# Fileset [09 Kuagai-Tanuma-Powell_AMS.pdf](https://mdr.nims.go.jp/filesets/92600618-8526-452a-996f-885d26ef42fc/download) ## Creator KUMAGAI, kazuhiro, [TANUMA, Shigeo](https://orcid.org/0000-0003-2628-9941), C.J. Powell ## Rights [Creative Commons BY-NC-ND Attribution-NonCommercial-NoDerivs 4.0 International](https://creativecommons.org/licenses/by-nc-nd/4.0/) ## Other metadata [Energy Dependence of Electron Stopping Powers in Elemental Solids over the 100 eV to 30 keV Energy Range](https://mdr.nims.go.jp/datasets/8821b484-0f4e-400d-9d19-818e6c5eaef4) ## Fulltext 09 Kuagai-Tanuma-PowellAuthor Manuscript: Published in final edited form as: Nuclear Instruments and Methods in Physics Research B 267 (2009) 167–170.https://doi.org/10.1016/j.nimb.2008.10.094Energy Dependence of Electron Stopping Powers in Elemental Solids over the 100eV to 30 keV Energy RangeK. Kumagai*Graduate School of Pure and Applied Science, University of Tsukuba, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, JapanS. TanumaMaterials Analysis Station, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba,Ibaraki 305-0047, JapanC. J. PowellSurface and Microanalysis Science Division, National Institute of Standards andTechnology, Gaithersburg, MD, 20899-8370, USA*Corresponding author. Tel. +81-29-859-2725 Fax. +81-29-859-2729e-mail: kumagai.kazuhiro@nims.go.jp AbstractWe analyzed the energy dependence of electron stopping powers (SPs) calculated for 41elemental solids from experimental optical data for electron energies between 100 eV and 30keV. Our analysis was performed based on the Hill equation to represent a series of steps inplots of the slopes of Fano plots. The average root-mean-square difference between SPs fromfits with an equation derived from the Hill equation and the calculated SPs was 1.0 %. The1new equation can provide SPs over a wide energy range for Monte Carlo simulations ofelectron transport with the continuous slowing-down approximation.Keyword: Electron stopping power; Fano plot; Elemental solids; Energy dependence 2The electron stopping power (SP) is an important parameter in the modeling of electron transport in solids for many applications such as electron-probe microanalysis [1-3], Auger-electron spectroscopy [4], and dimensional metrology in the scanning electron microscope [5-7]. In Monte Carlo simulations for these and other applications, the continuous slowing-down approximation has often been utilized in which it is assumed that the electron energy is a continuous function of the trajectory length in a material. This approach is convenient because data for differential cross sections as a function of energy loss are not required and because computation time is reduced if inelastic-scattering events are not individually simulated [4]. It is, however, necessary to know the dependence of the SP on electron energy in solids over a wide energy range, typically 100 eV to 30 keV.SPs for Monte Carlo simulations have often been determined from the Bethe SPequation [8-10] and data for one material parameter, the mean excitation energy [11]. Thisequation is expected to be valid for electron energies greater than about 10 keV or forenergies much greater than the largest K-shell binding energy in the material of interest. SPscalculated from the Bethe equation are available from a National Institute of Standards andTechnology (NIST) database for electron energies of 10 keV and above [12]. Severalempirical SP equations [13-15] have been developed for energies less than 10 keV but theiruse is restricted to materials for which needed parameters are known. Furthermore, it isdifficult to adapt these equations to energies less than several hundred eV. Althoughexperimental determinations of the SP over a range of energies are available for a limited3number of materials [16], different sets of data for a particular material can disagreesignificantly.Tanuma