First principles inelastic mean free paths coupled with Monte Carlo simulation of secondary electron yield of Cu-Ni, Cu-Zn, and Mo-Li

Inelastic electron scattering due to incident electron bombardment and the resultant secondary electron emission^1,2 are phenomena observed in many scientific and industrial applications. While some applications exploit secondary electron emission, such as photoelectron spectroscopy, secondary electron spectroscopy and microscopy, or detection of charged particles,^3,4 in others it is an undesired effect. For example, high secondary electron yield (SEY) leads to the mutipactor effect, a deleterious electron avalanche that damages high power RF producing devices such as traveling wave tubes used in space communications and plasma heating systems used in magnetic confinement fusion devices.⁵ All applications affected by SEY necessitate fundamental understanding of secondary electron generation, transmission, and emission processes. Of key importance in this effort is the understanding of the electron scattering or mean free path (MFP) in the material. The inelastic mean free path (IMFP) is also a key parameter used in many surface electron spectroscopy techniques, such as x-ray photoelectron spectroscopy and Auger-electron spectroscopy. Therefore, the goal of this work is to explore the alloying effect and the influence of the chemical composition on the MFPs and SEY. More specifically, we are interested in understanding how alloying changes the key dielectric and electronic properties of the component materials and how these changes, in turn, affect the scattering of electrons and secondary electron emission.

To this end, this work uses quantum mechanical calculations of the energy- and momentum-(⁠$q$-) dependent energy loss functions (ELFs) to evaluate IMFPs in the energy range from Fermi level to 1 keV. IMFPs evaluated from quantum mechanical calculation are then coupled with a Monte Carlo (MC) approach to evaluate SEY as a function of incident electron energy. Furthermore, we hope the density functional theory-(DFT-) informed high-fidelity results presented here will provide a useful baseline for expected SEY from perfect materials and encourage gathering additional experimental data for robust validation. The need for this baseline is clear because experimental measurements have shown a large variability in the peak amplitudes and shapes of SEY curves, even for simple metals.^6–9 For instance, the comprehensive database of SEY data published by Joy in 2008⁸ shows the maximum SEY of copper varying between 1.0 and 2.1 and that of nickel varying between 1.0 and 1.4. The variations are even greater for some other elements, such as aluminum, where the maximum SEY reported in previous works varies from 0.60 to 3.5.

MC simulations have long been used to predict electron transport and inelastic amplitudes, as well as SEY.^10–18 Theoretical calculations of the SEY require quantitative description of electron scattering processes or MFPs. Quantitative knowledge of IMFPs at low energies is of particular interest due to the challenges in obtaining reliable experimental data.^3,19 For the description of inelastic scattering, multiple theoretical approaches have been utilized. The continuous slowing down approximation introduced by Bethe and its derivatives have been widely used;^6,20–22 however, these methods are limited to sufficiently high energy ranges.¹⁰ Ding and Shimizu proposed an alternative method that is free of fitting parameters.^10,11,13 In Ding and Shimizu’s MC approach, secondary electrons are produced through cascade multiplication processes, while the inelastic cross sections required for the simulations are evaluated using single-pole approximation,²³ full Penn algorithm,^18,24 or Mermin method.¹⁵ In these models, the $q$-dependent electron ELFs are derived from optical dielectric functions, which are often taken from experimental measurements such as reflection electron energy loss spectroscopy (REELS).^25–27 The approach introduced by Ding and Shimizu provides a unified and relatively simple treatment for both inelastic scattering and secondary electron generation processes; however, it still relies on the analytical extrapolation to approximate the $q$-dependence of the ELF. Furthermore, the materials studied in this manner are often limited by the availability and the accuracy of optical measurements. It has been recently demonstrated that the use of energy- and $q$-dependent dielectric functions calculated directly using quantum mechanical approaches provide an improved description of IMFPs and, therefore, of secondary electron generation and emission.²⁸ In this MC approach, the IMFPs are evaluated from energy- and $q$-dependent ELFs, which are calculated from DFT, while relativistic (Dirac) partial-wave calculations of elastic scattering are performed via a local central interaction potential.²⁹ In addition, other quantities necessary to calculate SEY, such as density of states (DOS) and work function (WF) are also obtained from quantum mechanical calculations based on DFT. The research presented here takes this approach.

In this work, we use DFT to calculate the energy- and $q$-dependent ELF, WF, Fermi energy, and the DOS of Cu-Ni, Cu-Zn, and Mo-Li alloys and their component metals. These are then used as input parameters for the evaluation of elastic and inelastic MFPs and MC simulation of secondary electron emission for incident energies over the 15 eV–1 keV range. Cu and Ni were chosen as an example of metals with similar properties, mainly the similarity in their ELF, while Cu and Zn were chosen as an example of metals with different dielectric and electronic structure properties. However, our SEY results show that Cu, Ni, and Zn have SEYs that span a relatively narrow $∼15%$ range; therefore, we included an additional alloy whose components have very different SEYs. Although the MoLi alloy with 1:1 atom ratio is not a thermodynamically stable alloy (only small amounts of Li are soluble in Mo at room temperature), the system can still serve as a good test case. It is also worth emphasizing that SEYs of simple metals have been simulated previously. Some examples include works;^12,14,16,17 however, MC simulations of more complex materials have been scarce.^18,30,31 Moreover, this is the first work we know of that studies the influence of chemical composition on the emission of secondary electrons with the use of the MC approach based on $q$-dependent ELFs calculated from first principles, which provides a more complete description of the inelastic scattering processes.

