InternationalX-ray absorption spectroscopy and related techniquesTables for Crystallography Volume I Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2024 |
International Tables for Crystallography (2024). Vol. I. ch. 2.9, pp. 100-106
https://doi.org/10.1107/S1574870722001586 ## Inelastic scattering of electrons in solids
An overview is given of inelastic electron scattering in solids, with an emphasis on calculations and measurements of electron inelastic mean free paths (IMFPs). There is generally good agreement between IMFPs calculated from optical data and IMFPs measured by elastic-peak electron spectroscopy for electron energies greater than 100 eV. For lower energies, however, there are significant differences between these calculated IMFPs and IMFPs determined from X-ray absorption fine-structure experiments. Keywords: electrons; inelastic mean free paths; inelastic scattering. |

Information on the inelastic scattering of electrons in solids is important in theories of X-ray absorption fine structure (Rehr & Albers, 2000) and for various applications ranging from radiation physics and radiation transport to thin-film analysis in the transmission electron microscope (TEM) and surface analysis by Auger-electron spectroscopy and X-ray photoelectron spectroscopy. A key parameter in these applications is the electron inelastic mean free path (IMFP) λ, which is simply related to the total cross section for inelastic scattering σ and the number density of atoms per unit volume in the solid *N*, We will give a brief summary of efforts to calculate and measure IMFPs after first giving some historical information that will guide the subsequent discussion.

In the mid-1950s, a number of groups were measuring the energy-loss spectra of 20–50 keV electrons transmitted through thin sample films (for example of 50 nm thickness) of various solids (Marton, 1956; Pines, 1956). At the time, three main theories had been proposed to describe the observed energy losses. Firstly, Leder *et al.* (1956) found numerical correlations between the energy losses measured for many solids and the positions of fine structure in near-edge X-ray absorption spectra. These correlations suggested that the energy losses were predominantly due to single-electron excitations that could be related to the band structure of each solid. Secondly, Pines and coworkers proposed that the energy losses were due to plasma oscillations of the valence electrons, *i.e.* to bulk plasmons (Pines, 1956). While good correlations were found between observed and predicted energy losses for some solids (for example, the so-called free-electron solids such as magnesium, aluminium and silicon), there were appreciable inconsistencies for other solids (for example transition and noble metals). Thirdly, a dielectric model for describing inelastic scattering had been proposed by Fano (1956), but the lack of reliable optical data for the region of interest (wavelengths between 50 and 400 nm) made interpretations difficult.

Subsequent work has shown that the dielectric model, as described by Fano (1956) and many others, provides a comprehensive description of the inelastic scattering of electrons in solids. It is now recognized that this model incorporates both excitations of bulk plasmons and single-electron excitations. The probability of inelastic scattering in the limit of zero momentum transfer is proportional to the energy-loss function (ELF), , where ɛ_{1} and ɛ_{2} are the real and imaginary parts of the complex dielectric constant ɛ. ɛ_{1} and ɛ_{2} are simply related to the conventional optical constants, the index of refraction *n* and the extinction coefficient *k* by ɛ_{1} = *n*^{2} − *k*^{2} and ɛ_{2} = 2*nk*. For valence-electron excitations (typically involving energy losses of less than about 100 eV), is usually appreciably different from unity and there is a large difference between energy-loss and optical absorption spectra. Maxima may also occur in the ELF at energy losses or optical frequencies near those where ɛ_{1} = 0 and ɛ_{2} ≪ 1; these maxima can be identified as bulk plasmons. There may also be maxima in the ELF near frequencies for which there are maxima in ɛ_{2}; these maxima can be identified as single-electron excitations or interband transitions. Parametric calculations show how the positions and intensities of structure in the ELF can be related to the magnitude of ɛ_{1} and to structure in ɛ_{2} (Powell, 1969). For inner-shell excitations (typically involving energy losses greater than about 100 eV), ɛ_{1} ≃ 1 and ɛ_{2} ≪ 1, so that Im(−1/ɛ) ≃ ɛ_{2} and energy-loss spectra are similar to X-ray absorption spectra.

