Strain mapping of three-dimensionally structured two-dimensional materials

Abstract

Strain plays a crucial role in tuning materials’ properties, influencing their optical, electrical, and chemical performances. In two-dimensional (2D) materials, applied stress often induces out-of-plane deformation, resulting in a more intricate three-dimensional (3D) topography, where mapping the strain remains a challenge due to the limitations of conventional characterization techniques. In this work, we introduce BRIGHT (Bragg-Rod Informed, Gradient-based Height-mapping Technique), an integrated method for reconstructing both the topography and planar strain profile of 3D-structured 2D materials using nanobeam four-dimensional scanning transmission electron microscopy (4D-STEM). We apply BRIGHT to a MoS₂-MoSe₂ transition metal dichalcogenide (TMD) lateral heterojunctions exhibiting built-in strain and out-of-plane ripples and show that varying heterojunction widths lead to distinct surface morphologies and corresponding changes in the planar strain distribution. These results establish a foundation for more effective strain engineering in 2D materials by accounting for out-of-plane structural features, thereby enabling more precise control of strain-dependent properties.

Advances in electron microscopy enable quantifying structure and strain in highly deformed 2D materials.

INTRODUCTION

Strain engineering has emerged as a powerful approach to tailor the properties of two-dimensional (2D) materials, which, because of their atomically thin nature, can tolerate considerable stress. The extreme mechanical flexibility of 2D lattices means that any applied strain inevitably induces out-of-plane deformations, such as ripples, wrinkles, and buckling (1–3). These structural modifications alter the local atomic configuration, directly affecting the electronic band structure, excitonic properties, and phonon dynamics (4–6). This remarkable mechanical adaptability has been actively leveraged in strain engineering, where controlled deformations are used to fine-tune material properties for emerging applications, including flexible electronics (7–11), single-photon emitters (12–17), and reconfigurable nanomechanical systems (2, 7, 9, 18). These advancements enable expanded capabilities in tunable optics, sensors, and microactuators.

For 2D materials, the most relevant strain components can be described by a 2D strain tensor, with basis vectors lying in the material’s plane. However, inevitable corrugation or strain-induced rippling in 2D materials introduce spatial variations in the planar strain tensor, resulting in a complex 3D profile. Characterizing this strain distribution in 3D-structured 2D materials remains a challenge due to the limitations of conventional measurement techniques. Although methods such as geometric phase analysis (GPA) (19–22) and atomic-resolution electron tomography (23) provide insights into strain distributions, their small field of view (FOV), typically only a few nanometers, prevents them from fully resolving large-scale strain variations and 3D structures. In contrast, nanobeam four-dimensional scanning transmission electron microscopy (4D-STEM), which collects diffraction patterns from each scan position, has enabled micrometer-scale mapping of lattice structures, such as projected strain tensors (24–27) and stacking order information (28–30). However, its ability to quantify strain profiles in 3D-structured, deformed 2D materials remains elusive.

Here, we present BRIGHT (Bragg-Rod Informed, Gradient-based Height-mapping Technique), an integrated approach that simultaneously reconstructs 3D topography and maps the tilt-aware planar strain field of a rippled 2D film (Fig. 1; see also fig. S1). The 3D structure is reconstructed from a single nanobeam 4D-STEM dataset, mitigating electron radiation damage and misalignment artifacts, which are common challenges in electron tomography. Using this recovered topography, we compute the strain tensor directly on the actual 3D-structured surface. This approach is readily adaptable to other 2D materials, advancing the understanding of planar strain distributions on nonplanar (3D-structured) membranes and enabling precise strain modulation in applications such as flexible electronics and quantum computing.

Fig. 1. Data pipeline for strain mapping of 3D-structured 2D materials.

An overview of the processing steps for reconstructing strain maps from nanobeam 4D-STEM data. An electron beam is raster-scanned across the surface of a deformed TMD sample, and diffraction patterns are collected at each point using an EMPAD with dimensions k_y × k_x. These diffraction datasets are processed to extract tilt information, which is then converted into local surface gradients. A correction algorithm is applied to account for out-of-plane tilt distortions. The resulting gradient maps are used to reconstruct the 3D surface topography, enabling the generation of a spatially-resolved 3D strain map that visualizes the distribution of mechanical deformation across the suspended 2D TMD film. The scale prism in the 3D strain map is 25 nm by 25 nm by 25 nm; the height direction is stretched by a factor of 5 for visualization purposes.