This article is organized as follows. Section II summarizes the computational details. Section III presents the results and discussions. In the first part of Sec. III we discuss the prominent features of the frequency-dependent ELFs as well as the frequency- and $q$-dependent ELFs of Cu, Ni, Zn, Mo, Li and CuNi, CuZn, and MoLi alloys. In the second part of Sec. III we discuss the simulated electron MFPs and SEY of the alloys as compared to their component metals. The conclusions are presented in Sec. IV.

All DFT calculations were performed using the full-potential all-electron code Exciting, version nitrogen-14.³² Ground state and time-dependent calculations were performed within the generalized gradient approximation. CuNi and CuZn alloys were modeled using a two-atom $L11$ FCC primitive unit cell³³ and the first Brillouin zone was sampled with a $k$-point mesh of $30×30×15$⁠. For the $Cu3Ni$⁠, $CuNi3$⁠, $Cu3Zn$⁠, and $CuZn3$⁠, which were modeled using a four-atom $L12$ FCC conventional unit cell,³³ a $20×20×20$ $k$-point mesh was used. MoLi alloy was modeled using a two-atom conventional cell of BCC B2 structure with a $20×20×20$ $k$-point mesh. For the FCC and BCC component metals, one-atom primitive cells was used with a $30×30×30$ $k$-point mesh. The cell parameters of the studied systems were first optimized using the Perdew–Burke–Ernzerhof functional revised for solids (GGA-PBEsol).³⁴ The convergence criterion for energy was $10−6$ Hartree. Where applicable, the structures were further relaxed until forces acting on atoms were smaller than $2×10−4$ Hartree $Bohr−1$⁠. The DOS and frequency- and $q$-dependent ELF were calculated using the Perdew–Burke–Ernzerhof functional (GGA-PBE).³⁵ The ELFs were calculated within the linear-response formalism of time-dependent DFT³⁶ via the random-phase approximation kernel.³⁷ Intraband transition were included in the calculation of the density response function. The $rgkmax$ parameter, the product of the smallest muffin-tin radius and the maximum length for the $G+k$ vectors, was set to 10. The maximum length of $|G|$ for expanding the interstitial density and potential was set to 22. Up to 300 empty eigenstates per atom were included in the ELF calculations. The momentum dependence of the ELF was calculated along the crystallographic direction 111, denoted as [111], with the following increments: $|q→|=0.02Bohr−1$ between 0 and $1.90Bohr−1$⁠, $|q→|=0.05Bohr−1$ between 1.9 and $4.30Bohr−1$⁠, and $|q→|=0.2$ between 4.3 and $18.15Bohr−1$⁠.

The work function was calculated for the surfaces with Miller indices 111 and 110, denoted as (111) and (110), which correspond to the lowest energy surface of the FCC or BCC structures. Work function for all materials was calculated as the difference between the vacuum energy level and the Fermi level.³⁸ The model slabs included 6 layers of atoms and 8 layers of vacuum. For the alloys, these structures correspond to the layered structures in which every layer, including the top surface layer, has 1:3, 1:1, or 3:1 atom ratio of component metals. The vacuum energy level was calculated from the electrostatic potential in the $z$-direction, normal to the surface. Electrostatic potentials were evaluated on a $48×48×250$ grid using the PBE functional.

SEY calculations were performed using the MAST-SEY code, version 4.0.²⁸ The input values, other than the ELFs (volume of the unit cell, WF, and Fermi energy) used as inputs for the MC simulations are given in Table I and Table S1 in the supplementary material. The Fermi energy was defined as the energy from the bottom of the valence band to the DFT calculated Fermi level. For example, the volume of a unit cell per atom, WF for the (111) surface, and the Fermi energy in the case of CuNi are calculated to be $10.81Å3$⁠, 5.22 eV, and 9.46 eV, respectively. For CuZn, the corresponding values are $12.42Å3$⁠, 4.65 eV, and 10.82 eV.

As a part of the process of the SEY calculation, MAST-SEY evaluates the elastic and inelastic scattering properties. The elastic mean free paths (EMPFs) and IMFPs shown in this work were calculated in the energy range from Fermi level to 1 keV using a logarithmic scale grid with 500 points. The integrals of differential inelastic cross sections were evaluated using DFT-calculated $q$-dependent ELFs over 1000 points in momentum space. For the description of elastic scattering, Dirac partial-wave calculations were performed as implemented in the $elsepa$ code.²⁹ Following Ref. 29, the nuclear charge distribution was defined with the use of the Fermi model, while the electron distribution was determined by using the Dirac-Fock electron densities. The exchange was treated using the Furness–McCarthy exchange potential. To obtain the joint density of states (JDOS), which play a key role in determining the initial energy of the secondary electrons during the MC simulation, we used the DOS as calculated using the PBE functional (shown in Figs. S1, S2, and S3 in the supplementary material).

