Electron Inelastic Mean Free Paths Research Articles

Low-energy electron inelastic mean free path for monolayer graphene

The electron inelastic mean free path (IMFP) is an important quantity for electron spectroscopy and microscopy techniques. At present, there are very few data of IMFPs for two-dimensional (2D) materials. Here, we determine the IMFP at energies below 100 eV for monolayer graphene both experimentally and theoretically. The experimental determination is based on a data-driven spectral analysis technique, including the virtual substrate method and the reverse Monte Carlo method. The theoretical determination is performed within the dielectric formalism, using the energy-loss function calculated in the time-dependent density functional theory. The experimental and theoretical results show that the IMFP for monolayer graphene is almost constant (about 1 nm) in the energy range of 6–100 eV. This study suggests a general and reliable approach to determine low-energy IMFPs for 2D materials.

Measurement of the Low-Energy Electron Inelastic Mean Free Path in Monolayer Graphene

Measuring the electron-transport properties of substrate-supported nanomaterials with the traditional two-point comparison method is difficult at electron energies below 50 eV, where core-level signals are too feeble to be detected against the strong secondary-electron background. Herein, a data-driven spectral analysis technique is used to study the low-energy electron-transport properties of substrate-supported target nanomaterials, while eliminating the influence of the substrate signal. Applying this technique, the electron-transport properties of the effective attenuation length and the inelastic mean free path (IMFP) can be determined with extremely high efficiency over the entire measured energy range of 6--600 eV. Further, these results show excellent agreement with other experimental and theoretical results. Significant differences are observed between monolayer graphene and the bulk graphite IMFP, which illustrates the importance of the nanometer effect in the electron-transport properties of the material. Furthermore, this technique is readily applicable to any ultrathin material that can be transferred onto a polycrystalline gold substrate.

Electron Inelastic Mean Free Paths for LiF, CaF2, Al2O3, and Liquid Water from 433 keV down to the Energy Gap.

We report new calculations, which include the influence of the band gap and exciton states, of the electron inelastic mean free path (IMFP) for liquid water, LiF, CaF2, and Al2O3 from the band gap to 433 keV. Among compounds, liquid water is the most studied due to its role in radiobiological research, whereas LiF and CaF2 are the most widely used thermoluminescent dosimeters in environmental monitoring and medical and space dosimetry. Due to its sensitivity, the optically stimulated luminescent dosimeter, Al2O3, has recently begun to be used for personnel monitoring. Previous treatments have modified the integration domain to consider the indistinguishability between the incident electron and the ejected one or the bandgap energy for nonconductors but not to accommodate exciton states within the band gap, and no published IMFP data are available for CaF2. Our calculation was carried out using an electron-beam–solid-state interaction model through the relativistic full Penn algorithm. Integration limits that consider the band gap, the valence band width, and exciton interactions have been used. The results suggest that, at electron energies below 100 eV, the different choices of models for integration limits and the exciton interaction can affect the IMFP by 9–29%. At higher energies, the differences associated with the choice of energy-loss function and other input parameters are around 2.5–7.5%.

Raw-to-repository characterization data conversion for repeatable, replicable, and reproducible measurements

