Strain, stress and rotation fields, and energetic features of twisted 2D materials.

Twisted two-dimensional (2D) materials exhibit remarkable quantum properties due to Moiré-pattern-induced electronic band structure change, highly sensitive to nanoscale deformation from atomic-scale reconstruction. The absence of an analytical model linking deformation to twist angle limits property tunability. We developed a theoretical model characterizing deformation and energetics of twisted 2D materials. As the twist angle increases, Moiré patterns evolve from triangular partial-dislocation networks to hexagonal domains with domain walls. Using anisotropic dislocation theory, we derived analytical expressions for local rotation, strain and stress fields at small twist angles, and a non-linear formula relating energy density to twist angle, capturing the transition from rapid growth to near saturation (0°-30°). Theoretical predictions agree well with previous experimental and computational studies and our atomistic simulations for twisted bilayer graphene, hexagonal boron nitride and trilayer graphene. This work provides a theoretical foundation for twist-angle control of quantum properties, enabling design of 2D quantum devices.