- Email: [email protected]
Determination of the electron inelastic mean free path for samarium
Surface Science 595 (2005) 1–5 www.elsevier.com/locate/susc
Determination of the electron inelastic mean free path for samarium A. Jablonski a, B. Lesiak a, J. Zemek b
b,*
, P. Jiricek
b
a Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, Poland Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic
Received 20 May 2005; accepted for publication 12 August 2005 Available online 31 August 2005
Abstract Calculations of the inelastic mean free path of electrons (IMFP) from the predictive formulas TPP-2M require knowledge of the number of valence electrons, Nv, for atoms constituting the solid. Comparison of the IMFPs obtained from TPP-2M with the IMFPs derived from the elastic electron backscattering can be a tool, in principle, to verify the criteria for selecting a proper number of valence electrons for elements, Nv. In particular, the role of the 4f electrons in the case of lanthanides is presently verified on example of samarium. We found a substantial difference between IMFPs calculated from the TPP-2M formula and from elastic peak electron spectroscopy. This difference decreases only slightly when the 4f electrons are included in Nv. 2005 Elsevier B.V. All rights reserved. Keywords: Monte–Carlo simulations; Samarium; Electron spectroscopy; Electron–solid scattering, elastic
1. Introduction A convenient tool for determination of the IMFP for any solid is the set of equations known under the acronym TPP-2M [1]. This formalism is a modification of the predictive formula published earlier [2,3]. The IMFP, for any material, can be
calculated from the Bethe equation, modified as follows: k¼
E E2p ½b lnðcEÞ ðC=EÞ þ ðD=E2 Þ
ð1Þ
where E is the electron energy (in eV). Remaining parameters are defined by Ep ¼ 28:8ðN v q=MÞ1=2
*
Corresponding author. Tel.: +420 220 318 526; fax: +420 233 343 184. E-mail address: [email protected] (J. Zemek).
b ¼ 0:10 þ
0:50
c ¼ 0:191q
0039-6028/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2005.08.008
0:944=ðE2p
ð2Þ þ
E2g Þ1=2
0:1
þ 0:069q
ð3Þ ð4Þ
2
A. Jablonski et al. / Surface Science 595 (2005) 1–5
C ¼ 1:97 0:91U D ¼ 53:4 20:8U
ð5Þ ð6Þ
U ¼ N v q=M ¼ E2p =829:4
ð7Þ
where Nv is the number of valence electrons per atom or molecule, q is the material density (in g/cm3), and M is the atomic or molecular weight, and Eg is the band-gap energy (in eV) for nonconductors. There may be a controversy concerning selection of a proper number of valence electrons Nv for elements. The binding energy limit initially suggested was less than 30 eV, while Seah and Gilmore [4] indicated that the cut-off of 14 eV is more proper. Much attention has been devoted to rare-earth elements, for which a question has been debated if the 4f electrons should be included in Nv. Measurements of Auger electron signal intensity by Seah and Gilmore [4] have supported the conclusion that the 4f electrons are to be ignored. This conclusion was supported by Tanuma et al. [5,6]. However, more recently, Seah et al. [7] suggested that the 4f electrons should be included in Nv for lanthanides. Specific recommendations concerning the number of valence electrons for over 80 elements, including lanthanides, have been published by Tanuma et al. [5], and Powell and Jablonski [8,9]. In the NIST IMFP database published in 1999, the number of valence electrons for rareearths was assumed to be equal to the chemical valence, as summarized by Netzer and Matthew [10]. In that case, the value of Nv for rare-earths varied between 2 and 3. It has been indicated, however, that six electrons 5p should also be considered [5,9]. Thus, the recommended values of Nv for lanthanides vary between 8 and 9. In the present work, we analyse the influence of 4f electrons on the IMFP calculated for samarium, i.e. the rare-earth element with partly populated 4f subshell. The electronic structure of samarium is [Xe] 4f66s2, and the recommended value of Nv is 9 [5,9]. In the present work, an attempt is made to establish the proper value of Nv by comparison of the measured IMFPs, determined from elastic peak electron spectroscopy for samarium, with the IMFPs resulting from the predictive formula TPP-2M.
