IOP

Journal of Physics: Condensed Matter

Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter J. Phys.: Condens. Matter 31 (2019) 055901 (11pp) https://doi.org/10.1088/1361-648X/aaf363

31 Secondary electron emission and yield

2019 spectra of metals from Monte Carlo

© 2018 IOP Publishing Ltd simulations and experiments

JCOMEL Martina Azzolini1,2, Marco Angelucci3, Roberto Cimino3 , Rosanna Larciprete3,4, Nicola M Pugno2,5,6 , Simone Taioli1,7 055901 and Maurizio Dapor1

1 European Centre for Theoretical Studies in Nuclear Physics and Related Areas (*ECT*-FBK) and Trento M Azzolini et al Institute for Fundamental Physics and Applications (TIFPA-INFN), Trento, Italy 2 Laboratory of Bio-Inspired and Graphene Nanomechanics—Department of Civil, Environmental Secondary electron emission and yield spectra of metals from Monte Carlo simulations and experiments and Mechanical Engineering, University of Trento, Trento, Italy 3 Laboratori Nazionali di Frascati (INFN), Frascati (RM), Italy 4 Institute for Complex System (CNR), Rome, Italy Printed in the UK 5 School of Engineering and Materials Science, Materials Research Institute, Queen Mary University of London, London, United Kingdom 6 Ket Lab, Edoardi Amaldi Foundation, c/o Italian Space Agency, Rome, Italy CM 7 Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

E-mail: [email protected], [email protected] and [email protected] 10.1088/1361-648X/aaf363 Received 12 September 2018, revised 16 November 2018 Accepted for publication 23 November 2018 Paper Published 19 December 2018

Abstract 1361-648X In this work, we present a computational method, based on the Monte Carlo statistical approach, for calculating electron energy emission and yield spectra of metals, such as copper, silver and gold. The calculation of these observables proceeds via the Mott theory with a Dirac Hartree Fock spherical potential to deal with the elastic scattering processes, and by 5 – – using the Ritchie dielectric approach to model the electron inelastic scattering events. In the latter case, the dielectric function, which represents the starting point for the evaluation of the energy loss, is obtained from experimental reflection electron energy loss spectra. The generation of secondary electrons upon ionization of the samples is also implemented in the calculation. A remarkable agreement is obtained between both theoretical and experimental electron emission spectra and yield curves. Keywords: secondary electron emission, Monte Carlo method, noble metals, yield (Some figures may appear in colour only in the online journal)

1. Introduction with low work function photocathode or by increasing the local curvature of the surface by steep edges. The emission of secondary electrons plays a fundamental role On the other hand, in other applications the emission of in materials characterization techniques [1, 2], such as scan­ secondary electrons must be suppressed, e.g. in particle accel­ ning electron microscopy [3, 4], and in affecting the perfor­ erators. Indeed, detrimental effects on the machine stability, mance of a variety of electron devices, such as the detectors which might result in beam loss [7–12], can be caused by the based on electron multipliers [5, 6]. These techniques in par­ so-called multipactor effect. This phenomenon appears when ticular seek a high value of the electron yield to reach a low the current of re-emitted electrons grows uncontrollably due noise-to-signal ratio for enhancing the image quality. High to presence of electronic charges in proximity of the vacuum electron yield can be achieved by coating the photomultiplier tube walls. The latter are accelerated by the primary beam,

