Probing strong coupling in core--shell nanoparticles with fast electron beams

Abstract

Collective optical excitations, such as localized surface plasmons in metallic nanoparticles and Mie resonances in high-index dielectrics, play a central role in nanoscale light--matter interactions. When such optical modes interact with electronic transitions in matter under suitable conditions, they can couple strongly, analogous to two coupled harmonic oscillators, forming hybrid light--matter states. In this work, we probe this coupling in core--shell nanoparticles using fast electrons in electron energy-loss (EEL) and cathodoluminescence (CL) spectroscopy. Owing to their highly localized fields, fast electrons can excite modes inaccessible with light-based spectroscopies, including higher-order nonradiative modes, which offer greater field confinement and potentially stronger coupling. Here, we develop an analytical framework to calculate the EEL and CL probabilities for spherical core--shell nanoparticles under aloof and penetrating electron trajectories. This formalism is applied to two representative systems: an excitonic core with a metallic shell, and a silicon core with an excitonic shell. Our main focus is to examine how the electron beam position and velocity affect our ability to probe this coupling. Depending on the electron beam parameters, we find that the spectral signature of strong coupling remains robust in plasmonic nanospheres. In contrast, it can be significantly suppressed or even completely obscured in dielectric nanospheres. Our developed formalism enables a deeper understanding of the coupling mechanisms in electron--light--matter interactions, thereby accelerating progress in single-nanoparticle-based polaritonic studies.

Figures (4)

• Figure 1: Schematic illustration of an NP that consists of a core (region I) with radius $R_\text{c}$, surrounded by a shell (region II) of outer radius $R_\text{s}$, embedded in a host medium (region III). The three regions are characterized by their optical parameters $\varepsilon_i$ and $\mu_i=1$ with $i=1,2,3$. In all calculations, we consider air as the host medium with $\varepsilon_3=1$. An electron beam propagates parallel to the $z$–axis with impact parameter $b$. (a) Aloof trajectory: the electron remains entirely outside the NP ($b > R_\text{s}$). (b) Shell–penetrating trajectory: the electron enters the shell at $z=-z_\text{s}$ and exits at $z=z_\text{s}$ ($R_\text{s}>b>R_\text{c}$). (c) Core–penetrating trajectory: the electron additionally traverses the core, entering at $z=-z_\text{c}$ and exiting at $z=z_\text{c}$ ($R_\text{s}>R_\text{c}>b$).
• Figure 2: (Top) Schematic of a silver core--exciton shell NP with $R_\mathrm{c}=51.2$ nm and $R_\mathrm{s} = 60$ nm. All line plots have been calculated with $v = 0.5c$. (a) EEL probability of a silver shell decomposed to surface, bulk, and Begrenzung contributions for $b = 40$ nm. (b) EEL and (c) CL probability of the isolated silver shell for the different electron trajectories displayed in the schematic. (d) CL probability of a silver shell decomposed to its multipolar content---electric dipole (ED), electric quadrupole (EQ), and electric octupole (EO)---for $b = 40$ nm. (e) EEL and (f) CL probability of the exciton core for different electron trajectories. (g) CL probability versus reduced velocity $\beta=v/c$ considering the core--shell NP for $b = 40$ nm, The white dotted lines indicate the hybrid plexciton modes. (h) EEL and (i) CL probability along with the EQ contribution for the core--shell NP for $b=40$ nm.
• Figure 3: (Top) Schematic of a silicon core--exciton shell NP with $R_\mathrm{c}=85$ nm and $R_\mathrm{s} = 105$ nm. All plots have been calculated with $v = 0.7c$. (a) EEL probability of a silicon NP decomposed to surface, bulk, and Begrenzung contributions for $b=10$ nm. (b,c) EEL probability of (b) the isolated silicon core and (c) the exciton shell for the different electron trajectories displayed in the schematic. (d) CL probability of a silver shell decomposed to its multipolar content---magnetic dipole (MD), dashed green line; ED, solid blue line; EQ, solid red line; magnetic quadrupole (MQ), dashed purple line---for $b=10$ nm. (e,f) CL probability of (e) the isolated silicon core and (d) the exciton shell for the different electron trajectories. (g-i) EEL and (j-l) CL probability along with the MD contribution for the core-shell NP for the different electron trajectories indicated in the inset.
• Figure 4: Strong coupling in a silicon core--exciton shell NP with $R_\mathrm{c}=87$ nm and $R_\mathrm{s} = 105$ nm. All plots have been calculated with $v = 0.7c$. (a,b) EEL probability along with the EQ contribution for (a) an aloof and (b) a penetrating electron trajectory. (c,d) CL probability along with the EQ contribution for (c) an aloof and (d) a penetrating electron trajectory. (e,f) EEL probability versus core radius $R_\mathrm{c}$ showing the anticrossing between the EQ mode and the exciton-polaritons for (e) an aloof and (f) a penetrating electron trajectory.