et al. [17, 18] recently reported SPs for 41 elemental solids over the 100 eV to30 keV energy range that were calculated from experimental optical data using the Pennalgorithm [19]. Jablonski et al. [20] analyzed these results and proposed an empiricalpredictive SP formula for the 200 eV to 30 keV energy range.We report a new analysis of the calculated SPs of Tanuma et al. [17, 18] and propose anew SP formula that can be applied over the 100 eV to 30 keV energy range. Our analysis ofthe SP energy dependence for each solid was carried out using Fano plots [21, 22] in whichthe product of the SP, S, and electron energy, E, is plotted versus energy on a logarithmicscale. As examples, the solid circles in Figs. 1(a) and (b) show Fano plots of Si and Au usingSPs calculated from optical data [17, 18] (hereafter referred to as optical SPs). Our Fanoplots show optical SPs for energies between 10 eV and 100 eV to illustrate trends, althoughthese SPs should be regarded only as semi-quantitative guides [18].The nonrelativistic Bethe SP equation is [8-10]:S =784.6ZρEA ln (1.166EI ) (in eV/Å), (1)where Z is the atomic number, ρ is the density (in g/cm3), A the atomic weight, I is the meanexcitation energy (in eV), and E is expressed in eV. The dashed lines in Figs. 1(a) and (b)show plots of the product SE from Eq. (1) (with I values as recommended in Ref. [11]) as afunction of E. While the Fano plots from the optical SPs for Si and Au show linear regions,4for energies greater than 500 eV for Si and greater than 10 keV for Au, their slopes are largerthan those for the Fano plots from the Bethe equation. At lower energies, SE from the opticalSPs gradually decrease and approach zero with decreasing energy. That is, as is well known,the Bethe SP equation is not valid for energies less than about 10 keV [11-18].It is convenient to rearrange Eq. (1) for a Fano plot:SEk =ln(1 .166 /I )+ ln E (in eV2/Å), (2)where k =A /784 .6Zρ . If the Bethe equation were valid over a given energy range, thecorresponding plot of SEk versus lnE would be linear with a slope of unity.The slopes of the Fano plots for Si and Au from the optical SPs are shown as solidcircles in Figs.1 (c) and (d) as a function of electron energy. We see that these slopes show aseries of steps with increasing energy. Each of these steps corresponds to contributions to thestopping power from a particular electronic shell (valence-band, L-shell, and K-shell for Siin Fig. 1(c) and valence-band, O-shell, N-shell, and M-shell for Au in Fig. 1(d)). As theenergy increases to our limit of 30 keV, the slopes approach unity as expected from Eq. (2).We have chosen to fit plots of slopes of Fano plots using the Hill equation [23]. Inprinciple, any sigmoid function would be appropriate but the Hill equation is a simpleexpression that can be easily integrated. This equation can be expressed as y=a E nE n +b n, (3)5where a is the height of a step, b is the electron energy at the center of a step, and ndetermines the steepness of the slope at a step. Although this equation is empirical, we canassociate physical meanings to the parameters a, b, and n.If a plot of the slope of a Fano plot for a given element has m steps, we can describethe plot by k d (SE )d lnE =∑i =1ma i E n iE n i +b n i. (4)The solid lines in Figs. 1(c) and (d) show fits of the Fano-plot slopes for Si and Au with Eq.(4), and we see reasonable agreement. Although there are some deviations of the fits fromthe plotted points (e.g., for Si above 600 eV), these do not significantly affect our laterresults. We find it more convenient to fit the Fano plots directly with an integration of Eq.(4):SE =784.6ZρA ∑i =1m [ a in i ln(En i +b n ib n i )] (in eV2/Å). (5) We have fitted Fano plots with the optical SPs for our group of 41 elementalsolids with Eq. (5). The solid lines in Figs. 2(a) and (b) show examples of these fits for Siand Au. Direct comparisons of the optical SPs and the values derived from the fits with Eq.