Electronic properties of incommensurate atomic layers

We present a brief theoretical overview of electronic properties of incommensurate multilayer systems, i.e., a pair of two-dimensional atomic layers stacked in an arbitrary orientation. We introduce the general theoretical scheme to describe the interlayer interaction between incommensurate crystal structures, and apply the formula to two specific examples, the twisted bilayer graphene and graphene-hBN composite bilayer. In each case, we calculate the electronic band structure and demonstrate that the low-energy electronic properties are significantly modified by the interlayer interaction, particularly when the two lattice structures are close to each other. We also study the energy spectrum and the quantum Hall effect in magnetic fields, where we see that the spectral structure exhibits a fractal nature, as known as the Hofstadter butterfly. We argue about the optical absorption properties of the twisted bilayer graphene and show that the interlayer interaction gives rise to the characteristic spectroscopic features in zero magnetic field and also in strong magnetic field.