Inelastic mean free path of low‐energy electrons in condensed media: beyond the standard models

The most established approach for ‘practical’ calculations of the inelastic mean free path (IMFP) of low‐energy electrons (~10 eV to ~10 keV) is based on optical‐data models of the dielectric function. Despite nearly four decades of efforts, the IMFP of low‐energy electrons is often not known with the desired accuracy. A universal conclusion is that the predictions of the most popular models are in rather fair agreement above a few hundred electron volts but exhibit considerable differences at lower energies. However, this is the energy range where their two main approximations, namely, the random‐phase approximation (RPA) and the Born approximation, may be invalid. After a short overview of the most popular optical‐data models, we present an approach to include exchange and correlation (XC) effects in IMFP calculations, thus going beyond the RPA and Born approximation. The key element is the so‐called many‐body local‐field correction (LFC). XC effects among the screening electrons are included using a time‐dependent local‐density approximation for the LFC. Additional XC effects related to the incident and struck electrons are included through the vertex correction calculated using a screened‐Hubbard formula for the LFC. The results presented for liquid water reveal that XC may increase the IMFP by 15–45% from its Born–RPA value, yielding much better agreement with available experimental data. The present work provides a manageable, yet rigorous, approach to improve upon the standard models for IMFP calculations, through the inclusion of XC effects at both the level of screening and the level of interaction. Copyright © 2015 John Wiley & Sons, Ltd.     

Surface excitations in electron spectroscopy. Part I: dielectric formalism and Monte Carlo algorithm

The theory describing energy losses of charged non‐relativistic projectiles crossing a planar interface is derived on the basis of the Maxwell equations, outlining the physical assumptions of the model in great detail. The employed approach is very general in that various common models for surface excitations (such as the specular reflection model) can be obtained by an appropriate choice of parameter values. The dynamics of charged projectiles near surfaces is examined by calculations of the induced surface charge and the depth‐ and direction‐dependent differential inelastic inverse mean free path (DIIMFP) and stopping power. The effect of several simplifications frequently encountered in the literature is investigated: differences of up to 100% are found in heights, widths, and positions of peaks in the DIIMFP. The presented model is implemented in a Monte Carlo algorithm for the simulation of the electron transport relevant for surface electron spectroscopy. Simulated reflection electron energy loss spectra are in good agreement with experiment on an absolute scale. Copyright © 2012 John Wiley & Sons, Ltd.     

Monte Carlo simulation of the inelastic scattering of electrons and positrons using optical-data models

The algorithms implemented in the Monte Carlo codes LEEPS and PENELOPE for the simulation of the inelastic scattering of electrons and positrons are described. Both algorithms are based on the first Born approximation, in which the inelastic cross section is proportional to the generalized oscillator strength. This quantity is obtained by extrapolating the optical oscillator strength into the non-zero momentum transfer region using suitable extension algorithms. The calculated inelastic mean free paths and stopping powers are compared to other theoretical and experimental data available from the literature. The stability of PENELOPE's mixed simulation procedure under variations of the cutoff energy, which separates hard from soft collisions, is also analyzed. Finally, angular deflections of the projectile in inelastic collisions are considered.     

Low-energy electron distributions from the photoionization of liquid water: a sensitive test of electron mean free paths

The availability of accurate mean free paths for slow electrons (<50 eV) in water is central to the understanding of many electron-driven processes in aqueous solutions, but their determination poses major challenges to experiment and theory alike. Here, we describe a joint experimental and theoretical study demonstrating a novel approach for testing, and, in the future, refining such mean free paths. We report the development of Monte-Carlo electron-trajectory simulations including elastic and inelastic electron scattering, as well as energy loss and secondary-electron production to predict complete photoelectron spectra of liquid water. These simulations are compared to a new set of photoelectron spectra of a liquid-water microjet recorded over a broad range of photon energies in the extreme ultraviolet (20–57 eV). Several previously published sets of scattering parameters are investigated, providing direct and intuitive insights on how they influence the shape of the low-energy electron spectra. A pronounced sensitivity to the escape barrier is also demonstrated. These simulations considerably advance our understanding of the origin of the prominent low-energy electron distributions in photoelectron spectra of liquid water and clarify the influence of scattering parameters and the escape barrier on their shape. They moreover describe the reshaping and displacement of low-energy photoelectron bands caused by vibrationally inelastic scattering. Our work provides a quantitative basis for the interpretation of the complete photoelectron spectra of liquids and opens the path to fully predictive simulations of low-energy scattering in liquid water.