The National Institute for Materials Science, Japan, has been developing a materials data platform linked with a materials data repository system for rapid new material searching using materials informatics. Data conversion from raw data to a human-legible/machine-readable data file is one of the key preparation techniques prior to data analysis, where the converted data file should include metainformation. The tools developed by the authors can convert raw data to a structured data package that consists of mandatory and measurement-characterization metadata, primary and raw parameters, and formatted numerical data (FND). The FND are expressed as a matrix type with robust flexibility. This flexibility is achieved by applying the data analysis architecture of schema-on-read, not schema-on-write based on de jure standards, such as ISO documents. The primary parameters are carefully selected from raw parameters, and their vocabularies are transformed from instrument-dependent terms to general terms that everyone can readily understand. The converted data are linked with, for example, specimen information, process information, specimen handling records, and the electronic laboratory notebook. Using this raw-to-repository (R2R) conversion flow, the authors demonstrated that they can generate and store interoperable data files of x-ray photoelectron spectroscopy (XPS) spectra and depth profiles, powder x-ray diffraction patterns, (scanning) transmission electron microscope images, transmission electron diffraction patterns, electron energy-loss spectroscopy spectra, and calculated electron inelastic mean free path data. Linking measurement data to other required information ensures experimentally repeatable, replicable, and reproducible results. The mandatory and characterization metadata are used for quick electronic searching, and primary and raw parameters are convenient for setting up measurement conditions and useful for reproducibility/replicability and replicability/repeatability, respectively. The FND are human legible and machine readable using parser software, leading to a long lifetime of data utilization. The authors also developed a system to allow the semiautomatic data transfer from an instrument-controlling personal computer (PC) isolated from the communication network by adopting a Wi-Fi-capable secure digital card’s scripting capability while keeping the PC offline. They are developing further software for on-demand data manipulation after R2R data conversion. To date, it has been possible to perform XPS peak separation using an automated information compression technique without any a priori assumption. By combining R2R conversion with a high-throughput data collection system and automated data analysis routine, highly reproducible data acquisition and data analysis could be achieved, where human interaction is minimized. At this early stage, the authors demonstrate automated peak separation processing for XPS C 1s and O 1s narrow spectra of polyethylene terephthalate with very high reproducibility.

Electron inelastic mean free path in water.

Liquid phase transmission electron microscopy (LPTEM) is rapidly developing as a powerful tool for probing processes in liquid environments with close to atomic resolution. Knowledge of the water thickness is needed for reliable interpretation and modelling of analytical studies in LPTEM, and is particularly essential when using thin liquid layers, required for achieving the highest spatial resolutions. The log-ratio method in electron energy-loss spectroscopy (EELS) is often applied in TEM to quantify the sample thickness, which is measured relative to the inelastic mean free path (λIMFP). However, λIMFP itself is dependent on sample material, the electron energy, and the convergence and divergence angles of the microscope electronoptics. Here, we present a detailed quantitative analysis of the λIMFP of water as functions of the EELS collection angle (β) at 120 keV and 300 keV in a novel nanochannel liquid cell. We observe good agreement with earlier studies conducted on ice, but find that the most widely used theoretical models significantly underestimate λIMFP of water. We determine an adjusted average energy-loss term Em, water, and characteristic scattering angle θE, water that improve the accuracy. The results provide a comprehensive knowledge of the λIMFP of water (or ice) for reliable interpretation and quantification of observations in LPTEM and cryo-TEM studies.

Unveiling the principle descriptor for predicting the electron inelastic mean free path based on a machine learning framework

ABSTRACT The TPP-2M formula is the most popular empirical formula for the estimation of the electron inelastic mean free paths (IMFPs) in solids from several simple material parameters. The TPP-2M formula, however, poorly describes several materials because it relies heavily on the traditional least-squares analysis. Herein, we propose a new framework based on machine learning to overcome the weakness. This framework allows a selection from an enormous number of combined terms (descriptors) to build a new formula that describes the electron IMFPs. The resulting framework not only provides higher average accuracy and stability but also reveals the physics meanings of several newly found descriptors. Using the identified principle descriptors, a complete physics picture of electron IMFPs is obtained, including both single and collective electron behaviors of inelastic scattering. Our findings suggest that machine learning is robust and efficient to predict the IMFP and has great potential in building a regression framework for data-driven problems. Furthermore, this method could be applicable to find empirical formula for given experimental data using a series of parameters given a priori, holds potential to find a deeper connection between experimental data and a priori parameters.

Principles of High-Resolution Dopant Profiling in the Scanning Helium Ion Microscope, Image Widths, and Surface Band Bending

Dopant mapping using secondary electrons (SEs) in the scanning helium ion microscope is correlated with the surface charge distributions and electron inelastic mean free path. Numerical results show that the doping contrast is mainly a function of bulk built-in voltages modified by surface band-bending effects within a 3-nm-thick surface layer. The strong surface work function sensitivity of the SE yield and highly suppressed SE2/SE1 ratio militate the higher specificity to acceptors (donors), with surface charging constraining the threshold sensitivity to around 1017 cm−3 in thicker p-layers, compared to ~1015 cm−3 commonly attainable in the scanning electron microscope.