(5) are shown in Figs. 2(c) and (d) where we see excellent agreement with the optical SPs.The root mean square (rms) difference of the fitted values from the optical values of SE were0.8 % for Si and 1.0 % for Au. The average of the rms differences for the 41 solids was 1.06% over the 100 eV to 30 keV energy range. This value is superior to those found in fits of theoptical SPs with other empirical equations over the same energy range: 4.52 % with the Joy-Luo equation [14] and 3.04 % with the Jablonski-Tanuma-Powell [20] equation. Our new expression for the SP can be obtained by rearranging Eq. (5):S =784.6ZρEA ∑i =1m a in iln [1+( Eb i )n i ] (in eV/Å). (6)Values of the parameters in Eq. (6) for each of our 41 solids will be reported elsewhere [24].For E much larger than all bi, Eq. (6) becomesS =784.6ZρEA ∑i =1ma i ln ( Eb i ) (in eV/Å). (7)If a single electronic shell contributed to the SP (i.e., m = 1), Eq. (7) would become the BetheSP equation (Eq. (1)) if a1 = 1 and b1 = I/1.166. The optical SPs for 41 elements [17, 18] showed systematic changes as a function ofatomic number. Multiple peaks and shoulders were seen with shapes and widths that variedwith Z [18]. These changes were interpreted in terms of the varying contributions of valence-electron excitations and inner-shell excitations to the total SP. Our analysis of the optical SPswith the Hill equation has yielded insight into these separate contributions. The dashed linesin Figs. 2(a) and (c) show the valence-electron contributions to the SP for Si (i = 1), the L-shell contribution (i = 2), and the K-shell contribution (i = 3). While the electronic structureof Au is more complex than that of Si (with overlapping contributions from N- and O-7shells), we can similarly identify (at least approximately) the contributions of valence-electron excitations (i = 1), the O-shell (i = 2), the N-shell (i = 3), and the M-shell (i = 4) inFigs. 2(b) and (d). We now consider the physical significance of the parameters in Eq. (6). Figure 3 showsplots of the bi values derived from the fits of the optical SPs with Eq. (5) for the 41 solids asa function of binding energies of the K-shell and of the L3-, M3-, and N3-subshells. We seeclear near-linear dependences for bK, bL, bM, and bN, although there is larger scatter for the bNplot in Fig. 3(d). The slopes of linear fits to the plots of Figs. 3(a), (b), (c), and (d) are 3.4,3.4, 2.6, and 1.5, respectively. These slopes indicate that the values of bi for a given elementand shell can be regarded as an average excitation energy for that shell. It is reasonable thatthis average excitation energy should be roughly three times the corresponding inner-shellbinding energy. The smaller slope found for the bN plot in Fig. 3(d) is probably associatedwith the large range of binding energies for the various N subshells for medium- and high-Zelements and with the overlapping electronic excitations from N- and O-shells for high-Zelements.The parameter ai represents the height of a step in plots of the Fano-plot slopes such asFigs. 1(c) and (d). For E much larger than all bi, the total of all step heights must be unity inorder for the asymptotic slope to converge to the expected Bethe value (unity in the units ofFigs. 