2. Elastic peak electron spectroscopy The IMFP values can be derived from the probability of elastic electron backscattering. This method is developed since Eighties of the last century [11], and is considered as a reliable source of the IMFPs. Experiments comprise measurements of the elastic peak intensity for the studied sample and for the standard material for which the IMFP is known. The peak ratio is assumed to be proportional to ratio of elastic backscattering probabilities. Experiments are accompanied with calculations of this ratio assuming different input IMFPs for the sample. The theoretical models and the relevant algorithms were extensively reviewed in the literature [12,13]. In the present work, calculations were performed using the software packet EPESWIN [14]. The crucial parameter needed in calculations is the elastic scattering cross sections for the target atoms. These cross sections were taken from the NIST database [15]. They were found to agree well with experimental cross sections, as shown in an extensive compilation [16]. Four elemental solids were recommended as materials for which the IMFPs are known with highest accuracy: Ni, Cu, Ag and Au [13]. In the present work, copper was selected as a standard material. The so-called ‘‘recommended’’ IMFPs for copper were taken from Table 10 in Ref. [13].
3. Experimental The XPS and EPES measurements were carried in the ADES-400 spectrometer (VG Scientific, UK) equipped with a high-energy resolution hemispherical analyser, X-ray photoelectron source (Mg Ka, Al Ka), a Varian Auger electron gun (model 981–2455), and the Ar+ ion source. The analyser operated with an acceptance angle of ±4.1 at the pass energy of 100 eV and 5 eV for the XPS and EPES measurements, respectively. The electron beam with current of 0.1–1.0 lA in the energy range 200–1000 eV were set during the measurements. The Ar+ source operated at ion energy of 5000 eV, beam density 10 lA cm2 at the incidence angle of 60 with respect to the surface normal.
A. Jablonski et al. / Surface Science 595 (2005) 1–5
Samarium sample (Goodfellow, UK) was in the form of the rolled foil of purity 99.9%, thickness 0.1 mm, and dimensions 12 · 10 mm. The Cu standard was prepared by evaporating Cu on a Si wafer; the thickness of the Cu overlayer was estimated to be equal to 200 nm. The Sm and Cu surfaces were sputter-cleaned at room temperature prior to measurements. The XPS spectra were recorded for Sm and Cu to verify their cleanliness. Occasional check after EPES measurements did not indicate any major increase of contamination. The EPES measurements were carried for electron kinetic energies of 200 eV, 300 eV, 350 eV, 400 eV, 500 eV, 750 eV and 1000 eV with the primary electron beam normal to the surface. The elastically backscattered electrons were recorded at 35 with respect to the surface normal. The full width at half maximum (FWHM) of the elastic peak was 60.5 eV.
4. Results The IMFPs calculated for samarium from the TPP-2M predictive formula are shown in Fig. 1. In these calculations, the number of valence electrons, Nv, was assumed to be 3 (chemical valency), 9 (valency, and 5p6 electrons), and 15 (valency, 5p6 electrons, and the 4f 6 electrons). Although the value of Nv = 14 is recommended when the 4f electrons are included [6], we selected 15 to check the maximum effect of the parameter Nv on the IMFP. We see that the IMFP depends strongly on the number of valence electrons in the range of small Nv. For Nv varying between 9 and 15, the corresponding variation of the IMFP is much less pronounced. We also see that the IMFP values resulting from the the EPES method deviate distinctly from the IMFPs obtained from the predictive formula using the assumed Nv values. Attention should be drawn to the fact that the IMFPs calculated from the optical data are valid for the bulk of the solid, while the IMFPs calculated from the elastic peak intensity are influenced by the electron energy losses in the surface region of the solid [17–19]. The electrons entering the analyser, after traveling a given trajectory length
3
Fig. 1. Energy dependence of the IMFP for samarium. Dashed line: the IMFPs calculated from TPP-2M for Nv = 3; solid line: the IMFPs calculated from TPP-2M for Nv = 9; dot-dashed line: the IMFPs calculated from TPP-2M for Nv = 15; full circles: IMFPs, derived from EPES, uncorrected for SEE; squares: IMFPs, derived from EPES, corrected for SEE assuming Nv = 3; open circles: IMFPs, derived from EPES, corrected for SEE assuming Nv = 9; triangles: IMFPs, derived from EPES, corrected for SEE assuming Nv = 15.
