Progress on band structure engineering of twisted bilayer and two-dimensional moir\'e heterostructures

Artificially constructed van der Waals heterostructures (vdWHs) provide an ideal platform for realizing emerging quantum phenomena in condensed matter physics. Two methods for building vdWHs have been developed: stacking two-dimensional (2D) materials into a bilayer structure with different lattice constants, or with different orientations. The interlayer coupling stemming from commensurate or incommensurate superlattice pattern plays an important role in vdWHs for modulating the band structures and generating new electronic states. In this article, we review a series of novel quantum states discovered in two model vdWH systems -- graphene/hexagonal boron nitride (hBN) hetero-bilayer and twisted bilayer graphene (tBLG), and discuss how the electronic structures are modified by such stacking and twisting. We also provide perspectives for future studies on hetero-bilayer materials, from which an expansion of 2D material phase library is expected.