1361-648X/19/055901+11$33.00 1 © 2018 IOP Publishing Ltd Printed in the UK J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al producing an avalanche of secondary electrons. In this regard, curves from very low primary energies to about 1000 eV chemical treatment, coating [13, 14] and patterning of the [9–12, 29, 30]. The SEY, i.e. the ratio of the number of elec­ target surface [15–18] were used to overcome the harmful trons leaving the sample surface (Is) to the number of incident effects brought about by this phenomenon. Analogous criti­ electrons (Ip) per unit area, was determined experimentally cality concerns microwave and RF components for space by measuring I and the total sample current I = I I , so p t p − s applications that find one of their most important functional that δ = 1 It/Ip. For the SEY measurements, the electron limitations in the multipactor and corona breakdown dis­ beam was set− to be smaller than 1 mm2 in transverse cross- charges [19]. sectional area at the sample surface. To measure the current Despite these remarkable attempts, several issues could of the impinging primary electrons, a negative bias voltage be bypassed by developing an efficient and accurate method ( 75 V) was applied to the sample. The SEY measurements to calculate the electron yield of the investigated material, in were− performed at normal incidence, by using electron beam order to predict the secondary emission and thus to tailor the currents of a few nA. A SpectraLEED Omicron LEED/Auger solution according to the application sought for. In this regard, retarding field (RF) analyser system was specially modified analytic descriptions of the secondary electron energy yield to be able to collect angle integrated EDC with RF filtering have been developed over the years [20–24]. and computer control while using the gun in LEED mode, In this work, we present an accurate computational i.e. with a low-energy focused beam. The e− gun provided approach, based on the Monte Carlo method [25], to simu­ a small and stable (both in current and position) beam spot late electron trajectories leading to secondary electron on the sample, in the energy range from 30 to 1000 eV. This emission and we compare theoretical lineshapes with our set-up can measure angle integrated EDC with the limitation recorded experimental yield spectra. Within this approach, typical of any RF analyzers. While the secondaries are consis­ elastic col­lision and inelastic scattering processes are care­ tently measured at all primary energies, elastic peaks broaden fully evaluated. In the former case this means to assess the due to increasingly poor resolution at higher primary energies, angular deviation along the path of the electrons in their way showing an additional strong asymmetry on the low energy out of the solid, while in the latter the electron energy loss. side due to the integration of the background [31, 32]. Despite In particular, the elastic scattering is treated within the Mott those known limitations, the data can be used in this work to theory [26] using a Dirac–Hartree–Fock spherical potential, extract the relevant information needed. The sample can be while the inelastic scattering events are dealt with the Ritchie transferred from air into UHV conditions and can be cooled di­electric theory [27]. The Monte Carlo method is used to down to 10 K, exposed to various types of gases and cleaned simulate the secondary spectra of three metallic targets, that by subsequent cycles of Ar sputtering. All the polycrystalline is copper, silver and gold. The dielectric functions used in the metallic samples here studied were cleaned by repeated Ar+ 6 assessment of the energy loss of these materials are obtained sputtering cycles at 1.5 KeV in Ar pressure of 5 10− mbar from reflection electron energy loss (REEL) experiments [28]. until no signal of C and O was observed in the XPS× spectrum. This treatment thus takes into account the contribution to the secondary emission spectra of both bulk and surface plasmon 3. Computational details excitations, increasing the accuracy of the computed data. This paper is structured as follows: in the following sec­ 3.1. Monte Carlo tion 2 the experimental procedures used in the measurements of the secondary emission and yield spectra of all metals are The Monte Carlo approach models the spectral distribution of described. Afterwards, a detailed discussion of the computa­ energy transferred to a specimen bombarded with a perpend­ tional Monte Carlo approach is presented in section 3. Finally, icularly incident electron beam by following the electron in section 4 we present a thorough comparison between simu­ trajectories within the target. The electron trajectories result lated and experimental data of secondary emission and yield from the elastic and inelastic interactions undergone by the spectra. electrons scattered by the nuclei and the electron clouds of target atoms, respectively. Secondary electrons are generated via inelastic interac­ 2. Experimental details tions through the ionization of the target’s atomic centers. In the latter process, by measuring the kinetic energies of the The experimental apparatus used to study the secondary escaping charges, the electron emission energy spectra can electron yield (SEY) and angle integrated energy distribu­ be recorded. In the Monte Carlo simulation of this emission tion curves (EDC) is hosted in the ‘Material Science’ labo­ mechanism the trajectories of secondary electrons, similarly ratory of LNF-INFN, Frascati (Rome). For our experiments to those generated by elastic and non-ionizing inelastic scat­ we used a specially built UHV µ-metal chamber with less tering, are followed by using a statistical algorithm. At vari­ than 5 mGauss residual magnetic field at the sample position, ance, the path of electrons with kinetic energies below the pumped by a CTI8 cryopump to ensure a vacuum better than value of the work function or of electrons emitted by the spec­ 10 10− mbar. The set-up has been designed to limit the residual imen is considered terminated. magnetic field near the sample, which can deviate low-energy In Monte Carlo simulations, the probability of elastic and electrons. Ion pumps are not used due to their detrimental inelastic scattering events is assessed by comparing random stray magnetic field. This set up is routinely measuring SEY numbers with the correspondent probability distributions.

