# Fileset [IMFP_VII_pv.pdf](https://mdr.nims.go.jp/filesets/9cadba3e-d82b-4337-9a64-6fd34d1ead96/download) ## Creator [Tanuma, Shigeo](https://orcid.org/0000-0003-2628-9941), [Powell, C. J.](https://orcid.org/0000-0001-8990-2286), Penn, D. R. ## Rights ## Other metadata [Calculation of Electron Inelastic Mean Free Paths (IMFPs). VII. Reliability of the TPP-2M IMFP Predictive Equation](https://mdr.nims.go.jp/datasets/ae2a9fdd-62b2-4b92-bd45-df14c2c70252) ## Fulltext Nv in TPP-2M Ms. (Vers. 2) 1 Calculation of Electron Inelastic Mean Free Paths (IMFPs). VII. Reliability of the TPP-2M IMFP Predictive Equation S. Tanuma,1 C. J. Powell,2 and D. R. Penn2 1Materials Analysis Station, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan 2National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Abstract We report comparisons of electron inelastic mean free paths (IMFPs) determined from our predictive IMFP equation TPP-2M and reference IMFPs calculated from optical data. These comparisons were made for values of the parameter Nv (the number of valence electrons per atom or molecule) that we have recommended and those that were recommended in a recent paper by Seah et al. [Surf. Interface Anal. 31, 778 (2001)]. The comparisons were made for eight elemental solids (K, Y, Gd, Tb, Dy, Hf, Ta, and Bi) and two compounds (KBr and Y2O3) for which there were appreciable differences in the recommended Nv values from the two sources and for which optical data were available for the IMFP calculations. The average of the root-mean-square (RMS) deviations for the ten materials between IMFPs from the TPP-2M equation with our Nv values and the reference IMFPs was 11.0 % while the corresponding average with the Seah et al. Nv values was 20.2 %. The larger average in the latter comparison was mainly due to large (> 20 %) RMS deviations for four materials (K, Hf, Ta, and KBr). For the other six materials, the RMS deviations with the Seah et al. values of Nv were similar to those with our 2 values of Nv. Based on the comparisons for these ten materials, we believe it is preferable to use our values of Nv in the TPP-2M equation. INTRODUCTION The electron inelastic mean free path (IMFP) is a key material parameter in Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS) as well as in other techniques involving electron scattering or emission at solid surfaces.1,2 This parameter together with the experimental configuration affects the surface sensitivity of AES and XPS measurements3 and is needed for modeling of signal-electron transport. We previously reported calculations of electron IMFPs from experimental optical data for 27 elemental solids, 15 inorganic compounds, and 14 organic compounds as well as analyses of these results.4-9 The optical data were checked for internal consistency using two sum rules,10 and these checks revealed that the available data for the group of elements and the group of organic compounds were more reliable than for the group of inorganic compounds. We therefore analyzed IMFPs for the groups of elements and organic compounds to derive an equation, designated TPP-2M, that could be used to estimate IMFPs for other materials.8 The TPP-2M predictive equation for the IMFP, l, is: (in Å) (1a) (1b) (1c) (1d) (1e) )]/()/()ln([ 22 EDECEEEp +-=gbl1.02/122 069.0)(944.010.0 rb +++-= gp EE2/1191.0 -= rgUC 91.097.1 -=UD 8.204.53 -= 3 (1f) where is the free-electron plasmon energy (in eV), Nv is the number of valence electrons per atom (for elements) or molecule (for compounds), r is the density (in g/cm3), M is the atomic or molecular weight, and Eg is the bandgap energy (in eV). We have discussed the appropriate choice of values for the parameter Nv in Eq. (1) for different elements, and have pointed out that there can be ambiguity in choosing a value of this parameter for some elements.4-7,11 The value of Nv is normally computed from the number of electrons in the valence band for the particular solid. For example, in an elemental solid such as Al there are three valence electrons and , while in a compound such as Al2O3 there are six valence electrons from the Al and eighteen