We also want to note that, in the case of alloys, IMFPs were calculated directly from DFT in the same manner as for the component metals (detailed equations can be found in Ref. 28), while the elastic scattering properties were obtained from the virtual crystal approximation.³⁹ In this approach, the differential elastic cross sections (DECS) are obtained independently for each element of the alloy, and then a composition weighted average is calculated to produce a single DECS energy dependent curve. This curve is then used to calculate the EMFP for an alloy.

In order to obtain the SEY curves, we ran the MC simulations by varying the energy of the incident electrons in the 0.15–1 keV range, with an increment of 15 eV. The number of incident electrons for each incident energy was set to $106$⁠. DFT and MC parameters were extensively tested to make sure that the SEY curves are converged. Some of the convergence tests will be discussed later in the text.

The top part of Figs. 1 and S4 in the supplementary material represents the frequency-dependent ELF of Cu, Ni, and CuNi calculated for $q→0$⁠. The origin of the peaks in the ELF of Cu have been extensively discussed in previous works.^40–45 Similar analysis of the real and imaginary parts of the dielectric functions was performed herein to investigate the origin of the main peaks in the respective ELFs (Fig. S5 in the supplementary material). According to the classical model of Drude-Lindhard oscillators,⁴⁶ peaks in ELF that have a small corresponding value in the real part of the dielectric function can be classified as plasmon resonances. Peaks which are classified as interband transitions have corresponding peaks in the imaginary part of the dielectric function. Furthermore, the interband transitions that correspond with maxima of the ELF can be shown to have an inverse relationship with the value of the real part of the dielectric function, at the same energy as the peaks in the loss function.^40,46

Detailed analysis in previous work have shown that the two peaks at 4.4 and 10.5 eV in the ELF of Cu are due to plasmon resonances while the peaks at energies greater than 10 eV and below $∼60eV$⁠, including the two most prominent peaks at 19.5 and 28 eV, correspond to interband transitions from Cu $3d$ states. The ELF of Ni and CuNi show many similarities to the ELF of Cu. All three materials have a low intensity plasmon peak at around 4.5 eV; however, the peak for Ni has the largest intensity. Cu, Ni, and CuNi also have a second plasmon peak at $∼10$ eV, which is located at 10.5 eV for Cu, 7.7 eV for Ni, and 9.7 eV for CuNi, respectively. The ELFs of all three materials are futher characterized by two main peaks at roughly 20 and 28 eV that manifest themselves with slightly differing intensities. The two highest intensity peaks in all three materials are due to interband transitions from $3d$ states as corresponding peaks in the imaginary part of the dielectric function can be identified. In contrast to Cu, both Ni and CuNi have a peak at $∼11.5eV$ that is due to interband transitions and a sharp peak at 64.0 eV, which is known to correspond with the onset of the $M2,3$ edge and is responsible for the excitations from the $3p$ states.

The ELFs of Zn and CuZn in the middle of Fig. 1 exhibit completely different behavior than those of Cu, Ni, or CuNi. The ELF of Zn has two main peaks between 10 and 20 eV and a smaller peak around 26 eV. The peak at 13.1 eV can be classified as a plasmon resonance as it corresponds to a small value of the real part of the dielectric function of Zn (see Fig. S5 in the supplementary material). The peaks at roughly 17.9, 25.9, and 29.6 eV can be classified as interband transitions from $3d$ states. CuZn has three prominent peaks that decrease in magnitude as the energy increases between 18 and 30 eV. A shoulder at 12.3 eV can be classified as a plasmon resonance, while the peaks at $∼18eV$⁠, $∼23$ eV, and $∼28$ eV originate from interband excitations.

The ELFs of Mo, Li, and MoLi are shown in the lower part of Figs. 1 and S4 in the supplementary material. For Mo, the ELF is complex due to the more complex electronic structure of Mo. Analysis of the real and imaginary parts of the dielectric function (Fig. S6 in the supplementary material) shows that the peaks at 10.8 and 24.2 eV are plasmon peaks, while a small peak at 15 eV could correspond to transitions between occupied $4d$ states and unoccupied states above the valence band.⁴⁷ The peaks at 42.6 and 49.8 eV have been proposed to occur due to transitions between occupied Mo $4p$ states and unoccupied states above the valence band.^47,48 As expected, the ELF of Li has less features. There is one major peak around 6.6 eV that has been known to be plasmonic in nature,^49,50 which is further confirmed by the small value in the corresponding real part of the dielectric function (see Fig. S6 in the supplementary material). This plasmon peak has a shoulder feature at roughly 4 eV, which could correspond to the so-called zone-boundary collective state.^49,51 Considering MoLi, the features in the ELF of MoLi are more similar to those of Mo than Li; however, the peak positions are shifted to lower energies as compared to the ELF of Mo. The lowest energy peak at $∼9.2$ eV is a plasmon peak, which can be determined based on the small value in the real part of the MoLi dielectric function (Fig. S6 in the supplementary material). However, in contrast to Mo none of the higher energy peaks in MoLi, such as the peak at approximately 20.7, 24.8, 37.9, and 40.9 eV, are plasmonic in nature and may correspond to interband transitions or core excitations. This can be concluded based on the existence of corresponding peaks in the imaginary part of the MoLi dielectric function (see Fig. S6 in the supplementary material). In addition, except for the plasmon peak at 9.2 eV, the energy loss in MoLi is decreased as compared to Mo. Similarly, the plasmon peak at 9.2 eV has a significantly smaller intensity than the plasmon peak in Li.