2 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al These probability distributions are calculated by using the and absorption corrections in the calculation of the differential elastic and inelastic cross sections, computed via the Mott elastic cross-section, which in ELSEPA are lumped together [26] and the Ritchie dielectric theories [27], respectively. in an effective optical-model potential (‘static-field approx­ These models and the relevant distribution probabilities will imation’) [37]. While for energies higher than about 1 keV, be presented in detail in the sections 3.2 and 3.3. the (relative) magnitudes of the polarization and absorption The incident electron beam direction is set perpendicular corrections decrease when E increases, they become appre­ to the specimen surface and the kinetic energy is increased by ciable for relatively slow projectiles. Thus, the reliability of the specific target work function χ. Upon elastic and inelastic a semi-empirical optical-model potential below 50 eV might scattering events, the electron trajectory can reach the target be questioned due to the increasing importance of the correla­ surface. Finally, electrons can be emitted from the sample pro­ tion effects [37]. Nevertheless, being our goal the simulation vided that the following condition is satisfied: of the secondary electron emission typically characterized

2 by low energies, one needs to follow the electron path up to E cos θ χ, (1) kinetic energies as low as 1 eV. Thus, unfortunately ELSEPA where θ is the angle between the scattering direction inside cross sections are unsafe in the needed energy range and we the material and the normal to the target surface, and E is the must rely on the Mott’s model using the Dirac–Hartree–Fock electron kinetic energy. This condition stems from the fact that potential (Mott+DHF). We notice that in this case we do not the target-vacuum interface represents an energy barrier to be include solid-state effects. Thus we do not use potentials that overcome by the escaping electrons at the interface. Finally, take into account the presence of a charge distribution dif­ the electron emission is determined by assessing the transmis­ ferent from free atoms, which usually results in smaller differ­ sion coefficient t, which can be obtained by assuming that the ential cross sections in the former case than in the latter. electrons feel a model step potential at the surface [33]: From the knowledge of the function σel(E), one can calcu­ late the elastic scattering mean free path (λ ) at a given kinetic 4 1 χ/(E cos2 θ) el t = − . energy as follows: 2 (2) 2 1 + 1 χ/(E cos θ) 1 − λ (E)= , (4)   el In our Monte Carlo approach the transmission coefficient is Nσel(E) compared with a random number µ , uniformly sampled in 2 where N is the atomic density. In our MC model, the elastic the interval [0,1] so that for µ < t the electrons is emitted 2 scattering events lead to a change in the direction of the elec­ from the surface with a kinetic energy decreased by the work tron path. The scattering angle θ after an elastic collision can function χ, otherwise electrons are elastically reflected back be evaluated by calculating the cumulative elastic scattering and continue their path within the solid target. Here we notice probabilities P (θ, E) for different values of the electron that χ plays a key role in the electron emission process, and el kinetic energies E (see figure 2): we do expect a dramatic dependence of the electron yield from this observable. This effect will be discussed thoroughly 2π θ dσ (E) P (θ, E)= el sin θdθ. (5) in section 4.3. el σ (E) dΩ el 0 Finally, θ can be assessed by equalizing Pel at a given elec­ 3.2. Elastic scattering tron kinetic energy with a random number µ3, which is sam­ The elastic scattering between electrons and target nuclei is pled uniformly in the interval [0, 1]. described by the Mott theory [26]. The differential elastic scattering cross section (dσel/dΩ) can be written as: 3.3. Inelastic scattering and secondary electron generation dσ el = f 2 + g 2 (3) Inelastic scattering between the electrons in the beam and in dΩ | | | | the target atoms slows down the charge motion along the path. where f and g are the scattering amplitudes [33, 34], which The electrons moving within the solid may transfer a fraction can be obtained by solving the Dirac equation in a central of their kinetic energy to the target’s atomic electron cloud, field. For the materials under investigation in this work, that is producing both excitations and ionizations. These processes Cu, Ag and Au, the calculation of the elastic scattering cross can be fully described by the dielectric theory of Ritchie section was performed using the analytic formulation of the [27]. The dielectric function (W,q), which depends on trans­ atomic potential proposed by Salvat [35]. The total elastic ferred energy W and momentum q, describes the ‘tendency’ scattering cross section σel(E) is obtained by integrating equa­ of a solid to be polarized by an incoming charged particle or tion (3) in the solid angle. The results are shown in figure 1 and electro­magnetic wave. The dielectric function can be assessed compared with the tabulated values of [36]. We notice that the by both experiments and simulations [38, 39]. Recently, it is elastic cross sections obtained by using the Mott theory with possible to acquire emission electron energy loss spectra with a Dirac–Hartree–Fock (DHF) potential and those from the acceptable energy resolutions. This allows the calculation of NIST database differ significantly at low energies for all inves­ surface and bulk plasmon excitations as reported by Narayan tigated cases, that is below 100 eV. This is due to the treatment et al [40, 41]. In this case, we decided to use the dielectric of the exchange potential, and of the correlation-polarization function obtained from experimental reflection electron

3 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 1. Total elastic scattering cross sections of (a) Cu, (b) Ag, and (c) Au as a function of the electron kinetic energy. Black lines report tabulated values from [36].

