Source-linked AI summary
Simulating electron energy loss spectroscopy with the MNPBEM toolbox
Ulrich Hohenester
TL;DR
The paper addresses practical simulation of plasmonic EELS and presents a BEM implementation integrated into the MNPBEM toolbox. It demonstrates spectra, maps, induced fields, and cathodoluminescence-related calculations across nanoparticle geometries. The simulations reproduce analytic nanosphere results and resolve multiple nanodisk plasmon modes.
Problem
The paper targets simple and efficient computation of EELS spectra and maps for plasmonic nanoparticles and other nanophotonic structures.
Method
The implementation combines eelsret and eelsstat Matlab classes with the MNPBEM boundary element method to compute electron-induced potentials, charges, losses, fields, and scattered light.
Results
The simulations show good agreement with analytic Mie results for a silver nanosphere and resolve nanodisk dipolar, quadrupolar, breathing, and bulk-loss peaks at 2.6, 3.1, 3.5, and 3.8 eV.
Takeaways & Limitations
The toolbox supports EELS spectra and maps for metallic nanospheres, nanodisks, and nanotriangles, as well as electron-induced electromagnetic-field calculations.
Takeaways & Limitations
The approach assumes electron kinetic energy greatly exceeds plasmon energies and is not suited for low electron energies or thick samples.
Abstract
from arXiv · showhide
Within the MNPBEM toolbox, we show how to simulate electron energy loss spectroscopy (EELS) of plasmonic nanoparticles using a boundary element method approach. The methodology underlying our approach closely follows the concepts developed by Garcia de Abajo and coworkers [for a review see Rev. Mod. Phys. 82, 209 (2010)]. We introduce two Matlab classes that allow in combination with our recently developed MNPBEM toolbox for a simple, robust, and efficient computation of EEL spectra and maps. The classes are accompanied by a number of demo programs for EELS simulation of metallic nanospheres, nanodisks, and nanotriangles, and for electron trajectories passing by or penetrating through the metallic nanoparticles. We also discuss how to compute electric fields induced by the electron beam and cathodoluminescence.
Program summary
The MNPBEM toolbox is supplemented by Matlab demo files for simulating electron energy loss spectroscopy.
- The toolbox includes a collection of demo files for EELS simulations.The software is implemented in Matlab 7.11.0 and is vectorised or parallelized.
1. Introduction
EELS enables spatial and spectral investigation of plasmonic fields beyond conventional optical resolution. The paper integrates BEM-based EELS classes with the MNPBEM toolbox to compute spectra, maps, fields, and scattered light.
- EELS maps localized plasmonic fields by measuring electron energy loss as a function of beam position.High-energy electrons pass by or penetrate nanoparticles, excite plasmons, and lose part of their kinetic energy.
- The BEM approach discretizes nanoparticle boundaries and solves Maxwell’s equations using artificial surface charges and currents.
- The software provides eelsret and eelsstat classes for EEL spectroscopy and mapping in combination with the generic MNPBEM Maxwell solver.The implementation is intended for simple and efficient computation of EELS for plasmonic nanoparticles and other nanophotonic structures.
- A typical workflow defines particle boundaries and dielectric media, initializes the electron beam, computes external potentials, and solves the BEM equations.The classes store beam positions and velocity and return external scalar and vector potentials for a given loss energy.
- The EELS classes are embedded in a new MNPBEM toolbox version that also updates Mie classes for comparison with analytic results.Programs from the old version are expected to remain compatible with the new version.
2. Getting started
Getting started involves adding the toolbox and subdirectories to Matlab, generating help pages, and running supplied demos. The demos produce spectra and spatial maps for triangular nanoparticles through a workflow that initializes geometry, excitation, and BEM loss calculations.
- The toolbox is added to the Matlab search path with addpath(genpath(mnpbemdir));.
- Running makemnpbemhelp after changing to the toolbox directory generates help pages accessible through Matlab’s help browser.
- The demoeels directory contains demo programs for EELS simulations, including triangular silver nanoparticle spectra and spatial maps.
- The nanotriangle maps are computed at plasmon resonance energies identified in the corresponding spectra.
- The triangle spectrum demo evaluates beam positions at a corner, middle, and edge, using an eelsret excitation with specified velocity and broadening.
