Fig. 3: Calculated results of the distorted TBG structure with \({\boldsymbol{\theta }}=0.4{8}^{\circ }\).

a Schematic model of the moiré pattern. b Absolute magnitude of different in-plane atomic displacements (top panel) and out-of-plane displacements for the deformed system along the path MNP defined in a. For the in-plane displacements, the parameters (\(\Delta D\), \({l}_{D}\), \({\sigma }_{D}\)) are: A (0.04 nm, 5.91 nm, 1.14), B (0.07 nm, 5.91 nm, 1.14), C (0.07 nm, 3.70 nm, 1.14), D (0.07 nm, 3.70 nm, 11.44). For the out-of-plane displacements, the parameters (\(\Delta Z\), \({l}_{Z}\), \({\sigma }_{Z}\)) are: E (0.0115 nm, 4.93 nm, 0.7), F (0.023 nm, 4.93 nm, 0.7), G (0.0058 nm, 4.93 nm, 0.7). c, d Maps of the absolute magnitude of the in-plane atomic displacement \(| \Delta d|\) and out-of-plane displacement \(\Delta z\) upon deformation of rigidly TBG calculated from Eq. (1), respectively. e, f Local density of states in the AA region of deformed systems with different in-plane and out-of-plane displacements, respectively.