Figure 2. Cumulative elastic scattering probabilities of (a) Cu, (b) Ag, and (c) Au as a function of the scattering angle for several kinetic energies.

Figure 3. Real (red) and imaginary (blue) components of the dielectric functions of (a) Cu (a), (b) Ag, and (c) Au, as a function of energy loss for vanishing transferred momentum, obtained by using the fitting parameters reported in [28].

Figure 4. Energy loss functions of (a) Cu, (b) Ag, and (c) Au as a function of energy loss and for vanishing transferred momentum obtained using the fitting parameters provided in [28]. energy loss spectra by Werner et al [28], as it includes both the the real and imaginary parts of the dielectric functions of Cu, bulk and the surface contributions to the electronic excitation. Ag and Au for vanishing transferred momentum. The real and imaginary components of the dielectric func­ From the knowledge of the dielectric function, one can tion were fitted via Drude–Lorentz functions, which mimic obtain the key quantity in charge transport Monte Carlo simu­ plasmon oscillations, whose fitting parameters are provided in lations, that is the energy loss function (ELF) defined by the [28]. In figure 3 we show the dependence on the energy loss of following relation:

4 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 5. Elastic (blue line) and inelastic (red line) scattering probabilities of (a) Cu (a), (b) Ag, and (c) Au as a function of the electron energy.

Figure 6. Inelastic mean free paths of (a) Cu, (b) Ag, and (c) Au as a function of the electron kinetic energy. Dashed lines show the data obtained by Tanuma et al [42].

1 Im[ ] The knowledge of the ELF can be used for calculating the ELF = Im = . (6) − (q, W) Re[ ]2 + Im[ ]2 differential inverse inelastic mean free path (DIIMFP), which  is defined as:

Using the formula (6), the ELF of Cu, Ag, and Au were cal­ 1 q dλ− 1 + dq 1 culated and are shown in figure 4. While our electron loss inel = Im (8) dW πa0E q q −(q, W) functions are ultimately derived from reflection EELS mea­ −   surements, so that multiple scattering and surface losses where a0 is the Bohr radius and the integration limits are overlap with the bulk electron loss function, we notice that q = √2mE 2m(E W), with W the transferred energy. ± ± − recent transmission electron loss measurements [40, 41] may The total inelastic mean free path λ (IMFP) is obtained inel have enough resolution and accuracy to separate the different by integrating the DIIMFP in the energy loss range [0, E/2], components. where E/2 is the largest value that an electron undergoing To extend the dielectric function to transferred momenta q inelastic collisions can lose. In figures 6(a)–(c) we present different from zero we apply the following dispersion law to the IMFPs of Cu, Ag, and Au respectively compared with the the characteristic energies of the oscillators Wi(q): simulations performed by Tanuma et al [42]. The tendency of electrons to undergo elastic or inelastic ( q2) Wi(q)=Wi(q = 0)+α (7) collisions can be assessed by their respective distribution 2m probabilities, pel = λtot/λel and pinel = λtot/λinel, while the where m is the electron mass, q the modulus of the trans­ total mean free path λtot is defined by the following relation: ferred momentum, and α a dispersion coefficient that 1 1 1 λ− (E)=λ− (E)+λ− (E). (9) depends on the energy scale. Indeed, the Drude–Lorentz tot inel el theory was developed to describe excitations in the low The MC scheme applied to the electron transport within solids energy region, corresponding to energy losses lower than the proceeds in the following way: a random number µ4 uniformly semi-core transition energies. To extend properly the Drude– distributed in the range [0, 1] is generated and compared with Lorentz theory to higher energies the α dispersion parameter pinel. Should the condition µ4 < pinel be satisfied, the collision must be tuned. According to [28], we set α = 1 for oscillator is classified as inelastic, otherwise is elastic. In figure 5 we energies Wi(q = 0) lower than the characteristic energies plot the elastic (blue line) and inelastic (red line) probabilities of the semi-core transitions. For larger oscillators energy, for different values of the electron kinetic energy. α was set to 0.5, a value which ensures the best agreement As a results of the inelastic scattering event, the impinging with experimental measurements [28]. The threshold ener­ electron loses a fraction W of its kinetic energy. In the MC

gies of the semi-core transitions are E3p3/2 = 75.1 eV for calculation, the value of W is determined for each inelastic

