values are\n94 Å extracted from the noise-free (infinite-dose) tf-ADF\nc-Si\ntf-iDPC signal (blue line). MEP yields the smallest edge blur (22\nÅ), while tf-ADF (40 Å) and tf-iDPC (57 Å) exhibit\n57Å\nsubstantially larger axial broadening, consistent with\n5 Å 93 Å\ntheir reduced ability to localize features in depth.\nSupplementary Figure 1 provides selected depth slices\ncomparing the three imaging techniques.”\nIn revising Figure 3 and expanding the discussion of depth resolution, we identified an\nindexing error in the plotting script that made the true 66 Å Si segment appear as ~58 Å\nin the earlier figure. The underlying multislice simulations and all reconstructions were\nunaffected; only the displayed reference profile was mis-indexed. After correcting the\nvisualization and re-running the automated fits, the apparent thickness values shift\nslightly: MEP changes from 64 Å to 66 Å, tf-iDPC from 95 Å to 93 Å, and tf-ADF from 93\nÅ to 94 Å. These 1-2 Å differences simply reflect which pixel of the column is selected for\nthe width measurement; we have now standardized this by consistently identifying the\nbrightest pixel in the column. The corrections do not alter any conclusions but now allow\nus to present a clearer quantitative comparison of depth performance across the three\nmethods.\nWe have also performed a direct experimental estimation of the MEP depth blur. Unlike\nthe simulated test case, we cannot assume the experimental dataset contains truly\n“sharp” interfaces: the lamella surfaces are affected by FIB damage, amorphization, and\nredeposition, making column-edge-based measurements unreliable. Instead, we\n9\nmeasured isolated single atoms (a consequence of redeposition during the FIB milling)\npresent at the entrance and exit surfaces of the MEP reconstructions and extracted their\naxial intensity profiles. Gaussian + linear-background fits to these peaks, presented in\nFigure 4 and a new Supplementary Figure 5, yield a consistent depth blur of Δz = 40 ± 7\nÅ (s. d.) across six measurements. We have added text to the main part of the paper that\ndiscusses this (lines 207-208).\nEquivalent experimental depth-blur measurements could not be performed for tf-ADF or\ntf-iDPC. Under the experimental dose conditions, the atomic columns in the through-focal\nseries are too noisy to localize reliably, and surface atoms are not stably positioned\nbetween defocus frames due to beam-induced motion (lines 208-212). We were not able\nto increase the dose without damaging the sample, again illustrating the limitations of\nthese methods.\nFor ease of reference, we have now added Supplementary Table 3, which consolidates\nall theoretical and experimental depth-resolution values in one place:\nSupplementary Table 3. Depth-resolution comparison for MEP and through-focal\nimaging. The theoretical values correspond to multislice simulation-based estimates\nshown in Figure 3. Experimental depth resolutions were measured from the\nreconstruction shown in Figure 5 and Supplementary Figure 5. Through-focal ADF and\niDPC did not yield experimental depth-resolution estimates because the through-focal\nseries were too noisy to localize individual atomic columns reliably.\nTheory Experiment\nMEP 22 40 ± 7 Å\ntf-ADF 40 -\ntf-iDPC 57 -\nAlthough broader than the simulated MEP blur (22 Å for the idealized simulated c-Si test\ncolumn), this is expected from the stronger regularization setting (0.5) that was used in\nthe experimental reconstructions compared to 0.1 used in the simulation ones, to ensure\nconvergence of the reconstructions, which inherently degrades the depth resolution (lines\n222-224).\n(See also Responses 12–14 for acquisition and reconstruction parameters and depth\nresolution constraints imposed by β, Nyquist sampling, dose, and regularization.)\n8. Line 111 and 114 - \"these issues\", \"these limitations\". Could the authors phrase the\nissues/limitations more explicitly. e.g., \"the primary issue is multiple scattering... and\n10\nthe with MEP we overcoming multiple scattering through multi-slice ptychography,\nwhich models the multiple scattering process.\"\nResponse:\nWe have removed the sentence on line 111 that started with “these issues”, since now\nthere is discussion of multiple scattering and channeling explicitly (see response to\ncomment 10). We have rephrased the sentence to now read (lines 131-135):\n“MEP uses the same optical configuration as tf-ADF and tf-iDPC (Figure 2a) but\novercomes their limitations—probe–sample coupling and the lack of any correction\nfor multiple scattering in thicker specimens—by recording a momentum-resolved\nscattering distribution and reconstructing both the probe and the sample potential38,39.\nThe multiple scattering is accounted for using a multislice forward model28,40,41 (Figure\n2b). ….”\n9. Line 115 - first mention of 4D-STEM dataset, please explain here or earlier. Potentially\nduring the description of ptychography and that it acquires 4D-STEM data.\nResponse:\nWe have now clarified what is meant by a 4D-STEM dataset at its first mention and have\nbriefly described how it is acquired (lines 135-138):\n“…. In practice, this is achieved by acquiring a four-dimensional scanning\ntransmission electron microscopy (4D-STEM) dataset: a defocused probe is raster-\nscanned over the sample, ensuring overlap between adjacent illuminated regions,\nand recording a two-dimensional diffraction pattern at each probe position in real\nspace. ….”\n10. Line 117 - \"multiple scattering and channeling effects\". I am not familiar what are the\nchanneling effects and how multi-slice ptychography can overcome them. Could the\nauthors explicitly mention these artefacts in the earlier section of the text and explain\nwhy multiple-scattering happens and what are channeling effects. Potentially around\nLine 111. Please be clear on the challenges and how they are solved.\nResponse:\nWe have added a new sentence that explicitly defines multiple scattering and electron\nchanneling and describes the imaging artifacts they produce (see also response to\ncomment 6; text lines 121-130):\n“Both methods rely on a linear, weak-phase imaging approximation—i.e., that the probe\nshape remains unchanged from its free-space solution as it propagates through the\nspecimen—an assumption that fails for device-relevant thicknesses (tens of nanometers)\ndue to multiple scattering. At these thicknesses, the probe wavefront is repeatedly\naltered as it interacts with successive atomic planes (multiple scattering) and can\nbe guided along the positively-charged columns of atomic nuclei (channeling)37,\nbreaking the direct relationship between image intensity and the underlying atomic\n11\npotential. These nonlinear probe-sample interactions produce well-documented\nartifacts10,11,30,36, including axial elongation, contrast reversals, and positional\ninaccuracies, which lead to mis-localized interfaces and spurious depth-dependent\nfeatures (Figure 2e–g).”\nWe have also expanded our explanation on how MEP inherently accounts for these\neffects (lines 138-142):\n“The MEP reconstruction from these datasets explicitly accounts for the probe shape and\nits evolution—including multiple scattering, channeling, partial coherence, and sample\ntilts—by iteratively propagating the recovered probe though successive slices of the\nspecimen (Figure 2c), enabling quantitatively accurate 3D reconstructions even for ~40\nnm-thick specimens.”