Effective Dielectric Function of a van der Waals Heterostructure Made of Two-Dimensional Materials Separated by Insulating Layers

Electron energy loss spectroscopy (EELS) is a commonly used experimental technique for investigating electronic and plasmonic properties of two-dimensional (2D) materials and van der Waals (vdW) heterostructures [1-3]. Following Ref. [4], we use the continued fraction (CF) method to derive a general expression for the effective 2D dielectric function of a vdW heterostructure made of 2D materials separated by insulating (isotropic or anisotropic) layers. The first objective is to obtain the EEL spectrum of such materials and (if possible) compare it with the available experimental data. We have extensive experience in the theoretical modeling of the experimental EELS data for free-standing (single and multilayer) graphene sheets obtained by scanning transmission electron microscope [5,6], as well as in the theoretical modeling of the experimental EELS data for monolayer graphene supported by different substrates [7-9]. As an implementation of the CF method, we evaluate the wake potential produced by a particle moving parallel to two graphene sheets separated by an isotropic insulator. For the first time, we assign the finite size of the gap between graphene and insulator in a graphene-insulator-graphene composite system. The second objective is to explore the effects of the graphene-insulator distance on the hybridization between the plasmon modes in graphene and phonon modes in the insulating substrate.