Cu, E5p3/2 = 57.2 eV for Ag, and E4p3/2 = 58.3 eV for Au collision by comparing a random number µ5 (uniformly respectively [28]. distributed in the interval [0, 1]), with the correspondent

5 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 7. Left panels: cumulative inelastic scattering probabilities of (a) Cu, (b) Ag, and (c) Au as a function of the energy loss for different kinetic energies. Right panels: relevant zoom showing the shoulders due to surface and bulk plasmons.

cumulative probability distribution Pinel(E, W). These prob­ Table 1. Characteristic quantities of the target materials: density (second column) and mean ionization energy B (third column). abilities are calculated for different values of the electron  3 kinetic energies E (see figure 7), as: Metal Density (g cm− ) B (eV)  W 1 dλinel− (E, W) Cu 8.96 [43] 7.726 Pinel(E, W)=λinel(E) dW. (10) Ag 10.5 [42] 7.576 0 dW Au 19.32 [44] 9.226 The value of W for which the value of Pinel is equal to µ5 is the energy loss upon one inelastic collision. Thus, the electron kinetic energy will be decreased by this value. The angular electron from the outern electron shell of the target atom), deviation due to inelastic scattering is evaluated according to a secondary electron is emitted with kinetic energy equal to the classical binary collision theory. W¯ B. After the ionization event, the generated secondary − Moreover, should the energy loss be larger than the first electron moves inside the solid target as any other particle, ionization energy B (that is, the energy required to extract one due to the indistinguishability of the electrons.

6 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 8. (a) Experimental elastic peak of copper; (b) Cumulative probability distribution of the experimental elastic peak.

4. Results and discussion to E +∆E and the Monte Carlo evaluation of its trajectory within the solid resulting from elastic, inelastic and ionizing In our MC simulations we considered copper, silver and gold events can be pursued. bulk metals as test cases of our method. In table 1 the char­ acteristics of these materials are reported. In the next sec­ tions we present both electron emission spectra and yield 4.2. Full energy emission spectra of Cu curves of these three metals. MC simulations were performed to calculate the full energy emission spectrum of bulk copper for different energies E of 4.1. Determination of initial kinetic energy of the electron the primary beam. The number of electrons in the beam was beam set to 107 to obtain stable results. The sample work function was set to 5.4 eV. The starting point of charge transport Monte Carlo calcul­ The initial electron kinetic energy is distributed according ations is to determine the initial kinetic energy and the cumu­ to the experimental elastic peak of copper reported in the pre­ lative distribution probability of the primary beam impinging vious figure 8. Emitted electrons are collected as a function on the surface target. The kinetic energy can be obtained from of their kinetic energies. Theoretical spectra are compared our experiments by considering the energy distribution of with our experimental data for different initial kinetic ener­ the elastic peak, generated by the electrons which have been gies in figure 9. The panels on the left side of this figure report elastically reflected by the target surface. These electrons are the spectra normalized at a common height of the secondary recorded around a sharp peak at energies E ∆E, where E is ± electron (SE) emission peak, while the right panels show the the energy of the peak maximum and ∆E takes into account spectra normalized at a common area. the width of the kinetic energy distribution. In particular, in As it is reported above, we notice that the experimental this work we used the experimental elastic peak distribution spectra were acquired with a (RF) analyzer which is known of copper f (∆E), which is reported in figure 8(a). This elastic to cause a characteristic broadening of the elastic peak due to distribution is conventionally centered in zero. Then, the poor resolution at high energies and a strong asymmetry on cumulative distribution probability (see figure 8(b)) is com­ the low energy side due to the integration of the background puted as a function of the energy correction by using the fol­ [31, 32]. Thus, the integrated area of the whole elastic peak lowing expression: is always higher than that obtained by our MC simulations 1 ∆E [45, 46]. This is the reason why the MC secondary electron P(∆E)= f (∆E )d(∆E ) (11) peaks are more intense than the experimental ones when the Area E − spectra are normalized at a common area (see the panels at where the total area of the peak (Area) was calculated by the right side of figure 9). Nevertheless, by normalizing to integrating the experimental f (∆E) curve in the symmetric a common height of the secondary electron emission peak, energy interval [E ; E+], with E = 2.5 eV as follows: a remarkable agreement is obtained between the secondary − ± ± E+ electrons MC and the experimental lineshapes. Area = f (∆E )d(∆E ). (12) E − 4.3. Electron yield The value of the correction ∆E to the maximum E of the elastic peak has to be determined to set the initial kinetic The electron yield is defined as the total number of emitted energy for each electron in the primary beam. This is accom­ electrons divided by the number of electrons in the beam. plished by generating a random number µ1, uniformly distrib­ This quantity was calculated in the case of Cu, Ag and Au uted in the interval [0,1], and by finding the value of ∆E for for different initial kinetic energies. To calculate the electron 6 which P(∆E) in equation (11) is equal to µ1. Finally, the ini­ yield spectra, we set to 10 the number of electrons in the tial kinetic energy of each electron in the primary beam is set beam. In figure 10 we compare the theoretical electron yield