\n11. Line 128 - \"Unlike tf-ADF and tf-iDPC (or tomography approaches), which require\nmultiple scans, only a single 4D-STEM dataset is needed for MEP, which can reduce\nthe total electron dose (Figure 2b-d, Extended Data Figure 3).\" Please clarify what the\nauthors mean by \"scan\", since 4D-STEM is also produced by raster-scanning the\nsample. The authors should clarify that 3D information in MEP is obtained from the\nsame dataset that would be acquired for a 2D sample structure recovery i.e., depth\ninformation basically comes \"for free\". With other approaches the 3D information is\ndoes not come \"for free\" and instead required acquisition of many 2D images e.g.,\nmultiple through-focus slices or rotation angles. Without this information some readers\nmight be confused.\nResponse:\nWe thank the reviewer for this helpful suggestion, which we have incorporated it in the\nnew text (lines 147-152):\n“Unlike through-focal methods (or tomography approaches), which require multiple\nseparate x-y raster scans, each representing a separate exposure of the sample to\nthe electron beam, only a single x-y scan and its resulting 4D-STEM dataset is\nneeded for MEP, from which 3D structural information is recovered (Figure 2b-d).\nIn this sense, the depth information in MEP comes essentially “for free” with the x-\ny raster (Figure 2e), while the use of a single dataset also reduces the total electron\ndose (Figure 2b-d).”\n12. Line 141-148 - I am missing details on the imaging methods. How many slices are the\nauthors reconstructing with MEP? How many through focus images are the authors\nacquiring? This would give directly the expected axial resolution for a given sample\nthickness. What are the performance differences between MEP compared to the\nthrough-focus methods. Could the authors expand?\nResponse:\nnew line 202. We have also provided a new summary table that details all the\nexperimental conditions for ease of comparison (Supplementary Table 1). All the\nexperimental datasets used in this study are also provided in our Zenodo repository\ntogether with detailed experimental conditions.\nAs described in the MEP Reconstruction subsection of the Methods, in this work, the MEP\nreconstructions were performed using a 1 nm slice thickness. Given the experimentally\nrealized depth blur of approximately 4 nm (see Response 7), this slice spacing satisfies\nthe Nyquist criterion and ensures accurate sampling of the 3D electrostatic potential. For\nthe ~38 nm-thick GAA lamella, this corresponds to 42 reconstructed slices (we include\nadditional slices to ensure the full structure is encompassed). Importantly, the slice\nthickness in MEP does not define the achievable axial resolution; as long as the sampling\nsatisfies Nyquist, the axial resolution is instead determined by the probe convergence\nangle, detector collection angle, and signal-to-noise ratio. We also included a brief\nsupplementary comparison showing that using finer vs. coarser MEP slice spacings\nproduces negligible differences for reconstructing the same dataset (Supplementary\nFigure 2). This simply illustrates that, once Nyquist-sampled, changing the z-slice\nthickness (or defocus step size) does not improve the achievable axial resolution.\nWe have also created Supplementary Table 2 that summarizes all the simulation\nconditions used, again for ease of comparison.\nFor the through-focal (tf) methods, we have also added to the Methods section the\nnumber of defocus planes used in experiments (quoted there for clarity). As with MEP,\nthe defocus step size is chosen to at least Nyquist-sample the expected depth resolution,\nrather than to set it.\nThe relative performance of MEP and the through-focal techniques are now discussed in\ndetail in the main text as described in Response 7.\n13. Line 166 - \"this enhanced resolution is limited by the small diffraction collection angle\nrequired for the ~38 nm thick section, whereas the poorer resolution of ADF and iDPC\nis primarily dose-limited, due to radiation damage concerns\". This explanation is not\nclear. The collection angle for MEP is higher, which allows acquisition of a wider\nscattering leading to higher resolution, as one would expect. Plus the authors have\ndose limitations for ADF and DPC leading to even lesser resolution. My questions\nwould be: is the resolution difference explained by the detection acceptance angles\nbeta and alpha alone? Or it also requires the additional discussion on radiation dose?\nCould the authors please discuss the resolution from both aspects.\nResponse:\ndiscussion paragraph to more clearly indicate the factors limiting resolution. The\nachievable resolution in all three imaging modes is set by two main factors: (i) the\nexperimental geometry, which defines the maximum spatial frequencies that can be\nrecorded (set by the probe convergence semi-angle α for tf-ADF/tf-iDPC and by the\ndetector collection angle β for MEP), and (ii) the signal-to-noise ratio, which is limited by\nthe maximum electron dose the sample can tolerate without damage. In practice, the\nregularization used for the MEP reconstruction (for better convergence) can degrade the\nachievable depth resolution.\nIn our experiments, the achievable β for MEP was limited by the ~38–40 nm specimen\nthickness and the finite detector pixel count; increasing β further would require\nproportionally higher dose to maintain SNR at the largest scattering angles, exceeding\nthe damage threshold, and would undersample the diffraction patterns. These combined\ngeometric and SNR considerations explain the relative resolution performance observed\nin the experiments.\nTo further clarify the role of β and sampling, we have added a new supplementary figure\ncomparing simulated MEP reconstructions for β = 3α versus 2α and for different slice\nspacings (5 Å and 10 Å). As shown in Supplementary Figure 2, once Nyquist conditions\nare satisfied and the available dose is fixed, expanding β beyond 2α yields only marginal\nimprovement under our experimental dose constraints.\nRegarding depth resolution, we experimentally estimated the depth blur in MEP by fitting\nisolated single-atom features, obtaining a consistent FWHM of 40 ± 7 Å. Performing an\nequivalent analysis for tf-ADF and tf-iDPC was not feasible because individual atoms are\ntoo noisy and often mobile between defocus frames, consistent with the behavior seen in\nthe simulated through-focal datasets. Finally, the experimental MEP blur is broader than\nthe simulated value (22 Å) because a stronger regularization (0.5 vs 0.1 in simulation)\nwas used for stable convergence, which necessarily suppresses high-frequency detail.\nThe revised text (lines 207–224) now explicitly discusses geometry (α vs β),\ndose/SNR, and regularization as the three key contributors to the observed resolution:\n“From isolated single-atom features in the experimental MEP reconstructions, we\nmeasured an axial blur of 40 ± 7 Å (Figure 4 and Supplementary Figure 5).\nEquivalent estimates were not possible for tf-ADF or tf-iDPC because finite-dose\nimages are too noisy (consistent with limitations seen in simulations) and\nindividual atoms likely move between defocus frames. Thus, MEP provides\nsufficient detail for quantitative 3D analysis, while tf-iDPC and tf-ADF could not\nreliably recover depth information under comparable conditions (Supplementary\nTable 3).\nfor tf-ADF and tf-iDPC, and by the detector’s outer collection angle β for MEP.