valence electrons from the O so that . Ambiguity in the choice of Nv arises for certain elements in the periodic table that occur after those elements in which an atomic subshell is filled. There is then a large change in Nv and electrons that were formerly in the valence band now occupy a subshell with a "small" binding energy (BE). The question then arises as to how large this "small" BE can be so that the subshell does not contribute significantly to the IMFP. For example, elemental copper has ten 3d electrons and one 4s electron in its valence band, and so we expect for Cu. For the neighboring element Zn in the periodic table, there are two 4s electrons in the valence band and the ten 3d electrons have an average BE of about 10 eV.12 For the two following elements in the periodic table, Ga and Ge, the numbers of electrons in their valence bands are three and four, respectively, while the average BEs of the ten 3d electrons are about 18.7 eV and 29.7 eV, respectively.12 While the ten 3d electrons are clearly contributing to the strength of inelastic scattering (more precisely, to the energy-loss function or the differential inelastic-scattering cross section for energy losses between 0 eV and about 50 eV, the energy-loss region that has a large contribution to the total inelastic-scattering cross section) in copper, they have a progressively weaker influence as the BE increases with atomic number (that is, for Ga, Ge, and following elements). Similar considerations apply to other elements in the periodic table as valence sub-shells are filled and become core levels. Although the IMFP from Eq. (1) depends on Nv in a complicated way on the values of Ep, b, C, and D, the IMFP value for some elements and 4.829// 2pv EMNU == r2/1)/(8.28 MNE vp r=3=vN24=vN11=vN 4 compounds fortunately does not depend sensitively on the choice of Nv.4-7,11 Additional examples will be presented here of the extent to which IMFPs depend on Nv. We show in Table 1 the values of Nv for each element that we have recommended in previous papers4-8 or that have been recommended in a NIST IMFP database;13 these Nv values will hereafter be referred to as the TPP values. For the rare-earth elements, the values of Nv have been determined from the sum of the number of valence electrons for the solid state, either 2 or 3 as discussed by Netzer and Matthew,14 and the six 5p electrons that contribute strongly to the energy-loss function.15,16 Although the 5p electrons for the rare-earth elements have binding energies between 18 eV and 27 eV,17 they have been included in the calculation of Nv because of the strong 5p-6d excitations that occur close to the threshold energy for excitation (that is, the BE).15-16 The number of 4f electrons, which increases with atomic number through the rare-earth series, has not been included in the Nv calculation. While these 4f electrons have binding energies of less than 9 eV,18 they contribute weakly to the energy-loss function because of the substantial "delayed onset" in plots of the photo-absorption cross section versus photon energy.15 Similar delayed onsets have been found in plots of the yields of N67VV Auger electrons in Au, Pb, and Bi as a function of incident electron energy.19 Seah, Gilmore, and Spencer (hereafter SGS) recently reported an analysis of Auger-electron intensities for 61 elemental solids and of photoelectron intensities for 58 elemental solids (in AES and XPS experiments, respectively).20 These intensities were compared with predicted intensities for which the TPP-2M equation was used to obtain the IMFPs for the relevant elements and electron energies. Seah et al. found systematic differences between the experimental and predicted intensities from which they concluded that there was either an error in their intensity-measurement procedure or a systematic error in IMFPs from the TPP-2M equation. The latter possible systematic error was associated with the choice of Nv in Eq. (1). Seah et al. found that the systematic differences between the measured and calculated intensities could be minimized by computing Nv from the number of electrons with BEs