The ELFs calculated in this work for Cu, Ni, and Mo can be compared to the loss functions extracted from REELS;⁴⁴ see Fig. S7 in the supplementary material. For Cu, the locations of the first four peaks match well with the features observed in the experimental spectra. After the peak at 28 eV, the Cu data derived from REELS are generally decreasing with no sharp features; however, there is a very broad peak around 60 eV. In contrast to experimental data extracted from REELS spectra, DFT calculations can resolve fine features of the ELF past 28 eV. The locations of the peaks in the ELF of Ni at 4.6, 7.7, 11.5, and 20.7 eV compare well with the experimental data. The largest discrepancy is observed for the peak at 29.4 eV, which manifests itself in the experimentally derived ELF at around 26.5 eV. In addition, the onset of the $M2,3$ edge at 64.0 eV matches very well the experimental peak at 64.5 eV. The ELF of Mo extracted from REELS spectra is characterized by four main peaks that match well with the position of the DFT calculated ELF peaks at 10.8, 24.2, 42.6, and 49.8 eV. The position of the DFT plasmon peak at 10.8 eV is overestimated for 0.4 eV as compared to the experimental value, but the position and the intensity of the plasmon peak at 24.2 eV are well reproduced. The peak at $∼24$ eV in the experimental data has a shoulder feature at $∼13.5$ eV, which could correspond to the DFT calculated peak at 15 eV. The same peak is observed at 15.0 and 14.5 eV in other experimental work.^47,52

Figure 2 shows the DFT calculated frequency- and $q$-dependent ELF of metals Cu, Ni, Zn, Mo, and Li, and of alloys CuNi, CuZn, and MoLi calculated in the [111] direction. The value of the magnitude of the momentum transfer, $|q→|$⁠, varies from $0.04Bohr−1$ to $1.88Bohr−1$ and is plotted incremented by a value of $0.04Bohr−1$⁠. The general trend seen on all ELFs shown in Fig. 2 is that the absolute value of the loss function decays with increasing momentum transfer. The $q$-dispersion of peaks’ position depends on the origin of the peak.³² 

The $q$-dependent ELFs of Cu, Ni, and CuNi show very similar behavior. In all three materials, plasmon peaks between $∼4$ and 10 eV show quadratic dependencies of the peak position. As the momentum increases to roughly $0.6Bohr−1$⁠, the position of the peaks’ maximum shift to larger energies, whereas for momentum transfers greater than $0.6Bohr−1$⁠, the positions of the peaks begin to shift to lower energy values and slowly diminish for greater values of $q$⁠. In all three materials, the plasmonic peak at $∼4eV$ broadens as $|q→|$ increases and reaches a value smaller than 0.1 for $|q→|$ larger than $0.6Bohr−1$⁠, while the plasmonic peak $∼10eV$ begins to broaden at approximately $|q→|=0.4Bohr−1$ and decays to values below 0.1 for $|q→|$ larger than $1.5Bohr−1$⁠. The two main peaks of Cu, Ni, and CuNi that correspond to interband transitions from $3d$ states, at $∼20$ and 28 eV, show very little change in the position of their maximum as $|q→|$ increases. The peak at $∼20eV$ begins to broaden at $∼0.7Bohr−1$ whereas the peak $∼28eV$ begins to broaden at $∼0.9Bohr−1$⁠. Both peaks decay with increasing momentum transfer; however, the intensity of the peak at 20 eV dampens at a faster rate than the peak at 28 eV. Similarly, for the $M2,3$ edge at 64 eV in ELF of Ni and CuNi, the position of the edge does not shift significantly with increasing momentum transfer. For Ni, the intensity of this edge decays slowly from a value of roughly 0.8 to 0.3, at $|q→|=2.0Bohr−1$⁠. Likewise, for CuNi, the intensity of this edge slowly decreases from roughly 0.6 to 0.2 at $|q→|=2.0Bohr−1$⁠.