\nIn our experiments, our choice of β was constrained by the ~40 nm specimen\nthickness and the detector’s finite pixel count, which limited the attainable\ncollection angle without undersampling. Second, dose can also limit the achievable\nresolution: because each method accesses a different fraction of the scattered\nelectrons (Figure 2a), not all have enough signal to reach their geometric limits.\nMEP collects nearly all scattered electrons (~99%), whereas tf-ADF uses <1%, and\ntf-iDPC—despite the high SNR in the bright-field disk—cannot convert this advantage\ninto reliable depth localization due to uncorrected multiple scattering. As shown in\nFigure 2g, tf-iDPC’s axial profile is broadened to nearly the full sample thickness,\nreflecting its practical limitation. Additionally, the experimental MEP depth\nresolution is broader because a stronger regularization (0.5 vs. 0.1 in simulation)\nwas used for stable convergence, which inherently suppresses high-frequency\ndetail.”\n14. Moreover, is the dose for a 2D image acquisition in both MEP and DPC/ADF is the\nsame? ADF/DPC requires many through-focus slices which multiplies the dose\nrequirements. Did the authors reduce the dose for each of the through-focus slices in\nDPC/ADF or kept it the same? It would be good to clearly establish at the beginning\nwhat are the fundamental data acquisition differences between DPC/ADF vs MEP,\nin terms of dose, data acquisition etc.\nResponse:\nIn the text, we report the total dose used for each imaging modality to obtain the 3D\nstacks, this means that for MEP this value is for the single 4D-STEM dataset, while for\nthe tf-ADF/tf-iDPC it is the total dose for the whole series of the images, since here we\nare answering the question of, if we have X total electrons we can use to obtain a 3D\nimage of the structure, how does each method do? The total doses used are stated in the\ncaption of each comparison figure (Figure 2-3, Supplementary Figure 1-4). This was also\nstated in the Methods section, under Data Acquisition (lines 511-525):\n“Data Acquisition\n…. A total dose of 0.5 - 2.5 × 105 e-/Å2 was used for the 4D-STEM datasets.\nThe DPC images were acquired on the Thermo Fisher Scientific Panther detector (8\nsections) and the ADF images were acquired using the Thermo Fisher Scientific ADF\ndetector, with total through focal series dose of 1 × 105 e-/Å2 using an automated\nacquisition method outlined here72. ….”\nIn the experiments, the tf-ADF/tf-iDPC images were allocated a higher dose since at the\nsame dose as MEP the images would simply be too noisy to interpret in any meaningful\n15\nway, and, in our experience, this was the maximum dose before substantial sample\ndamage. We have made the wording more explicit in the paragraph in question to specify\nthat the doses refer to a single MEP dataset and a full tf-ADF/tf-iDPC series (lines 204-\n206):\n“MEP achieves a lateral information limit of 0.49 Å, compared to 0.66 Å for tf-iDPC and\n0.83 Å for tf-ADF (Supplementary Figure 4), using only half the electron dose (0.5 × 105\ne-/Å2 for the single MEP dataset versus 1 × 105 e-/Å2 for the full tf-iDPC and tf-ADF\nimage series).”\nFor the full experimental parameters, also see response to comment 12.\n15. Line 196 - The following paragraph discusses strain, but it is not explained how they\nauthors obtain the strain information. The resulting MEP reconstruction is simply the\n3D structure of the sample. How is the strain recovered from the acquired data?\nResponse:\nWe thank the reviewer for the comment and clarify that the procedure for obtaining strain\nfrom the MEP reconstruction is described directly in the paragraph beginning at Line 210\n(of the originally submitted manuscript, line 255 in revised manuscript) and detailed in\nthe Methods section. Briefly, strain is quantified by tracking the position of silicon atomic\ncolumns in each depth slice using Atomap (see Methods, Supplementary Figure 7, and\nSupplementary Movies 1–3). The atomic positions are grouped into bilayers from the c-\nSi/a-SiO interface inward, and Si–Si spacings are measured per bilayer to extract the\n2\nstrain profile, as shown in Figure 6.\n16. Line 226 - \"In both cases two fitting parameters are used...\". Could the authors please\nelaborate what is actually being fitted? I would assume that the authors are fitting the\nextracted atoms and their positions using Atomap, but it is just a guess on my part.\nResponse:\nWe appreciate the reviewer’s attention to this point. As described in the Methods section\n(“Interface Roughness Quantification and Mobility Estimation”, line 618 onwards) and\nshown in Supplementary Figure 8, the fitting refers to the autocorrelation function of the\nmeasured interfacial roughness, not to the atomic positions themselves. The RMS\namplitude (Δ) and correlation length (Λ) are obtained by fitting the autocorrelation\nenvelope with exponential and Gaussian models. We have added a brief description to\nthe main text for clarity (lines 272-274):\n“…. In both roughness models two fitting parameters are used: RMS amplitude Δ and\ncorrelation length Λ, extracted by fitting the autocorrelation function of the measured\nroughness with exponential or Gaussian models (see Methods and Supplementary\nFigure 8). …..”\n16\n17. Line 249 - \"In the long-wavelength limit...\". Could the authors briefly explain what is\nthe wavelength referring to here and how it relates to the carrier mobility mentioned\npreviously.\nResponse:\nThe “wavelength” here refers to the spatial wavelength of the charge carriers (or\nequivalently, the inverse of their momentum k) relative to the characteristic length scale\nof interface roughness features. In the long-wavelength limit, the carrier wavelength is\nmuch larger than the roughness correlation length Λ, such that carriers experience the\ninterface as an averaged potential variation. Under this approximation, standard interface-\nroughness scattering formalisms simplify to the mobility scaling of μ ∝ 1/(Δ2Λ2). We have\nclarified this point in the revised text by explicitly stating what the wavelength refers to\nand how this limit connects to carrier mobility (lines 296-301):\n“These quantitative values of Δ and Λ can be linked to carrier mobility using established\nscattering formalisms. In the long-wavelength limit, where the carrier wavelength is\nmuch larger than the characteristic interface roughness scale Λ, the carriers\nexperience the interface as a smooth averaged potential, and interface roughness\nscattering rates scale approximately as 1/(Δ2Λ2) and reduce mobility as μ ∝ 1/(Δ2Λ2)68. At\nshorter wavelengths, the dependence is more complex and requires full evaluation of the\nroughness power spectrum.”\n18. Line 271 - \"Importantly, for the information gained, MEP is relatively time- and\nresource-efficient: from sample preparation to a fully reconstructed 3D image takes\nonly about two days, providing rapid atomic-scale structural feed\". Could the authors\nexpand a bit more on the time distribution for each step? E.g., how much slower is the\nsample preparation compared to imaging itself and the data reconstruction.\nResponse:\nWe appreciate the reviewer’s interest in the practical timing of the workflow. These details\nhave been added to the Methods section to clarify the typical time distribution (lines 553-\n561).