In an interesting application, Werner *et al.* (2009) have shown that the optical properties of solids for photon energies between 0.5 and 70 eV can be derived from reflection energy-loss spectra acquired at two different primary energies. Measurements of inner-shell excitations in energy-loss spectroscopy of electrons transmitted through thin films, particularly in the TEM, have now become a useful method of thin-film analysis (Egerton, 2011).

We describe several approaches that have been used to calculate IMFPs.

With the use of the first Born approximation, the differential cross section (DCS) for inelastic scattering per atom or molecule for energy loss ω and momentum transfer *q* in condensed matter is given by (Schnatterly, 1979; Powell, 1984)where *E* is the electron energy and Im[−1/ɛ(ω, *q*)] is the ELF, which is now defined by the complex dielectric constant ɛ(ω, *q*). We have used Hartree atomic units in equation (2), in which *m*_{e} = *e* = ℏ = 1, where *m*_{e} is the electron rest mass, *e* is the elementary charge and ℏ is the reduced Planck constant.

The key material parameter in equation (2) is the ELF. If this is known, the IMFP at an electron energy *E* can be calculated from integration of the DCS,The integration domain *D* for conductors is determined from the maximum and minimum energy losses and the largest and smallest kinematically allowed momentum transfers for a given energy *E* and ω,where *E*_{F} is the Fermi energy of the material of interest.

Unfortunately, the ELF is not known for most materials. For a limited number of materials, optical data are available over a sufficiently large range of photon energies that the optical ELF, Im[−1/ɛ(ω, *q* = 0)], can be determined. We then have to use an extension algorithm to determine the *q*-dependence of the ELF. We now briefly describe three of these algorithms.

Penn (1987) proposed an algorithm now known as the full Penn algorithm (FPA) for the calculation of Im[−1/ɛ(*q*, ω)] in equation (3). The FPA is based on the Lindhard model dielectric function (Lindhard, 1954) and the use of measured optical ELF data for a given material. The ELF in equation (2) is then given bywhere ɛ^{L} denotes the Lindhard model dielectric function for a free-electron gas with plasmon energy ω_{p} [= (4π*n*)^{1/2}], *n* is the electron density and *g*(ω_{p}) is a coefficient introduced to satisfy the condition Im[−1/ɛ(*q* = 0, ω)] = Im[−1/ɛ(ω)]. The coefficient *g*(ω_{p}) is given byThe ELF from the FPA in equation (5) can be described as the sum of two contributions, one associated with the plasmon pole and the other with single-electron excitations (Mao *et al.*, 2008), Details of the calculation procedures for each term in equation (7) have been published by Shinotsuka *et al.* (2015). This algorithm has been used to calculate IMFPs for energies from 50 to 200 keV.

A second algorithm for IMFP calculations was proposed by Abril *et al.* (1998). They recommended the use of the Mermin (1970) dielectric function, ɛ_{M}(*q*, ω), since this approach (unlike the Lindhard dielectric function used in the FPA) takes the plasmon lifetime into account.

The ELF in equation (3) is separated into two components, one for valence-electron excitations and the other for inner-shell excitations:The former component is fitted with a selected number, *L*, of Drude functions,where *A _{i}*,

*ω*and

_{pi}*γ*are fitting parameters that represent the oscillator strength, peak energy and width of the

_{i}*i*th oscillator, respectively, so thatThat is, a satisfactory fit has to be made with equation (10) to a set of optical ELF data, typically from about 1 eV to the threshold energy for the lowest inner-shell excitation.

For inner-shell excitations, Heredia-Avalos *et al.* (2005) utilized atomic generalized oscillator strengths (GOSs) from hydrogenic calculations for the second term of equation (8), where is the GOS for the (*n*, *l*) subshell of the *j*th element.

