Electron Inelastic Mean Free Paths Research Articles

Electron backscattering coefficients for Cr, Co, and Pd solids: A Monte Carlo simulation study

We have calculated electron backscattering coefficients, η(Ep), at primary electron energies Ep of 0.1–100 keV for three elemental and intermediate atomic number solids, Cr, Co and Pd, with an up-to-date Monte Carlo simulation model. A relativistic dielectric functional approach is adopted for the calculation of the electron inelastic cross section, where several different datasets of optical energy loss function (ELF) are adopted. The calculated backscattering coefficient is found to be substantially affected by the ELF, where the influence can be seen to follow the f- and ps-sum rules and the resultant energy dependence of electron inelastic mean free path. To understand the uncertainties involved in a comparison with experimental data both the theoretical uncertainty due to the elastic cross-section model and the experimental systematic error for the contaminated surfaces are investigated. A total of 192 different scattering potentials are employed for the calculation of Mott's electron elastic cross section and this theoretical uncertainty is confirmed to be small. On the other hand, the simulation of contaminated Co and Pd surfaces with several carbonaceous atomic layers can well explain the experimental data. The present results indicate that accurate backscattering coefficient data should be either measured from fully cleaned surfaces or obtained from modern Monte Carlo theoretical calculations involving reliable optical constants data. With the recent progress in the accurate measurement of optical constants by reflection electron energy loss spectroscopy technique, constructing a reliable theoretical database of electron backscattering coefficients for clean surfaces of elemental solids is highly hopeful.

Energy Dissipation of Fast Electrons in Polymethylmethacrylate: Toward a Universal Curve for Electron-Beam Attenuation in Solids between ∼0 eV and Relativistic Energies.

Spectroscopy of correlated electron pairs was employed to investigate the energy dissipation process, as well as the transport and the emission of low-energy electrons on a polymethylmethacrylate surface, providing secondary electron spectra causally related to the energy loss of the primary. Two groups are identified in the cascade of slow electrons, corresponding to different stages in the energy dissipation process. The characteristic lengths for attenuation due to collective excitations and momentum relaxation are quantified for both groups and are found to be distinctly different: λ_{1}=(12±2) Å and λ_{2}=(62±11) Å. The results strongly contradict the commonly employed model of exponential attenuation with the electron inelastic mean free path as characteristic length, but they essentially agree with a theory used for decades in astrophysics and neutron transport, albeit with characteristic lengths expressed in units of angstroms rather than light-years.

Monte Carlo Simulation of Free-standing Thin Films under Low Energy Electron Bombardment: Electron Inelastic Mean Free Path (IMFP) Determination Using Elastic Peak of the Transmitted Electrons

Abstract: The electron energy spectra of transmitted scattered electrons from free-standing films are simulated using a Monte Carlo computational approach. Elastic scattering is simulated using Mott cross-sections and inelastic scattering via discrete processes determined from dielectric function data. This allows one to simulate the secondary electrons as well as the loss peaks near the elastic (zero-loss) peak. The current study suggested a directed approach for determining the electron inelastic mean free path (IMFP) of materials at low primary electron energies. The IMFP of the reference material is not necessary for the suggested technique. The suggested technique uses the ratio between the transmitted elastic peak intensity and the background intensity of backscattered electrons. Free-standing films of Si, Cu, and Au were studied with thicknesses varying from 2 to 12 nm. Primary electron energies of 1, 3, and 5 keV were applied. The results appeared very good, with the percentage error range being between 5% and 25%. We also investigated the proportion of the first and second plasmon peak intensities to the elastic peak intensity. We believe that the latter could provide a directed method of measuring the IMFP of materials. Keywords: Inelastic mean free path, Monte Carlo simulation, Geant4, Free-standing film, Zero-loss peak.