\n“Typical MEP workflow duration\nThe total measurement time for MEP experiments is comparable to standard high-\nquality TEM workflows. Focused ion beam (FIB) sample preparation for a <50 nm\nlamella typically requires half a day for an experienced user, followed by\napproximately half a day of microscope time (including alignment and tuning).\nEach 4D-STEM dataset (256 x 256 scan positions) is acquired in about 7 seconds\non the EMPAD-G2 detector. Reconstruction typically takes from an hour to a day,\ndepending on user familiarity, computational resources, degree of convergence\nneeded for imaging vs quantification, and the algorithm used. These times are\nrepresentative of our setup and will vary between laboratories.”\n17\n19. Line 271 - The paragraph starting here seems to be related more to the \"general\ndiscussion\" and seems unrelated to the findings within the current section. Consider\nintegrating into the discussion.\nResponse:\nWe agree with the reviewer that the final paragraph of this section functioned more as\ngeneral discussion. This paragraph has been moved to the Discussion section (starting\non line 332).\nMinor comments\n1. Line 37 - explain FET abbreviation (Field effect transistor).\nResponse:\nWe have added the full form field-effect transistor (FET) upon first mention in line 38:\n“GAA transistors, with their nanosheet-based design, provide superior electrostatic\ncontrol by completely surrounding the channel with the gate electrode, unlike planar field-\neffect transistors (FETs) or FinFETs, which lack full gate-channel coupling1-5.”\n2. Line 45 - \"X-ray ptychographic tomography\" rather than just \"X-ray ptychography\".\nResponse:\nRevised to “X-ray ptychographic tomography” (line 48):\n“X-ray ptychographic tomography provides 3D imaging at deep sub-micron scales, …”\n3. Line 54 - \"projection\" to \"violation of the projection approximation\".\nResponse:\nWe have made this change (line 58).\n4. Line 79 - if crystalline-Si is c-Si, please add the abbreviation in brackets, which\nappears later in the text.\nResponse:\nAdded the abbreviation “c-Si” after “crystalline Si” and “a-SiO ” after amorphous-SiO for\n2 2\nconsistency:\nLine 85: “…. Using MEP on early-stage GAA test structures from Interuniversity\nMicroelectronics Centre (imec), we directly image stacking faults, pinholes, interface\nroughness, and strain relaxation in the crystalline-Si (c-Si) channel, capturing buried\ndetrimental structural features well before electrical testing is possible.”\nLine 91: “…. Comparing these findings to a conformal amorphous-SiO (a-SiO ) layer\n2 2\non epitaxial c-Si, we find that the strain relaxation is process-dependent and can serve as\na metric for interface quality.”\n5. Line 81 - could the authors please refer here (and int his paragraph in general) to\nFigure 1 b such that it is clear what the text is referring to. Otherwise for people not\nfamiliar with the GAA structure it is unclear.\nWe have added a reference to Figure 1b in this paragraph to help orient readers with the\nGAA device structure (lines 89-91):\n“In these GAA devices, in the c-Si channel (labelled c-Si in Figure 1b), we measure\nthe structural transition from strained interfacial silicon to bulk-like silicon to span over 2\nnanometers of the 5-nanometer-wide channel (over 40%).”\n6. Line 111 - replace \"distortions in contrast\" to \"contrast distortions\".\nResponse:\nThis entire paragraph has been revised to better explain the effects of non-linear probe-\nsample interactions and the original phrasing of \"distortions in contrast\" has been\nreplaced by a more comprehensive description of the resulting artifacts, including axial\nelongation, contrast reversals, and positional inaccuracies (lines 121-130).\n7. Line 132 - \"This enables resolution beyond the diffraction limit imposed by the probe\nconvergence semi-angle α\". The probe convergence does not impose a \"diffraction\nlimit\", only the wave propagation physics imposes a \"diffraction limit\".\nResponse:\nWe thank the reviewer for this opportunity to clarify. The sentence has been revised (lines\n161-165):\n“ …. By capturing scattering to high angles (β, Figure 2a), these detectors extend the\naccessible spatial frequencies, enabling lateral and depth resolution beyond the\ndiffraction limit arising from the limited probe-forming-aperture semi-angle, α.\nUnder high-dose conditions, the ultimate resolution in MEP is therefore bounded by the\ndetector’s maximum collection angle β (> α) rather than α alone (Figure 2a) 28,47-49.”\n8. Line 144, 223 - \"S/TEM\" and \"(S)TEM\". Please use a consistent naming convention.\nResponse:\nWe have standardized the terminology to use “STEM” when referring to specific\nexperiments performed in scanning TEM mode, and “(S)TEM” only in general statements\napplicable to both TEM and STEM (e.g. when discussing sample preparation or general\nelectron microscopy principles).\n9. Line 149 - would be good to mention that \"c\" stands for crystalline and \"a\" for\namorphous.\nResponse:\nWe have clarified in the text that “c” and “a” denote crystalline and amorphous,\nrespectively (lines 175-176).\n“Figure 3 summarizes the simulated 3D benchmark of a rough crystalline-silicon (c-\nSi)/amorphous-hafnium oxide (a-SiO2)/amorphous-hafnium oxide (a-HfO2) interface,\n….”\n19\n10. Line 197 - \"depth-resolving power\" to \"depth-resolving capabilities\".\nResponse:\nThe sentence has been revised (line 242):\n“Building on MEP’s demonstrated depth-resolving capabilities, we now apply it …. ”\n11. Line 223 - \"for the first time determining\" to \"determine\"\nResponse:\nChanged to “determine” (line 281).\n12. Line 533 - \"due to linear atomic number contrast, all sections of the structure are\nvisible.\" what does this mean exactly?\nResponse:\nWe have clarified the caption to indicate that “linear atomic number contrast” means MEP\nallows both light (Si, O) and heavy (Hf) atoms to be simultaneously visible, making all\nlayers of the structure apparent (lines 379-381):\n“…. d MEP depth slices from the region in b-c (light and dark orange planes)\nshowing both light (Si, O) and heavy (Hf) atoms, visible due to the linear atomic-\nnumber contrast. ….”\n13. Line 623 - \"3D rendering of GAA 2’s surface\", what is the \"2's surface\"?\nResponse:\nThe text has been clarified to indicate that “GAA-2” refers to the second of the two GAA\nregions analyzed in this study: GAA-1 and GAA-2 (lines 479-480):\n“…. d 3D rendering of the second GAA (GAA-2), showing the full channel morphology.\n….”\n14. Figure 1 b - it is unclear what the text is pointing to exactly. Could the authors highlight\nand assign the layers better by e.g., a contour for each layer.Otherwise it is unclear.\nResponse:\nWe have revised Figure 1b to make the layer assignments clearer. Specifically, we added\nthree arrows indicating the extents of the c-Si, a-SiO , and a-HfO layers, and updated\n2 2\nthe labels to match these arrows. We also simplified the on-image text to avoid\novercrowding. The caption has been revised to explicitly list each layer and describe the\narrows (lines 371-375):\n20\na\nSEM b ADF\ntop\nview\nc-Si\na-SiO\n2\na-HfO\n2\n3 µm\nMetal Gate 5 nm\n0.5 µm TEM cross section c MEP d MEP\n!=10 &' 3 nm\n50 nm !=16 &' 3 nm\n“…. b ADF image of a single gate-all-around structure with labeled components\n(crystalline-silicon (c-Si)/amorphous-hafnium oxide (a-SiO )/amorphous-hafnium\n2\noxide (a-HfO ); thin black and white arrows indicate the extents of the Si channel\n2\nand surrounding amorphous layers). Hafnium oxide intrusions into the silicon channel\nare visible, one of them highlighted with a red arrow. ….”