3D mapping of defects and moiré corrugations via electron ptychography atomic coordinate retrieval

Abstract

Defects and reconstructions in two-dimensional (2D) moiré materials cause out-of-plane deformations that strongly modify their electronic properties but are difficult to experimentally access. Here, we solve the 3D atomic coordinates of twisted bilayer WSe₂ with picometer-scale accuracy using multislice electron ptychography (MEP) acquired from a single orientation. The resulting atomic models individually visualize each of the six atomic planes, revealing the curvature of each WSe₂ layer, variations in the interlayer spacing, and the 3D locations of individual vacancies, which lie exclusively in the outer Se planes. We also observe an unexpected type of structural disorder consisting of mixed bending- and breathing-type moiré-induced corrugations that should strongly affect the emergent electronic properties. Broadly, our methods generate 3D atom-by-atom models of a 2D heterointerface from data acquired in about 30 seconds, methods that should unlock routine access to 3D atomic information in 2D systems and catalyze design methods to control out-of-plane deformations.

3D atomic coordinates from single-orientation electron ptychography reveal disordered corrugations in a 2D twisted bilayer.

INTRODUCTION

Three-dimensional (3D) structural relaxations have profound effects on the properties of interfaces across materials systems. For example, in moiré materials such as bilayers of graphene or transition metal dichalcogenides (TMDCs) with small interlayer twists, the atoms displace in-plane to increase regions of low-energy stacking and reduce areas of the higher-energy stacking (1–6). These structural relaxations strongly modify the properties of 2D moiré systems from those of rigid layers; they have been correlated to the emergence of flat bands (7, 8) and exotic electronic phases such as superconductivity in graphene (9) and twisted bilayer WSe₂ (10, 11). The in-plane relaxations are also disordered on length scales of tens to hundreds of nanometers (4, 12), creating spatial variations in the atomic and electronic structures that are unavoidable in moiré systems (13).

Yet, in-plane behavior does not fully capture the structural relaxations of moiré systems. Modeling of 2D twisted bilayers predicts out-of-plane structural relaxations, periodic patterns of corrugation and interlayer spacing variations (3, 14–17) that accompany the in-plane displacements. Experimentally, these out-of-plane displacements are difficult to access. Scanning probe methods such as scanning tunneling microscopy (STM) can detect corrugations only in the topmost atomic layer (18, 19). Electron tomography (ET), where 3D structures are reconstructed from a series of scanning transmission electron microscopy (STEM) images acquired from multiple tilt angles, can achieve angstrom-scale resolution in 3D but reaches its highest 3D resolution for specific sample geometries, such as nanoparticles or needle-shaped specimens (20–22). For 2D materials, ET has been used to probe buckling at defects in monolayers (23, 24), but the planar geometry of 2D materials ensures the presence of missing wedge effects, limiting ET’s out-of-plane resolution (25) and making it difficult to probe subangstrom variations in interlayer spacing.

Advances in multislice electron ptychography (MEP) present an alternate route for obtaining 3D atomic structure (26, 27). Electron ptychography enables exceptionally high spatial resolutions, up to 0.2 Å in-plane (27, 28), with depth resolutions of 2 to 4 nm (27, 29–31). While the typical nanoscale depth resolution of MEP is not sufficient for atomic resolution in 3D, standard MEP uses a completely general approach without any prior information about the sample. By augmenting 3D information from MEP with simple assumptions such as atomicity (that the sample is composed of discrete atoms) and known atomic spacings, it should be possible to extract considerably more 3D information, unlocking fast and powerful routes to 3D atomic structures.