Determination of electron inelastic mean free path and stopping power of hafnium dioxide

We present inelastic mean free path (IMFP) and stopping power data of the hafnium dioxide applying the relativistic dielectric response theory. The energy loss function (ELF) derived from reflection electron energy loss spectroscopy spectrum with the reverse Monte Carlo method was used for the first time to obtain the IMFP and stopping power of HfO2. The probability of the energy loss is determined by the dielectric response function εq,ω as a function of the frequency ω and the wavenumber q of the electromagnetic disturbance. Two algorithms, namely the full Penn algorithm (FPA) and the super-extended Mermin algorithm (SMA), were employed to expand the optical energy loss function, Im-1/ε0,ω, into the q,ω-plane. The results indicate that the IMFP and the stopping power obtained by using both algorithms are consistent at high electron energies, but show differences at low electron energies (less than ∼ 70 eV). This discrepancy arises from the consideration of the finite plasmon lifetimes effect in the SMA model, while it is neglected in the FPA model. Additionally, we observed that the band gap has a significant influence on the IMFP and the stopping power at low electron energies. Typically, the inclusion of the band gap leads to an increase in the IMFP, because transition channels with energies larger than E-Eg-Ev are prohibited.

Calculations of electron inelastic mean free paths (IMFPs). XIV. Calculated IMFPs for LiF and Si3N4 and development of an improved predictive IMFP formula

We report inelastic mean free paths (IMFPs) of Si3N4 and LiF for electron energies from 50 eV to 200 keV that were calculated from their optical energy‐loss functions using the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. Our calculated IMFPs, designated as optical 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 root mean square (RMS) deviations in these fits were less than 1% for Si3N4 and LiF. The IMFPs were also compared with the relativistic version of our predictive Tanuma–Powell–Penn (TPP‐2M) equation. We found that IMFPs calculated from the TPP‐2M equation are systematically larger than the optical IMFPs for both LiF and Si3N4. The RMS differences between IMFPs from the TPP‐2M equation and the optical IMFPs were 49.3% for LiF and 17.3% for Si3N4 for energies between 50 eV and 200 keV. These RMS differences are much larger than those for most of the inorganic compounds in our previous IMFP calculations where the average RMS difference was 10.7% for 42 inorganic compounds. We also report the development of an improved predictive IMFP formula, which we designate as the JTP equation. This formula is a refinement of the TPP‐2M equation and is based on the recent IMFP calculations for 100 materials including the present IMFPs for Si3N4 and LiF (41 elemental solids, 45 inorganic compounds, and 14 organic compounds) for 83 electron energies between 50 eV and 200 keV. Our predictive JTP equation gave satisfactory results in comparisons of optical IMFPs and IMFPs calculated from the JTP equation. The RMS difference between the 8300 optical IMFPs used for optimization and the IMFPs calculated from the JTP equation was 10.2%. This value is appreciably less than the RMS difference of 16.0% found in a similar comparison of the optical IMFPs and IMFPs from the TPP‐2M equation. Furthermore, IMFPs from the JTP equation were compared with measured IMFPs for energies between 50 eV and 200 keV for 16 elemental solids and 37 inorganic compounds. We found that the JTP equation gave satisfactory results that were comparable with previous comparisons of the optical IMFPs and measured IMFPs. We believe that the JTP equation will be applicable to a wider range of materials than the TPP‐2M equation.

Electron inelastic mean free path in UO2 and (U, Pu)O2 fuels

Inelastic mean free path (IMFP) of UO2 and (U, Pu)O2 fuels at 0, 20, 50 and 95 at.% of Pu were determined for the first time using a combination of convergent beam electron diffraction (CBED) and electron energy loss spectroscopy (EELS) on TEM FIB lamellae. Non-linear least squares fit was performed on the CBED intensity profiles to only obtain the crystalline thickness of the sample with a precision of up to 0.2% [1] while EELS spectra were used to calculate the relative thickness of the sample: crystalline and amorphous contribution. The IMFP were determined to be 119 – 129 nm for UO2, 107 – 117 nm for (U, Pu)O2 at 20 at.% of Pu, 116 – 125 nm for (U, Pu)O2 at 50 at.% of Pu and 123 – 137 nm for (U, Pu)O2 at 95 at.% of Pu using collection semi-angles β = 23.4 – 11.6 mrad, convergence semi-angle α = 8 mrad and accelerated voltage of 200 keV on a Thermofischer TALOS F200X TEM in the LECA STAR hot laboratory (CEA Cadarache). A comparison with the most relevant models in the literature shows that for compounds with a comparable quantity of light and heavy elements it is better to use a mean atomic number instead of an effective one as proposed by different authors. The importance of accurate density determination is also emphasized if the IMFP values are to be predicted by the existed models. As a second part, a strategy is developed to calculate the effective refractive index at the nanometer scale for the fuels allowing to easily find the IMFP under different measurement parameters such as those in energy filtered transmission electron microscopy (EFTEM) maps.

