Journal of Electron Spectroscopy and Related Phenomena

Since the emission of secondary electrons for any incident energy always involves the formation and emission of a cascade of slow electrons (<50eV) the secondary electron yield (SEY) for arbitrary energies depends sensitively on the inelastic mean free path (IMFP) values at low energies (below 100eV). 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 (<100eV). This is done by varying the energy dependence of the IMFP at low energies (<100eV) 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 χ 2 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.


Introduction
The secondary electron yield (SEY) δ represents the number of emitted secondary electrons (SEs) per incident primary electron. Since it is impossible to experimentally distinguish between primary (in particular inelastically backscattered primaries) and secondary electrons, those electrons in the electron spectrum with an energy below 50 eV are usually designated as secondary electrons. This is based on the assumption that the number of inelastically backreflected primaries with energies below 50 eV is low in comparison with those in the SE cascade, which is mainly made up of low energy electrons. The SEY measurements found in the literature are usually performed for incident energies ranging from several eV to several keV. To theoretically describe the secondary electron emission from solids one needs quantitative knowledge of electron scattering processes including those at energies below 100 eV. In particular, quantitative knowledge of IMFP values at low energies are needed. In the past 30 years, the determination of IMFP values at energies above 200 eV has been an active area of research and nowadays commonly accepted values are calculated using dielectric response theory, employing optical constants [1,2]. It is important to emphasize that different approaches give a similar IMFP in this energy range and are in a good agreement with experimental results. However, different models give widely different values for the IMFP at low energies as illustrated in Fig. 1(a), which shows the IMFP according to the formalism by Penn implying the simplified single-pole approximation (SSPA) (represented by the dot-dashed green curve) [3,4] and the IMFP theoretically calculated using the Mermin dielectric function, which gives significantly lower values (represented by the dashed black curve). Data points show experimentally measured or theoretically calculated values found in the literature [5][6][7]3,[8][9][10]. Note that IMFP values based on the Penn SSPA algorithm are known to be less reliable for energies below 100 eV, this approach is used in the present paper for comparison only. The Mermin dielectric function q ( , ) M [11] represents an improvement over the Lindhard dielectric function since it includes broadening due to the finite lifetime of excitations, which makes this model a more realistic and widely-used to calculate the IMFP [12][13][14]. Below 100 eV it is complicated to conceive any experimental method to verify theoretical calculations. Elastic peak electron spectroscopy (EPES) technique is usually employed for the experimental determination of IMFPs [8], although, it becomes less reliable at energies below 100 eV. There is a number of other experimental techniques but all of them require measurements at low https://doi.org/10.1016/j.elspec.2019.02.003 Received 5 October 2018; Received in revised form 17 January 2019; Accepted 1 February 2019 energies, which is essentially a big challenge [15]. In the present work, an approach is presented to retrieve the low energy IMFP values from high energy SEY experiments. Fig. 1(b) shows results of Monte Carlo (MC) calculations of the SEY for Au assuming the two extreme sets of IMFP data addressed above. Note that the IMFP data in Fig. 1(a) are identical for energies above 100 eV. It is clearly seen that even at high energies the SEY values depend very strongly on the IMFP values at energies below 100 eV. Fig. 2 conveys this idea clearly. Each panel of Fig. 2 shows 20 trajectories of primary electrons impinging on a gold target and concurrently the secondary electron cascade for different energies (50, 500 and 5000 eV) for the two extreme energy dependencies of the IMFP shown in Fig. 1(a). The upper panel corresponds to the IMFP data calculated using the Mermin dielectric function and the lower panel corresponds to the data calculated by the Penn algorithm [19]. The penetration of incident electrons into the solid is seen to be deeper for higher incident energies due to the increasing of the IMFP ( Fig. 1(a)). However, the main difference is the fact that at all incident energies (50, 500 and 5000 eV) the number of outgoing electrons is also very different for the two assumed low energy dependencies of the IMFP: for higher IMFP values a higher fraction of electrons within the cascade is indeed emitted from the surface. Therefore, even the high energy SEY data contain information on the low energy IMFP.
In the present work, MC calculations were performed using different energy dependencies of the IMFP values below 100 eV ranging between the two extreme cases shown in Fig. 1(a) and determining those IMFP values which give the minimum least squares deviation of the simulated SEY curves with experimental data available in the literature. In Section 2 a Monte Carlo simulation model for the SEY calculations is described. Section 3 describes the input parameters needed for the MC simulation of the SEY. In Section 4 results of comparisons of the simulated SEY data with available experimental literature data [16] are given as well as a comparison with individual extensive data sets produced by Bronstein [17] and Zadrazil [18].

