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Interlayer rotation angle couples strongly to the electronic states oftwisted van der Waals layers. However, not every angle is energeticallyfavorable. Recent experiments on rotation-tunable electronics reveal theexistence of a discrete set of angles at which the rotation-tunable electronicsassume the most stable configurations. Nevertheless, a quantitative map forlocating these intrinsically preferred twist angles in twisted bilayer systemhas not been available, posing challenges for the on-demand design of twistedelectronics that are intrinsically stable at desired twist angles. Here wereveal a simple mapping between intrinsically preferred twist angles andgeometry of the twisted bilayer system, in the form of geometric scaling lawsfor a wide range of intrinsically preferred twist angles as a function of onlygeometric parameters of the rotating flake on a supporting layer. We revealthese scaling laws for triangular and hexagonal flakes since they frequentlyappear in chemical vapor deposition growth. We also present a general methodfor handling arbitrary flake geometry. Such dimensionless scaling laws possessuniversality 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 thatintrinsically prefers zero-approaching sequence of multiple magic-angles inbilayer graphene system can be revealed.

Selection rules of twistronic angles in 2D material flakes via dislocation theory