Surface response function of van der Waals heterostructures involving two-dimensional materials with the out-of-plane polarization for applications in electron energy loss spectroscopy
Phys. Rev. B 114, 105404 – Published 7 August, 2026
DOI: https://doi.org/10.1103/bxr4-359m
Abstract
We derive an analytical expression for the surface response function (SRF) of a large-area van der Waals heterostructure (vdWH) comprising an arbitrary sequence of atomically thin two-dimensional (2D) materials separated by insulating dielectric layers. The SRF is expressed as a continued fraction that incorporates the electronic conductivity components of each 2D material, accounting for both in-plane and out-of-plane (OOP) polarization induced by local macroscopic electric fields. Using generalized boundary conditions within the nonretarded limit, our approach aligns with established nanophotonic modeling of 2D materials. Our theory is primarily designed for electron energy loss spectroscopy (EELS) with nonpenetrating trajectories to probe the hybridization of plasmon modes in 2D materials with phonon modes in (potentially anisotropic) dielectric layers. Furthermore, our SRF is applicable to tip-based scanning near-field optical microscopy (s-SNOM). We illustrate potential applications for vdWHs containing one or two 2D materials and for a semi-infinite periodic superlattice. For the periodic superlattice, we derive explicit expressions for the surface plasmon mode and the bulk plasmon band, specifically highlighting the influence of OOP polarization of the constituent 2D materials.