Establishment and validation of an electron inelastic mean free path database for narrow bandgap inorganic compounds with a machine learning approach.

Narrow bandgap inorganic compounds are extremely important in many areas of physics. However, their basic parameter database for surface analysis is incomplete. Electron inelastic mean free paths (IMFPs) are important parameters in surface analysis methods, such as electron spectroscopy and electron microscopy. Our previous research has presented a machine learning (ML) method to describe and predict IMFPs from calculated IMFPs for 41 elemental solids. This paper extends the use of the same machine learning method to 42 inorganic compounds based on the experience in predicting elemental electron IMFPs. The in-depth discussion extends to including material dependence discussion and parameter value selections. After robust validation of the ML method, we have produced an extensive IMFP database for 12 039 narrow bandgap inorganic compounds. Our findings suggest that ML is very efficient and powerful for IMFP description and database completion for various materials and has many advantages, including stability and convenience, over traditional methods.

Experimental effective attenuation length on solids for electron kinetic energies between 1 and 14 KeV: Determination of a simple practical equation

The effective attenuation length is a fundamental parameter for the correct evaluation of XPS and AES spectra. For electron kinetic energies below 2 KeV, a large database with experimental EALs for most organic and inorganic compounds exists. In addition, several predictive equations valid in the low electron kinetic energy range are available. However, for HAXPES applications in which electron kinetic energies up to 14 KeV are used, the lack of experimental EAL data and predictive equations difficult the spectra analysis and quantification. In this work, we obtain experimental EAL values in the kinetic energy range between 1 and 14 KeV for Cu, Au, Ti, Si, Ge, TiN, SiO2, HfO2, Fe3O4, SrTiO3 and La0.7Sr0.3MnO3. The validity of the predictive equations developed for soft X-ray XPS are evaluated for high electron kinetic energies using the obtained experimental data. We find that the λfit formulae developed by Jablonski and Tougaard reproduces the experimental EAL data up to electron kinetic energies of 14 KeV within the experimental errors. Based on a fitting to the obtained experimental data for all materials used in this work we propose a simple practical equation for the evaluation of EAL for solids in the kinetic energy range between 1 and 14 KeV. The proposed equation, in comparison with the λfit predictive formulae, is only dependent on the compound effective atomic number eliminating tedious calculations based on electron inelastic mean free paths and transport mean free paths. The accuracy of the proposed practical equation reproduces within the experimental errors the experimental EALs presented in this work and the data available in the literature.

Multiple elastic scattering of electrons in condensed matter (new version announcement)

Multiple elastic scattering of electrons in condensed matter (new version announcement)

Evaluation of dielectric function models for calculation of electron inelastic mean free path

This work investigates the detailed difference between dielectric function models, the Mermin model and the full Penn algorithm (FPA) model, for the determination of an electron inelastic mean free path (IMFP) with optical energy loss function (ELF), as an extension of our previous study [Da et al., Surf. Interface Anal. 51, 627 (2019)] by using the simple Drude-type ELF. In the conventional normal Mermin (NM) model, the approximations of ELF by the Drude equation will introduce inevitable fitting error. In order to enhance the accuracy of the NM model, our previous proposed extended Mermin model [Da et al., Phys. Rev. Lett. 113, 063201 (2014)], which is renamed as a super-extended Mermin algorithm (SE-MA) now, is employed to eliminate the error by expanding the definition of Drude oscillators used in the NM. In the SE-MA, the Drude-like oscillators allow the existence of negative strengths to express the fine structures of phonon–electron scattering and the plasmon lifetime broadening effect. Because in our previous study, the simple Drude-type ELF cannot include these complex structures, in this work, the electron IMFPs are calculated for five realistic materials, Al, Si, Cu, Au, and MgO. The difference between IMFPs calculated by the SE-MA model and the FPA model is material dependent and is significant in the low energy region, which is analyzed by using the Fano plot. This is due to the more important role played by the plasmon lifetime broadening effect.