of 28 eV or less; for the lanthanide series, they recommended that the number of 4f electrons for each element be included. Table 1 shows the values of Nv recommended by Seah et al.20,21 Figure 1 is a plot of the Nv values from Table 1 versus atomic number, Z. For 45 of the 75 elements for which there are both TPP and SGS recommendations of Nv, there is no difference in 5 the Nv values from the two sources. We also note that, of the remaining 30 elements, Seah et al. measured and analyzed AES or XPS data for all elements except F, K, Rb, and Cs. We report here comparisons of IMFPs calculated from the TPP-2M equation for eight elemental solids (K, Y, Gd, Tb, Dy, Hf, Ta, and Bi) for which there were significantly different values of Nv (Table 1) and for which we had calculated IMFPs from experimental optical data. The latter IMFPs are considered as reference values for each solid, and we make comparisons between these values and the corresponding IMFPs from the TPP-2M equation with the two proposed values of Nv for each solid. We also make similar comparisons for two compounds, KBr and Y2O3, for which the Nv values are 8 and 24, respectively, from the TPP values in Table 1 for the constituent elements, and 14 and 36, respectively, from the SGS recommendations in Table 1. RESULTS AND DISCUSSION Figures 2 through 11 show plots of IMFPs from experimental optical data (solid lines) for K,15 Y,15 Gd,15 Tb,15 Dy,15 Hf,5 Ta,5 Bi,5 KBr,15 and Y2O3,15 respectively, for electron energies between 50 eV and 2,000 eV. The long-dashed lines are IMFPs from the TPP-2M equation (Eq. (1)) with our recommended values of Nv while the short-dashed lines are IMFPs from this equation with the SGS recommendations for Nv. The other material parameters needed for the evaluation of TPP-2M are listed in Table 2. In order to provide a quantitative description of the results in Figs. 2-11, we have calculated percentage deviations between IMFPs calculated from the TPP-2M equation, with the TPP and SGS values of Nv, and the corresponding reference IMFP values at 10 eV intervals between 50 eV and 200 eV and at 100 eV intervals between 200 eV and 2,000 eV. The root-mean-square (RMS) deviations for each material are shown in Fig. 12. The average of the RMS deviations in Fig. 12 with the TPP values of Nv is 11.0 % while the corresponding average with the SGS values of Nv is 20.2 %. We note here that the average RMS deviation of 11.0 % with our values of Nv is close to the values found in similar comparisons of 10.2 % for the original group of 27 elements that we investigated and of 8.5 % for our group of 14 organic compounds.8 The larger average of the RMS deviations in Fig. 12 for the SGS values of Nv (20.2 %) than for the TPP values of Nv (11.0 %) is mainly due to the large (> 20 %) RMS deviations for four materials (K, Hf, Ta, and KBr). For these four materials, the TPP values of Nv give IMFPs 6 that are appreciably closer to the reference IMFPs than the SGS values of Nv. For the other six materials, the average of the RMS deviations with the SGS values of Nv is 10.3 %. This average is comparable to that found with the TPP values of Nv for the same materials (11.8 %). We thus cannot make a meaningful choice between the TPP and SGS values of Nv for these six materials. We also note that, for the three rare-earth elements considered here, the average RMS deviations are 9.2 % and 7.8 % for the TPP and SGS values of Nv, respectively; inclusion or exclusion of the 4f electrons in Nv does not make an appreciable difference in the IMFPs from TPP-2M for these materials. For K, Hf, Ta, and KBr, however, the TPP values of Nv give IMFPs that are in clearly better agreement with the reference IMFPs than IMFPs obtained from the SGS values of Nv. We now consider whether the RMS deviations in Fig. 12 might be correlated with uncertainties of the energy-loss functions derived from optical data and in some cases from inelastic-electron-scattering data that were used to calculate the reference IMFPs.4-8 Table 3 shows the errors in the f-sum rule and the Kramers-Kronig (KK) sum rule for the energy-loss functions of our