A third algorithm has been proposed by Da *et al.* (2014). They developed what they called an extended Mermin (EM) method to avoid separate IMFP calculations for valence-band and inner-shell electrons. The EM method employs an unlimited number of Mermin oscillators and allows negative oscillators [*i.e.* negative values of *A _{i}* in equations (9) and (10)] to take into account not only a generally small number of valence-electron excitations for observed structure in the ELF, as is common with the usual Mermin algorithm, but also infrared transitions and inner-shell excitations that can display `edges' and continua.

We briefly mention several other IMFP calculations. IMFPs can be obtained from first-principles calculations of hot-electron lifetimes, typically for energies of less than 4 eV, although some calculations have been made for energies up to 100 eV (Chulkov *et al.*, 2006). That is, there was no use of optical ELF data. Nagy & Echenique (2012) reported what they term lower bounds for copper IMFPs at energies of less than 75 eV based on Fermi liquid (FL) theory. Nguyen-Truong (2016) calculated the copper ELF using the adiabatic local-density approximation in the framework of time-dependent density-functional theory (DFT) for energy losses of up to 60 eV. This ELF agreed well with an earlier DFT calculation by Werner *et al.* (2009). The ELF was then used to compute IMFPs for copper at energies between 1 and 100 eV (Nguyen-Truong, 2016). Finally, de Vera & Garcia-Molina (2019) have reported IMFPs for water, aluminium, copper and gold based on higher-order corrections to the first Born approximation.

The first three algorithms described above require optical data, typically for photon energies between about 1 eV and an upper limit that can be as high as 200 keV (Shinotsuka *et al.*, 2015), from which the optical ELF (*i.e.* for *q* = 0) can be calculated. The required data come from a variety of sources, as described by Tanuma *et al.* (2011) and Shinotsuka *et al.* (2015). It is important to check the reliability or internal consistency of optical ELFs using two useful sum rules (Tanuma *et al.*, 1993, 2011).

ELFs for valence-electron excitations have also been derived from density-functional theory by Werner *et al.* (2009) and by Chantler & Bourke (2014).

Tanuma *et al.* (2011) reported IMFP calculations for 41 elemental solids (lithium, beryllium, graphite, diamond, glassy carbon, sodium, magnesium, aluminium, silicon, potassium, scandium, titanium, vanadium, chromium, iron, cobalt, nickel, copper, germanium, yttrium, niobium, molybdenum, ruthenium, rhodium, palladium, silver, indium, tin, caesium, gadolinium, terbium, dysprosium, hafnium, tantalum, tungsten, rhenium, osmium, iridium, platinum, gold and bismuth) for electron energies between 50 eV and 30 keV using the FPA, while Shinotsuka *et al.* (2015) reported similar calculations for the same solids for energies between 50 eV and 200 keV using a relativistic version of the FPA. Fig. 1 shows a summary plot of the IMFPs calculated by Shinotsuka and coworkers for the 41 elemental solids as a function of electron energy between 10 eV and 10 keV where the energy is expressed relative to the Fermi level. While the FPA is expected to give reliable results for energies above 50 eV (Tanuma *et al.*, 1991), we show IMFPs at lower energies in Fig. 1 to indicate trends.

Fig. 2 shows IMFPs for 42 inorganic compounds (AgBr, AgCl, AgI, Al_{2}O_{3}, AlAs, AlN, AlSb, cubic BN, hexagonal BN, CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF_{2}, MgO, NaCl, NbC_{0.712}, NbC_{0.844}, NbC_{0.93}, PbS, PbSe, PbTe, SiC, SiO_{2}, SnTe, TiC_{0.7}, TiC_{0.95}, VC_{0.76}, VC_{0.86}, Y_{3}Al_{5}O_{12}, ZnS, ZnSe and ZnTe) that were calculated by Shinotsuka *et al.* (2019) for energies between 10 eV and 10 keV where the energy is expressed relative to the bottom of the conduction band. These IMFPs were calculated for energies up to 200 keV using the relativistic FPA algorithm of Shinotsuka *et al.* (2015) that was modified to include the effect of the bandgap energy for nonconductors on the integration domain *D* of equation (1). We note that the ELFs for many of these compounds were calculated using the *WIEN*2*k* and *FEFF* computer codes (Shinotsuka *et al.*, 2021).