RESULTS

Multislice ptychography with subnanometer depth resolution

In this work, we use MEP augmented with priors to reveal the 3D atomic structure of twisted bilayer WSe₂. Figure 1 shows how we use MEP to distinguish individual atoms in 3D and quantify the resulting depth resolution. We fabricate 6°-twisted bilayer WSe₂ (0° refers to parallel or AB stacking) using gold-assisted monolayer exfoliation followed by bilayer assembly using a polydimethylsiloxane lens pickup method using the methods described in (28) (see the Supplementary Text). Next, we collect 4D-STEM data with the sample oriented perpendicular to the electron beam (illustrated in Fig. 1A) and conduct an MEP reconstruction (27, 32) (see Materials and Methods). In MEP, the sample is represented as a stack of thin slices, and the reconstruction solves for a 3D object whose phase represents the sample potential.

Fig. 1. Distinguishing individual atoms using subnanometer depth-resolved MEP from a single orientation.

(A and C) Schematics of 2D projected atomic coordinates of a region of bilayer WSe₂ with near-AA stacking before (A) and after (C) tilting. Only six atomic columns are resolved in the 2D projection without tilt, while all 18 atoms are resolved in the 2D projection with tilt. (B and D) Max-phase image of twisted bilayer WSe₂ from an experimental multislice electron ptychographic reconstruction without (B) and with (D) a 15° sample tilt. Gray hexagons mark sample regions similar to those in (A) and (C). (E, F, H, and I) Depth slices from ptychographic reconstructions without [(E) and (F)] and with [(H) and (I)] tilt, where the contrast is dominated by either the top [(E) and (H)] or the bottom [(F) and (I)] WSe₂ layer. Scale bars, 5 Å. (G and J) Cross sections of the region marked by dashed rectangles in (B) and (D), respectively. The labels indicate the atomic plane and layer number (e.g., Se₁₁ represents a Se atom in top atomic plane in layer 1 and W₁ represents a W atom in layer 1). In (G), atoms in different WSe₂ layers can be distinguished by their shifts in depth, but Se planes in the same WSe₂ layer are not resolved. In contrast, in (J), all six atomic planes can be distinguished. Because the depth resolution is considerably worse than the lateral resolution, the atoms in the cross section appear elongated in the z direction.

Figure 1B shows an MEP image of 6°-twisted bilayer WSe₂ (full field of view images shown in fig. S1). Specifically, Fig. 1B plots a “max-phase” image, a 2D projection of the 3D phase where each (x, y) pixel shows the highest phase value in its depth stack. This image has an in-plane resolution of 0.31 Å, as measured from the information transfer limit (see fig. S2), comparable to the highest quality reconstructions of 2D materials (28, 33, 34). By leveraging the depth resolution of MEP, we separately image each WSe₂ monolayer. Figure 1 (E and F) shows MEP slices at different z depths, where the +z direction is defined along the electron beam propagation (see fig. S3 for all slices). Unlike the full 2D projection (Fig. 1B), the individual depth slices distinguish the two WSe₂ layers: Fig. 1E is dominated by the top layer, and Fig. 1F by the bottom one.

To visualize the relative z positions of the WSe₂ layers, Fig. 1G plots a cross section of the 3D phase. We estimate a depth resolution of 7.5 Å by measuring the full width at half maximum in the z direction for W atoms (see fig. S5). Notably, our depth resolution significantly exceeds prior experimental MEP (2 to 4 nm) (27, 29–31). We attribute this high depth resolution to the thinness of our sample, which enables high-quality ptychographic reconstructions from diffraction patterns with higher maximum collection angles, conditions predicted to improve MEP depth resolution (35, 36). Yet, we still cannot directly resolve pairs of Se atoms, which are separated by 3.3 Å along the crystal’s c axis and occupy the same lateral (x, y) position, so Se atom pairs appear as a single continuous column.

To individually locate each atom, we tilt the sample by 15° before acquiring 4D-STEM data (Fig. 1C). By tilting, we leverage the high lateral resolution of MEP to compensate for its lower depth resolution. Figure 1 (A and C) illustrates this concept by comparing 2D projections of an AA-stacked bilayer viewed top-down (1A) versus at a tilt (1C). Whereas Se atoms are directly on top of one another when viewed top-down, tilting displaces the atoms in x, y. Figure 1 (D, H, and I) shows the experimental max-phase image of the tilted sample (1D) and depth slices distinguishing the two WSe₂ layers (1H and 1I; all slices are shown in fig. S4). In the tilted view, individual Se atoms can now be distinguished. Figure 1J shows a cross section of the 3D phase, where each atom can be individually located and unambiguously assigned to one of the six atomic planes using shifts in their relative z positions.