Monte Carlo simulation model
An MC code has been developed to simulate electron scattering processes in solids, including the production of the secondary electron cascade, based on a commonly employed algorithm [20][21][22]. This common model describes the secondary electron emission process implying three steps: primary electron transport in the solid, production of a secondary electron, transport and escape over the surface potential barrier of the produced secondary electron.

Electron transport
In the present algorithm trajectories of individual electrons are modeled by assuming that their paths can be described by straight lines between scattering processes. The considered scattering processes comprise elastic and inelastic interaction with the solid. Elastic scattering occurs by the interaction with the ionic subsystem of the solid and leads to a change in the direction of motion by the (screened) Coulomb field of the nucleus with negligible energy loss. Inelastic scattering occurs through interaction with the valence band electrons and ionisation of core electrons. These two inelastic processes are responsible for change of the energy and creation of a secondary electron, but the initial direction of motion is assumed to remain unchanged. In the present model surface excitations are not considered, although it is believed that they have a significant influence on the SEY [23]. The description of elastic scattering is based on the Mott elastic scattering cross section [24]. Linear response theory based on empirical optical constants is used to describe inelastic scattering [3,22]. In the present work, elastic and inelastic scattering processes are assumed to be independent [20]. Although this assumption is questionable in principle for low energy electrons, it is expected to not significantly affect the SEY. A measure for the momentum transfer along the initial direction which is referred to as the transport mean free path (TRMFP) during inelastic scattering is usually one order of magnitude larger than the transport mean free path for elastic scattering. This fact allows neglecting the deflections during inelastic collisions.

Production of secondary electrons
It is assumed that at each inelastic interaction the energy loss ω, experienced by the primary electron, is transferred to a secondary electron. The SE then has the energy E SE = E F + ω in the case of a valence electron loss, where E F is the Fermi energy. Here it is assumed that SEs are excited only from the Fermi level. While this is a commonly adopted approach the authors of Ref. [21] took into account the band structure effect, within which the probability distribution of electron excitation in the valence band is assumed to be proportional to the density of states of a free electron gas. This causes a decreasing of the SEY values.
In the case of ionisation of an inner shell electron a secondary electron with the energy E SE = ω − E b is produced, where E b is the binding energy of the ionised shell.
During the production of SEs the momentum transfer is neglected. The initial angular distribution of SEs is assumed to be isotropic in the present model since by the time the electron comes out it completely 'forgets' its initial momentum due to the momentum relaxation after multiple elastic scatterings. Therefore, this is believed to be a  [16][17][18] for normal electron incidence and emission into the entire hemisphere above the sample.
reasonable approximation for the present purposes.