For energies between 1 and 10 keV, there is at least a factor of 5 difference between the smallest and largest IMFPs (at a given energy) for the elemental solids in Fig. 1 and a factor of 2.2 difference for the inorganic compounds in Fig. 2. These variations in IMFP magnitudes are mainly due to variations in bulk densities. Minima in the IMFP plots occur at energies between about 10 eV and about 100 eV. These variations are associated with differences in the shapes of the ELFs for the solids.

Shinotsuka *et al.* (2022) calculated IMFPs for 14 organic compounds [26-*n*-paraffin, adenine, β-carotene, diphenylhexatriene, guanine, Kapton, polyacetylene, poly(butene-1-sulfone), polyethylene, polymethylmethacrylate, polystyrene, poly(2-vinylpyridine), thymine and uracil] and liquid water for electron energies between 50 eV and 200 keV. These IMFPs were also calculated with the relativistic FPA and consideration of the bandgap energy of each compound.

Tanuma *et al.* (1994) developed the following predictive IMFP formula from an analysis of calculated IMFPs for their original group of 27 elemental solids (Tanuma *et al.*, 1991) and for a group of 14 organic compounds for energies between 50 eV and 2 keV,where *E* is the electron energy (in eV), *E*_{p} = 28.82(*N*_{v}ρ/*M*)^{1/2} is the bulk plasmon energy (in eV), *N*_{v} is the number of valence electrons per atom or molecule, ρ is the bulk density (in g cm^{−3}) and *M* is the atomic or molecular weight. Simple expressions were found for the four parameters in equation (12*a*) in terms of material properties,where *E*_{g} is the bandgap energy (in eV) for nonconductors and *U* = *N*_{v}ρ/*M*. Equation (12) is the TPP-2M formula of Tanuma *et al.* (1994) that can be used to estimate IMFPs in other materials. In later work, Tanuma *et al.* (2011) found that the TPP-2M formula was useful for energies up to 200 keV. The relativistic version of equation (12*a*) iswhereand β, γ, *C* and *D* are the same parameters as given in equations (12*b*), (12*c*), (12*d*) and (12*e*). We note that the relativistic term α(*E*) (equation 13*b*) decreases from unity by 0.3%, 0.6%, 1.4%, 2.9% and 5.6% for energies of 1 keV, 2 keV, 5 keV, 10 keV and 20 keV, respectively.

Shinotsuka *et al.* (2015) found that the average root-mean-square (r.m.s.) difference between the calculated IMFPs for 41 elemental solids and the corresponding values from equation (13) was 11.9%. Relatively large r.m.s. differences (of up to 71%) were found for diamond, graphite and caesium, as found by Tanuma, Powell *et al.* (2005). If these three elements were excluded from the comparisons, the average r.m.s. difference was 8.9%. In similar comparisons, Shinotsuka *et al.* (2019) found an average r.m.s. difference of 10.7% between their calculated IMFPs for 42 inorganic compounds and the values from equation (13), although relatively large differences were found for cubic BN (66%) and hexagonal BN (34%). In a most recent paper, Shinotsuka *et al.* (2022) reported an average r.m.s. differences of 7.2% between their calculated IMFPs for 14 organic compounds and liquid water and the IMFPs from equation (13). They concluded that the relativistic TPP-2M equation (equation 13) is useful for estimating IMFPs in solid materials for electron energies between 50 eV and 200 keV, although the accuracy of these estimates is likely to be poorer for energies less than about 200 eV.

We briefly describe four experimental approaches that have been used to obtain IMFPs. All of these approaches rely on models of the relevant physical processes and auxiliary data.