Retrieving 3D atomic coordinates for twisted bilayer WSe₂

Next, we develop and benchmark a workflow for 3D atomic coordinate retrieval for twisted bilayer WSe₂ from a single, tilted acquisition. We create an initial guess of the 3D atomic coordinates by finding peaks in the 3D phase. Figure 2B shows top and oblique views of coordinates from an MEP reconstruction of a simulated 4D-STEM dataset of 6°-twisted bilayer WSe₂. The 4D-STEM simulations use coordinates obtained from molecular dynamics (MD) simulations (see Materials and Methods and the Supplementary Text). By comparing retrieved coordinates against the ground truth, we see that atom-finding from MEP produces high-quality (x, y) atomic coordinates with a root mean square (RMS) error of 4.5 pm (Fig. 2D, top left) but noisy z coordinates with a RMS error of 81 pm (Fig. 2D, top right). The larger z errors are expected given our limited z resolution compared to x, y.

Fig. 2. 3D coordinate retrieval achieves picometer-scale precision in all dimensions after refinement.

(A and B) Top-down view (top) and oblique view (bottom) of raw atomic coordinates taken directly from the ptychographic phase (A) and after refinement (B). Atoms are colored by the atomic layer that they belong to. A corresponding ptychography image is shown in fig. S6. Scale bars, 5 Å. Grid spacing in the oblique views is 1 nm. (C) Comparison of the true (circles, radius of 30 pm) and refined (dots) atomic positions for the region corresponding to the dashed gray box in (B). These data show close correspondence between the true and recovered positions. (D) Histograms of the in-plane and z errors of the 1733 refined atomic coordinates, before (top row) and after (bottom row) coordinate refinement. The x and y directions are perpendicular to the beam direction. In-plane error is defined as $\sqrt{d{x}^2+d{y}^2}$, where dx and dy are the errors in the x and y directions. Before refinement, the in-plane RMS error is 4.5 pm and the z RMS error is 81 pm. After refinement, the in-plane RMS error is 4.5 pm and the z RMS error is 5.3 pm.

Next, we refine the 3D coordinates to be both physically plausible and consistent with our ptychography data. The refinement minimizes a loss function using the x, y, and, to a lesser extent, the z coordinates from ptychography as well as prior knowledge about the structure of the sample, such as the mean W─Se bond length and average interlayer spacing (see the Supplementary Text for details and uncertainty quantification). Because the x, y coordinates are tightly constrained by the ptychography data, the refinement mainly affects the z coordinates, which change by as much as angstroms while the x, y coordinates shift by picometers or less. Our approach is relatively insensitive to the accuracy of the prior knowledge about the crystal structure (see the Supplementary Text and figs. S13 to S15 for robustness analysis).

Figure 2D (bottom left) shows the errors of the x, y coordinates after refinement. Their 4.5-pm RMS error is nearly identical to before the refinement, indicating that the x, y coordinates maintain high quality during refinement. Meanwhile, the refinement greatly improves the RMS error of the z coordinates to 5.3 pm (Fig. 2D, right column), comparable to that of the in-plane coordinates and an order of magnitude better than prior single-orientation methods (37, 38). These accuracy values represent optimistic upper bounds because our 4D-STEM simulations do not include experimental factors that degrade ptychography reconstructions, such as sample drift and incoherence. Nevertheless, the simulations show that our methods can obtain 3D atomic coordinates with picometer-level accuracy using data acquired from a single-view direction, demonstrating the power of augmenting MEP data with simple materials constraints. By coupling the 3D information from MEP with our refinement process, we recover 3D atomic coordinates from a single dataset acquired in about 30 s, whereas ET often requires dozens of images acquired over the course of hours. Our coordinate extraction and refinement process could be further streamlined by advancements in electron ptychography reconstruction methods or more advanced loss functions, to even further improve its speed, level of automation, and accuracy.

