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.     

Local inverse inelastic mean free path at interface

We calculated a local inverse inelastic mean free path (local-IIMFP) for electrons crossing a medium–medium interface, considering various incident electron energies, crossing angles and combinations of materials. We used an extension of a classical dielectric model developed by Li and co-workers for an electron crossing a surface (interface vacuum-medium). Moreover, the integration over the distance of the local-IIMFP allows to obtain the interface excitation parameter (or IEP) characterizing the change in excitation probability for an electron crossing an interface once caused by the presence of the interface in comparison with an electron for which only volume excitations are considered. We perform these calculations for angles between 0° and 80°, for electron energies between 500 and 2500 eV and for various pairs of materials, as Al/In for its academic interest or Au/Si and SiO₂/Si for their technological importance. Small but not negligible variations of the local-IIMFP and the IEP were observed for metal–metal or metal–semiconductor interfaces, while quite significant variations are obtained when one of the materials is a insulator.     

Variation of the electron inelastic mean free path during depth profiling of the Fe/Si interface as determined by quantitative REELS

The aim of this work is to determine the dependence of the electron inelastic mean free path (IMFP) at the Fe/Si interface during depth profiling by sputtering with 3 keV Ar⁺ ions. In order to estimate the variation of the IMFP at the interface, reflection electron energy‐loss spectroscopy (REELS) measurements were performed after different sputtering times at the Fe/Si interface with three different primary electron energies (i.e. 0.5, 1 and 1.5 keV). Even though it is highly likely that a compound (i.e. Fe_xSi) is formed at the interface, all the experimental REELS spectra could be analysed as a linear combination of those corresponding to pure Si and Fe. Using the model developed by Yubero and Tougaard for quantitative analysis of these REELS spectra we could estimate the IMFP values along the depth profile at the interface. The resulting IMFPs are observed to vary linearly with the average composition (as determined by REELS) at the Fe/Si interface as it is sputter depth profiled. The energy dependence of the IMFP for different compositions is presented and discussed. For completeness, we have determined the energy‐loss functions as well as the IMFPs of the pure elements (i.e. Fe and Si). Copyright © 2004 John Wiley & Sons, Ltd.     

An accurate and simple universal curve for the energy‐dependent electron inelastic mean free path

The values of inelastic mean free paths (IMFPs) calculated from optical data for the three material categories of elements, inorganic compounds and organic compounds are re‐assessed to provide a simple equation giving an estimate of the IMFP, knowing only the identities of the elements in an analysed layer and the atomic density of that layer. This simple equation is required for quantification of the thicknesses for layers of mixed elements in which the required parameters for use of the popular equation, TPP‐2M, are insufficiently known. It describes the published values, calculated from optical data for energies above 100 eV, to a similar root mean square (RMS) deviation as that for TPP‐2M in the three material categories. The RMS deviation for all three categories averages 8.4%, provided the inorganic data are ‘corrected’ for the published sum rule errors. If, in an analysed layer, only elements are identified and the atomic density is unknown, i.e. only the average Z value of the layer is known, a simpler relation is provided for the IMFP in monolayers with only one unknown parameter Z that exhibits an RMS deviation from the IMFPs calculated from optical data of 11.5%. Copyright © 2011 Crown copyright.     

Dependence of the recombination kinetics of geminate electron-ion pairs on the mean free path of electrons

We have investigated how the kinetics of geminate recombination of electron-ion pairs is influenced by the mean free path of electrons. By use of the Monte Carlo method previously developed the decay of the pair survival probability was numerically calculated for a variety of values of the mean free path of electrons. When the mean free path is negligibly small compared with the Onsager length, our result is in good agreement with the Hong-Noolandi result based on the Smoluchowski equation. However, as the mean free path increases, it deviates from their result increasingly. This indicates that the treatment based on the Smoluchowski equation is inadequate when the mean free path of electrons is not negligible compared with the Onsager length. We have found that for mean free paths comparable with or larger than the Onsager length the decay kinetics of the pair survival probability approximately follows the first-order kinetics except at very short and very long times. The dependence of the escape probability on the mean free path of electrons is also briefly discussed.     

Fluorescence quantification for surface plasmon excitation

Surface plasmon resonance and surface plasmon fluorescence spectroscopy in combination have the potential to distinguish multicomponent surface processes. However, surface intensity variations from resonance angle shifts lead to a nonlinear response in the fluorescence intensity. We report a method to account for surface intensity variations using the experimentally measured relationship between fluorescence and reflectivity. We apply this method to monitor protease adsorption and proteolytic substrate degradation simultaneously. Multilayer protein substrates are prepared for these degradation studies using a layer-by-layer technique.     

Calculation of the surface excitation parameter for Si and Ge from measured electron backscattered spectra by means of a Monte-Carlo simulation.

A new Monte Carlo method has been developed for simulating backscattered electron spectra, and this was applied for determining the surface excitation parameter (SEP). The simulation is based on direct tracking of electron trajectories in the solid, taking into account elastic and inelastic events. The elastic scattering cross sections are taken from literature, while inelastic cross section data are obtained by a fitting procedure. After some iterations, the program produces electron spectra fitting well to the experimental ones. Si and Ge electron spectra were simulated and SEP values were calculated. The SEP values are compared to other ones from literature.     

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.     

Influence of the mean free path of reactant particles on the kinetics of diffusion-controlled reactions: Escape probability of geminate recombination

The escape probability of geminate particles between which no long-range force acts is calculated as a function of their mean free path. The escape probability increases significantly with an increase in their mean free path, especially when their initial separation is small.     