RESULTS

Topography mapping by single-scan nanobeam 4D-STEM

Our experiments began with the development of BRIGHT to map the topography of a deformed monolayer 2D transition metal dichalcogenide (TMD) in the out-of-plane direction. Figure 2A illustrates a 3D model of the TMD, with distinct flat and tilt regions highlighted. Variations in tilting change how the Ewald sphere samples the Bragg rods (Fig. 2B), which modulates the integrated intensities of the diffraction peaks. Kinematic simulations confirm tilt sensitivity across first-, second-, and third-order reflections (fig. S2). In flat regions, the intersections are uniform, whereas in tilted areas, they vary, leading to observable differences in Bragg peak intensities within the diffraction patterns. Notably, the second-order Bragg peaks exhibit the most pronounced intensity changes upon tilting (Fig. 2C), which is also evident in virtual dark-field images (fig. S3). In addition, the intensity of second-order spots is less sensitive to polarity, which introduces intensity differences in conjugate spots (31), whereas the second-order Bragg rods of TMDs remain symmetric about the out-of-plane axis, providing a reliable reference for quantifying tilt and topography (26). For these reasons, we use the second-order Bragg peaks for TMD materials in the BRIGHT method.

Fig. 2. 3D topographic reconstruction of TMDs through nanobeam 4D-STEM diffraction intensities.

(A) 3D deformed TMD surface model highlighting a flat and a tilted region. (B) Schematic representations (top and bottom) of the Ewald sphere intersecting six second-order Bragg rods. The top schematic shows uniform intersections (flat region), whereas the bottom schematic shows varied intersections (tilted region). (C) Comparisons of diffraction patterns for the flat and tilted regions are separated into “simulation” (left) and “experiment” (right). The decreased Bragg peak intensities for Bragg peaks farthest from the tilt axis illustrate the impact of tilt. (D) Definitions of the tilt descriptors: The elevation angle (θ) is measured out-of-plane from the xy plane, whereas the azimuthal angle (ϕ) is measured in-plane from the y axis. (E) Workflow for recovering tilt parameters (θ, ϕ) by comparing experimental diffraction intensities with a simulation library. A “match” in intensity vectors yields the corresponding best-fit tilt angles. (F) Gradient mapping and 3D reconstruction: Tilt angles (θ, ϕ) extracted from each point in the sample are used to compute the local x and y gradients, which are then integrated to form a topographic map of the TMD surface.

To quantify the local tilt in TMDs, we define two angular descriptors: the out-of-plane elevation angle (θ) and the in-plane azimuthal angle (ϕ) (Fig. 2D). For each diffraction pattern in the 4D-STEM dataset, we extract the integrated intensities of the second-order Bragg peaks (see Materials and Methods). Using the kinematic diffraction analytic model from (26), we then generate a simulation library that maps specific tilt parameters (θ, ϕ) to the corresponding vector of Bragg peak intensities. This library spans tilt values from 0° to 25° in elevation and 0° to 360° in azimuth, with a resolution of 0.1°. Each experimental intensity vector is compared against the simulated vectors using a loss function—Euclidean distance (L² norm; see Materials and Methods for comparison with cosine similarity)—to identify the best match, which provides the corresponding experimental tilt parameters (θ, ϕ) (Fig. 2E). By extracting these tilt values for every diffraction pattern, we obtain a real-space tilt map of the sample that serves as a crucial foundation for reconstructing its surface topography.

To reconstruct the surface topography, we first convert the extracted tilt maps (elevation and azimuth) into local gradient information using Eq. 1. These gradients represent the directional derivatives of the surface height and serve as the fundamental inputs for our reconstruction process. Next, the gradient data are fed into an iterative gradient-correction algorithm, which integrates the measured gradients to generate a continuous 3D height map of the TMD surface (Fig. 2F; see the Supplementary Materials for algorithm details and fig. S4). This iterative procedure refines the height map by correcting both the direction (sign) and the absolute magnitude of the gradients, thereby minimizing discrepancies between the calculated and measured values. As a result, the reconstructed topography faithfully captures the actual surface morphology.