Electron inelastic mean free path (IMFP) values of Kapton, polyethylene (PE), polymethylmethacrylate (PMMA), polystyrene (PS) and polytetrafluoroethylene (PTFE) measured with elastic peak electron spectroscopy (EPES)

Elastic peak electron spectroscopy (EPES) was employed to measure the inelastic mean free path (IMFP) for energies between 500 and 1600 eV for five insulating organic compounds: Kapton, polyethylene (PE), poly(methyl methacrylate) (PMMA), polystyrene (PS) and polytetrafluoroethylene (PTFE). A Ni and a Si sample were used as reference materials to avoid measurement of the elastic reflection coefficient in absolute units. Correction of experimental elastic peak intensities for surface excitations was performed which turned out to be essential. The results are compared with recent evaluations of optical constants to yield the IMFP in the literature giving satisfactory agreement, with deviations generally below 20%. Investigation of the kinematics in an electron reflection experiment shows that the dispersion coefficient used in REELS data analysis cannot be identified with the true plasmon dispersion.

Combined first-principles-Monte Carlo analysis to evaluate the effect of surface hydrogen on the secondary electron yield of nickel

Secondary electron yield (SEY) modeling of Ni(110) surface has been carried out with and without the inclusion of wavevector-dependent harmonic corrections (which alter both the inelastic mean free path and stopping power) and is compared to available experimental data. The correction is shown to improve predictions of the inelastic electron mean free path in Ni and yield better agreement with experimental SEY data. It is found that the SEY is strongly dependent on the presence of adsorbates on surfaces. An increase of hydrogen on the surface, for example, is predicted to result in a significant enhancement in the secondary electron yield, with the positional placement of hydrogen layers on or near the Ni surface influencing the SEY. Using first-principles calculations, the permittivities work function and adsorption energy of various Ni systems have also been calculated, and have shown to compare favorably with available experimental data, and have been used in the present Monte Carlo calculations of electron transport.

Measurement of the surface excitation parameter of Kapton, polyethylene (PE), polymethyl methacrylate (PMMA), polystyrene (PS) and polytetrafluoroethylene (PTFE)

Reflection electron energy loss spectra (REELS) were measured for five insulating organic compounds: Kapton, polyethylene (PE), poly(methyl methacrylate) (PMMA), polystyrene (PS) and polytetrafluoroethylene (PTFE), as well as for Ni and Si, in the energy range between 200 and 1600 eV. The average number of surface excitations for a single surface crossing were determined from the experimental data and were found to be considerably smaller than for earlier studied materials, which mainly consisted of elemental metals [Surf. Sci. 486(2001)L461]. The surface excitation parameter, a material parameter used to quantify the relative intensity of surface losses in (photo)electron spectroscopy, was extracted from the data and compared with values found in the literature. The results indicate that surface excitations only have a minor influence on quantification of XPS spectra of polymers. On the other hand, a correction for surface excitations turns out to be essential for measurements of the electron inelastic mean free path of polymers when a metal is used as reference material.

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.

Electron Inelastic Mean Free Path in UO2 and (U, Pu)O2 Fuels

Electron Inelastic Mean Free Path in UO2 and (U, Pu)O2 Fuels

Low-Energy Electron Inelastic Mean Free Path of Graphene Measured by a Time-of-Flight Spectrometer.

The detailed examination of electron scattering in solids is of crucial importance for the theory of solid-state physics, as well as for the development and diagnostics of novel materials, particularly those for micro- and nanoelectronics. Among others, an important parameter of electron scattering is the inelastic mean free path (IMFP) of electrons both in bulk materials and in thin films, including 2D crystals. The amount of IMFP data available is still not sufficient, especially for very slow electrons and for 2D crystals. This situation motivated the present study, which summarizes pilot experiments for graphene on a new device intended to acquire electron energy-loss spectra (EELS) for low landing energies. Thanks to its unique properties, such as electrical conductivity and transparency, graphene is an ideal candidate for study at very low energies in the transmission mode of an electron microscope. The EELS are acquired by means of the very low-energy electron microspectroscopy of 2D crystals, using a dedicated ultra-high vacuum scanning low-energy electron microscope equipped with a time-of-flight (ToF) velocity analyzer. In order to verify our pilot results, we also simulate the EELS by means of density functional theory (DFT) and the many-body perturbation theory. Additional DFT calculations, providing both the total density of states and the band structure, illustrate the graphene loss features. We utilize the experimental EELS data to derive IMFP values using the so-called log-ratio method.