The $q$-dependent ELF of Zn and CuZn show more complex behavior than that of Cu, Ni, and CuNi. Both Zn and CuZn have a plasmonic peak $∼13$ eV which exhibits quadratic dispersion for increasing $|q→|$-values. For Zn, the peaks at $∼18$⁠, $∼26$⁠, and $∼30$ eV corresponding with interband transitions show very little change in the position of their maximum at lower $|q→|$ values; however, for higher momentum transfers the positions of the maxima shift toward higher energies in a quadratic manner, potentially influenced by the plasmonic resonances. The same can be observed for the interband transitions of CuZn at 18, 26, and 30 eV. Furthermore, at high $|q→|$-values, the plasmon resonance in both Zn and CuZn appears to merge with the neighboring interband peak, which occurs at roughly 18 eV for both Zn and CuZn. For Zn, the merging of the energy loss peaks occurs for a $|q→|$ value of around $0.5Bohr−1$⁠, and this effect results in a maximum of the energy loss for Zn, which reaches a value of $∼1.8$ at $|q→|=0.62Bohr−1$⁠. For CuZn, the merging of the plasmonic and interband peaks occurs $∼0.7Bohr−1$⁠. This peak then shows the quadratic dispersion as the $|q→|$-value increases, while its intensity decreases until it is damped for $|q→|$-values larger than $2.0Bohr−1$⁠.

The assignment of the peaks observed in the ELF of Mo, Li, and MoLi at low momentum transfer is further confirmed by analyzing their behavior at higher $|q→|$-values. Peaks in the ELF of Mo and MoLi show a rather complex behavior with increasing $|q→|$-values, but it can be observed that similarly to Cu, Ni, Zn, and their alloys, the plasmon peaks in the ELF of Mo at 10.8 and 24.2 eV and the plasmon peak of MoLi at 9.2 eV show quadratic dispersion for increasing $|q→|$-values. Namely, as momentum increases, the plasmon peak maxima shift to lower energies in a quadratic manner while peaks broaden as they decay. Similarly to CuZn and CuNi, the peaks at higher energies that correspond to interband transition or core excitations show very little change in the position of their maximum for high $|q→|$-values. We also observe that, in general, plasmon peaks decay much faster than peaks due to interband transitions or core excitations. This is particularly true for the peak at 42.6 eV in the ELF of Mo and the peaks at $∼37.9$ and 40.9 eV in the ELF of MoLi. In particular, the intensity of the plasmon peak at 9.2 eV in the ELF of MoLi decreases from 0.9 for $|q→|=0.002Bohr−1$ to 0.24 at $|q→|=1.00Bohr−1$⁠, while the intensity of the peaks at 37.9 and 40.9 eV decreases from 1.67 and 1.52 at $|q→|=0.002Bohr−1$⁠, to 0.91 and 0.89 at $|q→|=1.00Bohr−1$⁠, respectively. Peaks located at $∼4eV$ and 6.6 eV for small momentum values in the ELF of Li show quadratic dispersion for increasing $|q→|$-values; however, while the peak at $∼4eV$ decays, the intensity of the plasmon peak at 6.6 eV increases from 1.7 at $|q→|=0.002Bohr−1$ to 2.2 at $|q→|=0.32Bohr−1$⁠. Furthermore, as momentum transfer increases the plasmon peak at 6.6 eV broadens and at $∼0.6Bohr−1$ splits to two peaks with maximum at 10.1 and 11.9 eV. Additional features in the ELF of Li emerge for energies between 10 and 35 eV. Similar features are observed and analyzed in more detail in the ELF of Li obtained using first-principles pseudopotential calculations.⁵¹ 

The top half of Fig. 3 shows the EMFPs (solid lines) and IMFPs (dashed lines) for Cu, Ni, and CuNi as calculated with the MAST-SEY code using the energy- and $q$-dependent ELFs discussed in Sec. III B. The MFPs shown in Fig. 3 can also be found in Tables S2–S4 in supplementary material. Note that the zero energy is at the bottom of the valence band as determined through DOS calculations. The differences in the IMFPs of Cu, Ni, and CuNi are most pronounced for energies below roughly 30 eV, while at higher energies the curves merge closer together. In the entire energy range the IMFP of CuNi lies between that of Cu and Ni but is closer to the IMFP of Ni. The shortest IMFPs of approximately 4.2 Å are determined for energies of 63.9, 66.3, and 65.7 eV for Cu, Ni, and Zn respectively. The EMFP of CuNi also lies between that of Cu and Ni, but is closer to the EMFP of Ni. The EMFP curves for these three materials in 10 eV $−1$ keV energy range also show a distinct local maximum and minimum. Similarly, the middle part of Fig. 3 shows MFPs of Cu, Zn, and CuZn. As in the case of CuNi, both the EMFP and IMFP curves of CuZn lie between that of the component metals, Cu and Zn, but are closer to those of Zn. However, the IMFP of Zn and CuZn exhibit slightly different behavior at low energies. The IMFP of Zn has a shoulder between roughly 15 and 30 eV while CuZn does not display this feature. Both IMFPs of Zn and CuZn exhibit a minimum at 58.8 and 63.9 eV with IMFPs of 4.1 and 4.2 Å, respectively. The EMFPs of Cu, Zn, and CuZn have a similar shape with one local maximum and one local minimum. Finally, the bottom part of Fig. 3 shows the MFPs of Mo, Li, and MoLi. Similar to CuNi and CuZn, the IMFP of MoLi lies between that of Mo and Li but is closer to the one of Mo, especially for the energies larger than $∼90$ eV. The IMFP curves of Mo, Li, and MoLi are characterized with one minimum. In particular, the shortest IMFPs of approximately 4.0 Å can be determined for MoLi and Mo and they correspond to energies of 74.9 and 101.7 eV, respectively. The shortest IMFP of 3.2 Å was determined for Li at the energy of 22.0 eV. It is also interesting that the IMFP curve of Li lies below that of Mo and MoLi for energies smaller than $∼50eV$⁠, while for higher energies the IMFPs of Li are longer than those of Mo or MoLi, indicating that the inelastic scattering of electrons with high energy is less likely to occur on Li atoms. As in the case of the IMFP, the EMFP curve of MoLi lies closer to that of Mo. The EMFP curves of Mo and MoLi have a distinct local maximum and minimum, while the EMFP of Li increases monotonically and reaches close to 75 Åat 1 keV.

