The derivation of the one-dimensional analytical GSP model for calculating the EELS signal is presented in detail in
Supplementary Note 1.
The EELS signal from the top and bottom surface plasmons (
Fig. 2a) is calculated using a 2D model in
COMSOL Multiphysics (version 5.1), where the electron beam is set to travel parallel to the metal-dielectric interfaces. The electron beam is modelled as an out-of-plane line current with wave number
ke=
ω/
v, where
v is the electron speed. Although the electron beam travels perpendicular to the metal-dielectric interfaces in the experiments, our only interest is in evaluating the EELS signal stemming from the excitation of surface plasmons at their non-retarded frequencies. This can be achieved using the 2D implementation by setting a low electron velocity (here,
v=0.5
c) and positioning the electron beam to always travel in the vacuum part of the domain (a distance of 5 nm to the outer interface is chosen). In this manner, the main contribution
to the theoretical EELS signal will be from the excitation of surface plasmons at their non-retarded energies, just as in the experiments. In all calculations, we use the permittivity for silver from ref. 55 , while the permittivity for silicon nitride is from ref. 63 . A thickness of 10 nm is used to model the silicon nitride layer from the TEM membrane.
Raza S., Esfandyarpour M., Koh A.L., Mortensen N.A., Brongersma M.L, & Bozhevolnyi S.I. (2016). Electron energy-loss spectroscopy of branched gap plasmon resonators. Nature Communications, 7, 13790.
