magic angle

Two-dimensional (2D) materials are crucially important nanomaterials because of their exciting multi-functional properties. However, a single-layer of 2D materials may not have a certain property adequately , or multiple application-specic properties simultaneously to the desired and optimal level. For mitigating this lacuna, a new trend has emerged recently to develop nano-scale engineered heterostruc-tures by stacking multiple layers of dierent 2D materials, wherein each of the layers could also be twisted. The vast advantage of combining single layers of dierent 2D materials with dierent twisting angles has dramatically expanded this research eld well beyond the scope of considering a 2D material mono-layer, leading to a set of multifunctional physical properties corresponding to each possible combination of number of layers, dierent 2D materials therein, stacking sequence and the twisting angle of each layer. Eective mechanical properties such as Young's moduli are generally of utmost importance for analysing the viability of such engineered nano-heterostructures in various nanoelectromechanical applications. We have proposed efficient closed-form generic formulae for the eective Young's moduli of twisted multi-layer heterostructures. Based on this physics-based analytical approach, a wide range of insightful new results are presented for twisted heterostructures, covering mono-planar and multi-planar congurations with homogeneous and heterogeneous atomic distributions.