To ensure reconstruction fidelity, acquisition parameters (scan step and FOV) must be set to satisfy the sampling criteria from our geometric analysis (figs. S5 to S8): a sampling ratio of about 0.20 (five or more scan steps across the smallest feature to be resolved) delivers ~90% mean reconstruction accuracy, whereas achieving ≥95% typically requires a sampling ratio of ≤0.13 (at least eight steps across the feature). In the later section, the experimental sampling conditions satisfied this geometric accuracy threshold, yielding roughly 6 scan steps across the smallest ripples and up to 30 across the largest ones. The results indicate that, with appropriate parameters, the approach can faithfully capture the actual surface morphology and provides a basis for further strain analysis

$$\frac{\mathrm{\partial }z}{\mathrm{\partial }y}=\text{tan} \mathrm{\theta } \text{cos} \mathrm{\varphi }; \frac{\mathrm{\partial }z}{\mathrm{\partial }x}=\text{tan} \mathrm{\theta } \text{sin} \mathrm{\varphi }$$ (1)

Beyond TMDs, our method also has the potential for broad adoption. Simulations demonstrate that other few-atom-thick 2D materials, such as In₂Se₃ and GeSe, also exhibit diffraction intensities that are highly sensitive to tilt angles (figs. S9 to S12). These results suggest that integrated Bragg-spot intensity can serve as a reliable parameter for calculating local tilts in other types of 2D materials. Consequently, the workflow for topography reconstruction and strain mapping from the measured tilts can be directly extended to other 2D material systems as well with consistent reliability.

Error analysis of topography mapping

To validate the BRIGHT method, we conducted an error analysis using a simulated ripple-like MoSe₂ 4D-STEM dataset generated with abTEM (32) (Fig. 3A). This benchmark dataset, featuring predetermined height and tilt parameters, allowed for controlled assessment of reconstruction accuracy (see figs. S13 and S14). Diffraction patterns simulated at 219 × 219 pixels resolution were processed using the BRIGHT algorithm (see figs. S15 to S17). Height maps were then reconstructed and their accuracy quantified against ground truth using mean absolute error (MAE) and the absolute L² norm. For infinite dose, the MAE maintains ~0.09 nm in the nonplanar region. To approximate realistic experimental conditions, Poisson shot noise was added at the raw diffraction-count level across a broad electron-dose series spanning 10¹ to 10⁴ e⁻/Ų (see figs. S18 to S20 for summary of results). Figure 3B directly compares the reconstructed surface from the dose condition of 10³ e⁻/Ų with the ground truth, showing an MAE of ~0.20 nm in the same region. By comparing different dose conditions, these results show that ripple morphology and quantitative accuracy are preserved down to ~10³ e⁻/Ų, which represents a reasonable dose for 2D materials. Below 10² e⁻/Ų, the algorithm begins to show degradation. The analysis confirms the reliability of our approach in capturing TMD surface topography and angular variations with a reasonable electron dose range.

Fig. 3. Simulated nanobeam 4D-STEM data and performance of the tilt-mapping approach.

(A) 3D, ripple-like TMD surface geometry used for simulating a nanobeam 4D-STEM dataset. The 4D dataset below is represented as a 5 × 5 array of diffraction patterns for illustration purposes. (B) Comparison of the ground-truth surface and the reconstructed one (simulated with an electron dose of at 10³ e⁻/Ų). The vertical scale (z height) in (A) and (B) is magnified by a factor of 7 for clarity; scale bars, 40 nm. (C) Definition of tilt axes used for assessing the accuracy of the tilt-mapping model, ϕ = 60°k and ϕ = 30°(2k+1), illustrated with simulated diffraction patterns with a tilt of 20°. Dashed lines indicate the tilt axes relative to second-order Bragg peaks. (D) Tilt error plotted against the ground-truth tilt angle for each azimuthal tilt axis shown in (C). The error band is for ±1 SD of the curve fit. (E) Influence of electron dose on tilt-mapping accuracy for θ_GroundTruth values of 9° and 18°, plotted over a dose range of 10³ to 10⁷ e⁻/Ų. Each point is colored to reflect the azimuthal error (ϕ_Error).