in the solid, are passing the surface region twice, which increases the probability of an energy loss, as compared with a trajectory of the same length traveled in the bulk of the solids. The probability of passing a given trajectory length without energy loss is decreased by the following factor [20] in out out ; EÞ fs ¼ exp½P in s ða ; EÞ exp½P s ða
ð8Þ
where Ps(a, E) is the surface excitation parameter (SEP) [17–19], and the superscripts ‘‘in’’ and ‘‘out’’ indicate the direction of an electron with respect to the surface. Werner and coworkers [18,19] proposed a simple empirical expression describing the SEP for different materials P s ða; EÞ ¼
1 0:173aH E
1=2
cos a þ 1
ð9Þ
where aH is the material dependent parameter. To introduce the surface energy losses into the algorithm for calculating the IMFP, we need to divide the measured ratio of intensities by the surface electronic excitation correction (SEE)
4
K¼
A. Jablonski et al. / Surface Science 595 (2005) 1–5
fsSm fsCu
ð10Þ
In the present analysis, we use the value of aH equal to 2 for copper [18,19]. The value of aH for samarium has been estimated from the approximate expression proposed by Werner and coworkers [18,19] aH ¼ 0:039Ep þ 0:4
ð11Þ
where Ep is the generalized plasmon frequency defined by Eq. (2). As follows from Eqs. (8)–(11), the correction factor K, and consequently the corrected IMFP resulting from EPES, is a function of the number of the valence electrons, Nv. For samarium, the parameter aH is equal to 0.835, 1.154, and 1.373 for Nv equal to 3, 9, and 15, respectively. The IMFP values resulting from EPES, uncorrected and after correction for surface energy losses, are compared in Fig. 1 with IMFPs resulting from the predictive formula. We see that all the IMFPs obtained from EPES are smaller than the IMFPs calculated from the TPP-2M formula. Correction for the surface energy losses increases the IMFPs from EPES and thus improves agreement with the predictive formula, however the extent of this increase is insufficient. To quantify the dependence of the IMFP resulting from the TPP-2M predictive formula on the parameter Nv, the following percentage deviations were calculated: Dk ¼ 100
kTPP ðN v ¼ mÞ kTPP ðN v ¼ 9Þ kTPP ðN v ¼ 9Þ
ð12Þ
where m is equal to 3 or 15. Results of calculations are shown in Fig. 2(a). We see that the influence of the parameter Nv on k increases with energy up to about 200 eV; at higher energies Dk is practically independent of energy. Increase of Nv from 3 to 9 leads to the decrease of the IMFP of about 45%. Further increase of Nv from 9 to 15 leads to the change of the IMFP by much less, i.e. 10%. Let us additionally consider the percentage difference between the IMFPs obtained from the EPES method, kEPES, and the IMFPs calculated from the TPP-2M, kTPP
Fig. 2. Percentage differences between IMFP values. Panel (a): differences between IMFPs resulting from TPP-2M calculated from Eq. (12); dashed line: Nv = 3 and Nv = 9; dotdashed line: Nv = 15 and Nv = 9. Panel (b): differences between IMFPs resulting from TPP-2M and EPES calculated from Eq. (13); squares: Nv = 3; circles: Nv = 9; triangles: Nv = 15.
Dk ¼ 100
kEPES kTPP kTPP
ð13Þ
As follows from Fig. 2(b), the percentage deviations are considerable, varying in the range from 19% to 55%. We see that the increase of Nv leads to improving agreement between the IMFPs resulting from EPES and TPP-2M. The smallest percentage deviation, varying from 19% at 200 eV to 36% at 1000 eV, is observed for Nv = 15. This is an apparent indication that the 4f electrons in samarium should be accounted in the TPP-2M formula. However, one should notice that the increase of Nv from 9 to 15 improves the agreement only slightly.