Figure S8 in the supplementary material compares IMFPs for Cu, Ni, Mo, and Li calculated in this work with those obtained in previous work that aimed at deriving IMFPs from experimental measurements. Data for IMFPs from Refs. 53, 54, and 55 are measured for Cu and Mo from elastic-peak electron spectroscopy or using high-precision x-ray absorption fine structure (XAFS) measurements, while the ones from Refs. 15 and 56 are calculated for Cu, Ni, and Mo from experimental optical ELFs by applying single-pole approximation, full Penn algorithm, or Mermin model on measured or fitted optical ELFs. Data for Cu, Ni, Mo, and Li from Ref. 44 represent IMFPs calculated from REELS data using Penn’s algorithm. The comparison shows excellent agreement between IMFPs in this work and the data from previous works in the case of Cu, Ni, and Li. The exception are IMFPs at low energies for Cu where large variation of theoretical models is observed. In the case of Mo, good agreement exists between this work and the data obtained in Refs. 44 and 56, but discrepancies exist with the data obtained from elastic-peak electron spectroscopy⁵³ at energies below $∼200$ eV and with the data obtained with XAFS technique below $∼80$ eV.⁵⁵ However, it is worth emphasizing that these techniques become less reliable at low incident electron energies.^3,54,55

Figure 4 shows the SEY curves, calculated using the MAST-SEY code, based on EMFPs and IMFPs shown in Fig. 3. The top shows the SEY curves of Cu, Ni, and CuNi. We notice that similarly to the IMFPs, the SEY curve of CuNi lies between the SEY of component metals Cu and Ni; however, it is closer to the SEY of Ni than it is to the SEY of Cu. The middle and bottom of Fig. 4 shows the SEY curves for Cu, Zn, and CuZn followed by Mo, Li, and MoLi. In a similar fashion to the CuNi alloy, the SEY curve of CuZn and MoLi lie between that of component metals but SEY of CuZn is closer to SEY of Zn, while SEY of MoLi is closer to SEY of Mo.

To determine the maximum SEY (⁠$δm$⁠), corresponding primary energy (⁠$Em$⁠), and the first crossover energy (⁠$E1$⁠), we calculated a line of best fit which follows the universal secondary yield curve derived from an empirical double power law.⁵⁷ The equation is of the form

where $E$ is the primary electron energy, $δ$ is the SEY, $δm$ is the maximum value of the SEY, and $Em$ represents the energy of primary electrons that corresponds to the SEY peak, respectively. $(n1−1)$ and $(n2−1)$ correspond to the slopes of the low and high energy asymptotes of the SEY curve on a log–log plot. As summarized in Table II and shown in Fig. S9 in the supplementary material, our calculations show that Cu, Ni, and CuNi have similar SEY curves and the largest SEY of $119%$⁠, $123%$⁠, and $122%$ will occur at incident energy of 439.9, 409.5, and 423.5 eV, respectively. For Zn and CuZn, the maximum SEY is calculated as $106%$ and $111%$ at incident energy of 472.8 and 457.5 eV, respectively. The highest SEY was calculated for Mo and the lowest SEY was calculated for Li. The SEY curve for Mo has a maximum value of 141% at the incident energy of 332.7 eV while the maximum SEY for Li of 36% should occur at the incident energy of 52.0 eV. Our simulations therefore show that the maximum SEY of the studied materials should follow the trend $Mo>Ni≈CuNi≈Cu>CuZn≈MoLi≈Zn>Li$⁠. A similar trend is observed for the first crossover energy (⁠$E1$⁠). Namely, the first crossover energy is predicted to be the highest for Zn and lowest for Mo and is calculated as 101.6, 142.1, 179.5, 188.0, 204.2, 255.4, 297.6 eV for Mo, MoLi, Ni, CuNi, Cu, CuZn, and Zn, respectively.