Further analysis reveals that the accuracy of tilt mapping also depends on the orientation of the tilt axis. Figure 3C compares two sets of tilt axes: one set defined by angles ϕ = 60°k, corresponding to axes oriented between second-order Bragg peaks, and another set defined by ϕ = 30°(2k+1), corresponding to axes passing directly through the peaks. In addition, we analyze an intermediate set of axes defined by ϕ = (2k+1) × 15°, with results shown separately in figs. S20 and S22. Quantitative results in Fig. 3D indicate that the tilt extraction error (θ_Error) remains <1° for ground-truth tilts up to about 18°. At a 20° ground-truth tilt, the error is ~1.2° for ϕ = 60°k axes and around 0.2° for ϕ = 30°(2k+1). However, when the tilt increases to 25°, the error for ϕ = 30°(2k+1) rises sharply to over 3°, whereas the error for ϕ = 60°k decreases slightly to about 1°. These results underscore that local tilt axis variations in deformed TMDs affect the tilt-mapping accuracy, especially at elevation angles above 18°. In addition, these axis-dependent trends are not substantially influenced by electron dose. Figure 3E demonstrates that electron doses above 10³ e⁻/Ų generally yield good accuracy for both θ and ϕ, with higher doses providing only marginal improvement (<0.2° for elevation and <1° for azimuth)—see figs. S22 and S23 for additional insights into how electron dose influences tilt-mapping accuracy. In summary, the method is generally accurate when the out-of-plane tilt angle is below 20°, which encompasses most planar-deformed 2D thin films. However, more severe deformations will require alternative approaches for quantification, such as tomography from tilt series.

Reconstruction of 3D strain

Beyond topography, our approach provides a robust framework for reconstructing 3D profiles of the 2D strain tensors by correcting sample tilt effects. The process begins with a conventional strain measurement (25), yielding an apparent strain (ε_apparent) that reflects both the true lattice deformation (ε_real) and tilt-induced artifacts—where increased tilt results in additional artificial compressive strain. To address this, we first extract the tilt parameters (θ, ϕ) from each diffraction pattern. Using plane-strain equations (eqs. S1 to S3), we then reorient the strain components so that the component perpendicular to the tilt axis can be corrected according to Eq. 2 (see figs. S24 and S25). Last, the corrected strain components are transformed back to the original coordinate system, providing an accurate representation of the true lattice strain

$${\mathrm{\epsilon }}_{\text{real}}=\frac{{\mathrm{\epsilon }}_{\text{apparent}}-1}{\text{cos} \mathrm{\theta }}+1$$ (2)

We incorporated the corrected strain data into an interactive visualization workflow facilitated by our interactive graphical user interface (GUI), as illustrated in Fig. 4. This GUI enables real-time examination of 3D maps of strain, topography, tilt, and phase. Key interactive features include sliders to adjust the vertical exaggeration of the surface and rotate strain measurement axes via the plane-strain equations, radio buttons for selecting specific strain or rotation components (ε_xx, ε_yy, ε_xy, and ε_rot), and a toggle button for switching between corrected and uncorrected maps. Figure 4 (B and C) highlights how ripple formation in MoSe₂ alleviates compressive strain, a phenomenon readily visualized through toggling the strain correction feature. In addition, Fig. 4 (D and E) demonstrates how rotating the coordinate axes affects the interpretation of shear strain distributions, a useful feature when exploring anisotropic features interactively. Last, Fig. 4F showcases alternative visualization overlays, including tilt-phase, compositional contrast [high-angle annular dark-field (HAADF)–STEM], and height maps, enriching the analysis of structural characteristics. Together, this GUI-based approach enhances both qualitative visualization and quantitative analysis of strain mechanisms in TMDs, providing a powerful platform for continued investigation. For a demonstration of the GUI’s features and its real-time visualization capabilities—including the interactive manipulation of strain components, coordinate rotation, and correction toggling—readers are encouraged to view movie S1.