Our study reveals the detailed influence of elastic and inelastic mean-free paths on the complete photoelectron spectra of liquid water, including the low-energy electron distributions and the reshaping of the primary photoelectron bands.     

Calculations of electron inelastic mean free paths. XI. Data for liquid water for energies from 50 eV to 30 keV

We calculated electron inelastic mean free paths (IMFPs) for liquid water from its optical energy‐loss function (ELF) for electron energies from 50 eV to 30 keV. These calculations were made with the relativistic full Penn algorithm that has been used for previous IMFP and electron stopping‐power calculations for many elemental solids. We also calculated IMFPs of water with three additional algorithms: the relativistic single‐pole approximation, the relativistic simplified single‐pole approximation, and the relativistic extended Mermin method. These calculations were made by using the same optical ELF in order to assess any differences of the IMFPs arising from choice of the algorithm. We found good agreement among the IMFPs from the four algorithms for energies over 300 eV. For energies less than 100 eV, however, large differences became apparent. IMFPs from the relativistic TPP‐2M equation for predicting IMFPs were in good agreement with IMFPs from the four algorithms for energies between 300 eV and 30 keV, but there was poorer agreement for lower energies. We calculated values of the static structure factor as a function of momentum transfer from the full Penn algorithm. The resulting values were in good agreement with results from first‐principle calculations and with inelastic X‐ray scattering spectroscopy experiments. We made comparisons of our IMFPs with earlier calculations from authors who had used different algorithms and different ELF data sets. IMFP differences could then be analyzed in terms of the algorithms and the data sets. Finally, we compared our IMFPs with measurements of IMFPs and of a related quantity, the effective attenuation length. There were large variations in the measured IMFPs and effective attenuation lengths (as well as their dependence on electron energy). Further measurements are therefore required to establish consistent data sets and for more detailed comparisons with calculated IMFPs. Copyright © 2016 John Wiley & Sons, Ltd.     

Stopping powers and inelastic mean free path from quantitative analysis of experimental REELS spectra for electrons in Ti,Fe, Ni,and Pd

Stopping power (SP) and inelastic mean free path (IMFP) of electrons in Ti, Fe, Ni, and Pd have been determined by using dielectric models. We have used energy loss function (ELF) determined from quantitative analysis of experimental reflection electron energy loss spectroscopy (REELS) spectra as the input parameter for this model. ELF in this study was determined from the previously published quantitative analysis of REELS spectra. The SP of Fe, Ni, Pd, and Ti was compared with several calculation methods for energies from 100 eV to 10 keV and shows SP in this study, which are in best agreement for medium to high energies (greater than or equal to 300 eV). The IMFP obtained in this study shows the best agreement with online database TPP2M and NIST and also calculation by Tanuma with a root mean square (rms ) less than 12%. The present approach shows ELF from quantitative analysis of REELS spectra has a high potential for the experimental determination of SP and IMFP of metals.     

Analysis of Fe-Si layered structures by reflected electron energy loss spectroscopy and inelastic scattering cross-section

This paper reports on our study of the formation of an interface of layered structures in the Fe-Si system by reflected electron energy loss spectroscopy (REELS). Quantitative element analysis was performed using the product of the mean length of the inelastic free path by the inelastic scattering cross-section of electrons. It is shown that the Fe-Si interface is quite uniform.     

Semi‐empirical inelastic mean free paths for positrons

Previously, we developed a semi‐empirical method for determining the inelastic mean free paths of positrons in a wide variety of materials. This work is heavily based on the earlier work of Tanuma, Powell and Penn on the inelastic mean free paths of electrons in the 50–2000 eV energy range. One of the remaining questions still to be answered was the validity of ignoring terms of the order of the inverse energy squared in the denominator of our final expression. In this paper, we investigate this question in some detail by comparing our approximations with calculated values of the positron inelastic mean free paths based on experimental optical data. We conclude that the exclusion of the higher order terms is consistent with the other approximations in this methodology. Copyright © 2005 John Wiley & Sons, Ltd.     

Chemical‐Reaction‐Induced Hot Electron Flows on Platinum Colloid Nanoparticles under Hydrogen Oxidation: Impact of Nanoparticle Size

Generation of hot electron flows and the catalytic activity of Pt nanoparticles (NPs) with different sizes were investigated using catalytic nanodiodes. We show that smaller Pt NPs lead to higher chemicurrent yield, which is associated with the shorter travel length for the hot electrons, compared with their inelastic mean free path. We also show the impact of capping on charge carrier transfer between Pt NPs and their support.     