Additional analysis was performed in an effort to explain some of the observed trends. Cu and Ni have very similar electronic structure and dielectric properties; therefore, the similarities in their SEYs are not surprising; however, the ELF and DOS of Zn are different than those of Cu and Ni. Table III shows the energy of the newly generated secondaries in Cu, Zn, CuZn, and Mo, Li, MoLi analyzed as contributions from inelastic scattering events (ELFs) and JDOS. The table shows that the energy gained by secondary electrons in inelastic scattering events (energy transferred from the incident electrons) follows the trend $Zn>CuZn>Cu$ at incident electron energies of 100 and 600 eV. On average, the secondary electrons from Zn will gain 12.64 eV in an inelastic scattering event with 100 eV primary electrons as compared to the secondary electrons in Cu, which are expected to gain on average 11.70 eV. However, if we look at the JDOS of Zn in Fig. S10 in the supplementary material, we can see that the secondary electrons in Zn are generated from valence electrons with smaller energy as compared to Cu (3.82 eV as compared to 6.10 eV). This is due to the fact that Zn has most of its valence electrons at energies closer to the bottom of the valence band (see Fig. S1 in the supplementary material). The average total energy of the secondary electrons generated in the inelastic scattering event is then defined by the sum of the two contributions (denoted as $E$ gained; total) and generated secondary electrons in Cu and CuZn, on average, have $∼1$ eV larger energy than those in Zn. For the Mo–Li system, all quantities in Table III follow the same trend and the average total energy of the newly generated secondaries decreases in the order $Mo>MoLi>Li$ both at incident electron energy of 100 and 600 eV; however, MoLi values are closer to those of Mo than Li. It is also worth emphasizing that, although we find this analysis very useful, the SEY is influenced by other quantities as well. In the MC simulations, newly generated secondaries travel through the material and can undergo additional inelastic scattering events, which are governed by the IMFP. In addition, if the internal secondary electrons reach the surface, they need to overcome the barrier in order to escape the surface, a process which is determined by the transmission function as defined in Ref. 28. For instance, newly generated secondary electrons in CuZn, on average, have energy closer to secondary electrons in Cu than in Zn; however, the WF of CuZn surface is higher than that of Zn leading to the larger barrier electrons have to overcome on the surface of CuZn. Similarly, the newly generated secondary electrons in Cu are, on average, higher in energy than in Mo; however, the surface of Mo has a smaller WF than Cu, leading to a smaller barrier for escape at the surface of Mo. We can, therefore, conclude that the observed trends in the SEYs of studied materials are a result of relationships between their dielectric properties and electronic structure, such as WF, DOS, and Fermi energy.

An example of secondary electron energy spectra at 500 eV incident energy has also been calculated (Fig. S11 in the supplementary material). The spectra show that most true secondary electrons emitted from Cu, Ni, Zn, Mo, Li, and their alloys have energies between $∼2–3.5$ eV. A sharp feature at 500 eV corresponding to elastically back-scattered electrons is also visible.

Due to practical interest in these alloys, the SEY was further probed for Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 atom ratio. The $q$-dependent ELFs and other MC input parameters are given in Fig. S12 and Table S1 in the supplementary material. The resulting IMFPs are shown in Fig. S13 and given in Tables S5 and S6 in the supplementary material, while the corresponding SEYs are shown Fig. S14 in the supplementary material. The maximum SEY, the corresponding energy of incident electrons, and the first crossover points are tabulated in Table S7 and plotted in Fig. S9 in the supplementary material in comparison to composition weighted average values. The results show that the SEYs of Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 composition vary between SEYs of component metals, but the variations are much smaller in the case of Cu–Ni alloys as Cu and Ni have similar SEY curves. Namely, the compositions $Cu3Ni$⁠, $CuNi3$⁠, $Cu3Zn$⁠, and $CuZn3$ have maximum SEY values of 122%, 122%, 113%, and 107% and occur at energies of 424.7, 411.5, 443.0, and 456.8 eV, respectively. The first cross over energy value of these compositions follows the order $CuZn3>Cu3Zn>Cu3Ni>CuNi3$ and have values of 281.4, 237.1, 189.1, and 181.3 eV, respectively.