A. Jablonski et al. / Surface Science 595 (2005) 1–5
5
5. Discussion and conclusions
Acknowledgements
The percentage deviation between kEPES and kTPP for any value of Nv is larger than the percentage variation of kTPP due to the increase of Nv from 9 to 15. Thus, the EPES method does not seem to be a proper criterion for a decision if the 4f electrons should be included in parameter Nv. Although the deviation between kEPES and kTPP decreases with Nv, this is hardly a proof that the 4f electrons should be taken into account. We may expect that the IMFPs obtained from the EPES method is burdened with experimental and systematic errors, which lead to a significant scatter of the published data. Extensive discussion of the accuracy of the EPES method has been published by Powell and Jablonski [13]. One of the contributions to the observed difference between the EPES and TPP-2M data may be due to difficulties in determining the density of the Cu overlayer in the standard sample. It has been shown that the mean percentage deviation of the IMFPs obtained from the EPES method (uncorrected for SEE) from the calculated IMFPs is 17.4%. The present results seem to confirm this observation. Furthermore, Eq. (11) is an approximate guide providing the coefficient aH, and consequently the SEE correction for samarium may have a limited accuracy. One should also be aware that the TTP-2M formula is burdened with systematic errors, which are difficult to estimate for samarium. As shown in Fig. 2(b), the largest difference between the IMFPs calculated from TPP-2M and determined from elastic backscattering probability, reaching and exceeding 50%, has been found for assumption of Nv equal to 3. This may be an indication that this assumption is incorrect. The IMFPs for samarium in the NIST database, version 1.0 [8] were calculated for Nv = 3, and these values should be considered with some caution. Unfortunately, the precision of IMFPs from EPES is insufficient to decide which assumption, Nv = 9 or Nv = 15, is justified. It seems to be advisable to check if similar conclusions can be obtained if other standard materials are used instead of Cu. Furthermore, similar studies seem to be useful for other lanthanide materials. Such work is planned for the future.
One of the authors (A.J.) would like to acknowledge partial support by the Foundation for Polish Science. The research work at the Institute of Physics is supported by Institutional Research Plan No. AV0Z10100521.
References [1] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 21 (1994) 165. [2] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 17 (1991) 911. [3] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 17 (1991) 927. [4] M.P. Seah, I.S. Gilmore, Surf. Interf. Anal. 26 (1998) 908. [5] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 35 (2003) 268. [6] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 37 (2005) 1. [7] M.P. Seah, I.S. Gilmore, S.J. Spencer, Surf. Interf. Anal. 31 (2001) 778. [8] C.J. Powell, A. Jablonski, NIST Electron Inelastic-MeanFree-Path Database, Version 1.0, Standard Reference Data Program Database 71, US Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 1999. [9] C.J. Powell, A. Jablonski, NIST Electron Inelastic-MeanFree-Path Database, Version 1.1, Standard Reference Data Program Database 71, US Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 2000. [10] F.P. Netzer, J.A.D. Matthew, Rep. Prog. Phys. 49 (1986) 621. [11] A. Jablonski, Surf. Sci. 151 (1985) 166. [12] A. Dubus, A. Jablonski, S. Tougaard, Prog. Surf. Sci. 63 (2000) 135. [13] C.J. Powell, A. Jablonski, J. Phys. Chem. Ref. Data 28 (1999) 19. [14] A. Jablonski, Surf. Interf. Anal., in press. [15] A. Jablonski, F. Salvat, C.J. Powell, NIST Electron ElasticScattering Cross-Section Database, Version 3.1, Standard Reference Data Program Database 64, National Institute of Standards and Technology, Gaithersburg, MD, 2003, Available from:. [16] A. Jablonski, F. Salvat, C.J. Powell, J. Phys. Chem. Ref. Data 33 (2004) 409. [17] Y.F. Chen, Surf. Sci. 380 (1997) 199; Y.F. Chen, Surf. Sci. 519 (2002) 115. [18] W.S.M. Werner, W. Smekal, C. Tomastik, H. Stori, Surf. Sci. 486 (2001) L461. [19] W.S.M. Werner, Surf. Interf. Anal. 31 (2001) 141. [20] S. Tanuma, S. Ichimura, K. Goto, J. Surf. Anal. 5 (1999) 48.