Report on the 42nd IUVSTA workshop ‘Electron scattering in solids: from fundamental concepts to practical applications’

A summary is given of the workshop entitled ‘Electron Scattering in Solids: from fundamental concepts to practical applications,’ which was held in Debrecen, Hungary, on July 4–8, 2004, under the sponsorship of the International Union of Vacuum Science, Technique, and Applications (IUVSTA). This workshop was held to review the present status and level of understanding of electron‐scattering processes in solids, to identify issues of key importance (hot topics) in the light of the most recent scientific results, and to stimulate discussions leading to a deeper understanding and new solutions of current problems. This report contains summaries of presentations and discussions in sessions on elastic scattering of electrons by atoms and solids, inelastic scattering of electrons in solids, modeling of electron transport in solids and applications, and software. The principal areas of application include Auger‐electron spectroscopy (AES), X‐ray photoelectron spectroscopy, elastic‐peak electron spectroscopy (EPES), reflection electron energy‐loss spectroscopy (REELS), secondary‐electron microscopy, electron‐probe microanalysis (EPMA), and the use of coincidence techniques in electron‐scattering experiments. A major focus of the workshop was determination of the inelastic mean free path of electrons for various surface spectroscopies, particularly corrections for surface and core‐hole effects. Published in 2005 by John Wiley & Sons, Ltd.     

Calculations of electron inelastic mean free paths. XIII. Data for 14 organic compounds and water over the 50 eV to 200 keV range with the relativistic full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 14 organic compounds (26-n-paraffin, adenine, β-carotene, diphenyl-hexatriene, guanine, Kapton, polyacetylene, poly (butene-1-sulfone), polyethylene, polymethylmethacrylate, polystyrene, poly(2-vinylpyridine), thymine, and uracil) and liquid water for electron energies from 50 eV to 200 keV with the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants and from calculated atomic scattering factors for X-ray energies. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.17%. The IMFPs were also compared with a relativistic version of our predictive Tanuma–Powell–Penn (TPP-2M) equation. The average RMS deviation in these comparisons was 7.2% for energies between 50 eV and 200 keV. This average RMS deviation is smaller than that found in a similar comparison for our group of 41 elemental solids (11.9%) and for our group of 42 inorganic compounds (10.7%) for the same energy range. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations for energies between 200 eV and 10 keV. We also found reasonable agreement between our IMFPs for organic compounds and measured IMFPs for energies between 50 eV and 200 keV. Substantial progress for IMFP measurements for liquid water has been made in recent years through the invention of liquid water microjet photoelectron spectroscopy and droplet photoelectron imaging. We found that the IMFPs from these experiments and the associated analyses were larger than our IMFPs by factors between two and four for energies between about 30 eV and 1000 eV. The energy dependences of the measured IMFPs are, however, similar to that of our IMFPs in the same energy range. Since IMFPs calculated from the same algorithm for a number of inorganic compounds agree reasonably well with measured IMFPs for energies between 100 eV and 200 keV, the large differences between IMFPs for water from recent experiments and our results are surprising and need to be resolved with additional experiments.     

Single nanoporous gold nanowire sensors

Chemisorption from the gas or liquid phase can result in a measurable resistance change in a metallic material when at least one dimension is smaller than the mean free path for electrons. Here we report on the fabrication of single nanoporous gold nanowires and demonstrate that adsorption of an alkanethiol can be monitored in real time. Single nanowire devices were fabricated by in situ etching of Au0.18Ag0.82 alloy nanowires in dilute nitric acid. The evolution of the porous structure was characterized by monitoring the resistance change and comparing to cross-sectional images. The feature size of about 10 nm is less than the mean free path for electrons in bulk gold, and hence the resistance is dominated by surface scattering. Adsorption of a monolayer of octadecanethiol onto the nanoporous gold nanowire results in a resistance change of about 3%. The sensitivity factor of 1.0x10(-16) cm2 is comparable to values reported for adsorption at ultrathin films.     

Effective depths for surface excitation derived by reflection electron energy‐loss spectroscopy analysis

A method of estimation is proposed for determining the effective depth of surface excitation. For this, the effective differential inverse inelastic mean free path (DIIMFP) is presumed to be represented as a linear combination of theoretical DIIMFPs for surface and bulk excitation, which are derived by the use of optical dielectric constants. The effective DIIMFP in the approach is derived by a reflected electron energy‐loss spectroscopy analysis based on the extended Landau approach. The present analysis for 1 kV electrons has led to a simple estimation of the effective depth for surface excitations (～14.5 Å for Al and ～21 Å for Ag), agreeing well with an estimation given by υ/ω _s, where υ and ω _s are the velocity of the primary electrons and the average surface plasmon frequency, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Momentum transfer effects in near surface electron energy loss