materials.10 The absolute values of the average sum-rule errors for these materials are shown in Fig. 12. The average value of these absolute errors, 4.8 %, is comparable to the corresponding value of 5.4 % for our original group of 27 elements.4 We also see from Fig. 12 that the sum-rule errors for K, Hf, Ta, and KBr are similar to those for the other materials. In addition, there is no obvious correlation in Fig. 12 between the RMS deviations for the different choices of Nv and the corresponding absolute values of the average sum-rule errors. It is interesting to point out the extent to which IMFPs from Eq. (1) depend on the choice of Nv. Figure 13 shows a plot of ratios of IMFPs at a representative energy, 1000 eV, determined from Eq. (1) with the SGS and TPP values of Nv versus the ratio of these Nv values for the ten materials considered here. Although there is some scatter in the plotted points, we see that the IMFP ratio depends inversely on the Nv ratio. Even for a seven-fold change in the value of Nv (for K), the IMFP changes by less than a factor of two. For the two materials for which the Nv ratio is 3, Y and Bi, the IMFPs with the SGS Nv values are less than those with the TPP values by 25.1 % and 29.4 %, respectively. For the five materials for which the Nv ratio is (Y2O3, KBr, Gd, Tb, and Dy), the change in IMFPs is less than 16 %. This latter uncertainty is larger than the uncertainty of IMFPs from the TPP-2M equation for our group of 27 elements (10.2 %)5 and for 2£ 7 our group of organic compounds (8.5 %)8 but may be sufficiently small for some applications of TPP-2M. Seah et al.20,22-25 have actually presented three different recommendations concerning the choice of Nv in Eq. (1). Each recommendation was based on comparisons of experimental and calculated intensities for certain sets of elemental AES and XPS spectra but involved different and increasingly refined procedures for determining peak intensities from the spectra. Initially, they recommended that Nv be computed from the number of electrons with BEs of 14 eV or less, and that the 4f electrons for the lanthanide series be excluded.22-24 In another paper, they concluded that Nv should be obtained from the number of electrons with BEs up to a cutoff energy between 14 eV and 28 eV and that the 4f electrons for the lanthanides be included.25 Finally, as mentioned previously, they recommended that Nv be determined from the number of electrons with BEs of 28 eV or less and that the 4f electrons for the lanthanides be included.20 In the latter paper, Seah et al. also comment that they analyzed a subset of their XPS data with IMFPs obtained from optical data5 rather than from the TPP-2M equation, and obtained marginally poorer results. The recommendations of Seah et al.20-23 are valuable because they provide guidance on the choice of Nv from large sets of experimental data that were obtained and analyzed in a consistent manner. As noted earlier, there has been ambiguity in the choice of this parameter for some elements.4-7,11 It is difficult, however, to develop and apply simple rules for the determination of Nv based on BE considerations alone. For the lanthanide elements, in particular, the contributions of the 4f electrons to the IMFP are expected to be weak while the effects of the 5p electrons are much stronger.15,16 Nevertheless, substantial changes of Nv in the TPP-2M equation do not lead to appreciable changes in the resulting IMFPs, as shown here for Gd, Tb, and Dy in Figs. 4-6 and for the entire lanthanide series by Seah et al.22 when Nv is larger than about 8. Three elemental solids (K, Hf, and Ta) and one compound (KBr) have been identified here as materials that give substantially (> 20%) smaller IMFPs from the TPP-2M equation with the SGS choices of Nv than the corresponding IMFPs calculated from optical data. While the optical data used in the IMFP calculations for these four materials is of similar quality as those used in the IMFP calculations for our group of 27 elemental solids (as judged by the sum-rule consistency checks),4 further experimental tests with these materials are needed. It would clearly 8 be desirable to obtain independent values of the IMFP by elastic-peak electron spectroscopy (EPES).1,2 It would also be desirable to assess the procedures20,22-25 used to determine AES and XPS peak intensities of these materials using the reference IMFPs obtained from optical data as well as independent IMFP measurements when available. Since K oxidizes rapidly and it may be difficult to prepare stoichiometric surfaces of KBr, it is suggested that these tests be made with Hf and Ta. Independent determinations of IMFPs by EPES should also be made of those elemental solids in Table 1 for which there are substantial differences in the TPP and SGS values of Nv (e.g., Ca, Ga, Sr, In, Sn, Cs, Ba, Tl, and Pb). SUMMARY We have investigated the uncertainty in IMFPs derived from the TPP-2M predictive equation (Eq. (1)) associated with different choices of the parameter Nv. We compared IMFPs from this equation using Nv values recommended by us (TPP)4-8,13 and by Seah et al. (SGS)20 with reference IMFPs calculated from optical data. These comparisons were made for eight elemental solids (K, Y, Gd, Tb, Dy, Hf, Ta, and Bi) and two compounds (KBr and Y2O3) for which there were appreciable differences in the recommended Nv values (Table 1) and for which optical data were available for the IMFP calculations. We found that the average of the RMS deviations for the ten materials between IMFPs from Eq. (1) with the TPP Nv values and the reference IMFPs was 11.0 % while the corresponding average with the SGS Nv values was 20.2 %. The larger average for the SGS Nv values was mainly due to large (> 20 %) RMS deviations for four materials (K, Hf, Ta, and KBr). For the other six materials, the RMS deviations with the SGS values of Nv were similar to those with the TPP values of Nv. Based on the comparisons for these ten materials, we believe it is preferable to use the TPP values of Nv in the TPP-2M equation. Further experimental tests (particularly for K, Hf, Ta, and KBr) are needed of IMFPs calculated from optical data as well as from the TPP-2M equation. Such tests could also be used to assess the procedures used for AES and XPS peak-intensity measurements by Seah et al.20 Acknowledgments 9 The authors thank Drs. M. P. Seah and I. S. Gilmore for providing the SGS values of Nv shown in Table 1 and Prof. J. A. D. Matthew for useful discussions. 10 REFERENCES 1. Powell CJ, Jablonski A. J. Phys. Chem. Ref. Data 1999; 28: 19. 2. Gergely G. Prog. Surf. Science (in press). 3. Jablonski A, Powell CJ. (to be published). 4. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1988; 11: 577, Paper I in this series. 5. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1991; 17: 911, Paper II in this series. 6. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1991; 17: 929, Paper III in this series. 7. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1993; 20: 77, Paper IV in this series. 8. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1994; 21: 165, Paper V in this series. 9. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1997; 25: 25, Paper VI in this series. 10. Tanuma S, Powell CJ, Penn DR. J. Electron Spectrosc. Relat. Phenom.. 1993; 62: 95. 11. Tanuma S, Powell CJ, Penn DR. Acta Phys. Polonica A 1992; 81: 169. 12. Evans S. Surf. Interface Anal. 1985; 7: 299. 13. Powell CJ, Jablonski A. NIST Electron Inelastic-Mean-Free-Path Database, SRD 71, Version 1.1. US Department of Commerce, National Institute of Standards and Technology: Gaithersburg, MD 2000. 14. Netzer FP, Matthew JAD. Rep. Prog. Phys. 1986; 49: 621. 15. Tanuma S, Powell CJ, Penn DR (to be published). 16. Matthew JAD (private communication). 17. Wagner CD, Naumkin AV, Kraut-Vass A, Allison JW, Powell CJ, Rumble JR. NIST X-ray Photoelectron Spectroscopy Database, SRD 20, Version 3.2 (2002) (http://srdata.nist.gov.xps. 18. Cardona M, Ley L (eds). Photoemission in Solids I. Springer-Verlag: Berlin, 1978. 19. Smith DM, Gallon TE, Matthew JAD. J. Phys. B 1974; 7: 1255. 11 20. Seah MP, Gilmore IS, Spencer SJ. Surf. Interface Anal. 2001; 31: 778. 21. Seah MP, Gilmore IS (private communication). 