Simulation of positron and electron elastic mean free path and diffusion angle on DNA nucleobases from 10 eV to 100 keV

Positron and electron interaction collisions in living cells are efficiently simulated by Monte Carlo (MC) codes where huge data tables are needed. Present study provides detailed results of charged particles elastic interactions needed in MC on DNA nucleobases (adenine, thymine, cytosine and guanine, deoxyribose, and phosphoric acid). Indeed, electron and positron elastic cross sections, elastic mean free paths, and elastic angular distributions P(θ) are calculated from 10 eV to 100 keV using a corrected form of the independent atom method taking into account the geometry of the biomolecule. Our calculated results are compared with theoretical data available in the literature in absence of experimental data, in particular for positron. Moreover, our numerical results are presented in analytic format modeled to be used for fast sampling in the MC simulation of elastic collisions; particularly, we provide a useful analytic expression for sampling the elastic diffusion angle. For positron collisions on adenine, the relative error between numerical and analytic elastic diffusion angles is not exceeding 2% in the energy full range 10 eV to 100 keV.     

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.     

Formation of uniform ferrocenyl-terminated monolayer covalently bonded to Si using reaction of hydrogen-terminated Si(111) surface with vinylferrocene/n-decane solution by visible-light excitation

Electrochemically active self-assembled monolayers (SAM) have been successfully fabricated with atomic-scale uniformity on a silicon (Si)(111) surface by immobilizing vinylferrocene (VFC) molecules through Si-C covalent bonds. The reaction of VFC with the hydrogen-terminated Si (H-Si)(111) surface was photochemically promoted by irradiation of visible light on a H-Si(111) substrate immersed in n-decane solution of VFC. We found that aggregation and polymerization of VFC was avoided when n-decane was used as a solvent. Voltammetric quantification revealed that the surface density of ferrocenyl groups was 1.4×10(-10) mol cm(-2), i.e., 11% in substitution rate of Si-H bond. VFC-SAMs were then formed by the optimized preparation method on n-type and p-type Si wafers. VFC-SAM on n-type Si showed positive photo-responsivity, while VFC-SAM on p-type Si showed negative photo-responsivity.     

Systematic calculation of the surface excitation parameters for 22 materials

Quantitative surface analysis requires knowledge of surface excitations by electrons. These excitations are characterized by the surface excitation parameter (SEP) which represents the surface excitation probability while an electron moves across a solid surface. In this work, a systematic calculation of a SEP database has been performed for 22 materials, including metals, oxides and semiconductors, for electron energies between 100 eV and 5000 eV, and for angles α of incidence/emission between 0.5^o and 89.5^o with respect to the surface normal. Surface excitations are considered for both sides of a solid–vacuum interface when an electron is incident on or emitted from a surface. These SEPs represent not only the appearance of surface excitations but also the inhibition of bulk excitations. Four common SEP formulas are evaluated, and we present best‐fit parameters for the most satisfactory formula. SEP can then be readily determined for about 22 materials and various energies and electron incidence or emission angles. Copyright © 2012 John Wiley & Sons, Ltd.     

Higher order phase corrected transition amplitudes for time dependent semiclassical surface hopping calculations

A trajectory based, surface hopping expansion of the time dependent quantum propagator has recently been shown to satisfy the multi-state Schrodinger equation to all orders in . Higher order transition amplitudes for hops between states within an interval of fixed length along the trajectory are presented. These amplitudes include contributions from terms corresponding to any number of hops in the interval. They also account for the dependence of the phase associated with the trajectory and the time taken to cross the interval on the location of the hops within the interval. The higher order amplitudes allow for the use of wider intervals in numerical surface hopping calculations. More of the interference between different hopping trajectories is analytically accounted for when the higher order amplitudes are used with wider intervals. Monte Carlo procedures must generally be employed in deciding whether to hop or not in each interval for multi-dimensional problems. Numerical calculations on a model system indicate that the use of the higher order amplitudes can significantly improve the efficiency and accuracy of these methods.     

A surface hopping algorithm for nonadiabatic minimum energy path calculations

The article introduces a robust algorithm for the computation of minimum energy paths transiting along regions of near‐to or degeneracy of adiabatic states. The method facilitates studies of excited state reactivity involving weakly avoided crossings and conical intersections. Based on the analysis of the change in the multiconfigurational wave function the algorithm takes the decision whether the optimization should continue following the same electronic state or switch to a different state. This algorithm helps to overcome convergence difficulties near degeneracies. The implementation in the MOLCAS quantum chemistry package is discussed. To demonstrate the utility of the proposed procedure four examples of application are provided: thymine, asulam, 1,2‐dioxetane, and a three‐double‐bond model of the 11‐cis‐retinal protonated Schiff base. © 2015 Wiley Periodicals, Inc.     

Inclusion of exact exchange for self-interaction corrected H3 density functional potential energy surface

The effect of the inclusion of the exact exchange into self-interaction corrected generalized gradient approximation density functional theory (GGA-DFT) for the simplest hydrogen abstraction reaction, H + H₂ → H₃ → H₂ + H, is presented using a triple-zeta augmented 6-311++G(d,3pd) basis set. The introduction of the self-interaction correction has a considerably larger effect on molecular geometry and vibrational frequencies than the inclusion of the exact exchange. We investigate the influence of the self-interaction error on the shape of the potential energy surface around the transition state of the hydrogen abstraction reaction. The decomposition of the self-interaction error into correlation and exchange parts shows that the exchange self-interaction error is the main component of the energy barrier error. The best agreements with the experimental barrier height were achieved by self-interaction corrected B3LYP, B-LYP and B3PW functionals with errors of 1.5, 2.9 and 3.0 kcal/mol, respectively. Received: 13 August 1997 / Accepted: 14 November 1997