- Table 1 lists supplied EELS demo programs, typical runtimes, and brief explanations tested on a standard PC.
- For a given loss energy, the excitation returns external potentials; solving the BEM equations yields surface and bulk loss probabilities, spectra, and maps.Maps are obtained by evaluating a rectangular grid of beam impact parameters.
3. Theory
The theory combines boundary-element electromagnetic modeling with an electron-beam source to compute EELS from induced surface responses. It also introduces efficient boundary-only evaluation, trajectory smoothing near crossings, and a quasistatic approximation under stated energy and size assumptions.
- Boundary element formulation: BEM represents nanoparticle responses with scalar and vector potentials plus surface charge and current distributions enforcing interface boundary conditions.The potentials satisfy the governing equations away from particle boundaries, while discretized boundary elements determine the induced surface quantities.
- Electron-beam source: The electron trajectory is modeled as r(t) = r0 + vt with v = vẑ, charge −e, impact parameter R0, and q = ω/v.The high-energy assumption permits neglecting the small velocity change caused by plasmon excitation.
- Energy-loss calculation: EELS probability can be obtained from the work done by the electron against the induced field, with bulk loss included separately for propagation in a lossy medium.The induced field is computed from the surface charge and current distributions, while Γbulk accounts for bulk losses.
- Boundary-only evaluation: A more efficient final expression integrates over particle boundaries using available surface charge and current distributions instead of evaluating the induced field along the trajectory.For penetrating trajectories, crossing-point terms cancel because the potential is continuous across boundaries.
- Refined boundary integration: Near trajectory-boundary crossings, the logarithmic K0 singularity remains integrable, but numerical integration is difficult near element edges and corners.The method therefore replaces the delta-like transverse trajectory profile with a smooth distribution whose potential remains finite at small distances and matches the original potential at large distances.
- Quasistatic approach: The quasistatic approximation applies when kL ≪ 1, retaining scalar potentials and frequency-dependent permittivities while replacing the Helmholtz problem with Laplace or Poisson equations.The approximation assumes the nanostructure is significantly smaller than the light wavelength.
4. Results and detailed toolbox description
The toolbox applies BEM-based EELS simulations to electron trajectories passing by or through plasmonic nanoparticles, with demos covering spectra, maps, and induced fields. Results agree with analytic Mie theory for a silver nanosphere, resolve nanodisk plasmon modes, and show how beam broadening affects numerical stability.
- Electron beam passing by a nanosphere: BEM simulations of a silver nanosphere produce energy-loss probabilities in good agreement with analytic Mie results.The demomie.m example uses an 80 nm sphere, 200 keV electrons, and a 10 nm minimum beam-surface distance.
- EELS excitation and solver setup: The eelsret class defines electron-beam excitation from particle boundaries, impact parameters, beam width, and velocity, returning external potentials for BEM calculations.It supports arrays of impact parameters for EELS maps and computes trajectory-boundary crossing points during initialization.
- EELS excitation and solver setup: The workflow solves BEM equations for each loss energy, stores induced surface charges and currents, and evaluates surface and bulk loss probabilities.The toolbox converts electronvolts to nanometers internally, while electron kinetic energy can be converted to velocity in units of c.
- Electron propagation through nanoparticles and EELS maps: For electron beams passing through a nanodisk, EEL spectra show dipolar, quadrupolar, breathing, and bulk-loss peaks at 2.6, 3.1, 3.5, and 3.8 eV, respectively.The spectra and density maps vary across impact parameters and use a 60 nm diameter, 10 nm height silver nanodisk.
- Electron propagation through nanoparticles and EELS maps: Moderate beam-width parameters of 0.1 or 0.2 nm remove numerical spikes while preserving nearly indistinguishable EELS maps; excessive width smears features.The width should remain significantly smaller than the boundary-element size but large enough to avoid collocation-point artifacts.
- Electric field and cathodoluminescence: Computed electric fields are strongly enhanced near the nanodisk and very small inside it, consistent with evanescent plasmonic fields and free-carrier screening.The field is evaluated at specified points using a Green-function object after solving for the induced charges and currents.
- Quasistatic limit: Full-Maxwell and quasistatic simulations give very similar results, with differences mainly in the lowest-energy dipolar peak from retardation and radiation damping.The comparison is presented as a quasistatic-limit validation of the EELS calculations.