Electron elastic backscattering probability and inelastic mean free path

We study the angular distribution of electron elastic backscattering probability, using the Oswald–Kasper–Gaukler (OKG) model and Monte Carlo simulation. The present results are consistent with experimental data and other theoretical calculations. We also propose an approach that makes the OKG model applicable to determine electron inelastic mean free paths for compounds. We apply the OKG model with the proposed approach to zinc oxide (ZnO) and cerium dioxide (CeO2).

Low-energy electron properties: Electron inelastic mean free path, energy loss function and the dielectric function. Recent measurements, applications, and the plasmon-coupling theory

We review new self-consistent models of inelastic electron scattering in condensed matter systems for accurate calculations of low-energy electron inelastic mean free paths (IMFPs) for XAFS and low energy diffraction. The accuracy of theoretical determinations of the electron IMFP at low energies is one of the key limiting factors in current XAFS modeling and Monte Carlo transport. Recent breakthroughs in XAFS analysis show that there exist significant discrepancies between theoretical and experimental IMFP values, and that this can significantly impact upon extraction of other key structural parameters from both XANES and XAFS. Resolution of these discrepancies is required to validate experimental studies of material structures, and is particularly relevant to the characterization of small molecules and organometallic systems for which tabulated electron scattering data is often sparse or highly uncertain.Novel models implement plasmon coupling mechanisms for the first time, in addition to causally-constrained lifetime broadening and high-precision density functional theory, and enables dramatic improvements in the agreement with recent high profile IMFP measurements. We discuss a theoretical approach for IMFP determination linking the optical dielectric function and energy loss spectrum of a material with its electron scattering properties and characteristic plasmon excitations. We review models inclusive of plasmon coupling, allowing us to move beyond the longstanding statistical approximation and explicitly demonstrate the effects of band structure on the detailed behaviour of bulk electron excitations in a solid or small molecule. This interrogates the optical response of the material, which we obtain using density functional theory. We find that new developments dramatically improve agreement with experimental electron scattering results in the low-energy region (30 eV → 200 eV) where plasmon excitations are dominant. Corresponding improvements are therefore made in Low Energy Electron Transport, LEEM, theoretical XAFS spectra and detector modelling.

Comparison of the Mermin and Penn models for inelastic mean‐free path calculations for electrons based on a model using optical energy‐loss functions

We present detailed comparisons of calculated inelastic mean‐free paths (IMFPs) of electrons with energies up to 10 keV from the full Penn algorithm (FPA) model and the Mermin model using prototypical Drude‐type energy‐loss functions (ELFs). From systematic comparisons of IMFPs obtained from the prototypical ELFs using these two models, we obtain evidence that supports the Mermin model as the more reasonable model for electron IMFP calculations than the FPA model. The deviations between these two models are mainly because of finite plasmon lifetimes and show a strong dependence on the width of the ELFs and electron energy. For instance, the theoretical description of the ELF based on the Mermin model predicts lower values for the IMFPs than those obtained by the FPA approach in most cases. As for those prototypical ELFs with neither very sharp nor very broad structures that describe the ELF of actual materials, the Mermin IMFPs are smaller than the FPA IMFPs by about 5% for energies between 1 and 10 keV. We also found that over the 130 to 10 keV range, the modified Bethe equation fits IMFPs calculated with both algorithms within a 1% relative difference.

Low energy (1–100 eV) electron inelastic mean free path (IMFP) values determined from analysis of secondary electron yields (SEY) in the incident energy range of 0.1–10 keV

Since the emission of secondary electrons for any incident energy always involves the formation and emission of a cascade of slow electrons (<50 eV) the secondary electron yield (SEY) for arbitrary energies depends sensitively on the inelastic mean free path (IMFP) values at low energies (below 100 eV). This makes it possible to retrieve the information about the low energy IMFP from high energy SEY experiments. A Monte Carlo (MC) code has been developed to simulate SEY values and was employed to determine the IMFP at low energies (<100 eV). This is done by varying the energy dependence of the IMFP at low energies (<100 eV) during the MC simulation of the SEY between two extremes, calculated on the basis of the Mermin dielectric function and the Penn algorithm within the simplified single-pole approximation (SSPA). Those IMFP values that give the best χ2 fit of the simulated SEY values with experimental results are considered to be the most reliable. The described algorithm was employed for the investigation of Be, Al, Si, Ti, V, Fe, Ni, Cu, Ge, Nb, Mo, Pd, Ag, Ta, W, Pt, Au. For most materials these optimum IMFP values are found to be close to IMFP values based on the Mermin dielectric function.

