Excitation of Confined Bulk Plasmons in metallic nanoparticles by penetrating electron beams within a non-local analytical approach

TL;DR

The paper develops a nonlocal hydrodynamic-model framework to analytically compute the EEL probability for penetrating electron beams interacting with sub-5 nm spherical metallic nanoparticles. By expanding the response in spherical harmonics and separating the EEL signal into bulk, Begrenzung, and external contributions, the authors explicitly capture confined bulk plasmons (CBPs) that are inaccessible to local dielectric descriptions. Key findings include a blueshift of the bulk plasmon envelope (BPE) with decreasing NP size and with increasing beam impact parameter, plus a threshold impact parameter necessary to efficiently excite CBPs, all governed by multipolar symmetries and CBP radial nodes. The results provide a detailed, parameter-dependent map of LSP and CBP excitation that helps interpret EELS spectra and informs experimental design and atomistic simulations in nanoplasmonics.

Abstract

Using a linear hydrodynamic model (HDM) we investigate theoretically the interaction between penetrating electron beams and sub-5 nm metallic spherical nanoparticles (NPs), and provide an analytical expression of the electron energy loss (EEL) probability including non-local effects in the response of the confined electron gas. We focus on the characterization of the longitudinal plasmon excitations, or confined bulk plasmons (CBPs), which cannot be addressed within local dielectric frameworks, and show that their excitation is highly sensitive to the impact parameter and kinetic energy of the incident electron beam, as well as to the NP's size. In contrast to the local approach, our decription captures a blueshift of the bulk plasmon envelope (BPE) with decreasing NP size and a blueshift with increasing impact parameter. Moreover, it predicts a threshold impact parameter, or minimum electron path inside the NP, to efficiently activate a set of CBPs. Exploiting the multipolar description of the CBPs we identify the underlying symmetry rules governing their excitation by electron beams, and correlate the observed blueshift of the BPE for increasing impact parameters with the excitation of higher-order CBPs. Dispersion of the CBPs with decreasing NP sizes further increases this impact parameter dependent blueshift of the BPE and also explains the decrease in the impact parameter threshold.

Figures (7)

• Figure 1: a) Sketch of a metallic spherical nanoparticle of radius $a$ targeted by a penetrating electron beams with impact parameter $b$ and velocity $\textbf{v}$. The electron probe, described by the charge density, $\varrho^{\textrm{ext}}(\textbf{r}, \omega)$, creates a potential, $\phi^{\textrm{ext}}(\textbf{r}',\omega)$, that induces a charge density in the NP, $\varrho^{\textrm{ind}}(\textbf{r}',\omega)$, which produces an induced potential, $\phi^{\textrm{ind}}(\textbf{r},\omega)$, that acts back on the electron probe making it lose certain energy.
• Figure 2: Schematic illustration of the contributions adding to the total electron energy loss probability, $\Gamma_{\textrm{EELS}}$, in the frequency domain. The blue solid line represents the external charge density, $\varrho^{\textrm{ext}}(\textbf{r})$, which generates an external potential (blue arrow), $\phi^{\textrm{ext}}(\textbf{r}')$, which induces a charge density (red area), $\varrho^{\textrm{ind}}(\textbf{r}')$, within the nanoparticle (NP). In turn, this induced charge density creates an induced potential (green arrow), $\phi^{\textrm{ind}},(\textbf{r})$, that acts back on the external charge density (green solid line), $\varrho^{\textrm{ext}}(\textbf{r})$, resulting in the energy loss. The figure outlines the different contributions to $\Gamma_{\textrm{EELS}}$, based on the location of the electron beam path -whether inside or outside the NP- during both the excitation (blue line) and the energy loss (green line) processes: a) inside-inside, both excitation and energy loss occur within the NP (contributes to $\Gamma^{\textrm{Begr,i}}$); b) outside–inside, excitation occurs outside, while energy loss occurs inside the NP; c) inside–outside, excitation occurs inside, while energy loss occurs outside the NP (both b) and c) contribute to $\Gamma^{\textrm{Begr,o}}$); and d) outside–outside, both excitation and energy loss occur outside the NP (contributes to $\Gamma^{\textrm{ext}}$).
• Figure 3: a) Charge density isosurfaces associated to the lowest order eigenmodes of a spherical NP within the HDM in the absence of an external excitation source, obtained as a solution of Eq. \ref{['eq:modecond']}. The number $l$ is associated to the polar symmetry and $n$ to the number of nodes of the charge density associated to the mode in the radial direction. Red and blue surfaces show charge density of opposing phase. The contour of the spherical NP is highlighted in gray. b) Radial component [$j_{l}(\mu r)$, spherical Bessel function] associated to the charge density of the eigenmodes as a function of the reduced radius $r/a$.
• Figure 4: a) EEL probability (gray shaded area) and its components: the bulk term (blue curve), inner and outer Begrenzung terms (dark solid and clear dashed green curves, respectively) and external term (red curve) for multipole order cutoff $l_{\textrm{max}} = 20$. The bulk, Begrenzung and external terms are given by Eq. \ref{['eq:Gammabulk']} and Eqs. \ref{['eq:GammaBegr_i']}-\ref{['eq:Gammaext']}, respectively. The inset shows the full scale of the spectra. Vertical dashed gray lines mark the the local dipolar plasmon frequency, $\omega_{\textrm{DP}} = \omega_{p}/\sqrt{3}$, the local surface plasmon frequency, $\omega_{\textrm{SP}} = \omega_{p}/\sqrt{2}$, and the plasma frequency, $\omega_{p}$. b) Logarithmic plot of the data shown in (a). The Begrenzung terms are plotted as the full Begrenzung contribution, $\Gamma^{\textrm{Begr}}$, in a single curve (positive values are plotted as a green curve and negative values as an ocher curve, which is also highlighted by the labels at the bottom of the figure). The insets show the induced charge densities corresponding to the main LSP and CBP excitations for the axial trajectory, $(l,n) = (2,0)$ and $(0,1)$, respectively. Dashed red line represents the contribution of the $\Gamma_{l=2\,m=0}^{\textrm{ext}}$ term, and the peaks corresponding to the LSP and CBPs in this term are marked with red dots.
• Figure 5: a) Electron energy loss probability, $\Gamma_{\textrm{EELS}}$, of a spherical sodium NP ($r_\textrm{s} = 2.08 \textrm{ \AA}$, $\omega_{p} = 6.05 \textrm{ eV}$) of radius $a=1.5 \textrm{ nm}$, as a function of the energy lost by the electron probe, $\omega$, and the reduced impact parameter, $\widetilde{b}=b/a$, calculated within the HDM. The kinetic energy of the electron beam is $\mathcal{E}_{k} = 100 \textrm{ keV}$. The vertical purple straight line highlights the grazing trajectories ($b=a$) and the horizontal white dashed line highlights the plasma frequency $\omega_{p}$. The dots in colors point the positions of the LSP peaks and the black diamonds point to the bulk plasmon envelope (BPE) for spectra with reduced impact parameters $\widetilde{b} = 0, 0.2, 0.4 \textrm{ and } 0.6$ (vertical gray dashed lines). b)-e) Electron energy loss probability (black line) and the contribution from $l=0-10$ modes (we include the contribution from all $m$ for each $l$) to the total energy loss probability for normalized impact parameters $\widetilde{b} = 0, 0.2, 0.4 \textrm{ and } 0.6$. The main contributing modes are highlighted with dots and labeled as $(l,n)$. The gray shaded curve represents the electron energy loss probability calculated within the LRA. f) Induced charge density maps for selected energies $\omega$ corresponding to the position of the BPE.