A comparison between Monte Carlo method and the numerical solution of the Ambartsumian-Chandrasekhar equations to unravel the dielectric response of metals

Abstract

In this work we describe two different models for interpreting and predicting Reflection Electron Energy Loss (REEL) spectra and we present results of a study on metallic systems comparing the computational cost and the accuracy of these techniques. These approaches are the Monte Carlo (MC) method and the Numerical Solution (NS) of the Ambartsumian-Chandrasekhr equations. The former is based on a statistical algorithm to sample the electron trajectories within the target material for describing the electron transport. The latter relies on the numerical solution of the Ambartsumian-Chandrasekhar equations using the invariant embedding method. Both methods receive the same input parameters to deal with the elastic and inelastic electron scattering. To test their respective capability to describe REEL experimental spectra, we use copper, silver, and gold as case studies. Our simulations include both bulk and surface plasmon contributions to the energy loss spectrum by using the effective electron energy loss functions and the relevant extensions to finite momenta. The agreement between MC and NS theoretical spectra with experimental data is remarkably good. Nevertheless, while we find that these approaches are comparable in accuracy, the computational cost of NS is several orders of magnitude lower than the widely used MC. Inputs, routines and data are enclosed with this manuscript via the Mendeley database.

In search of the ground state crystal structure of Ta$_2$O$_5$ from ab-initio and Monte Carlo simulations

“…To run our Monte Carlo method for modelling the energy loss spectra of tantalum pentoxides, we need first to determine the inelastic and elastic scattering cross sections. The former can be obtained by knowing the dependence of the ELF on the entire spectrum of excitation energies W and momentum transfers q [2, 20,21,22,23]. Ab-initio calculations of the ELF over a large energy range are prohibitive owing to the high computational costs of including many electronic transitions to the excited states.…”

Electronic excitation spectra of cerium oxides: from ab initio dielectric response functions to Monte Carlo electron transport simulations

“…In principle, to determine the inelastic scattering cross-section one needs to know the dependence of the ELF over the entire spectrum of meaningful excitation energies W and momentum transfer q. 43,46,53,66 However, typically one has access only to a limited range of energies, corresponding to those of the valence electrons (t100 eV), owing to the prohibitive computational effort of including high-energy excitations as well as their momentum dispersion. Thus, to extend the excitation energy range beyond the valence regime, we propose to use the MELF-GOS model, [51][52][53][54] which implements a numerically effective and accurate method to compute the ELF over the entire Bethe surface (i.e., the momentum and energy transfer plane) by including both valence and inner shell electronic excitations.…”

“…The ELF, conveniently weighted and integrated, provides the electron inelastic scattering cross-section, 39,40 which is used, along with the elastic scattering cross-section derived from the relativistic Mott theory, 41 as an input to a Monte Carlo (MC) routine to model the transport of charged particles within these solids and predict the reflection electron energy loss (REEL) spectra in particular. [42][43][44][45][46][47][48][49] REEL spectroscopy is an analysis technique that uses an electron beam impinging with kinetic energy lower than 2 keV into thin films. Primary electrons penetrate a few nanometers into the material surface, lose their energy via inelastic collisions and some of them are eventually backscattered to the spectrometer, 50 resulting in spectra that are typically characterised by a number of structures attributed to both collective (plasmons) and single-electron excitations and can be directly benchmarked against the available experimental data.…”

Electronic excitation spectra of cerium oxides: from ab initio dielectric response functions to Monte Carlo charge transport simulations

“…In principle, to determine the inelastic scattering cross section one needs to know the dependence of the ELF over the entire spectrum of meaningful excitation energies W and momentum transfer q 42,45,52,65 . However, typically one has access only to a limited range of energies, corresponding to those of the valence electrons ( 100 eV), owing to the prohibitive computational effort of including high-energy excitations as well as their momen-tum dispersion.…”

“…The ELF, conveniently weighted and integrated, provides the electron inelastic scattering cross section 38,39 , which is used, along with the elastic scattering cross section derived from the relativistic Mott theory 40 , as input to a Monte Carlo (MC) routine to model the transport of charged particles within these solids and predict the reflection electron energy loss (REEL) spectra in particular [41][42][43][44][45][46][47][48] . REEL spectroscopy is an analysis technique that uses an electron beam impinging with kinetic energy lower than 2 keV into thin films.…”