Determination of electron inelastic mean free path of three transition metals from reflection electron energy loss spectroscopy spectrum measurement data

The energy loss functions (ELFs) of three transition metals (Cr, Co and Pd) have been derived from reflection electron energy loss spectroscopy spectrum with a theoretical analysis of the measured data [Xu et al., J. Appl. Phys. 123, 043306 (2018)]. In this work, we update our previous ELFs in a wider photon energy region (0–200 eV) with a better accuracy, which is verified by sum rules and a root-mean-square deviation. The electron inelastic mean free paths (IMFPs) of Cr, Co and Pd have been calculated with the obtained ELFs by adopting a dielectric response theory. We employ both the single-pole approximation and full Penn algorithm for the calculation of IMFPs, and the calculated results are compared with other references.

Electron Inelastic Mean Free Paths in Condensed Matter Down to a Few Electronvolts

A method is reported for a simple, yet reliable, calculation of electron inelastic mean free paths in condensed phase insulating and conducting materials, from the very low energies of hot electrons up to the high energies characteristic of electron beams. Through a detailed consideration of the energy transferred by the projectile in individual and collective electronic excitations, as well as ionizations, together with the inclusion of higher order corrections to the results provided by the dielectric formalism, inelastic mean free paths are calculated for water, aluminum, gold and copper in excellent agreement with the available experimental data, even at the elusive very low energy region. These results are important due to the crucial role played by low energy electrons in radiobiology (owing to their relevant effects in biodamage), and also in order to assess the not yet elucidated disagreement between older and recent measurements of low energy electron mean free paths in metals (which are relevant for low energy electron transport and effects in nanostructured devices).

Electron inelastic mean free paths in compounds

Electron inelastic mean free paths in compounds

Calculations of electron inelastic mean free paths. XII. Data for 42 inorganic compounds over the 50 eV to 200 keV range with the full Penn algorithm.

We have calculated inelastic mean free paths (IMFPs) for 42 inorganic compounds (AgBr, AgCl, AgI, Al2O3, AlAs, AlN, AlSb, cubic BN, hexagonal BN, CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF2, MgO, NaCl, NbC0.712, NbC0.844, NbC0.93, PbS, PbSe, PbTe, SiC, SiO2, SnTe, TiC0.7, TiC0.95, VC0.76, VC0.86, Y3Al5O12, ZnS, ZnSe, and ZnTe) for electron energies from 50 eV to 200 keV. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants for 15 compounds while calculated ELFs were utilized for the other 27 compounds. Checks based on ELF sum rules showed that the calculated ELFs were superior to the measured ELFs that we had used previously. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.60 %. 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 10.7 % for energies between 50 eV and 200 keV. This average RMS deviation is almost the same as that found in a similar comparison for a group of 41 elemental solids (11.9 %) although relatively large deviations were found for cubic BN (65.6 %) and hexagonal BN (34.3%). If these two compounds are excluded in the comparisons, the average RMS deviation becomes 8.7 %. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations and from experiments.

Electron inelastic mean free path in carbon and polycarbonate using a newly developed wide spectrum measurement method