1(c) and (d)). We found that the average value of the total step heights for the 41 solidswas 1.08±0 .07 (where the uncertainty represents one standard deviation). It is clear,however, from Figs. 1(c) and (d) that the Fano-plot slopes are decreasing with increasing8energy for E > 10 keV, and would become closer to unity for E > 30 keV. We can thereforeregard the product aiZ in Eqs. (6) and (7) as an effective number of electrons in each shellcontributing to the stopping power.The parameter ni is a measure of the steepness of steps in plots of the Fano-plot slopes.Comparison of Figs. 1(c) and (d) shows that the steps for Si are relatively sharp while thesteps for Au corresponding to the O-shell and N-shell steps are broader. These differencesare qualitatively reasonable because of the large range of binding energies of the subshells,57 eV to 762 eV, for the Au O-shell and N-shell [25].In summary, we believe that Eq. (6) will be useful for Monte Carlo simulations ofelectron transport in solids that utilize the continuous slowing-down approximation forelectron energies between 100 eV and 30 keV. Our fits of the optical SPs with the Hillequation have enabled us to identify the separate contributions of valence-band and inner-shell excitations to the stopping power, and of the trends in these contributions with Z. 9References[1] R. Gauvin, Surf. Interface Anal. 37 (2005) 875. [2] N. W. M. Ritchie, Surf. Interface Anal. 37 (2005) 1006. [3] X. Llovet, J. M. Fernández-Varea, J. Sempau, and F. Salvat, Surf. Interface Anal. 37(2005) 1054. [4] A. Jablonski, C. J. Powell, and S. Tanuma, Surf. Interface Anal. 37 (2005) 861. [5] C. G. Frase and W. Häßler-Grohne, Surf. Interface Anal. 37 (2005) 942.[6] J. S. Villarrubia, A. E. Vladár, and M. T. Postek, Surf. Interface Anal. 37 (2005) 951.[7] D. V. Gorelikov, J. Remillard, N. T. Sullivan, and M. Davidson, Surf. Interface Anal. 37(2005) 959.[8] H. Bethe, Ann. Physik (Leipzig) 5 (1930) 325.[9] H. Bethe, in: H. Geiger and K. Scheel (Ed.), Handbuch der Physik, Springer, Berlin,1933, vol. 24/1, p. 273.[10] H. A. Bethe and J. Ashkin, in: E. Segre (Ed.) Experimental Nuclear Physics, Wiley,New York, 1953, p. 166.[11] Stopping Powers for Electrons and Positrons, International Commission on RadiationUnits and Measurements Report No. 37, International Commission on RadiationUnits and Measurements, Bethesda, 1984.[12] M. J. Berger, J. S. Coursey, M. A. Zucker, and J. Chang, Stopping Power and RangeTables for Electrons, Positrons, and Helium Ions, (http://physics.nist.gov/PhysRefData/Star/Text/contents.html), 2005.10[13] T. S. Rao-Sahib and D. B. Wittry, J. Appl. Phys. 45 (1974) 5060. [14] D. C. Joy and S. Luo, Scanning 11 (1989) 176. [15] J. M. Fernández-Varea, R. Mayol, D. Liljequist, and F. Salvat, J. Phys.: Condens.Matter 5 (1993) 3593.[16] D. C. Joy, http://web.utk.edu/~srcutk/htm/interact.htm.[17] S. Tanuma, C. J. Powell, and D. R. Penn, Surf. Interface Anal (2005) 37, 978.[18] S. Tanuma, C. J. Powell, and D. R. Penn, J. Appl. Phys. 103 (2008) 063707.[19] D. R. Penn, Phys. Rev. B 35 (1987) 482.[20] A. Jablonski, S. Tanuma, and C.J. Powell, J. Appl. Phys. 103 (2008) 063708.[21] U. Fano, Phys. Rev. 95 (1954) 1198. [22] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297.[23] A. V. Hill, J. Physiol. (Lond.) 40 (1910) iv.[24] K. Kumagai, S. Tanuma, and C. J. Powell (to be published).[25] NIST X-ray Photoelectron Spectroscopy Database, Version 3.5 (2008); http://srdata.nist.gov/xps.11Figure captionsFig. 1. Fano plots ((a) and (b)) and slopes of Fano plots ((c) and (d)) for Si and Au. Theclosed circles are derived from optical SPs and the dashed lines in (a) and (b) show valuesfrom the Bethe equation [Eq. (1)]. The solid lines in (c) and (d) show fits to the Fano-plotslopes with Eq. (4), and the dashed lines indicate the value expected from the Betheequation.Fig. 2. The solid lines show fits of the Fano plots with the optical SPs (solid circles) usingEq. (5) for (a) Si and (b) Au. The solid circles in (c) and (d) are the optical SPs and the solidlines show the results of the fits. The dashed lines show the contributions of valence-bandand inner-shell excitations to SE and S.Fig. 3. Plots of values of bi (solid circles) derived from fits of the Fano plots with Eq. (5) for the 41 elemental solids as a function of (a) K-shell, (b) L3-subshell, (c) M3-subshell, and (d) N3-subshell binding energy. The solid lines show linear fits for each plot.12Fig.1.13Fig.2.14Fig.3.1516