Selection rules of twistronic angles in two-dimensional material flakes via dislocation theory

Interlayer rotation angles couple strongly to the electronic states of twisted van der Waals layers. However, not every angle is energetically favorable. Recent experiments on rotation-tunable electronics reveal the existence of a discrete set of angles at which the rotation-tunable electronics assume the most stable configurations. Nevertheless, a quantitative map for locating these intrinsically preferred twist angles in a twisted bilayer system has not been available, posing challenges for the on-demand design of twisted electronics that are intrinsically stable at desired twist angles. Here we reveal a simple mapping between intrinsically preferred twist angles and the geometry of the twisted bilayer system, in the form of geometric scaling laws for a wide range of intrinsically preferred twist angles as a function of only geometric parameters of the rotating flake on a supporting layer. We reveal these scaling laws for triangular and hexagonal flakes since they frequently appear in chemical vapor deposition growth. We also present a general method for handling arbitrary flake geometry. Such dimensionless scaling laws possess universality for all kinds of two-dimensional material bilayer systems, providing abundant opportunities for the on-demand design of intrinsic "twistronics."For example, the set of increasing magic sizes that intrinsically prefer a zero-approaching sequence of multiple magic angles in a bilayer graphene system can be revealed.