Determination of the electron inelastic mean free path for samarium A. Jablonski a, B. Lesiak a, J. Zemek b
b,*
, P. Jiricek
b
a Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, Poland Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 53 Prague 6, Czech Republic
Received 20 May 2005; accepted for publication 12 August 2005 Available online 31 August 2005
Abstract Calculations of the inelastic mean free path of electrons (IMFP) from the predictive formulas TPP-2M require knowledge of the number of valence electrons, Nv, for atoms constituting the solid. Comparison of the IMFPs obtained from TPP-2M with the IMFPs derived from the elastic electron backscattering can be a tool, in principle, to verify the criteria for selecting a proper number of valence electrons for elements, Nv. In particular, the role of the 4f electrons in the case of lanthanides is presently verified on example of samarium. We found a substantial difference between IMFPs calculated from the TPP-2M formula and from elastic peak electron spectroscopy. This difference decreases only slightly when the 4f electrons are included in Nv. 2005 Elsevier B.V. All rights reserved. Keywords: Monte–Carlo simulations; Samarium; Electron spectroscopy; Electron–solid scattering, elastic
1. Introduction A convenient tool for determination of the IMFP for any solid is the set of equations known under the acronym TPP-2M [1]. This formalism is a modification of the predictive formula published earlier [2,3]. The IMFP, for any material, can be
calculated from the Bethe equation, modified as follows: k¼
E E2p ½b lnðcEÞ ðC=EÞ þ ðD=E2 Þ
ð1Þ
where E is the electron energy (in eV). Remaining parameters are defined by Ep ¼ 28:8ðN v q=MÞ1=2
*
Corresponding author. Tel.: +420 220 318 526; fax: +420 233 343 184. E-mail address: [email protected] (J. Zemek).
b ¼ 0:10 þ
0:50
c ¼ 0:191q
0039-6028/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2005.08.008
0:944=ðE2p
ð2Þ þ
E2g Þ1=2
0:1
þ 0:069q
ð3Þ ð4Þ
2
A. Jablonski et al. / Surface Science 595 (2005) 1–5
C ¼ 1:97 0:91U D ¼ 53:4 20:8U
ð5Þ ð6Þ
U ¼ N v q=M ¼ E2p =829:4
ð7Þ
where Nv is the number of valence electrons per atom or molecule, q is the material density (in g/cm3), and M is the atomic or molecular weight, and Eg is the band-gap energy (in eV) for nonconductors. There may be a controversy concerning selection of a proper number of valence electrons Nv for elements. The binding energy limit initially suggested was less than 30 eV, while Seah and Gilmore [4] indicated that the cut-off of 14 eV is more proper. Much attention has been devoted to rare-earth elements, for which a question has been debated if the 4f electrons should be included in Nv. Measurements of Auger electron signal intensity by Seah and Gilmore [4] have supported the conclusion that the 4f electrons are to be ignored. This conclusion was supported by Tanuma et al. [5,6]. However, more recently, Seah et al. [7] suggested that the 4f electrons should be included in Nv for lanthanides. Specific recommendations concerning the number of valence electrons for over 80 elements, including lanthanides, have been published by Tanuma et al. [5], and Powell and Jablonski [8,9]. In the NIST IMFP database published in 1999, the number of valence electrons for rareearths was assumed to be equal to the chemical valence, as summarized by Netzer and Matthew [10]. In that case, the value of Nv for rare-earths varied between 2 and 3. It has been indicated, however, that six electrons 5p should also be considered [5,9]. Thus, the recommended values of Nv for lanthanides vary between 8 and 9. In the present work, we analyse the influence of 4f electrons on the IMFP calculated for samarium, i.e. the rare-earth element with partly populated 4f subshell. The electronic structure of samarium is [Xe] 4f66s2, and the recommended value of Nv is 9 [5,9]. In the present work, an attempt is made to establish the proper value of Nv by comparison of the measured IMFPs, determined from elastic peak electron spectroscopy for samarium, with the IMFPs resulting from the predictive formula TPP-2M.