\n15. Figure 2 c - \"Visualization of depth-resolved slices obtained by different methods\". I\nonly see three slices, which I assume are from MEP. Where are the other slices\n\"obtained by different methods\" which here should include also through focal series?\nDescription seems misleading.\nTo clarify, Figure 2c (and panels 2a–d) are schematic illustrations rather than\nexperimental data. Panel 2c is intended to conceptually show how depth-resolved slices\ncan, in principle, be obtained using either MEP or through-focal (tf-ADF/tf-iDPC)\napproaches. We have revised the caption to state this explicitly and added arrows from\npanel 2d to 2c to emphasize that these slices are illustrative and not independent\nexperimental reconstructions.\nTo improve completeness and better align with the manuscript’s revised flow, we have\nmerged the original Figure 4 into the new Figure 2, so that experimental depth-resolved\ncomparisons for all three methods (MEP, tf-ADF, tf-iDPC) now appear directly alongside\nthe schematic panels. This provides the reader with both the conceptual illustration and\nthe corresponding experimental realizations in a single figure, addressing the ambiguity\nnoted by the reviewer. The updated figure is presented below (lines 384-409):\n21\n[001]\n[\n)110]\n[110]\nc-Si\na-SiO\n2\ny a-HfO\nx 2\nz\na b c d\n$\n$\nSample Δ\"\n$ ! =$ $ e*+,\" # ! !\nΔ' #\nADF ! Δ\"\n$ =& Δ' ⨂$\nDetector %,,! !\n#\n# Δ\"\n#\n#\n#\niDPC $\ne*+,#\n=& Δ' ⨂$\n%,,\nMEP\nQuadrant\nPixelated\nDetector #\nDetector \"\n# \" Δ\" \"\nα​\nβ​ $\n%,,\n=& Δ' ⨂$\n%,(“(!)\ne*+,!\ne MEP f (tf-ADF)0.5 g tf-iDPC\na-HfO\n2\na-SiO\n2\nc-Si\n“Figure 2. Experimental configurations and depth-resolved imaging comparisons\nfor MEP, tf-iDPC, and tf-ADF. a STEM geometry illustrating the converged electron\nprobe (green, semi-angle α) raster-scanned (orange arrow) over an electron-transparent\nsample. Transmitted electrons are recorded using different detectors to generate distinct\nimaging modes: a high-angle annular dark-field (ADF) detector (blue), a quadrant detector\nfor iDPC (purple), and a pixelated detector (EMPAD, black, 128×128 pixels) that records\nmost scattered electrons up to the detector’s outer collection angle β. b Schematic of the\nmultislice electron ptychography (MEP) reconstruction, where the probe is untangled from\nthe sample potential to recover a stack of potential slices V …V . c Conceptual\n! \"\nillustration of depth-resolved slices reconstructed by different approaches, shown\nhere using an atomic model of a GAA device. Green arrows from panel b and black\narrows from panel d indicate that such depth resolved information can, in principle,\nbe obtained from either MEP (single dataset) or through-focal (n scans of tf-ADF/tf-\n22\nmn\n31=z\nmn\n22=z\nmn\n82=z\na-HfO\n2\na-SiO\n2\nc-Si\n2 nm\niDPC) data. d Through-focal imaging scheme for tf-iDPC and tf-ADF, where a series of\n2D images is recorded at different defocus values f and aligned to form a 3D\n\" representation of the sample convolved with the probe’s point-spread function. Unlike\nMEP, these methods assume a weakly interacting probe and therefore neglect probe\nevolution from multiple scattering, leading to characteristic artifacts. e–g Depth-specific\nexperimental comparison for the same GAA device region. e MEP reconstructed\nelectrostatic potentials show an intact channel at 13 nm depth, a partially missing channel\nat 22 nm, and a distorted channel shape at 28 nm (yellow dashed outlines). Total dose:\n0.5 × 105 e-/Å2. f tf-ADF images at corresponding defocus values (square-root contrast\nshown) obscure Si lattice contrast due to dose limitations and fail to recover the correct\nc-Si channel shape in depth. Total dose: 1 × 105 e-/Å2. g tf-iDPC images show the channel\nappearing intact at all depths, with poorly resolved lattice details and spurious depth\ninformation, missing the channel deformation. Total dose: 1 × 105 e-/Å2. Additional defect\ncomparisons and depth slices are provided in Supplementary Figure 4.”\nKarapetyan et al describe a carefully written study using multislice electron ptychography\n(MEP) to study the nanoscale local structure of an advanced gate-all-around (GAA)\nnanosheet transistor. GAA transistor structures are central to next generation\nmicroelectronics. Characterizing the nanoscale structure of these nanodevices is a critical\nmetrology challenge for the microelectronics industry. This report describes how MEP\ncan provide nanoscale interface roughness and directly measure the local strain in the\nsilicon channel of a single GAA transistor. They compare MEP to other electron\nmicroscopy modes and show how it improves what can be extracted from the microscope.\nBelow I detail a couple of minor improvements that can be made to the manuscript. The\nmanuscript will be of wide interest to the readership of Nature Communications.\nSeeing this level of detail on a single device is amazing and will be of great interest to the\nsemiconductor industry.\nWe thank the reviewer for their thoughtful and encouraging assessment of our work, and\nfor recognizing the significance of applying Multislice Electron Ptychography (MEP) to\nGate-All-Around (GAA) transistor metrology. We greatly appreciate the reviewer’s\npositive comments on both the technical rigor and the broader relevance of this study to\nthe semiconductor industry.\nThe authors should note that a chip has billions of transistors and that there is often\nvariation between each individual device. That variation is quite important for product\nperformance. The unaddressed point is statistical sampling. Measuring one out of a\npopulation of one billion is not statistically representative of the population. If they\nmeasured a second transistor it likely would be different from the first one. This makes\nthe discussion in the manuscript about speeding up the measurement and reconstruction\nalgorithm critical to the utility of the measurement so that it can be applied to many\ntransistors to improve the statistics. I should note that for the purposes of the manuscript\nit is acceptable that they only report on one transistor device.\nWe thank the reviewer for the insightful comment regarding statistical sampling and\ndevice-to-device variability and the recognition that for this manuscript it was not\nnecessary to measure multiple devices, although we did (Supplementary Figure 9 actually\nshows 3 device structures that have been fully analyzed – GAA-1, GAA-2 and a planar\nstructure). Nevertheless, the editor asked us to comment further on this. We agree that\nstatistical sampling across multiple devices is essential for capturing transistor-to-\ntransistor variability, although we should note that this is not the typical workflow for\nmicroscopy in metrology or failure analysis today. Usually, variability is captured by\nelectrical measurements from which bad devices are identified and individually targeted\n24\nfor cross-section TEM by FIB liftout–a process we can supplement or replace\nstraightforwardly with our approach.\nHere, we showed that 3D atomic-scale imaging of a single device is now achievable with\nMEP and that such measurements are becoming practical: what once took weeks for\nreconstruction can now be completed within an hour (our record is 40 minutes) to a day.