Escape over the surface potential barrier
A final aspect of the secondary electron emission is the escape of slow electrons over the surface potential barrier. Therefore, a simple model for the potential step between the vacuum and the solid is to assume a barrier consisting of a potential step with height equal to the so-called inner potential U i [25] (see Fig. 3). Disruption of the periodicity of the crystal lattice at the surface leads to the potential step, represented by the inner potential U i as the energy difference between the bottom of the valence band and the vacuum level. Note that here and below the bottom of the valence band is used as zero for the energy scale.
When an electron, considered as a plane wave, travels from the solid (with the inner potential U i ) with a given energy E s to vacuum at any angle other than zero from the surface normal it will be refracted at the solid surface boundary according to Snell's law for electrons. The perpendicular component of the momentum is reduced due to the presence of the surface potential barrier. The energy E s inside the solid will be decreased by U i on escaping through the vacuum-surface interface (see Fig. 3): where E v is the electron energy in vacuum. For an electron traveling from the solid to vacuum the angle v outside the solid after refraction is given by: where θ s is the polar angle inside the solid, both defined with respect to the surface normal. A phenomenon known as total internal reflection occurs if θ s > θ c , where θ c is the critical angle for total reflection, which represents the largest possible angle of incidence still resulting in a refracted trajectory and is given by: Hence, the trajectories inside the solid with a polar angle θ s > θ c are reflected back into the solid. In the case θ s = θ c the refracted beam travels along the boundary between the two media. The critical angle θ c determines an escape cone inside the solid. It means that in order to escape not only does the electron need to have enough energy, but it must also travel toward the surface-vacuum interface within the escape cone defined for a given energy as follows: The presence of the potential barrier strongly influences the escape probability of secondary electrons, because their energy is comparable with the value of U i . However, there are large differences between U i values for any given material reported in the literature [26]. The employed physical model implies a binary encounter approximation. This approach assumes that the volume occupied by an atom significantly exceeds the volume in which the electron interaction takes place [20]. Thus, electron wavelengths are considered to be smaller than the inter-atomic distance or the electron correlation lengths. This condition is generally satisfied for medium-energy electrons, whereas for low energy electrons this is no longer valid since the electron wavelength becomes larger than the inter-atomic distance by several   3. Model for the potential barrier at the surface-vacuum interface for metals. The bottom of the valence band is taken as zero energy scale. ϕ and E F together constitute to U i . Consequently, the energy in vacuum E v differs from the energy in solid E s just by the height of U i . Adapted from [25]. orders of magnitude. However, inside the solid this wavelength is decreased due to the potential step at the surface-vacuum interface as shown in Fig. 3 by the purple curve. Therefore, since only those electrons that can escape and then be detected are relevant inside the solid, the electron wavelength is always limited by the one corresponding to the inner potential, which is usually of the order of 15 eV.