2. Elastic peak electron spectroscopy The IMFP values can be derived from the probability of elastic electron backscattering. This method is developed since Eighties of the last century [11], and is considered as a reliable source of the IMFPs. Experiments comprise measurements of the elastic peak intensity for the studied sample and for the standard material for which the IMFP is known. The peak ratio is assumed to be proportional to ratio of elastic backscattering probabilities. Experiments are accompanied with calculations of this ratio assuming different input IMFPs for the sample. The theoretical models and the relevant algorithms were extensively reviewed in the literature [12,13]. In the present work, calculations were performed using the software packet EPESWIN [14]. The crucial parameter needed in calculations is the elastic scattering cross sections for the target atoms. These cross sections were taken from the NIST database [15]. They were found to agree well with experimental cross sections, as shown in an extensive compilation [16]. Four elemental solids were recommended as materials for which the IMFPs are known with highest accuracy: Ni, Cu, Ag and Au [13]. In the present work, copper was selected as a standard material. The so-called ‘‘recommended’’ IMFPs for copper were taken from Table 10 in Ref. [13].
3. Experimental The XPS and EPES measurements were carried in the ADES-400 spectrometer (VG Scientific, UK) equipped with a high-energy resolution hemispherical analyser, X-ray photoelectron source (Mg Ka, Al Ka), a Varian Auger electron gun (model 981–2455), and the Ar+ ion source. The analyser operated with an acceptance angle of ±4.1 at the pass energy of 100 eV and 5 eV for the XPS and EPES measurements, respectively. The electron beam with current of 0.1–1.0 lA in the energy range 200–1000 eV were set during the measurements. The Ar+ source operated at ion energy of 5000 eV, beam density 10 lA cm2 at the incidence angle of 60 with respect to the surface normal.
A. Jablonski et al. / Surface Science 595 (2005) 1–5
Samarium sample (Goodfellow, UK) was in the form of the rolled foil of purity 99.9%, thickness 0.1 mm, and dimensions 12 · 10 mm. The Cu standard was prepared by evaporating Cu on a Si wafer; the thickness of the Cu overlayer was estimated to be equal to 200 nm. The Sm and Cu surfaces were sputter-cleaned at room temperature prior to measurements. The XPS spectra were recorded for Sm and Cu to verify their cleanliness. Occasional check after EPES measurements did not indicate any major increase of contamination. The EPES measurements were carried for electron kinetic energies of 200 eV, 300 eV, 350 eV, 400 eV, 500 eV, 750 eV and 1000 eV with the primary electron beam normal to the surface. The elastically backscattered electrons were recorded at 35 with respect to the surface normal. The full width at half maximum (FWHM) of the elastic peak was 60.5 eV.
4. Results The IMFPs calculated for samarium from the TPP-2M predictive formula are shown in Fig. 1. In these calculations, the number of valence electrons, Nv, was assumed to be 3 (chemical valency), 9 (valency, and 5p6 electrons), and 15 (valency, 5p6 electrons, and the 4f 6 electrons). Although the value of Nv = 14 is recommended when the 4f electrons are included [6], we selected 15 to check the maximum effect of the parameter Nv on the IMFP. We see that the IMFP depends strongly on the number of valence electrons in the range of small Nv. For Nv varying between 9 and 15, the corresponding variation of the IMFP is much less pronounced. We also see that the IMFP values resulting from the the EPES method deviate distinctly from the IMFPs obtained from the predictive formula using the assumed Nv values. Attention should be drawn to the fact that the IMFPs calculated from the optical data are valid for the bulk of the solid, while the IMFPs calculated from the elastic peak intensity are influenced by the electron energy losses in the surface region of the solid [17–19]. The electrons entering the analyser, after traveling a given trajectory length
3
Fig. 1. Energy dependence of the IMFP for samarium. Dashed line: the IMFPs calculated from TPP-2M for Nv = 3; solid line: the IMFPs calculated from TPP-2M for Nv = 9; dot-dashed line: the IMFPs calculated from TPP-2M for Nv = 15; full circles: IMFPs, derived from EPES, uncorrected for SEE; squares: IMFPs, derived from EPES, corrected for SEE assuming Nv = 3; open circles: IMFPs, derived from EPES, corrected for SEE assuming Nv = 9; triangles: IMFPs, derived from EPES, corrected for SEE assuming Nv = 15.