IMFPs can be obtained from measurements of the ratios of elastically back-scattered electron yields from a sample of interest and a reference material for which the IMFP is known, a technique known as elastic-peak electron spectroscopy (EPES; Powell & Jablonski, 1999, 2009). These ratios, determined for a range of incident electron energies, are then compared with corresponding ratios from Monte Carlo simulations in which the sample IMFP is a parameter. Nickel, copper, silver or gold are often chosen as reference materials since these materials showed good consistencies in comparisons of IMFPs calculated from optical data and IMFPs determined from EPES experiments (Powell & Jablonski, 1999).

Werner *et al.* (2009) analyzed reflection electron energy-loss spectra (REELS) for 17 elemental solids and derived their optical constants for photon energies between 0.5 and 70.5 eV. With these results and atomic photoabsorption data for higher photon energies, Werner and coworkers computed ELFs and then IMFPs using the Penn algorithm for energies between 100 eV and 10 keV. The average r.m.s. deviation between these REELS IMFPs and the optical IMFPs of Tanuma and coworkers was 5.8%. Werner and coworkers also commented that their REELS IMFPs were `indistinguishably similar' to those obtained from optical constants calculated using density-functional theory.

Bourke & Chantler (2010) and Chantler & Bourke (2010) reported IMFPs of 5–120 eV electrons in copper and molybdenum that were derived from high-accuracy measurements of X-ray absorption fine structure (XAFS). The measured spectra were compared with corresponding calculated spectra (that included the effects of core-hole widths and thermal vibrations) and the IMFPs were obtained as measures of the additional broadening needed for the calculated spectra to agree with the measured spectra.

Knapp *et al.* (1979) determined photoelectron lifetimes from an analysis of photoelectron spectra (PES) from a Cu(100) surface for energies between about 11 and 14 eV. These lifetimes were converted to IMFPs using a group velocity for the excited electrons based on a free-electron model.

Wertheim *et al.* (1992) measured PES of lithium, sodium, potassium and rubidium for photon energies between 22 and 140 eV. These spectra showed separate peaks due to photoemission from the surface layer of atoms in each metal as well as other peaks arising from other atoms, *i.e.* the bulk of the solid. The samples were prepared by vacuum evaporation onto an Ni(100) substrate cooled to 78 K and the films had a (110) surface orientation. With the use of model descriptions of surface and bulk lineshapes, they determined the fractional intensity of bulk emission, *f*_{B}, for each spectrum and equated this fraction to exp(−*d*/λ), where *d* is the (110) layer spacing. Wertheim and coworkers could thus determine IMFPs for each metal as a function of photoelectron energy. Pi *et al.* (2012) used the same method to determine IMFPs in GaAs for energies between about 24 and 140 eV.

Two groups have reported IMFPs for elemental solids from EPES experiments, with Tanuma, Shiratori *et al.* (2005) providing IMFPs for 13 elemental solids for energies between 50 eV and 5 keV and Werner *et al.* (2000, 2001) providing IMFPs for 24 elemental solids for energies between 50 eV and 3.4 keV. The average r.m.s. difference between the optical IMFPs of Tanuma *et al.* (2011) and the EPES IMFPs of Tanuma, Shiratori *et al.* (2005) was 12% (for the 11 solids that were common to both data sets), while the average r.m.s. difference between the optical IMFPs of Tanuma *et al.* (2011) and the EPES IMFPs of Werner *et al.* (2000, 2001) was 15% (for the 17 solids that were common to both data sets).

We now show comparisons of calculated and measured IMFPs for an elemental solid (copper) and an inorganic compound (GaAs). Fig. 3 shows plots of copper IMFPs calculated using the FPA by Shinotsuka *et al.* (2015), IMFPs calculated using the EM method by Da *et al.* (2014), IMFPs from the FL calculations of Nagy & Echenique (2012), IMFPs from the DFT calculations of Nguyen-Truong (2016), IMFPs from the dielectric formalism including high-order corrections to the first Born approximation of de Vera & Garcia-Molina (2019) and IMFPs from the TPP-2M predictive formula (equation 11) as a function of electron energy together with IMFPs determined from XAFS experiments (Bourke & Chantler, 2010), PES experiments (Knapp *et al.*, 1979) and EPES experiments (Tanuma, Shiratori *et al.*, 2005). The copper IMFPs from the EM method in Fig. 3 agree well with those previously obtained with the Mermin method by Bourke & Chantler (2012).