Input data for the MC simulation
The above model has been implemented using the BRUCE software developed within the SIMDALEE2 (Sources, Interaction with Matter, Detection and Analysis of Low Energy Electrons) Marie-Curie Initial Training Network. For the MC calculation data are needed for the elastic and inelastic mean free paths, the differential elastic cross section (DECS), the differential inverse inelastic mean free path (DIIMFP), the inner potential U i and the work function ϕ. The DECS data have been generated with the ELSEPA code [27]. To calculate IMFP values using the Penn SSPA algorithm we used software provided by the authors of Ref. [19]. For this calculation, as well as for calculation of the DIIMFP, optical constants from Refs. [1,2] were used employing linear response theory. Experimental data for U i were used for all materials for which such values are available whereas for other materials U i values were calculated according to an empirical formula proposed by Ross and Stobbs [26]. Besides values for the IMFP the inner potential U i also is not accurately known or at least there is a big discrepancy in the literature. Fortunately, as explained further below, the value of U i does not critically influence the determination of the IMFP. Values used for the work function ϕ were taken from Ref. [28].
It is important first to consider the differences between the two sets of used IMFP data. In both cases IMFP values were calculated following the well-known relationship: where E 0 is the incident energy. The integration limits over momentum transfer q are given by: where atomic units have been used. Here q Im[ 1/ ( , )] is the energy loss function in the entire q ( , ) plane, which was determined on the basis of the Mermin dielectric function q ( , ) M and the Penn SSPA algorithm. In the former approach, the energy loss function in the optical limit q Im[ 1/ ( , 0)] [1,2] was fitted in terms of Drude--Lindhard oscillators [14]. The oscillator parameters are chosen to reproduce the main features in the optical ELF and to satisfy the perfect screening sum rule. The fitting parameters for the 17 materials investigated in this paper are given in the Appendix. In the case of Si the parameters were taken from Ref. [29]. The ELF fitted at q ≈ 0 was then extended to all values of q through the 'built-in' dispersion within the Mermin dielectric function, therefore no dispersion relation was needed.
The Penn algorithm is based on a modification of the statistical approximation [3] developed by Lindhard and can be employed within the full Penn algorithm (FPA) and the simple Penn algorithm or singlepole approximation (SPA) [30,31,4]. The SPA approach implying a quadratic dispersion relation to extend the experimental optical ELF to all values of q is referred to as the simplified single-pole approximation (SSPA). The Penn algorithm assumes the validity of the Born approximation, neglects the vertex correction, self-consistency, and exchange and correlation. It should also be mentioned that the present theory is expected to be reliable only for energies > 100 eV. Although the authors of [19] recommend not to use this approach for energies below 100 eV, it was used in the present work just in order to have large sample values for the IMFP at low energies.
The main difference in the two presented approaches is the description of plasmon damping. The inclusion of plasmon damping into the model dielectric function and an accurate description of the energy loss function (ELF) at low energy losses are known to have a significant influence on the low energy IMFP [32,14]. Since a low energy electron can no longer excite a plasmon, the IMFP increases with decreasing incident energy. However, for Penn IMFP values this increase is observed to be faster than in the case of the Mermin IMFP. In the Lindhard approach, which is used in the Penn algorithm, plasmon excitations are supposed to be undamped below the critical value of the momentum transfer. Whereas the Mermin approach implies an increase of plasmon broadening at any q, which makes the ELF peaks broader not only away from q = 0 but also at small values of q, giving a contribution to the DIIMFP intensity at low energy losses which is absent in Penn's theory. Therefore, the increased intensity of the DIIMFP at low energies causes the reduction of the IMFP values (Eq. (5)).
There are other ways to determine the momentum-dependent ELF. Chantler and co-authors used density functional theory (DFT) concurrently with developments in many-pole dielectric theory [32]. Nguyen-Truong used the Mermin dielectric function to include damping in the ELF within the Penn algorithm [33]. The advantage of this method is that it does not demand any fitting parameters but only the knowledge of the optical dielectric function. Comparison of the IMFP values for Au calculated in this work using the Mermin dielectric function with the IMFP data obtained by Chantler and Nguyen-Truong is shown in Fig. 4. All three data sets are seen to be similar demonstrating that the ELF accurately calculated in the entire q ( , ) plane from the first principles yields the IMFP values in a good agreement with those calculated in more specific ways.

Results and discussion
Determination of IMFPs was performed for the following materials: Be, Al, Si, Ti, V, Fe, Ni, Cu, Ge, Nb, Mo, Pd, Ag, Ta, W, Pt, Au by varying the energy dependence of the IMFP at low energies (< 100 eV) between two extreme values according to the following expression: where the number η ranges from 1 for the Mermin IMFP to 11 for Penn's SSPA IMFP. Then the SEY calculated using each of the IMFP values from Eq. (7) were compared with different data sets of SEY values from the literature. These data sets comprise: (a) collected literature data [16][17][18]; (b) Bronstein data [17]; (c) Zadrazil data [18]. Those IMFP values that give the best χ 2 fit of SEY values with experimental results are considered to be the most reliable ones. The χ 2 optimisation finds those model parameters that minimize the sum of quadratic residues between experimental and theoretical values.  [16][17][18]; (c) Bronstein data [17]; (d) Zadrazil data [18]. The collected literature data [16][17][18] in Figs. 5-9 (c) and (d) are shown as grey data points in order to highlight the chosen data set (red points) for the SEY comparison and to demonstrate its position with respect to all  [34][35][36][37]. Comparison of SEYs for Al calculated using IMFP values shown in Fig. 5(a) by lines with: (b) collected literature data [16,18]; (c) Bronstein data [17]; (d) Zadrazil data [18].  Table 1. There is a monotonic relation between η and the IMFP on one hand, and η and the SEY on the other hand. Thus, lower values of η correspond to lower SEY values and lower IMFP values at low energies.
The SEY is typically characterized in terms of the maximum SE yield δ m at the corresponding energy E m . Different values of δ m and E m are observed during the inspection of different SEY data sets for such materials as Al and Cu (Figs. 5 and 7). The reason for this spread in the experimental SEY data is likely due to the fact that these materials oxidize easily. Even a slight oxidation of a sample leads to a significant rise of the SEY. Since SEs are mainly emitted from the very surface region the SEY measurements, in particular at low energies, require strictly controlled vacuum conditions and, consequently, the purity of the sample surface. Bronstein and Fraiman [17] carried out particularly careful in-situ SEY measurements in the former Soviet Union. The  authors used different methods to obtain clean surfaces such as annealing at high temperatures during the measurements (refractory metals), evaporation of the investigated material in vacuum, electron, and ion sputtering and had a good control of vacuum conditions during their measurements. The value of η in Table 1 for this data set is considerably lower than for other data sets and the energy dependence of the SEY is in a good agreement with the MC results. The fact that the IMFP increases with the increased thickness of the oxidized layer on top of the sample and that Bronstein SEY data correspond to the lowest IMFP values points to a good cleanliness of samples during the measurements, which were documented in a very careful way. Therefore, we conclude that this set of SEY data is more reliable than the total collection of SEY data. Another extensive set of SEY measurements, selected for separate consideration, was carried out by Zadrazil and coauthors in ultra high vacuum (UHV) conditions with samples of a high purity (99.99% or better) as reported in [45,18]. The SEY values from these two data sets are believed to be the most reliable since they are always lower than other measurements, which might indicate bad vacuum conditions or a contamination of the samples in the case of other data.
As can be seen from Table 1 for most materials the value of η is close to those which corresponds to the IMFP values calculated using the Mermin dielectric function = ( 1). Figs. 5, 8, and 9 show the situation when η is equal or close to 1 for Al, Ag, and W. The opposite situation when η value is close to 11, which corresponds to the Penn SSPA IMFP, is shown in Fig. 6 for Fe. The case of an intermediate η value between 1 and 11 is demonstrated as an example for Cu in Fig. 7. Here, and also in the case of Mo (not shown), IMFP values predicted by XAFS Fig. 9. The same as Fig. 5 but for W. The literature data in Fig. 9(a) are taken from [8,10]. a Collected SEY data [16][17][18]. b Bronstein SEY data [17]. c Zadrazil SEY data [18]. measurements [15] are much lower than the Mermin IMFPs, which seems to be contradicting all available IMFP data including the results of this work. As seen from the upper panels of Figs. 5-9 there is always at least an order of magnitude difference between Mermin and Penn's SSPA IMFP values at the vacuum level in all cases with the exception for Fe, which is very important for quantitative understanding of the secondary electron emission process. It is also seen that most of the IMFP data found in the literature tend to be close to the Mermin IMFPs, which is in a reasonable agreement with the results obtained in this work. Note that most of the literature data for the IMFP shown in the upper panels of Figs. 5-9 are theoretical data, since it is a quite challenging task to determine IMFP values at low incident energies using any experimental technique. However, the present work demonstrates an approach to do so using experimental SEY curves.
In the case of transition metals, such as Fe and Pt, the obtained optimum η values are large compared with most of the investigated materials (see Table 1). In Ref. [45] the energy E m corresponding to the maximum SE yield δ m was shown to increase across each of the transition metal series. This dependence was explained by the concurrent increase of the IMFP with the number of electrons in the d band. However, since the authors did not use energy dependent IMFPs but only one IMFP value to describe the electron escape, the above statement does not explain the high η values. IMFP values close to the ones predicted by the Penn model are already large and do not limit the emission of the secondary electron cascade from the surface. It shows that the behavior of the SEY curves of the transition metals cannot be quantitatively explained by an appropriate choice of the low energy IMFP only. This means that further development of the SE emission model in the case of transition metals is needed with a detailed consideration of the electronic structure of these materials.
Finally, another important point concerning our approach needs to be discussed, namely the fact that the SEY not only depends on the IMFP at low energies which is not well known but also on the value of the inner potential U i , which is also not well known. For this reason, the influence of the value of U i on the outcome of our study has been investigated. Fig. 10 shows the results of the χ 2 fit for three different U i values for Au. It demonstrates that the error introduced by the uncertainty of the inner potential makes an influence of plus or −1 in average on the resulting value of η. A similar result was found for all materials investigated in the present study. Therefore the optimum value of the energy dependent IMFP (characterized by the value of η as explained before) does not depend critically on the inner potential.

Summary and conclusions
Determination of the inelastic mean free path (IMFP) at energies below 100 eV was performed by analyzing the secondary electron yield (SEY) in the incident energy range of 0.1-10 keV. A Monte Carlo (MC) model was employed to simulate SEYs for two different energy dependencies of the IMFP at low energies (below 100 eV), calculated using the Mermin dielectric function [11] and the Penn algorithm employed within the simplified single-pole approximation (SSPA) [3]. Subsequent comparison of the simulated SEYs based on these IMFPs with the experimental SEY data [16][17][18] give an estimate for the IMFP values at energies below 100 eV. The optimum values of η (ranging from 1 for IMFP values based on the Mermin dielectric function to 11 for IMFP values based on the Penn SSPA) for 17 investigated materials are presented in Table 1. The general conclusion that can be drawn on the basis of Table 1 is that for most materials, with the exception of the transition metals, the more realistic energy dependence of the IMFP at low energies is given by the Mermin model. These optimum IMFP values were shown to be not critically affected by the choice of the inner potential U i . The presented MC model has some deficiencies in that it does not consider surface excitations, the band structure effect, a quantum-mechanical representation of the potential barrier, etc. The main purpose of this work is to demonstrate the possibility of the presented approach to analyse high energy SEYs and to reverse engineer the IMFP at low energies. Further development of the MC model taking into account all possible considerations will allow obtaining more reliable IMFPs at low energies.

Acknowledgments
The authors would like to thank Shigeo Tanuma for providing the software to calculate the IMFP within the Penn algorithm. The authors also would like to thank Hieu T Nguyen-Truong for making a set of collected IMFP data available for us. Financial support by the FP7 People: Marie-Curie Actions Initial Training Network (ITN) SIMDALEE2 (Grant No. PITN 606988) is gratefully acknowledged. The computational results presented have been achieved using the Vienna Scientific Cluster (VSC).

Appendix A
Here the model dielectric functions that were used are briefly presented. The Mermin dielectric function q ( , ) M used for the IMFP calculations is given by [11]: where ω i is the binding energy, γ i corresponds to the width of the excitation, q ( , ) L is the Lindhard dielectric function. In the optical limit q ≈ 0 the Mermin ELF coincides with the Drude-Lindhard ELF while it becomes broader for larger values of q due to the plasmon damping included into the Mermin model [14]. Since it is more convenient to perform the fitting of the optical ELF as a sum of Drude-Lindhard oscillators, the latter approach was used: with A i related to the density of electrons with the binding energy ω i [14]. The IMFP was then calculated using the Mermin ELF with the Drude-Lindhard oscillators obtained for q = 0. Fig. 11 shows the ELF for Cu, Ag, and Au in the optical limit compared with the experimental optical ELF [1,2]. Tables 2-18 give the fitting parameters for the 17 investigated materials.