Finally, the effect of momentum transfer, $q$⁠, applied in different crystallographic directions was also tested. Cu and CuNi were chosen as representative materials for these tests. Figure S15 in the supplementary material shows the $q$-dependent ELFs while Fig. S16 in the supplementary material shows the IMFPs that are calculated using the $q$-dependent ELFs on Fig. S15 in the supplementary material. Figure S17 in the supplementary material shows the corresponding SEY curves. The momentum transfer was applied along the [111], [100], and [321] directions but no large differences in the corresponding IMFPs of Cu and CuNi were found. Similarly, there were small differences in the SEY curves; for example, the maximum SEYs differ by $∼3%$⁠. More specifically, for Cu the calculated maximum of SEY for the [111], [100], and [321] cases was determined as $119%$⁠, $117%$⁠, and $116%$⁠, which occur at incident energies of 439.9, 438.2, and 438.3 eV, respectively. For CuNi, the maximum SEY calculated based on the $q$-dependent ELF in [111], [100], and [321] directions was determined as $122%$⁠, $120%$⁠, and $118%$⁠, and occur at incident energies of 423.5, 421.8, and 421.0 eV, respectively. In the case of SEY curves shown on Fig. S17 in the supplementary material, MC simulations were performed using WF of the (111) surface. Although this paper does not discuss the details concerning the choice of WF and its influence on the SEY curves, it is important to note that the WF of the material has an impact on its SEY, which makes the comparison between the computational and experimental curves even more challenging. To illustrate this point, Fig. S18 in the supplementary material shows the range in the SEY of Cu, Ni, Zn, Mo, and Li as calculated using WF of polycrystalline surface as well as of a surface with the highest and lowest WF. The WF data for these evaluations are taken from Ref. 38.

In an effort to study the influence of chemical composition on the scattering of incident electrons and the emission of secondary electrons, quantum mechanical calculations based on DFT were carried out to obtain DOS, WFs, and frequency- and momentum-dependent ELFs. The DFT results were used to evaluate MFPs and MC simulations were performed to calculate the SEY of Cu-Ni and Cu-Zn alloys with 1:3, 1:1, and 3:1 atom ratios and of MoLi alloy in the 15 eV–1 keV range. The dielectric properties, electronic structure, electron MFPs, and SEYs of the alloys were compared to those of component metals Cu, Ni, Zn, Mo, and Li.

For all the alloys that were tested in this work, we find that the IMFPs and SEY curves lie in between the SEYs of their component metals, but are not a simple composition weighted average. These findings imply that alloying could be used as a way to modulate the inelastic electron scattering and secondary electron emission processes. However, the changes in the SEY of alloyed materials relative to the SEY of their components could be complex as they depend on the changes in electronic structure, dielectric properties, and work function of the material. Therefore, computational tools, such as the ones based on this DFT-MC approach, could prove particularly useful as a way to predict the secondary electron emission of complex materials without relying on experimentally determined properties. Further computational studies of the alloying effect with controlled MC input parameters could be useful to understand the intricate relationship between SEY curves and different properties of materials.

Figures S1–S3 show DFT calculated density of states for all compositions considered in this work; Fig. S4 shows DFT calculated frequency-dependent ELF of Cu, Ni, CuNi, Cu, Zn, CuZn, Li, Mo, and MoLi in the 100–1000 eV range. Figs. S5 and S6 show ELF plotted together with imaginary and real part of the dielectric function for Cu, Ni, Zn, Mo, Li and Cu–Ni, Cu–Zn, and Mo–Li alloys with 1:1 atom ratio; Fig. S7 shows DFT calculated ELF of Cu, Ni, and Mo plotted with data extracted from REELS; Fig. S8 shows calculated IMFPs of Cu, Ni, Mo, and Li plotted with available experimental data. Figure S9 shows the SEY maximum, the corresponding incident energy, and the first crossover point plotted against composition weighed averages; Fig. S10 shows JDOS for incident energies of 100 and 600 eV of Cu, Zn, and CuZn; Fig. S11 shows secondary electron energy spectrum of Cu, Ni, Zn, Mo, Li, CuNi, CuZn, and MoLi at incident energy of 500 eV. Figure S12 shows DFT calculated ELF of Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 atom ratio; Fig. S13 shows EMFP and IMFP of the Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 atom ratio; Fig. S14 shows SEYs of the Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 atom ratio; Figs. S15– S17 show ELF, MFPs, and SEY of CuNi and Cu, where momentum transfer is applied in the [111], [100], or [321] direction; Fig. S18 shows SEY of Cu, Ni, Zn, Mo, and Li calculated using WF of polycrystalline surface and WF of a surface with the highest and lowest WF. Table S1 shows MC input parameters for Cu, Ni, Zn and all Cu–Ni and Cu–Zn alloys; Table S2 shows calculated EMFPs and and IMFPS of Cu, Ni, and Zn; Table S3 shows calculated EMFPs and IMFPs of Mo and Li; Table S4 shows calculated EMFPs and IMFPs of CuNi, CuZn, and MoLi; Tables S5 and S6 show calculated EMFPs and IMFPs of the Cu-Ni and Cu-Zn alloys with 1:3 and 3:1 atom ratio; Table S7 shows values of the maximum SEY, the corresponding incident energy, the first crossover point and fitting parameters for the Cu–Ni and Cu–Zn alloys with 1:3 and 3:1 atom ratio.

We would like to acknowledge funding received from AFOSR MURI Grant No. FA9550-18-1-0062. We would also like to thank the UNM Center for Advanced Research Computing, supported in part by the National Science Foundation, for providing the high performance computing resources used in this work.

All the input and output data that support the findings of this study are available at https://doi.org/10.6084/m9.figshare.14161940.