in the solid, are passing the surface region twice, which increases the probability of an energy loss, as compared with a trajectory of the same length traveled in the bulk of the solids. The probability of passing a given trajectory length without energy loss is decreased by the following factor [20] in out out ; EÞ fs ¼ exp½P in s ða ; EÞ exp½P s ða
ð8Þ
where Ps(a, E) is the surface excitation parameter (SEP) [17–19], and the superscripts ‘‘in’’ and ‘‘out’’ indicate the direction of an electron with respect to the surface. Werner and coworkers [18,19] proposed a simple empirical expression describing the SEP for different materials P s ða; EÞ ¼
1 0:173aH E
1=2
cos a þ 1
ð9Þ
where aH is the material dependent parameter. To introduce the surface energy losses into the algorithm for calculating the IMFP, we need to divide the measured ratio of intensities by the surface electronic excitation correction (SEE)
4
K¼
A. Jablonski et al. / Surface Science 595 (2005) 1–5
fsSm fsCu
ð10Þ
In the present analysis, we use the value of aH equal to 2 for copper [18,19]. The value of aH for samarium has been estimated from the approximate expression proposed by Werner and coworkers [18,19] aH ¼ 0:039Ep þ 0:4
ð11Þ
where Ep is the generalized plasmon frequency defined by Eq. (2). As follows from Eqs. (8)–(11), the correction factor K, and consequently the corrected IMFP resulting from EPES, is a function of the number of the valence electrons, Nv. For samarium, the parameter aH is equal to 0.835, 1.154, and 1.373 for Nv equal to 3, 9, and 15, respectively. The IMFP values resulting from EPES, uncorrected and after correction for surface energy losses, are compared in Fig. 1 with IMFPs resulting from the predictive formula. We see that all the IMFPs obtained from EPES are smaller than the IMFPs calculated from the TPP-2M formula. Correction for the surface energy losses increases the IMFPs from EPES and thus improves agreement with the predictive formula, however the extent of this increase is insufficient. To quantify the dependence of the IMFP resulting from the TPP-2M predictive formula on the parameter Nv, the following percentage deviations were calculated: Dk ¼ 100
kTPP ðN v ¼ mÞ kTPP ðN v ¼ 9Þ kTPP ðN v ¼ 9Þ
ð12Þ
where m is equal to 3 or 15. Results of calculations are shown in Fig. 2(a). We see that the influence of the parameter Nv on k increases with energy up to about 200 eV; at higher energies Dk is practically independent of energy. Increase of Nv from 3 to 9 leads to the decrease of the IMFP of about 45%. Further increase of Nv from 9 to 15 leads to the change of the IMFP by much less, i.e. 10%. Let us additionally consider the percentage difference between the IMFPs obtained from the EPES method, kEPES, and the IMFPs calculated from the TPP-2M, kTPP
Fig. 2. Percentage differences between IMFP values. Panel (a): differences between IMFPs resulting from TPP-2M calculated from Eq. (12); dashed line: Nv = 3 and Nv = 9; dotdashed line: Nv = 15 and Nv = 9. Panel (b): differences between IMFPs resulting from TPP-2M and EPES calculated from Eq. (13); squares: Nv = 3; circles: Nv = 9; triangles: Nv = 15.