Mapping defects in each atomic layer

In Fig. 3, we extract 3D atomic coordinates from experiment and visualize individual vacancies. Figure 3 (A and B) shows face-on and edge-on views of refined atomic coordinates from 2.4°-twisted bilayer WSe₂ (corresponding ptychography reconstruction is shown in figs. S7 and S8). Figure 3C shows the atoms in each layer. Only the outer two Se planes, labeled Se₁₁ and Se₂₂, contain vacancies. These planes contain 16 vacancies each, representing 5.0% of the Se atoms in those layers.

Fig. 3. Layer-by-layer imaging and vacancy identification in 2.4°-twisted bilayer WSe₂.

(A and B) Face-on (A) and edge-on (B) views of the atomic structure extracted from MEP, after 3D coordinate refinement. These data are acquired at a 15° tilt but untilted for the visualization. The corresponding MEP-phase images are shown in figs. S7 and S8. (C) Face-on view of atoms in each layer. Se vacancies are visible in the exterior Se₁₁ and Se₂₂ layers, but not in the interior Se₁₂ and Se₂₁ layers. These data indicate that the vacancies are not intrinsic to the bulk source WSe₂ crystals but, instead, are generated during TEM sample preparation or imaging. Scale bars, 5 Å for all panels.

Over the last decade, substantial efforts have been made to understand point defects in 2D TMDCs because their type and density affect key properties such as the photoluminescence yield and charge doping and because point defects themselves have been used for quantum emission, and catalysis, and memory devices (39, 40). (S)TEM, STM, and other scanning probe microscopies can visualize point defects in TMDCs, each with advantages and drawbacks. For example, the identities, 2D locations, and strain fields of point defects in monolayer TMDCs have measured with annular dark-field STEM (41–43). These studies show that chalcogen vacancies are the most prevalent point defect in synthesized TMDCs, but understanding whether chalcogen vacancies observed in (S)TEM measurements are intrinsic is challenging because electron beam damage causes vacancy formation at the doses typically used for atomic resolution imaging. Although STM and conductive atomic force microscopy minimize sample damage, it remains challenging to use those techniques to assign defect structures (44–46).

Using the 3D information in our model, we can readily identify point defects and understand their origins. Because the vacancies occur only in the outer Se layers, we believe that they indicate sample damage during sample preparation or data acquisition (47), rather than intrinsic defects from the growth process. In the region shown, we used an accumulated dose of 1.1 × 10⁶ electrons/Ų, typical for atomic resolution (S)TEM and sufficient to create Se vacancies (48) (see Materials and Methods for discussion of vacancy counts). Although we operated below the knock-on damage threshold (49, 50), 2D material samples like WSe₂ are susceptible to additional damage mechanisms such as ionization (51), chemical etching (52), and a combination of ballistic energy transfer and electronic excitations (53). Because the individual WSe₂ layers are exfoliated from a bulk crystal grown by chemical vapor transport (CVT), intrinsic defects formed during the growth of the source crystal would be expected to occur with equal probability throughout all atomic layers. These results substantiate STM reports that intrinsic Se vacancies are relatively rare in CVT-grown crystals (44). The discrepancy with some STEM studies (42), which report a higher prevalence of Se vacancies, may arise because these measurements can be influenced by vacancies introduced during sample preparation (for example, due to chemical etching, annealing, or plasma cleaning) rather than reflecting the intrinsic defect density of the bulk crystal.