Fig. 4. Interactive GUI for 3D surface and strain visualization.

(A) Overview of the GUI, featuring interactive controls for visualizing and analyzing strain data. The “Toggle Strain Correction” button allows users to switch between uncorrected and tilt-corrected strain maps. Users can select different strain components—uniaxial (ε_xx, ε_yy), shear (ε_xy), or rotation (ε_rot). The “Height Stretch Factor” slider vertically exaggerates the displayed 3D surface, whereas the “Rotation Angle” slider allows real-time application of the strain tensor around the surface normal (z axis) using plane strain equations. (B) 3D visualization of a rippled MoSe₂ surface section with visible MoS₂ junctions, displaying projected (uncorrected) strain on the ripple walls. (C) Corrected strain visualization of the same region as in (B), after enabling the strain correction feature. This correction accounts for out-of-plane tilts, providing a more accurate representation of the strain distribution. (D) Overlay of the uncorrected shear-strain map (ε_xy) onto the 3D surface, oriented according to the GUI’s default coordinate axes. (E) Shear-strain map from (C) after a 45° rotation around the surface normal (z axis), illustrating how shear components realign when the measurement axes are rotated. (F) Three alternative 2D overlays accessible through the GUI: (left) a tilt-phase map encoding local surface tilt magnitude and direction, (middle) a HAADF intensity image highlighting compositional contrast, and (right) the original height map showing topographic elevation. Prism edges in (B) to (F) span 20 nm; vertical scaling is set to 3.7 by the “Height Stretch Factor” in (A).

3D strain in suspended monolayer 2D materials

To demonstrate a key application of BRIGHT, we investigate the 3D strain mapping of a suspended monolayer TMD heterostructure composed of alternating MoS₂ and MoSe₂ junctions, grown using a S/Se precursor flow-modulated method described in (33, 34). This approach achieves full lattice coherence despite lattice mismatch and enables strain-engineered optical properties via controlled junction widths. Because of the ~4% lattice mismatch between MoS₂ and MoSe₂, the built-in epitaxial strain induces uniaxial tensile and compressive strain at each junction. The compressive strain leads to ripple formation in MoSe₂, even when supported by its original substrate (25, 34). To investigate strain relaxation upon release from the substrate, we transferred the sample onto a holey carbon grid. The suspended regions over the holes relax to a complex 3D structure, exhibiting residual strain and suspended periodic ripples. Our sample consists of over 35 junctions with widths ranging from 10 to 150 nm, revealing the junction width dependence of strain relaxation and ripple morphology (fig. S26). This sample structure provides a unique platform for studying the mechanics of strain-engineered 2D materials.

Using the BRIGHT approach to investigate suspended monolayer lateral heterostructures, the reconstructed topography and phase maps (Fig. 5A) reveal distinct periodic ripples across three regions with varying junction widths. The mean MoSe₂ and MoS₂ junction widths in these regions, from top to bottom, are 19 and 38 nm (top), 137 and 69 nm (middle), and 15 and 10 nm (bottom), respectively, with the corresponding width ratios being ~1:2, 2:1, and 3:2. These variations reflect the precursor flow conditions established during growth. Gradient maps derived from the topography (Fig. 5B) indicate that a wider MoS₂ region (top) results in narrower vertical ripples in MoSe₂, whereas a wider MoSe₂ region (middle) leads to larger vertical ripples in MoSe₂ that gradually flatten in the MoS₂ region. When the MoSe₂ and MoS₂ widths are both below 15 nm (bottom), they form superlattices with relatively flat topography, except where large ripples are inherited from neighboring regions near a suspended edge (see fig. S27). The tilt-corrected strain maps (Fig. 5C) show a notable difference compared to the noncorrected strain maps (fig. S28), which exhibit tilt-induced artifacts in the projected strain tensor. The most notable change appears in the ε_xx strain component across the MoSe₂ junctions, where the profiles become much smoother along individual junctions, suggesting that rippling primarily serves to relax the ε_xx epitaxial strain induced by lattice mismatch. In addition, rotation maps reveal that ripple features introduce rotational dipoles at interfaces, similar to the role of misfit dislocations, ensuring the Burgers vector remains conservative.