\nMore importantly, once the microscope is aligned, a 3D data set for a single transistor\ntakes 7 seconds to record and over a hundred devices can be imaged from a single TEM\nFIB section. With a little automation, all these devices could be imaged in about 30\nminutes. The reconstructions can be processed in parallel on cluster, so the total\nreconstruction time still can be under hour. Even if we don’t improve the algorithm (and\nwe do expect to), Moore’s law alone will halve the reconstruction time every 18 months\nfrom improved processor speeds alone. This progress makes large-scale statistical\nstudies a realistic next step, and our results are a proof-of-concept that the method is\nready for that transition.\nBelow, we provide detailed, point-by-point responses addressing all reviewer comments.\nComment #1:\nIt would be helpful to non-microscopists to include some additional diagrams in the figures\nto help one understand where in the device the particular view/slice comes from. The\nslices sometimes can be non-intuitive, especially with the asymmetric resolution. For\nexample, on figure 1 it would be helpful to add marks on the 3D image in 1C highlighting\nwhere in z the slices shown in 1d come from. They should be color coded better. Making\n1D the same green as 1C is a little confusing.\nResponse:\nWe thank the reviewer for this helpful suggestion. We have updated Figure 1 to improve\nclarity for readers who may be less familiar with slice orientation in 3D reconstructions.\nSpecifically, we added light and dark orange markers in Figure 1c to indicate the z-\npositions of the slices shown in Figure 1d (also colored accordingly), and we revised the\ncaption to describe these additions (lines 363-383):\n25\na\nSEM b ADF\ntop\nview\nc-Si\na-SiO\n2\na-HfO\n2\n3 µm\nMetal Gate 5 nm\n0.5 µm TEM cross section c MEP d MEP\n!=10 &' 3 nm\n50 nm !=16 &' 3 nm\n“ …. c 3D MEP reconstruction showing interface roughness, a hafnium oxide intrusion\n(red arrow) and a step edge at the crystalline-silicon/amorphous-silicon oxide interface\n(green arrow). These buried features, important for device performance, while readily\napparent with MEP, can be inaccessible or easily missed with conventional imaging\nmethods. d MEP depth slices from the region in b-c (light and dark orange planes)\nshowing both light (Si, O) and heavy (Hf) atoms, visible due to the linear atomic-number\ncontrast. The hafnium oxide intrusion (top, light orange slice, z = 10nm, red arrow) is\nlocalized in depth and does not extend through the entire interface (bottom, darker\norange slice, z = 16nm).”\nComment #2:\nExtended data figure 4 - not clear from the caption what is different between the two\nimages. The figures also appear to have different scalings in the two directions. The\nvertical dimension is 20 nm according to the arrows and the horizontal scale bar is 1 nm.\nThe difference in scaling is about 3.67x.\nResponse:\nThe old Extended figure 4 showed two y-z depth slices through a planar device at 2\ndifferent locations in the x-y plane. We have updated the figure to add the xy area the\nsection came from. To address reviewer 2’s request for quantitative measurements of\ndepth resolution, we have further updated this figure and moved it to the main section\n(now Figure 4, see below, with an extended version in Supplementary Figure 5). The\ndifference in scaling reflects the difference between the lateral and depth resolutions and\nsamplings, now highlighted in the caption (lines 434-454).\n26\n[001]\n[\n)110]\n[110]\nc-Si\na-SiO\n2\ny a-HfO\nx 2\nz\na xy projection bzy depth c\nsection\n47 Å\n48 Å\na-HfO a-HfO\n2 2\n30 Å\na-SiO a-SiO\n2 2\n35 Å\nc-Si c-Si\n37 Å\n41 Å\n10 Å\n280 Å\nFigure 4. Experimental MEP depth sections from a planar c-Si/a-SiO /a-HfO\n2 2\ninterface and corresponding depth blur estimation. a. Projected xy view of the MEP-\nreconstructed electrostatic potential of a planar c-Si/a-SiO /a-HfO interface; the red\n2 2\ndashed line marks the location of the depth slice in panel b. b This MEP depth section\n(with entrance surface on the left, and exit surface on the right) shows internal features\nof the amorphous HfO layer, including nanoscale contrast variations consistent with\n2\nshort-range order or local density fluctuations. In contrast, the a-SiO portion appears\n2\nmore homogeneous, consistent with its lower atomic number and more uniform\namorphous structure. The characteristic triangular wedge geometry of the FIB-prepared\nTEM lamella is visible (outlined by dashed guidelines). A single atom located at the top\nsurface (red arrow) is identified as redeposited from the TEM sample preparation. These\nobservations demonstrate the sensitivity of MEP to both intrinsic and preparation-induced\nstructural features, highlighting its utility for detailed 3D characterization of amorphous-\ncrystalline interfaces. Note that the voxel size is anisotropic in the depth direction. c\nLocal axial intensity profiles along isolated single atoms from this reconstruction (see\nSupplementary Figure 5 for slice locations; the red arrow in panel b corresponds to the\nthird profile shown) together with Gaussian-plus-linear-background fits. The resulting full-\n27\nwidth-at-half-maximum (FWHM) values are listed below each profile. Across all six atoms,\nthe fits yield depth blurs of 30–48 Å, corresponding to an average depth blur of 40 ± 7 Å\n(s.d.). Equivalent measurements cannot be performed for tf-ADF or tf-iDPC because the\nfinite-dose images were too noisy and individual atoms often shifted between consecutive\nframes.\nComment #3:\nThe methods section does not describe the fabrication of the silicon, silicon oxide, and\nhalfnium oxide blanket film.\nResponse:\nWe have added a description of the blanket film preparation to the Device Fabrication\nsection of the Methods (lines 501-505):\n“The planar interface (c-Si/a-SiO /a-HfO ) blanket film sample used for comparison\n2 2\nwas fabricated separately. The Si portion of this sample corresponds to the starting\nSi wafer, while the SiO layer is the native oxide present on the wafer surface, which\n2\nserves as an effective nucleation layer for subsequent ALD deposition. The HfO\n2\nfilm was deposited at 300 °C using ALD with alternating HfCl and H O pulses and\n4 2\nN purge steps between pulses.”\nI recommend mentioning in the abstract that the GAA devices are fabricated by imec.\nWith the TSMC coauthors and no imec coauthors it is easy to assume the devices were\nmade by TSMC.\nResponse:\nWe appreciate the reviewer’s suggestion and have added a statement in the abstract to\nclarify that the prototype GAA devices were fabricated by imec (lines 22-24):\n“By performing electron ptychography on prototype gate-all-around transistors fabricated\nby imec we uncover and quantify distortions and defects at the interface of the 3D gate\noxide wrapped around the channel.”\nWe thank all reviewers for their thoughtful, constructive, and detailed feedback. We\nappreciate the reviewers’ positive assessment of the manuscript and their recognition of\nthe improvements made in clarity and accessibility. All remaining comments, primarily\nfrom Reviewer #3, have been carefully addressed through revisions to the manuscript\nand detailed point-by-point responses below. Our responses are shown in green and\nrevised or newly added text in the manuscript is shown in bold blue. Line numbers\nreferenced in our responses correspond to the revised clean manuscript. We have also\nclarified the figure captions (Figures 2 (lines 432-435) and 6 (lines 485-486)) to distinguish\nbetween shaded regions representing statistical variability (standard deviation) and\nwidths defined by physical metrics such as full width at half maximum from fitted profiles.