Excitation of confined bulk plasmons in metallic nanoparticles by penetrating electron beams within a nonlocal analytical approach

Technical Summary: Excitation of Confined Bulk Plasmons in Metallic Nanoparticles by Penetrating Electron Beams within a Nonlocal Analytical Approach

Problem Statement
The excitation of confined bulk plasmons (CBPs)—longitudinal collective oscillations of the electron density within a nanoparticle volume—by penetrating electron beams is a phenomenon that local response approximation (LRA) models fail to describe accurately. While LRA frameworks successfully model localized surface plasmons (LSPs) for nanoparticles larger than $\sim$10 nm, they break down in the sub-10-nm regime where nonlocal effects, such as finite electron compressibility, become significant. In the LRA, the excitation of bulk plasmons by penetrating probes leads to unphysical logarithmic divergences in the induced potential, typically requiring ad hoc momentum cutoffs to resolve. Furthermore, LRA models cannot account for the emergence of CBPs above the plasma frequency or the specific dependence of their excitation on the electron beam's impact parameter and kinetic energy. There is a lack of a transparent analytical expression for the electron energy loss (EEL) probability of a spherical nanoparticle that incorporates these nonlocal effects for penetrating trajectories.

Methodology
The authors employ a linear hydrodynamic model to describe the conduction electrons as a homogeneous free-electron gas confined within a spherical nanoparticle of radius $a$. This approach incorporates spatial dispersion through the finite compressibility of the electron gas, characterized by a hydrodynamic dispersion parameter $\beta$. The study derives an analytical expression for the EEL probability ($\Gamma_{EELS}$) for both penetrating ($b < a$) and external ($b > a$) electron beam trajectories, where $b$ is the impact parameter and $v$ is the electron velocity.

The derivation proceeds by:

1. Modeling the electron beam as a point charge moving with constant velocity along the $z$-axis.
2. Solving the inhomogeneous Helmholtz equation for the induced charge density inside the metal, subject to hydrodynamic boundary conditions (vanishing normal current at the surface).
3. Expanding the external and induced charge densities and potentials in terms of spherical harmonics to obtain a multipolar solution.
4. Decomposing the total EEL probability into three distinct physical contributions: bulk losses ($\Gamma_{bulk}$), the Begrenzung effect (surface corrections, $\Gamma_{Begr}$), and external losses ($\Gamma_{ext}$).
5. Applying the derived formalism to sodium (Na) nanoparticles as a canonical free-electron system, analyzing the dependence of the spectra on impact parameter, electron velocity, and nanoparticle size.

Key Contributions and Results
The paper provides a unified analytical framework that captures nonlocal effects without ad hoc momentum cutoffs, demonstrating rapid multipolar convergence. Key findings include:

• Emergence of CBPs and Spectral Blueshift: Unlike LRA predictions, which show a single bulk plasmon peak at the plasma frequency $\omega_p$, the hydrodynamic model predicts the emergence of multiple peaks associated with CBPs above $\omega_p$. The spectral envelope of these CBPs exhibits a pronounced blueshift as the nanoparticle size decreases and as the electron impact parameter increases from the center toward the surface.
• Impact Parameter Dependence and Thresholds: The study identifies a "bulk plasmon activation threshold" ($e_{bt}$), defined as the impact parameter where the CBP peak intensity drops to $1/e$ of its maximum. Contrary to LRA predictions where bulk losses scale with the path length inside the particle, the hydrodynamic model reveals that efficient CBP excitation requires the electron to penetrate deep into the nanoparticle. The threshold $e_{bt}$ decreases as the nanoparticle radius decreases.
• Selection Rules: Through multipolar analysis, the authors identify selection rules governing CBP excitation.
  • Parity Rules: For central trajectories ($b=0$), even-$l$ modes are predominantly excited due to phase-matching conditions. As $b$ increases, odd-$l$ modes are activated, contributing to the observed blueshift.
  • Radial Node Rules: A subtle selection rule related to the radial quantum number $n$ is identified. For central trajectories, modes with even $n$ are suppressed due to destructive interference between the electron beam and the CBP mode, while odd-$n$ modes are excited.
  • Velocity Dependence: The excitation of specific $(l, n)$ modes is maximized at specific values of the reduced wave number $q_e = \omega a / (\pi v)$, indicating that the electron beam velocity tunes the interference conditions for different CBP modes.
• Size Effects: As the nanoparticle radius decreases, the energy spacing between CBP modes increases due to dispersion. This leads to a reduction in the overlap of modes, causing the bulk plasmon envelope to resolve into distinct peaks and the activation threshold $e_{bt}$ to decrease. For very small nanoparticles ($a < 1.5$ nm), the evolution of the dominant peak energy with impact parameter exhibits abrupt jumps rather than a smooth shift.
• Physical Interpretation of Terms: The analytical expression explicitly separates surface, bulk, and Begrenzung contributions. The Begrenzung term is shown to be crucial for the excitation of CBPs, particularly the inner component ($\Gamma_{Begr,i}$) where the probe is inside the particle during both excitation and energy loss.

Significance
The paper claims to establish a unified analytical foundation for understanding the mechanisms of CBP excitation by penetrating electron beams. By providing a closed-form analytical expression within the hydrodynamic model, the work offers a practical tool for the design and interpretation of STEM-EELS measurements on sub-5-nm metallic nanoparticles. The results reproduce qualitative behaviors observed in experiments and ab initio calculations, specifically regarding the blueshift of the bulk plasmon envelope and the dependence on impact parameter. The study underscores that the hydrodynamic model captures essential nonlocal trends—such as the existence of a threshold impact parameter and the preferential excitation of higher-order CBPs near the surface—that are missed by local approaches, thereby providing a more accurate physical description of electron energy loss in the bulk plasmon energy range for small metallic systems.