The electron inelastic mean free path (IMFP) is commonly used in surface analysis techniques such as X-ray photoelectron spectroscopy. Though numerous studies have been conducted in the quest to define the IMFP value, it is still substantiated mainly on the basis of calculations and extrapolations. Experimental methods such as elastic peak electron spectroscopy tend to rely on indirect measurements. The main disadvantage of current methods is that the signal to noise ratio decreases with the increase in layer thickness or as the electron energy decreases.The newly developed wide energy spectrum (WES) method previously published by our group allows the IMFP to be directly evaluated for different electron energies. It exploits the attenuated electron multiple inelastic collisions (MIC) peak counts to calculate mean electron energy loss, rather than use the signal-electron peaks.In contrast, currently accepted methods focus on the signal electron in the spectrum and disregard the information contained in the lower-energy electrons. In understanding how to exploit this information, WES enhanced the relevant data collection by using a wide energy region upper bounded by the signal electron energy and lower bounded by the peak MIC energy in the spectrum.In this work, we have markedly expanded the validity of the WES method by assessing the IMFP values of a low Z polymer and by using a different substrate to produce the photoelectrons. The experiment was performed at the Spanish CRG Beamline (SpLine) at the European Synchrotron Radiation Facility (ESRF). Herein we present our initial results. In addition, we improved both the raw-data analysis and the background reduction processes used in WES to obtain enhanced and more accurate results.The IMFP values of carbon and polycarbonate were evaluated and compared with those predicted by the NIST. Small differences notwithstanding, the reported IMFP were shown to be well within the uncertainty boundaries.This work shows that the electron IMFP can be evaluated in solids by directly measuring the electrons that pass-through samples of the solids of interest. For the first time, there is an orderly path to experimentally evaluate electron IMFP by using the WES method. The new method was established to address the lack of experimental IMFP values for low energy electrons in solid materials.

The Chandrasekhar function for modeling photoelectron transport in solids

The objective of this study was to design an algorithm for calculating the Chandrasekhar function (H-function) dedicated for theoretical models of photoelectron transport in condensed matter. It has been shown that only the H-function values for non-conservative isotropic scattering are needed with the largest albedo, ω, values reaching 0.85. Different algorithms for calculating the H-function were analyzed to identify values of arguments for which an accuracy of 14 decimals is reached. It turned out that the most universal approach was an algorithm implementing the double-exponential rule which provided accurate values of the H-function for arguments varying over 10 orders of magnitude. However, the execution time was found to be shorter for algorithms implementing approximate analytical expressions in the region of small argument values, and the Stibbs and Weir algorithm (Stibbs and Weir, 1959) in the region of largest albedo values considered here. Based on these results, a mixed algorithm was created, and tested in calculations of integrals with integrands containing the H-function that are needed in the formalism for photoelectron transport. The execution time of calculations of the photoelectron current emitted from a solid or the photoelectron mean escape depth usually was very short, well below 1 s despite the fact that several such integrals were calculated, and the desired and attained accuracy was 13 decimals or better. Program summaryProgram title: H_FUNProgram Files doi: http://dx.doi.org/10.17632/xf4k25cnpk.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: Analytical formalism of photoelectron transport requires access to a program providing accurate values of the Chandrasekhar H-function within a possibly short execution time. However, the theory is based on non-conservative scattering; thus the albedo values associated with photoelectron transport vary in a limited range of values, not exceeding ω=0.85. Furthermore, the H-function in the range of small arguments is expressed by analytical formulas that can be implemented in programs providing the H-function with high speed and high accuracy. It would be of interest to develop an algorithm for calculating the H-function with the following features: (i) optimized with respect to execution time, (ii) providing the results with accuracy of 14 decimals, and (iii) taking into account the limited range of albedo values.Solution method: It has been proved that the H-function for both arguments smaller than about 10−5–10−4 are obtained with the desired accuracy from analytical formulas. In the remaining region of arguments, algorithms involving the integral representation need to be used. In recent reports, the integration method based on the double-exponential rule was recommended as an accurate and fast approach. Similarly, an algorithm implementing the Stibbs and Weir theory was also recently indicated as a very effective source of the H-function. In the present work, the execution times for the above algorithms were compared for different pairs of arguments. Finally, separate domains were ascribed to tested algorithms in such a way that the execution times were always the shortest. Based on these results, a program HFUNELE dedicated to calculations of photoelectron transport was developed.Additional comments:(i) Restrictions: The program HFUNELE optimized with respect to speed can be used for the directional variable, μ, varying in the range from zero to unity, while the allowed albedo, ω, range is from zero to 0.85, i.e., the range needed for applications in photoelectron transport theory.(ii) Unusual features: The enclosed test program demonstrates the fact that the typical integrals over the integrands containing the H-function reach or even exceed an accuracy of 13 decimals. Furthermore, application of the program HFUNELE is illustrated in exemplary calculations related to photoelectron transport theory (determination of the photoelectron signal intensity or the photoelectron mean escape depth). To facilitate the performance of the test program, several databases of parameters needed for the calculations are implemented using the DATA statement (electron inelastic mean free paths, albedo values for elements, atomic weights and densities), or included as accompanying text files: BAND_MG.TXT and BAND_AL.TXT (atomic subshells, subshell photoionization cross sections, electron binding energies, and asymmetry parameters).

