Lattice-Mismatch-Driven Small-Angle Moiré Twists in Epitaxially Grown 2D Vertical Layered Heterostructures.

Artificially introduced small twist angles at the interfaces of vertical layered heterostructures (VLHs) have allowed deterministic tuning of electronic and optical properties such as strongly correlated electronic phases and Moiré excitons. But creating a Moiré twist in van der Waals (vdWs) systems by manual stacking is challenging in reproducibility, uniformity, and accuracy of the twist angle, which hinders future studies. Here, it is demonstrated that contrary to the commonly believed 0°-orientation in vdWs epitaxy, these VLHs show small twist angles controlled by the low-order commensurate phase with low energy and local atomic relaxation. A commensurate multilevel map is proposed to predict possible orientations. Remarkably, high-mismatch VLHs show discrete and sometimes non-zero twist angles dependent on their natural mismatch value. Such framework is experimentally confirmed in five epitaxially grown VLHs under high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM), and can provide significant insights for large-scale engineering of twist angle in VLHs.