Fig. 5. 4D-STEM dataset analysis: Topography, strain, and deformation profiles.

(A) Topography and phase maps extracted from nanobeam 4D-STEM data. The left panel presents the height map, whereas the right panel represents the phase map, where brightness indicates tilt magnitude, and hue encodes tilt direction. (B) Gradient maps of the structure separated into the x gradient (left) and y gradient (right). Blue and red colors indicate positive and negative gradients, respectively, whereas white represents zero gradients. (C) Strain and rotation maps showing four distinct components: uniaxial (ε_xx, ε_yy), shear (ε_xy), and rotation (ε_rot). (D to F) Line profile extractions across regions containing large ripples (D), aligned “bumps” (E), and alternating or zigzag “bumps” (F). The top sections present experimental data, whereas the bottom sections provide a schematic representation. Line profiles compare vertical and horizontal deformations across undulated and nondeformed regions, examining their alignment both perpendicular and parallel to MoSe₂-MoS₂ junctions. Scale bars, 200 nm [(A) to (C)] and 100 nm [(D) to (F)].

To quantitatively evaluate the residual strain and ripple morphology depending on junction width, we further analyzed each region separately. In regions with larger junctions (middle region), our analysis reveals pronounced periodic rippling with an average ripple width of 77 nm (Fig. 5D). Line profile analyses (Fig. 5D, top) indicate that, despite the formation of large ripples, the overall strain relief remains partial, with horizontal profiles showing only a 1.55% difference in distance between the rippled MoSe₂ (blue solid line in Fig. 5D, top) and the adjacent nonrippled MoS₂ (blue dotted line in Fig. 5D, top), compared to an ~4% lattice mismatch. Vertical line profiles confirm that rippling induces minimal mismatch in distance (<0.1%), attributed to the MoSe₂ ripples forming hills and valleys as they pass over and under the MoS₂ plane (green solid and dotted lines in Fig. 5D, top), respectively (schematic illustration shown in Fig. 5D, bottom). In contrast, regions with smaller junctions in the top region exhibit horizontal bumps averaging 122 nm in length with 23 nm spacing (Fig. 5, E and F). The alignment of horizontal bumps follows two energetically equivalent configurations—either aligned (Fig. 5E) or zigzagged (Fig. 5F). The two geometries exhibit comparable horizontal lattice differences of 0.6% (blue solid and dotted lines in Fig. 5E, top) and 0.4% (blue solid and dotted lines in Fig. 5F, top), both indicating greater residual strain within the junctions—consistent with the tilt-corrected strain maps (fig. S29). In the vertical direction, the aligned configuration shows a 1% difference (green solid and dotted lines in Fig. 5E), whereas the zigzag configuration reduces this difference to 0.4% (green solid and dotted lines in Fig. 5F). These results reveal that the geometric arrangement induces an unconventional vertical lattice mismatch, arising from local tilting in the adjacent MoS₂ domain caused by the narrow MoSe₂ region. Although both geometries coexist, the zigzag configuration might be energetically favored due to the smaller lattice mismatch it introduces (schematic illustrations in Fig. 5, E and F, bottom).

Together, these results reveal distinct junction width–dependent strain profiles and strain relaxation mechanisms in suspended membranes. Such effects are crucial in shaping surface deformation and spatial strain distribution, providing a foundation for precision strain and morphology engineering in 2D materials.

DISCUSSION

In our study, we introduce BRIGHT, an integrated method that uses a single nanobeam 4D-STEM dataset to recover a membrane’s 3D topography together with a tilt-aware surface strain tensor. Applied to suspended MoS₂-MoSe₂ lateral heterostructures, BRIGHT reveals junction width–dependent ripple morphologies and strain-relaxation pathways. These results link morphology to strain in nonplanar 2D membranes and point to routes for precision strain engineering in 2D heterostructures.

A detailed comparison between BRIGHT and other established approaches is provided in table S1, clarifying outputs, resolution/FOV, dose efficiency, computational cost, and typical trade-offs. Within these bounds, BRIGHT offers a dose-efficient route to disentangle topography and surface strain in 2D materials.