The Influence of Photoelectron Escape in Radiation Damage Simulations of Protein Micro-Crystallography

Radiation damage represents a fundamental limit in the determination of protein structures via macromolecular crystallography (MX) at third-generation synchrotron sources. Over the past decade, improvements in both source and detector technology have led to MX experiments being performed with smaller and smaller crystals (on the order of a few microns), often using microfocus beams. Under these conditions, photoelectrons (PEs), the primary agents of radiation-damage in MX, may escape the diffraction volume prior to depositing all of their energy. The impact of PE escape is more significant at higher beam energies (&gt;20 keV) as the electron inelastic mean free path (IMFP) is longer, allowing the electrons to deposit their energy over a larger area, extending further from their point of origin. Software such as RADDOSE-3D has been used extensively to predict the dose (energy absorbed per unit mass) that a crystal will absorb under a given set of experimental parameters and is an important component in planning a successful MX experiment. At the time this study was undertaken, dose predictions made using RADDOSE-3D were spatially-resolved, but did not yet account for the propagation of PEs through the diffraction volume. Hence, in the case of microfocus crystallography, it is anticipated that deviations may occur between the predicted and actual dose absorbed due to the influence of PEs. To explore this effect, we conducted a series of simulations of the dose absorbed by micron-sized crystals during microfocus MX experiments. Our simulations spanned beam and crystal sizes ranging from 1μm to 5μm for beam energies between 9 keV and 30 keV. Our simulations were spatially and temporarily resolved and accounted for the escape of PEs from the diffraction volume. The spatially-resolved dose maps produced by these simulations were used to predict the rate of intensity loss in a Bragg spot, a key metric for tracking global radiation damage. Our results were compared to predictions obtained using a recent version of RADDOSE-3D that did not account for PE escape; the predicted crystal lifetimes are shown to differ significantly for the smallest crystals and for high-energy beams, when PE escape is included in the simulations.

Low-energy electron inelastic mean free paths for liquid water

We improve the Mermin–Penn algorithm (MPA) for determining the energy loss function (ELF) within the dielectric formalism. The present algorithm is applicable not only to real metals, but also to materials that have an energy gap in the excitation spectrum. Applying the improved MPA to liquid water, we show that the present algorithm is able to address the ELF overestimation at the energy gap, and the calculated results are in good agreement with experimental data.

Graphene Nanoreactors: Photoreduction of Prussian Blue in Aqueous Solution

Prussian dyes are characterized by interesting photomagnetic properties due to the photoinduced electron transfer involved in the Fe oxidation and spin state changes. Ferromagnetic Prussian blue (PB) in contact with titanium dioxide (TiO2) can be reduced to paramagnetic Prussian white (PW) upon UV band gap excitation of TiO2. This process is promoted by the presence of a hole scavenger, such as water, fundamental to ensure the overall charge balance and the continuity of the process. In order to clarify the photoinduced reduction mechanism and the role of water, an innovative system of graphene nanobubbles (GNBs) filled with a PB aqueous solution was developed, enabling the application of electron spectroscopies to the liquid phase, up to now limited by the vacuum required to overcome the short electron inelastic mean free path in dense medium. In this work GNBs formed on the photocatalytic substrate are able to act as “nanoreactors”, and they can control and take part in the reaction. The evolution of Fe L2,3 edge X-ray absorption spectra measured in total electron yield through the graphene membrane revealed the electron reduction from PB (FeIII–CN–FeII) to PW (FeII–CN–FeII) upon UV irradiation, shedding light on the photoinduced electron transfer mechanism in liquid phase. The results, confirmed also by Raman spectroscopy, unequivocally demonstrate that the reaction occurs preferentially in aqueous solution, where water acts as hole scavenger.