For energies over 300 eV, we see good agreement among the copper IMFPs calculated from the FPA and EM methods, the dielectric formalism including high-order corrections (HOC) and the TPP-2M formula with IMFPs from EPES experiments. For energies between 50 and 300 eV, IMFPs calculated from the EM method and from the TPP-2M formula generally agree well with IMFPs measured by the EPES and XAFS methods, although the FPA results are larger than the measured values. For energies between 50 and 300 eV, IMFPs calculated from the HOC are in excellent agreement with IMFPs measured by EPES and are slightly larger than IMFPs measured by the XAFS method, from EM methods and from the TPP-2M formula. IMFPs from the DFT calculations of Nguyen-Truong (2016) are consistent with IMFPs from the EM method, IMFPs from the TPP-2M equation and IMFPs from the XAFS experiments at about 100 eV, but are smaller than those from the FPA. For energies less than 20 eV, however, the DFT IMFPs become consistent with the FPA IMFPs, IMFPs calculated from the HOC method and the IMFPs measured by PES, but are much larger than the XAFS IMFPs. The XAFS IMFPs are also appreciably smaller than the IMFPs calculated by the FPA and EM methods for energies less than 50 eV. These systematic differences at low energies could be due to correlation and exchange effects that were not included in the IMFP calculations from the FPA (Powell & Jablonski, 1999) and/or to surface excitations (Powell & Jablonski, 2009), although correlation and exchange effects are included in the DFT calculations. The IMFPs calculated by Nagy & Echenique (2012) for energies between 10 and 75 eV are described as `lower bounds'. These IMFPs are numerically similar to the IMFPs from EXAFS experiments for energies between 10 eV and about 50 eV, but show a very different energy dependence.

We point out that Rundgren (1999) found good agreement between measured current–voltage plots in a low-energy electron-diffraction experiment with a Cu(111) surface and with calculated plots based on the FPA IMFPs of Tanuma *et al.* (1991) for energies between 30 and 190 eV. If the IMFPs in copper were appreciably smaller than the FPA values (as indicated by the XAFS IMFPs in Fig. 3 for energies less than 100 eV), the calculated current–voltage plots would have had much larger peak widths.

Fig. 4 shows comparisons of IMFPs for GaAs calculated from the FPA, from the TPP-2M equation and from EPES experiments (Krawczyk *et al.*, 1998) and derived from an analysis of PES experiments (Pi *et al.*, 2012). We see generally good agreement among the calculated and measured IMFPs for energies from 20 eV to 5 keV. While there is some scatter in the IMFPs from the photoemission experiments, there is no indication of the substantial systematic differences between IMFPs from the FPA and IMFPs from XAFS experiments that were seen for copper in Fig. 3.

Powell & Jablonski (2009) concluded that IMFPs calculated from optical data with the FPA have uncertainties of up to about 10%. Their analysis was made for electron energies between 50 eV and about 5 keV. It is now apparent that similar calculations with the Mermin (1970) dielectric function or the extended Mermin method (Da *et al.*, 2014) would be more reliable for energies of less than about 100 eV. Significant disagreements exist between IMFPs determined from XAFS experiments and PES experiments, and between IMFPs from XAFS experiments and calculated IMFPs, particularly for energies of less than about 100 eV, as indicated by the results for copper in Fig. 3. Additional experiments or analyses are therefore needed to resolve these differences.

A database is available that provides IMFPs calculated from optical data, IMFPs from EPES experiments and IMFPs from the TPP-2M predictive formula for electron energies from 50 eV to 10 keV (Powell & Jablonski, 2010).

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