Geometrical properties of strained and twisted moir\'e heterostructures

The experimental observations of many interaction-driven electronic phases in moir\'e superlattices have stimulated intense theoretical and experimental efforts to understand and engineer these correlated physics. Strain is a powerful tool for manipulating and controlling the geometrical and electronic structures of moir\'e superlattices. This review provides a comprehensive introduction to the geometry of strained moir\'e superlattices. First, starting from the linear elasticity theory, we briefly introduce the general formalism of small deformations in two-dimensional materials, and discuss the particular cases of uniaxial, shear and biaxial strain. Then, we apply the theory to twisted and strained moir\'e materials, mainly focusing on the hexagonal homobilayers, hexagonal heterobilayers and monoclinic lattices. Special moir\'e geometries, like the quasi-unidimensional patterns, square patterns and hexagonal, are theoretically predicted by manipulating the strain and twist. Finally, we review recently developed strain techniques and the special moir\'e geometries realized via these approaches. This review aims at equipping the reader with a robust understanding on the description and implementation of strain in moir\'e materials, as well as highlight some major breakthroughs in this active field.