We examine two formulations for the differential surface excitation parameter (DSEP): one provided by Tung et al. and the other given by the Chen–Kwei position‐dependent differential inverse inelastic mean free path integrated over the electron trajectory. We demonstrate that the latter converges to the former provided that the dielectric function of the solid does not depend on the momentum transfer or it depends on just the momentum transfer component parallel to the surface. Tung's DSEP represents therefore an approximation to the Chen–Kwei DSEP calculated for a dielectric function with no restrictions on the momentum dependence. The approximation is shown to work in the limit of small momentum transfer and to imply an error of 4%–5% for electrons traveling through the solid with energy E = 1 keV. Copyright © 2015 John Wiley & Sons, Ltd.     

Obtaining quantitative information on surface excitations from reflection electron energy‐loss spectroscopy (REELS)

A new analysis of reflection electron energy‐loss spectroscopy (REELS) spectra is presented. Assuming inelastic scattering in the bulk to be quantitatively understood, this method provides the distribution of energy losses in a single surface excitation in absolute units without the use of any fitting parameters. For this purpose, REELS spectra are decomposed into contributions corresponding to surface and volume excitations in two steps: first the contribution of multiple volume excitations is eliminated from the spectra and subsequently the distribution of energy losses in a single surface scattering event is retrieved. This decomposition is possible if surface and bulk excitations are uncorrelated, a condition that is fulfilled for medium‐energy electrons because the thickness of the surface scattering layer is small compared with the electron elastic mean free path. The developed method is successfully applied to REELS spectra of several materials. The resulting distributions of energy losses in an individual surface excitation are in good agreement with theory. In particular, the so‐called begrenzungs effect, i.e. the reduction of the intensity of bulk losses due to coupling with surface excitations near the boundary of a solid‐state plasma, becomes clearly observable in this way. Copyright © 2003 John Wiley & Sons, Ltd.     

Determination of amount of substance for nanometre‐thin deposits: consistency between XPS,RBS and XRF quantification

Quantification on the nanometre scale is a key task in quality control and for the development of new materials in nanotechnology. In this paper we have studied the consistency in the determination of the amount of substance found by XPS peak shape analysis, Rutherford backscattering spectroscopy (RBS) and x‐ray fluorescence spectrometry (XRF). To this end, ZnO was deposited by plasma‐enhanced chemical vapour deposition on the three substrates. Four different sets of samples were produced, with the amount of ZnO deposited in the range 1–10 nm. From XPS analysis it is found that ZnO grows in the form of islands on all three substrates. For each system, the analysis was done independently with two XPS peaks from the overlayer with widely different kinetic energy and one XPS peak from the substrate. The growth mechanism found from analysis of each of the three peaks was consistent and the total amount of determined ZnO material was identical to within 15%. The root‐mean‐square deviation from the XPS quantification of the relative AOS was 20% for XRF and 16% for RBS. Because the absolute amount of substance determined from analysis of the three XPS peaks for each sample was consistent, it is concluded that the energy dependence of the applied inelastic mean free paths (taken here from the empirical TPP‐2M formula) is correct. It was found that the absolute amounts of substance determined by RBS and XRF are consistently factors of 2.1 and 1.5 lower than the that determined by XPS peak shape analysis. It is suggested that the main reason for this large discrepancy is inaccuracy in the applied ‘effective’ inelastic electron mean free path. Copyright © 2003 John Wiley & Sons, Ltd.     

Memory effect on the inelastic interaction of electrons moving parallel to a solid surface

It is generally assumed that two successive inelastic interactions between an electron and a solid are independent of each other. In other words, the electron has no memory of its previous interaction. However, the previous interaction of the electron generates a potential that should influence its succeeding inelastic interaction. The aim of this work is to establish a model to account for the memory effect of an electron between two successive inelastic interactions. On the basis of the dielectric response theory, formulae for differential inverse inelastic mean free paths (DIIMFPs) and inelastic mean free paths (IMFPs) considering the memory effect were derived for electrons moving parallel to a solid surface by solving the Poisson equation and applying suitable boundary conditions. These mean free paths were then calculated with the extended Drude dielectric function for a Cu surface. It was found that the DIIMFP and the IMFP with the memory effect for electron energy E lay between the corresponding values without the memory effect for electron energy E and previous energy E₀. The memory effect increased with increasing electron energy loss, E₀ ? E, in the previous inelastic interaction. Copyright © 2005 John Wiley & Sons, Ltd.