Analytical Formula for the Electron Inelastic Mean Free Path

The electron inelastic mean free path (IMFP) is the average distance between successive inelastic collisions of an electron moving in a medium with a given energy. Within the dielectric formalism, the IMFP is defined by an integral. The problem is that the integration is rather complicated. In 2015, an analytical formula for IMFPs was derived from this definition (J. Phys. Chem. C 2015, 119, 23627–23631). However, the derived formula is restricted to energies above 500 eV. This limitation is overcome in this work. The lower limit of the present analytical formula for IMFPs is about 50 eV.

Machine learning approach for the prediction of electron inelastic mean free paths

The prediction of electron inelastic mean free paths (IMFPs) from simple material parameters is a challenging problem in studies using electron spectroscopy and microscopy. Herein, we propose a machine learning (ML) approach to predict IMFPs from some basic material property data. The ML model showed excellent performance based on the calculated IMFPs for a group of 41 elemental materials [Li, Be, C (graphite), C (diamond), C (glassy), Na, Mg, Al, Si, K, Sc, Ti, V, Cr, Fe, Co, Ni, Cu, Ge, Y, Nb, Mo, Ru, Rh, Pd, Ag, In, Sn, Cs, Gd, Tb, Dy, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Bi] from a previous paper [Shinotsuka et al., Surf. Interface Anal. 47, 871 (2015); Surf. Interface Anal. 47, 1132 (2015)]. which was comparable with that of the robust Tanuma-Powell-Penn (TPP-2M) formula. The developed ML model was then extended to materials that do not have reported IMFPs in the database. The IMFPs for 18 transition and lanthanide metals (Mn, Zn, Zr, Tc, Cd, La, Ce, Pr, Nd, Pm, Sm, Eu, Ho, Er, Tm, Yb, Lu, and Hg) were predicted by the ML model. In the comparison with full-Penn algorithm-calculated IMFPs through two newly found experimental energy loss functions (ELFs), i.e., Mn and Zr, the Gaussian process regression-predicted IMFPs not only agreed well with those calculated using the TPP-2M formula in the energy range >50 eV but were also consistent with the trend of IMFPs calculated through experimental ELFs in the range of 2.7--50 eV, where the TPP-2M formula cannot be used. Our findings suggest that ML is very powerful and efficient and has great potential to complete a database of IMFPs for materials that can prove solutions closer to reality than empirical models on materials with similar physical and chemical properties and can be applied to other different situations for correlated information prediction.

Low-energy electron inelastic mean free path and elastic mean free path of graphene

Based on a recent experimental data of elastic transmissivity and the elastic reflectivity measured on graphene, we have performed a theoretical study on electron inelastic mean free path (IMFP) and elastic mean free path (EMFP) by an improved approximation in a classical electron trajectory framework. A hump feature, which is considered to be owing to the out-of-plane mode of the π + σ plasmon, is clearly shown in our IMFP results of multilayer graphene while it is not seen in that of monolayer graphene. The obtained EMFPs are one order of magnitude greater than previously reported results. This work shows that the classical electron trajectory framework still works for revealing the physics picture of low-energy electron interaction with graphene, even for the transverse direction of monolayer graphene, which is the thinnest material.

Спектроскопия потерь энергии отраженных электронов γ-Fe-=SUB=-2-=/SUB=-O-=SUB=-3-=/SUB=-

The reflection electron energy losses spectra, obtained in a wide primary electron energy range of 200 - 3000 eV, are investigated. From these experimental spectra, for each primary electron energy, the spectra of the inelastic scattering cross section of electrons are calculated as the dependence of the product of the inelastic electron mean free path and the differential inelastic scattering cross section of electrons on the electron energy loss. The analysis of the fine structure of the reflection electron energy losses was carried out by decomposing the electron energy losses spectra in the region of energy losses of valence electrons into elementary peaks. A relationship is established between each of their elementary peaks with single and multiple energy losses due to the excitation of bulk and surface plasmons and interband transitions of electrons from the valence band to free states above the Fermi level. The analysis of the obtained results was carried out on the basis of experimental and theoretical literature data on the band structure of  Fe2O3.