\nReviewer #2 (Remarks to the Author):\nThe authors have addressed all of my comments and have made substantial\nimprovements to the manuscript. I greatly appreciate the hard work of the authors and\nwould be happy to see this work published. I have nothing to add or critique at this stage.\nWe thank the reviewer for their positive assessment of our work. We appreciate the\nreviewer’s recognition of the substantial improvements made in the revision and their\nsupport for publication.\nReviewer #3 (Remarks to the Author):\nThe authors have provided an extremely thorough and careful response to the reviews. I\nthank them for that. The clarity of the manuscript has improved a lot. I think it now would\nbe clear to both TEM people not familiar with advanced semiconductor devices and\nsemiconductor people that are not TEM experts. There are a few minor issues that would\nfurther improve the manuscript.\nWe sincerely thank the reviewer for their careful reading of the revised manuscript and\nfor their thoughtful and positive assessment and for the constructive suggestions, all of\nwhich we address in detail below and have incorporated into the manuscript to further\nimprove clarity and precision.\nComment #1:\nIn the introduction, the authors distinguish between “light” elements and “heavy” elements\nand changes in the contrast for different imaging methods associated with atomic number.\nLater in the manuscript in the caption of figure 1 they state that light for GAA devices\nmeans Si and O and heavy is Hf. It would be valuable to say that in the main text where\nlight and heavy are discussed in the introduction (page 4).\nResponse:\nWe thank the reviewer for this helpful suggestion. We have clarified the meaning of “light”\nand “heavy” elements in the context of GAA devices (lines 74-78):\n“Here, we show how multislice electron ptychography (MEP) addresses the critical\nmetrology gap in device characterization highlighted above, namely, the need for\nÅngström-scale resolution, three-dimensional imaging capabilities, and simultaneous\nsensitivity to both light and heavy atoms (in the context of GAA devices, “light” refers\nto Si and O, while “heavy” refers to Hf), thereby enabling direct measurement of atomic\nstructure, strain, and interface roughness in 3D.”\nComment #2:\nheavy elements” and sub-Angstrom resolution (lines 75-77 and 81-83). “the need for\nAngstrom-scale resolution three-dimensional imaging capabilities, and simultaneous\nsensitivity to both light and heavy atoms” and “MEP has been established as a technique\ncapable of sub-Angstrom in-plane resolution and nanometer-scale depth resolution, with\nsensitivity to both light and heavy elements.” They can combine those sentences to\nsmooth the flow.\nResponse:\nWe thank the reviewer for this suggestion. We have revised the paragraph to eliminate\nredundancy by incorporating the discussion of sub-Ångström resolution and sensitivity to\nlight and heavy elements, along with the corresponding citations, into the initial\ndescription of MEP (lines 74-84):\n“Here, we show how multislice electron ptychography (MEP) addresses the critical\nmetrology gap in device characterization highlighted above, namely, the need for\nÅngström-scale resolution, three-dimensional imaging capabilities, and simultaneous\nsensitivity to both light and heavy atoms27–29 (in the context of GAA devices, “light” refers\nto Si and O, while “heavy” refers to Hf), thereby enabling direct measurement of atomic\nstructure, strain, and interface roughness in 3D. By scanning a convergent electron probe\nacross the sample and collecting diffraction patterns from overlapping illuminated regions,\nMEP retrieves the phase shift introduced by the object’s atomically-resolved electrostatic\npotential, providing nanometer-scale depth information through post-processing of\nscanning diffraction data26. MEP has been established as a technique capable of sub-\nÅngström in-plane resolution and nanometer-scale depth resolution, with\nsensitivity to both light and heavy elements27–29. We demonstrate that MEP enables\nhigh-fidelity three-dimensional imaging of GAA transistors, validate its accuracy on\nsimulated structures and show experimentally that it outperforms conventional through-\nfocal STEM in resolution and dose efficiency. ….”\nComment #3:\nPage 9 (lines 195-199) – The transition between simulated and real data is not clear in\nthe text. They do clearly state that the GAA is experimental, but it would be good to say\nwhen they change from silicon/silicon oxide/hafnium oxide to the Ta-filled pinhole sample\nthat it is still simulated microscopy.\nResponse:\nWe thank the reviewer for pointing this out. We have revised the text to explicitly state\nthat the Ta-filled pinhole example is based on a simulation study (lines 197-201):\n“The improved depth fidelity of MEP translates directly to the ability to recover buried\nstructures: using a simulation study of a pMOS device model61, Supplementary Figure\n3 shows that MEP resolves an amorphous Ta-filled pinhole at 14 nm at the correct depth,\n3\nwhereas the feature is indistinct in tf-iDPC and barely visible in tf-ADF (with the silicon\nchannel completely obscured), despite the latter using more than twice the dose.”\nComment #4:\nFigure 4c – According to the caption, the 3rd slice in 4c corresponds to the single atom\nmarked by an arrow in 4b. The other slices correspond to single atoms shown in the\nsupplemental. I recommend they put a red box around the slice in 4c that corresponds to\nthe red arrow in 4b.\nResponse:\nWe have updated Fig. 4c to explicitly highlight the axial profile corresponding to the single\natom marked by the red arrow in Fig. 4b by enclosing it in a red box. The caption has\nbeen revised accordingly to clarify the correspondence between panels (lines 443, 458):\na xy projection bzy depth c\nsection\n47 Å\n48 Å\na-HfO a-HfO\n2 2\n30 Å\na-SiO a-SiO\n2 2\n35 Å\nc-Si c-Si\n37 Å\n41 Å\n10 Å\n280 Å\n“…. the red arrow in panel b corresponds to the third profile shown, boxed in red). ….”\nComment #5:\nOn page 5 (lines 95-100) and page 15 (lines 328-329) the manuscript mentions electrical\ntesting would require months of fabrication to produce a testable device. Full production\n4\nchip take months to produce, but during process development fabs make short loop\nwafers with a small number of metal layers to allow function testing after a much shorter\ntime (days to weeks). This is critical for development of new device structures/processes\nsince at that time they have no idea how to relate images of the devices to end\nperformance. The current wording is a little misleading since no fab would take multiple\nmonths to be able to etest structures during new device development. The wording seems\nto merge process monitoring during production where an early metrology step could save\na lot of time/cost and metrology during process development where it helps them know\nwhat they made and relate structure to electrical function. Separate from etest, 3D atomic\nscale resolution would speed up the development cycle by speeding up learning.