Although powerful, the present implementation reports the in-surface (planar) strain tensor on a nonplanar surface rather than the full 3D tensor, and performance is best in ultrathin, near-kinematic regimes with measurable Bragg-rod contrast and a stable set of strong reflections. According to abTEM simulations, accuracy can degrade at low dose (~10² e⁻/Ų) or high local tilt (>20°). Our experimental data were acquired near the boundary of these conditions, with a dose of ~1.7 × 10² e⁻/Ų and local tilts near 20°. Nevertheless, the topography reconstruction still shows well-converged results and physically reasonable structures. Minor contrast variations observed in strain maps near ripple walls are likely due to a slight underestimation of the tilt. For better quantitative analysis, a higher electron dose (~10³ e⁻/Ų) is recommended. For application to other ultrathin materials, reliable performance hinges on selecting six to eight strong reflections (second-order peaks are effective for TMDs), configuring acquisition to meet sampling criteria (≥5 to 8 scan steps across the smallest feature to faithfully resolve), and validating or generating the tilt-intensity library appropriate to the target material and thickness.

MATERIALS AND METHODS

Nanobeam 4D-STEM

Nanobeam 4D-STEM—also referred to as scanning electron nanodiffraction (SED) or nanobeam electron diffraction (NBED)—differs from the more common Å-probe, large-convergence form of 4D-STEM by using a small convergence angle (<2 mrad) and a nanometer-sized probe, producing sharp Bragg spots rather than disks. Experimental nanobeam 4D-STEM data for the MoS₂-MoSe₂ lateral heterojunction examined in this study were collected using an electron microscope pixel array detector (EMPAD) (35) (schematic shown in Fig. 1) installed on an FEI Titan Themis³ S/TEM operated at 80 keV. The convergence semiangle was 0.5 mrad, with a camera length of 720 mm, spot size setting 8, monochromator setting of 50, and condenser lens aperture set to 20 μm. Data were acquired at 150,000× magnification for 256 × 256 real-space scanning positions, with each probe position corresponding to a 128 × 128-pixel diffraction pattern recorded by the EMPAD. The total accumulated dose was ~1.7 × 10² e⁻/Ų. The 4D-STEM dataset (4.06 GB) can be downloaded through Zenodo: zenodo.org/records/17246822.

The 4D-STEM data were processed with the open-source 4Denoise (v1.0.0) Python package (available on GitHub through github.com/mirelesadan/4denoise/tree/v1.0.0), which implements the full BRIGHT pipeline: center-beam alignment, Bragg-peak detection, center-of-mass (c.m.) localization, intensity integration, tilt mapping using kinematic-diffraction simulation data, gradient-based height reconstruction, planar-strain mapping on the recovered surface, and tilt-correction of apparent strain. Before intensity integration, the integration window for each selected Bragg peak is recentered at the peak’s c.m. and integration is done within a sufficiently large circular aperture (with background subtraction when an amorphous film is present). The six integrated intensities from the second-order reflections formed the observation vector used to determine the best-fit elevation/azimuth angles by matching to the simulation intensity library; to assess the sensitivity of this step to the choice of loss function, we also tested alternative metric such as cosine similarity in place of the L² norm. The comparative results are presented in figs. S21 and S30.

Multislice simulations

Multislice electron diffraction simulations were conducted in the Python package abTEM (32) using an atomic structure model of MoSe₂ from The Materials Project (provided as a .cif file in the GitHub repository), expanded to form a simulation lattice measuring ~82 Å × 82 Å. The simulations were executed at an 80-keV electron beam energy, using a convergence semiangle of 1.5 mrad to replicate the experimental Bragg peak dimensions (refer to fig. S15). The computational parameters included a slice thickness of 0.5 Å, potential and probe sampling intervals both set to 0.2 Å, 300 grid points per slice, and a maximum scattering angle limited to 42°. All simulations were performed on a personal computer equipped with an NVIDIA GeForce RTX 4070 Laptop GPU and a 13th Gen Intel Core i7-13700H CPU operating at 2.40 GHz.

Supplementary Materials

The PDF file includes:

Supplementary Text

Figs. S1 to S30

Table S1

Legend for movie S1

Other Supplementary Material for this manuscript includes the following:

Movie S1