7 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 9. Electron energy emission spectra of Cu for different initial kinetic energies. The panels on the left side of the figure report spectra normalized at a common height of the secondary electron emission peak, while the panels on the right show spectra normalized at a common total area of the spectrum. The insets in each panel display the two main contributions to the spectra, that is the secondary electron (SE) emission peak (blue curve) and the back-scattered electrons (BE, red curve), respectively. The experimental data are shown as black lines.

8 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al

Figure 10. Electron yield curves of (a) Cu, (b) Ag, and (c) Au as a function of the initial primary electron beam kinetic energy, for different values of the work functions χ (left panels). Black lines report the experimental data [30]. In the right hand side we report the yield spectra for the value of the work function leading to the best agreement between simulations and experiments.

curves with the experimental data by Gonzales et al [30], all for χ = 5.4 eV for copper (experimental value is 4.6 eV [30]), recorded on polycrystalline atomically sputtered clean noble χ = 4.4 eV for silver (experimental value is 4.4 eV metals. We notice that the measured work functions fit well [30]), and χ = 4.7 eV for gold (experimental value is 5.3 eV with values reported in the literature [47]. In the left hand side [30]), respectively. We notice that for silver we are able to of figure 10 we tested the dependence of the yield upon a rea­ reproduce the yield experimental spectra with an exceptional sonable change of the work function χ: thus, different simula­ agreement of the work function between simulations and tions were carried out by changing this parameter. These results show that spectra calculated with a higher measurements. In the case of copper and gold this agreement value of χ display lower intensities than those obtained with is anyway rather good. Furthermore, in figure 11 we compare a lower χ. This finding reflects the fact that an increase of the the secondary electron yields of (a) Cu, (b) Ag, and (c) Au cal­ emission energy barrier, that is the work function, results in culated from Monte Carlo simulations using the Mott+DHF a decreasing number of electrons emerging from the surface. (filled blue circles) and the NIST (filled orange circles) [36] The best agreement with the experimental data [30] is obtained differential elastic cross sections, respectively.

9 J. Phys.: Condens. Matter 31 (2019) 055901 M Azzolini et al curves. We have simulated these characteristics for three dif­ ferent metals, that is copper, silver and gold. MC simulations were carried out also to investigate how tuning the metals’ work functions may affect the secondary electron yields. We found out that a remarkable agreement between simulated and exper­ imental yield spectra could be reached by setting the values of the work functions for the different metals very close to those obtained experimentally [30]. As a further improvement of our MC code suite, the possibility to model a tailored target surface morphology will be introduced, in order to investigate the effect that a given shape can have [48] in increasing or decreasing the electron yield according to the needed application.

Acknowledgments

NMP is supported by the European Commission under the Graphene Flagship Core 2 grant No. 785219 (WP14, ‘Composites’) and FET Proactive (‘Neurofibres’) grant No. 732344 as well as by the Italian Ministry of Education, University and Research (MIUR) under the ‘Departments of Excellence’ grant L.232/2016. The authors gratefully acknowledge the Gauss Centre for Supercomputing for funding this project by providing computing time on the GCS Supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC) [49]. Furthermore, the authors acknowledge FBK for providing unlimited access to the KORE computing facility.

ORCID iDs

Roberto Cimino https://orcid.org/0000-0002-7584-1950 Nicola M Pugno https://orcid.org/0000-0003-2136-2396 Simone Taioli https://orcid.org/0000-0003-4010-8000

References

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