\nResponse:\nWe thank the reviewer for prompting us to clarify the language regarding electrical testing\nduring process development. To remove any potential ambiguity, we have revised the\nmanuscript to more explicitly distinguish between short-loop electrical testing and the full\nfabrication flow, and to clarify the role of MEP during process development. Specifically,\nwe revised the text to emphasize that MEP performed on dedicated witness wafers\naccelerates the learning cycle during process development by enabling direct correlation\nbetween process changes and buried structural outcomes, complementary to short-loop\nelectrical testing (lines 94-102):\n\"Formation of the critical GAA structure occurs early in the overall CMOS fabrication\nprocess, which comprises roughly a thousand steps and typically takes 3-4 months to\ncomplete from start to finish. Early structural feedback from MEP performed on dedicated\nwitness wafers (test wafers fabricated alongside production wafers for process\nmonitoring, including by destructive analysis techniques) can provide rapid insight into\nstructural defects and interface quality early in the fabrication flow. This will accelerate\nthe learning cycle during process development by enabling direct correlation\nbetween process changes, structural outcomes, and complementary short-loop\nelectrical testing, reducing costly iterations.”\nWe also revised lines 328–329 to clarify that MEP can be applied after any fabrication\nstep during process development to provide early atomic-scale structural feedback,\nwithout implying that any electrical testing requires completion of the full fabrication flow.\nThe revised text now reads (lines 329-336):\n“MEP can be applied after any fabrication step during process development,\nproviding early three-dimensional atomic-scale structural feedback well before\ncompletion of the full fabrication flow. This information complements short-loop\nelectrical testing, which primarily probes aggregate electrical parameters such as\ncurrent, capacitance, and resistance, by instead directly characterizing buried\nstructural variations and process-dependent interfacial disorder that are not\naccessible through electrical measurements alone or through conventional\n5\nprojection-based TEM imaging. By providing this complementary structural\ninformation, MEP enables faster learning and refinement of fabrication processes.”\nIn addition, we note that while short-loop electrical testing is an essential part of process\ndevelopment, the additional processing steps required to enable electrical access\n(including metallization and associated thermal or annealing treatments) can themselves\nmodify the atomic-scale structure formed at earlier stages of fabrication, that are the steps\nof interest for process development. As a result, structural characterization performed\nimmediately after formation of the feature of interest, prior to subsequent thermal steps,\nprovides complementary and, in some cases, uniquely informative insight into the intrinsic\nstructural outcome of a given process step. This further highlights the value of MEP as an\nearly-stage, step-resolved metrology tool that accelerates learning during process\ndevelopment rather than replacing electrical validation.\nComment #6:\nHow does computation time scale with the size of the measured volume? Since the 4D\nSTEM measurement is pretty fast, end users would be interested in trying to get a larger\nreconstruction volume.\nResponse:\nWe thank the reviewer for this question. The reconstruction time in multislice electron\nptychography depends on several factors, including the number of probe positions,\ndetector pixels, number of reconstructed depth slices, and number of reconstruction\niterations, and therefore does not follow a single simple scaling law with volume. In\npractice, for the datasets considered here, reconstruction time scales approximately with\nthe total number of measured probe positions and the number of reconstructed depth-\nslices and can be effectively mitigated through the reconstruction of multiple datasets in\nparallel (this is the workflow used in our MEP reconstructions), i.e. it uses more compute\nresources, but does not take any longer. We have revised the Methods section to clarify\nthat while larger reconstruction volumes increase computational cost, the reconstruction\nworkflow is still practical for device-scale volumes (lines 565-577):\n“The total measurement time for MEP experiments is comparable to standard high-quality\nTEM workflows. Focused ion beam (FIB) sample preparation for a <50 nm lamella\ntypically requires half a day for an experienced user, followed by approximately half a day\nof microscope time (including alignment and tuning). Each 4D-STEM dataset (256 x 256\nscan positions) is acquired in about 7 seconds on the EMPAD-G2 detector.\nReconstruction typically takes from an hour to a day, depending on user familiarity,\ncomputational resources, degree of convergence needed for imaging vs quantification,\nand the algorithm used. These times are representative of our setup and will vary between\nlaboratories.\nnumber of probe positions (i.e. area) and with the number of reconstructed depth\nslices60; however, the iterative reconstruction is efficiently accelerated on GPUs\nand can be distributed across compute nodes. Thus, reconstructing a large area\ncan be done in a similar time, provided more compute resources are available,\nenabling practical times for larger volumes and multi-device studies.”\nComment #7:\nTypo (page 8, line 176) – “rough crystalline-silicon (c-Si)/amorphous hafnium oxide (a-\nSiO2)/amorphous-hafnium oxide (a-HfO2) interface.” The first hafnium oxide should be\nsilicon oxide.\nResponse:\nWe thank the reviewer for catching this typo. The text has been corrected (lines 177-178):\n“…. Figure 3 summarizes the simulated 3D benchmark of a rough crystalline-silicon (c-\nSi)/amorphous-silicon oxide (a-SiO2)/amorphous-hafnium oxide (a-HfO2) interface, ….”\nComment #8:\nThe author’s general response to reviewer #3 about measurement statistics appears to\nmix up two different uses for 3D atomic scale resolution imaging – failure/fault analysis\nand development testing/feedback. Both are important. It is very difficult to make what\nyou can’t measure, so fast data on process changes speed up the learning loop. In the\ncase of development testing/feedback statistics is very important because they aren’t\ntrying to look at a specific transistor to figure out why it didn’t work. They are trying to\nlearn what changed in the transistors for a processing variation. None of the review\nresponse discussion about failure analysis was in the manuscript so the comment is only\nto correct the review response.\nResponse:\nWe thank the reviewer for this clarification and agree with the distinction between\nfailure/fault analysis and development testing or process feedback. Our intent in the\nmanuscript is to emphasize the latter, namely, the use of three-dimensional atomic-scale\nimaging to accelerate learning during process development by enabling atomic-scale 3D\ninsight into structural changes arising from process variations, free from common\nprojective S/TEM imaging artifacts. We agree that this context is distinct from single-\ndevice failure analysis, and we confirm that the manuscript text is framed around\ndevelopment feedback and learning rather than failure analysis. No changes to the\nmanuscript were required.\n7"}]

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