Mapping out-of-plane structural reconstruction in twisted bilayer WSe₂

In Fig. 4, we investigate the out-of-plane structural reconstructions of 6°-twisted bilayer WSe₂. Figure 4 (A and E) shows top-down views of atomic structures obtained from experiment (Fig. 4A) and MD simulations (Fig. 4E). In each structure, we observe three AA-like stacking regions, indicated by dashed gray circles; in these regions, atoms in one layer nearly overlie atoms of the same type in the other layer. These regions are separated by AB stacking, where the metal atoms from one layer overlap the chalcogen atoms from the other, and solitons, stacking faults that localize shear and strain from the structural reconstruction (1–3, 16, 54). Figure 4 (B, D, and F) shows contour maps (Fig. 4, B and F) and corresponding line profiles (Fig. 4D) of the interlayer spacing through different stacking orders. In experimental data, the interlayer spacings are greatest in the AA regions, reaching up to 6.7 to 6.9 Å, 0.4 to 0.6 Å larger than that of the AB regions closest to them (6.2 to 6.4 Å). The solitons also exhibit a slightly larger (by 0.1 Å) interlayer spacing compared to AB. These data are in excellent quantitative agreement with MD simulations, where the interlayer spacings reach a maximum of 6.9 Å in the AA regions and minimum of 6.4 Å in AB regions (Fig. 4, E and F). Similar results have been reported in prior theoretical studies (3, 14–16) but have been difficult to validate because of a lack of methods to probe local changes in interlayer spacing. These variations occur because the AA and soliton stackings are energetically unfavorable compared to the AB stacking configuration (1, 2, 7, 55), resulting in local interlayer repulsion.

Fig. 4. Out-of-plane deformations of twisted bilayer WSe₂.

(A and E) 2D projected atomic coordinates of 6°-twisted WSe₂ obtained from experiment (A) and MD simulations (E). To determine the stacking order, all atoms are projected to a common surface defined by one of the W layers (see Materials and Methods). Locations with approximately AA stacking are labeled with dashed circles. (B and F) Contour plots of interlayer spacing between the two W layers in units of angstroms. AA stacked regions are labeled with dashed red circles. (D) A line profile shows that the interlayer spacing is greatest in the AA-like stacking regions, lowest in the AB-like stacking regions, and slightly larger in soliton regions in both simulation (dashed line) and experiment (solid line). (C and G) Mean curvature of top (top) and bottom (bottom) W atomic layers. The curvature of each AA region is also illustrated below (A) and above (E). In the MD-simulated structures, the mean curvature of AA regions takes opposite sign in the two layers, indicating that the AA regions bulge outward in a breathing mode. In experiment, we observe breathing modes (regions AA₁ and AA₃) as well as bending modes (region AA₂), where the two layers bend in the same direction. Scale bars, 5 Å for all panels.

Last, we probe out-of-plane corrugations in twisted bilayer WSe₂. To understand how the moiré structure bends to accommodate changes in interlayer spacing, we examine the sign of the mean curvature of the AA regions. The mean curvature measures how a surface locally bends; it is the average at each point of the curvature in two principal directions (see Materials and Methods for description of this analysis and fig. S11 for maps of the principal curvatures). We plot the mean curvature of the top and bottom W planes for experiment (Fig. 4C) and MD simulations (Fig. 4G). In MD simulations (Fig. 4G), the mean curvature at each of the three AA regions is negative (blue), indicating domes in the top WSe₂ layer, but positive (red), indicating bowls in the bottom WSe₂ layer. This behavior corresponds to an “out-of-phase” or “breathing mode” deformation, where the local increase in interlayer spacing at AA regions occurs by bulging the two layers outward. We qualitatively recover the same curvatures in ptychographic reconstructions of MD structures (fig. S12). Figure 4C shows the experimental curvatures for the 6°-twisted WSe₂. In the regions labeled AA₁ and AA₃, both WSe₂ layers bulge outward, similar to the MD simulations in Fig. 4 (E to G). In contrast, at AA₂, the top and bottom layers both appear red, indicating that they are curved in the same direction, forming an “in-phase” or “bending mode” deformation.

Theoretical studies on twisted bilayer 2D materials have predicted both in-phase and out-of-phase corrugations (17, 54, 56, 57) similar to those observed in our experiment. These predictions show that the twist angle and layer composition determine whether in- or out-of-phase modes represent the minimum-energy configuration (54, 56). Studies of twisted TMDC homobilayers generally predict out-of-phase corrugations (17, 58) at every AA region. In contrast, mixing of in- and out-of-phase modes in close proximity, as we observed, has neither been previously predicted nor experimentally observed. We attribute this mixing to the small energy difference between the two states. For example, for twisted bilayer graphene, this energy difference is ~0.01 meV per atom for few-degree twists (56). Furthermore, strain from sample fabrication and kinetic trapping may alter the sample from its thermodynamically stable configuration. These results are important because out-of-plane reconstructions have been tied to emergent superconductivity and changes in the electronic filling and vibrational states of moiré systems (56). The out-of-plane reconstructions are spatially nonuniform, representing a distinct type of disorder in moiré systems.

DISCUSSION

To conclude, we use MEP to retrieve 3D atomic coordinates of twisted WSe₂ bilayers, locate single vacancies in 3D, and map corrugations and subangstrom changes in the interlayer spacing due to moiré reconstructions. These methods demonstrate MEP’s ability to unlock previously inaccessible out-of-plane deformations at buried interfaces. Moreover, our methods markedly accelerate atomic-level 3D imaging for 2D materials, reducing acquisition times from hours to seconds and potentially unlocking in situ measurements of atomic structures and their transformations in all three dimensions. For 2D materials, our work uncovers disordered 3D corrugation modes that should be important to understanding and controlling the electronic properties of moiré systems.

MATERIALS AND METHODS

Sample preparation

Bulk WSe₂ was purchased from HQ Graphene. Monolayers of WSe₂ were exfoliated using a modified gold-assisted exfoliation method (59, 60), assembled into twisted bilayers, and suspended over TEM grids with ~2-μm-diameter holes. See the Supplementary Text for more details.

Data acquisition

We characterized the samples using a Thermo Fisher Scientific Themis Z aberration-corrected scanning transmission electron microscope. The microscope was operated at 80 kV to minimize electron beam damage. We used a defocused electron probe (7.5- and 10-nm overfocus) with a probe current of 7 to 15 pA and convergence semiangle of 25 mrad to acquire 4D-STEM datasets on a first-generation EMPAD detector with 128 × 128 scan positions using a scan step size of 0.43 Å and a dwell time of 1 ms. As timed with a stopwatch, a 4D-STEM dataset with 128 × 128 scan positions took ~33.5 s to record, corresponding to 2.0 ms per diffraction pattern: This is mostly accounted for by the 1-ms dwell time used and the 0.86-ms per-frame read time (61). We used these values to estimate an effective Se-vacancy generation cross section of 1.3 barns, which is similar to values previously reported for monolayer MoSe₂ (52). The camera length is 230 mm corresponding to a maximum scattering k-vector of 2.5 Å⁻¹. For the tilted experiments (Figs. 1D, 3, and 4A), an alpha tilt of 15° was applied to the sample holder.

Ptychography

Each 4D-STEM dataset consists of an array of convergent beam electron diffraction patterns. A multislice ptychographic reconstruction of the sample is produced by matching the experimental diffraction patterns to simulations using a forward model of electron scattering through a 3D sample. We performed ptychographic reconstructions using the fold_slice package (27, 62), a fork of the PtychoShelves package (63). For all reconstructions, the multislice maximum likelihood solver was used (32, 64). After every iteration of the reconstruction, the object was regularized according to the weighting function described in (29), with the layer regularization parameter γ set to 0.1. Diffraction patterns were padded to increase the real-space sampling of the ptychographic reconstruction. Probe position correction was enabled. For reconstructions on experimental data, multiple probe modes were used to model partial coherence (65).

Parameters for the final reconstructions are as follows:

1) Figure 1B: Diffraction patterns padded to 256 × 256 pixels; 8 probe modes, 19 slices, 1.535-Å spacing between slices.

2) Figure 3: Diffraction patterns padded to 192 × 192 pixels; 8 probe modes, 23 slices, 1.5-Å spacing between slices.

3) Figure 2 and fig. S12: Diffraction patterns padded to 192 × 192 pixels; 1 probe mode, 22 slices, 1.6-Å spacing between slices. This reconstruction was on simulated data.

4) Figures 1D and 4A: Diffraction patterns padded to 192 × 192 pixels; 10 probe modes, 22 slices, 1.6-Å spacing between slices.

The max-phase image of Fig. 1B was formed by taking the 7th through 19th object slices, which contained the WSe₂ sample and taking the maximum phase value along the z direction (the direction of the beam). The max-phase image of Fig. 1D was formed from the 3rd through 20th object slices.

Coordinate retrieval

Generally, the peaks in the phase of the 3D object function produced by multislice ptychography correspond to atoms. An initial list of peaks in the 3D phase is generated by listing voxels that are local maxima, that is, voxels that have the greatest phase value in their local 3 × 3 × 3-voxel neighborhood. Next, a threshold is applied to remove dim voxels that are only background noise. Then, for each of these peaks, a more precise peak position with subpixel precision is found by looking for a local maximum in the tricubic interpolation of the phase (see the Supplementary Text) around the voxel. To do this, the tricubic interpolation of the phase is represented as a piecewise function of 3D position, and our program compares the interpolated phase values of all nearby points where the gradient of this function is zero. The point with the highest phase is deemed to be the more precise peak position.

This approach works for locating most isolated atoms but fails for atoms that are overlapping in the 3D phase produced by ptychography, as well as for atoms that are close to the top or bottom face of the reconstruction volume. For such atoms, we resort to estimating atom positions by eye. We also manually add the locations of the lattice site wherever there is a vacancy.

Next, we label each position by the atomic layer that it belongs to. This is made possible by the tilted sample orientation and depth resolution of the multislice reconstruction. Last, the coordinates are refined as described in the Supplementary Text.

Analysis of coordinates

We constructed for each W layer a triangular mesh from the W coordinates using the Open3D package (66). In Fig. 4 (A and E), for the sake of illustrating the actual stacking order, each atom is projected to its nearest point within the mesh surface passing through the higher W layer. Rigidly rotating the atomic model is not enough to accurately show the stacking order for Fig. 4A because the WSe₂ layers are curved.

Interlayer spacing

To calculate interlayer spacing as a function of position, we sample a grid of points on the triangular mesh of the higher W layer and compute the closest distance between these points and the lower W mesh. Mean interlayer spacing is calculated as the average over this grid of points.

Estimate of curvature

For a point on a surface, there exist two principal directions where curvature is at a maximum or minimum, and, for each principal direction, there is a principal curvature value defined as the reciprocal of the radius of the osculating circle passing through the point along that direction. Starting from the triangular meshes of each W layer, we use the method described in (67) to calculate the normal vector, principal curvature values, and principal directions at each W atom. The mean curvature values (the average of the two principal curvature values) are interpolated to produce the plots in Fig. 4 (C and G) and fig. S12 (C and D).

MD simulations

We perform classical MD simulations via LAMMPS (68) using a hexagonal unit cell of 6°-twisted bilayer WSe₂. We use the Stillinger-Weber potential to describe the intralayer interactions and Kolmogorov-Crespi potential (55) for interlayer interactions. We use the parameters for the aforementioned force fields from (55). We first equilibrate the structure at 300 K for 100 ps with a Nose-Hoover thermostat using a time step of 0.5 fs. We apply a tether to the center of mass of the system to resolve the flying ice cube problem (69). After equilibration, the system is relaxed for another 100 ps to transition into microcanonical ensemble. The simulation continues for 3 ns, and we calculate an average atomic position.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S15

References