22. Seah MP, Gilmore IS. Surf. Interface Anal. 1998; 26: 908. 23. Seah MP. J. Electron Spectrosc. Relat. Phenom. 1999; 100: 55. 24. Seah MP, Gilmore IS, Spencer SJ. J. Electron Spectrosc. Relat. Phenom. 2001; 120: 93. 25. Seah MP, Gilmore IS, Spencer SJ. J. Vac. Sci. Tech. A 2000; 18: 1083. 12 Table 1. Values of Nv recommended by the present authors (TPP) and by Seah et al.20 (SGS) for the indicated elements and values of the atomic number, Z. ______________________________________________________________________________ Element Z Nv TPP SGS ______________________________________________________________________________ H 1 1 He 2 2 Li 3 1 1 Be 4 2 2 B 5 3 3 C 6 4 4 N 7 5 5 O 8 6 6 F 9 7 5 Ne 10 8 Na 11 1 1 Mg 12 2 2 Al 13 3 3 Si 14 4 4 P 15 5 5 S 16 6 6 13 Cl 17 7 7 Ar 18 8 K 19 1 7 Ca 20 2 8 Sc 21 3 3 Ti 22 4 4 V 23 5 5 Cr 24 6 6 Mn 25 7 7 Fe 26 8 8 Co 27 9 9 Ni 28 10 10 Cu 29 11 11 Zn 30 12 12 Ga 31 3 13 Ge 32 4 4 As 33 5 5 Se 34 6 6 Br 35 7 7 Kr 36 8 Rb 37 1 7 Sr 38 2 8 14 Y 39 3 9 Zr 40 4 4 Nb 41 5 5 Mo 42 6 6 Ru 44 8 8 Rh 45 9 9 Pd 46 10 10 Ag 47 11 11 Cd 48 12 12 In 49 3 13 Sn 50 4 14 Sb 51 5 5 Te 52 6 6 I 53 7 7 Xe 54 8 Cs 55 1 9 Ba 56 2 8 La 57 3 9 Ce 58 9 10 Pr 59 9 11 Nd 60 9 12 Sm 62 9 14 15 Eu 63 8 15 Gd 64 9 16 Tb 65 9 17 Dy 66 9 18 Ho 67 9 19 Er 68 9 14 Tm 69 9 15 Yb 70 8 22 Lu 71 9 17 Hf 72 4 18 Ta 73 5 19 W 74 6 6 Re 75 7 7 Os 76 8 8 Ir 77 9 9 Pt 78 10 10 Au 79 11 11 Hg 80 12 12 Tl 81 3 13 Pb 82 4 14 Bi 83 5 15 ______________________________________________________________________________ 16 Table 2. Material parameters needed for the evaluation of the TPP-2M equation (Eq. (1)). ______________________________________________________________________________ Material M r Eg (g/cm3) (eV) ______________________________________________________________________________ K 39.0983 0.89 0 Y 88.90585 4.47 0 Gd 157.25 7.9 0 Tb 158.925 8.23 0 Dy 162.50 8.55 0 Hf 178.49 13.3 0 Ta 180.9479 16.4 0 Bi 208.9804 9.79 0 KBr 119.0023 2.75 6.4 Y2O3 225.809 5.033 5.7 ______________________________________________________________________________ 17 Table 3. Errors in the f-sum rule and the Kramers-Kronig (KK) sum rule used to evaluate the energy-loss function for each material. ______________________________________________________________________________ Material f-sum error (%) KK-sum error (%) ______________________________________________________________________________ K -6.8 16 Y 1 4 Gd -5 2.1 Tb -0.2 6.6 Dy 0.6 -1.1 Hf -4 -16 Ta 1 3 Bi 6 -2 KBr -0.8 -4.7 Y2O3 4 7.7 ______________________________________________________________________________ 18 Figure Captions Fig. 1. Values of the parameter Nv in Eq. (1) recommended by Seah et al.20 (□) and the present authors (●) as a function of atomic number, Z. Fig. 2. Comparison of IMFPs calculated from experimental optical data (solid line) as a function of electron energy for potassium with IMFPs obtained from the TPP-2M equation (Eq. (1)) with values of the parameter Nv recommended by the present authors (long-dashed line) and by Seah et al.20 (short-dashed line). Fig. 3. IMFP results for yttrium; see caption to Fig. 2. Fig. 4. IMFP results for gadolinium; see caption to Fig. 2. Fig. 5. IMFP results for terbium; see caption to Fig. 2. Fig. 6. IMFP results for dysprosium; see caption to Fig. 2. Fig. 7. IMFP results for hafnium; see caption to Fig. 2. Fig. 8. IMFP results for tantalum; see caption to Fig. 2. Fig. 9. IMFP results for bismuth; see caption to Fig. 2. Fig. 10. IMFP results for potassium bromide; see caption to Fig. 2. Fig. 11. IMFP results for yttrium oxide; see caption to Fig. 2. Fig. 12. Root-mean-square (RMS) percentage deviation between IMFPs obtained from the TPP-2M equation (Eq. (1)), with the TPP (●) and SGS (□) values of Nv, and IMFPs calculated from experimental optical data for each material considered here. The absolute values of the average sum-rule errors (Table 3) found in the evaluations of the energy-loss functions for each material are also shown (▲). Fig. 13. Plot of ratios of IMFPs from Eq. (1) at 1000 eV for the SGS and TPP values of Nv versus the ratios of these Nv values for each material considered here. 19