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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-MoSe₂transition 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. 

Pentatwinned nanostructures are key to understanding the mechanical, chemical, and structural behavior of nanomaterials owing to their unique fivefold symmetry and lattice strain from a 7.35° disclination gap between {111} twin boundaries. However, the precise equilibrium strain distributions have remained unclear because of heterogeneity among individual particles, requiring statistical analysis across large sample populations. Here, we use nanobeam four-dimensional scanning transmission electron microscopy (4D-STEM) to extract averaged strain profiles from uniformly sized, shape-identical particles, achieving high-resolution, statistically robust insights beyond single-particle noise. The strain profiles reveal how tensile, shear, and rotational components collectively compensate for the angular deficit, with particle shape–dependent local variations highlighting the importance of morphological control in synthesis. By integrating in situ heating with 4D-STEM, we captured a previously unobserved strain relaxation pathway involving the formation of periodic partial dislocations that stabilizes the strain-relieved equilibrium state. This study establishes a quantitative framework for equilibrium strain in fivefold-twinned nanostructures and offers strain engineering strategies for tailored properties. 

Not AvailableTwo-dimensional (2D) superconductors are emerging platforms supporting both strongly correlated physics and quantum information science1,2. Their reduced dimensionality, atomically flat interfaces, and high-crystallinity are particularly attractive for realizing compact lumped-element devices in superconducting circuits3–5. However, large-scale synthesis of monolayer 2D superconductors remains challenging as they are easily oxidized in air6. Here, we report an “encapsulation epitaxy” mechanism that enables the growth of large-area (>1-inch), air-stable, monolayer NbSe2 films (1L-NbSe2) and explored their potential for superconducting quantum circuits. This work represents a distinct growth phenomenon in which a 2D encapsulation layer, such as graphene or hexagonal boron nitride (hBN), pre-deposited on a 3D substrate (e.g., SiO2, Si3N4) simultaneously serves as a template for the epitaxial growth of 1L-NbSe2 underneath it at encapsulation-substrate interface and as a protective capping layer against ambient degradation. The as-grown 1L-graphene/NbSe2 heterostructures exhibit robust superconductivity (superconducting transition temperature Tc ≈ 1 K) and enhanced charge density wave (CDW; CDW transition temperature TCDW ≈ 177 K). We further demonstrate the integration of 1L-NbSe2 into superconducting circuits by developing oxidation-free transfer and superconducting edge-contact techniques. The 1L-NbSe2 in these circuits feature a measured kinetic inductance LK ≈ 0.7 nH/□, making it suitable for quantum circuits requiring elements with high kinetic inductance. This encapsulation epitaxy methodology enables the production of air-stable 2D superconductors and van der Waals heterostructures, holding the promise for wafer-scale, monolithic fabrication of superconducting quantum circuitry.