Dk ¼ 100
kEPES kTPP kTPP
ð13Þ
As follows from Fig. 2(b), the percentage deviations are considerable, varying in the range from 19% to 55%. We see that the increase of Nv leads to improving agreement between the IMFPs resulting from EPES and TPP-2M. The smallest percentage deviation, varying from 19% at 200 eV to 36% at 1000 eV, is observed for Nv = 15. This is an apparent indication that the 4f electrons in samarium should be accounted in the TPP-2M formula. However, one should notice that the increase of Nv from 9 to 15 improves the agreement only slightly.
A. Jablonski et al. / Surface Science 595 (2005) 1–5
5
5. Discussion and conclusions
Acknowledgements
The percentage deviation between kEPES and kTPP for any value of Nv is larger than the percentage variation of kTPP due to the increase of Nv from 9 to 15. Thus, the EPES method does not seem to be a proper criterion for a decision if the 4f electrons should be included in parameter Nv. Although the deviation between kEPES and kTPP decreases with Nv, this is hardly a proof that the 4f electrons should be taken into account. We may expect that the IMFPs obtained from the EPES method is burdened with experimental and systematic errors, which lead to a significant scatter of the published data. Extensive discussion of the accuracy of the EPES method has been published by Powell and Jablonski [13]. One of the contributions to the observed difference between the EPES and TPP-2M data may be due to difficulties in determining the density of the Cu overlayer in the standard sample. It has been shown that the mean percentage deviation of the IMFPs obtained from the EPES method (uncorrected for SEE) from the calculated IMFPs is 17.4%. The present results seem to confirm this observation. Furthermore, Eq. (11) is an approximate guide providing the coefficient aH, and consequently the SEE correction for samarium may have a limited accuracy. One should also be aware that the TTP-2M formula is burdened with systematic errors, which are difficult to estimate for samarium. As shown in Fig. 2(b), the largest difference between the IMFPs calculated from TPP-2M and determined from elastic backscattering probability, reaching and exceeding 50%, has been found for assumption of Nv equal to 3. This may be an indication that this assumption is incorrect. The IMFPs for samarium in the NIST database, version 1.0 [8] were calculated for Nv = 3, and these values should be considered with some caution. Unfortunately, the precision of IMFPs from EPES is insufficient to decide which assumption, Nv = 9 or Nv = 15, is justified. It seems to be advisable to check if similar conclusions can be obtained if other standard materials are used instead of Cu. Furthermore, similar studies seem to be useful for other lanthanide materials. Such work is planned for the future.
One of the authors (A.J.) would like to acknowledge partial support by the Foundation for Polish Science. The research work at the Institute of Physics is supported by Institutional Research Plan No. AV0Z10100521.
References [1] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 21 (1994) 165. [2] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 17 (1991) 911. [3] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 17 (1991) 927. [4] M.P. Seah, I.S. Gilmore, Surf. Interf. Anal. 26 (1998) 908. [5] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 35 (2003) 268. [6] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interf. Anal. 37 (2005) 1. [7] M.P. Seah, I.S. Gilmore, S.J. Spencer, Surf. Interf. Anal. 31 (2001) 778. [8] C.J. Powell, A. Jablonski, NIST Electron Inelastic-MeanFree-Path Database, Version 1.0, Standard Reference Data Program Database 71, US Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 1999. [9] C.J. Powell, A. Jablonski, NIST Electron Inelastic-MeanFree-Path Database, Version 1.1, Standard Reference Data Program Database 71, US Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 2000. [10] F.P. Netzer, J.A.D. Matthew, Rep. Prog. Phys. 49 (1986) 621. [11] A. Jablonski, Surf. Sci. 151 (1985) 166. [12] A. Dubus, A. Jablonski, S. Tougaard, Prog. Surf. Sci. 63 (2000) 135. [13] C.J. Powell, A. Jablonski, J. Phys. Chem. Ref. Data 28 (1999) 19. [14] A. Jablonski, Surf. Interf. Anal., in press. [15] A. Jablonski, F. Salvat, C.J. Powell, NIST Electron ElasticScattering Cross-Section Database, Version 3.1, Standard Reference Data Program Database 64, National Institute of Standards and Technology, Gaithersburg, MD, 2003, Available from: