Angular and energy dependences of the surface excitation
parameter for electrons crossing a solid surface
C.M. Kwei
a, Y.C. Li
a, C.J. Tung
b,*aDepartment of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Nuclear Science, National Tsing Hua University, Hsinchu 300, Taiwan
Available online 17 April 2006
Abstract
When fast electrons cross a solid surface, surface plasmons may be generated. Surface plasmon excitations induced by electrons mov-ing in the vacuum are generally characterized by the surface excitation parameter. This parameter was calculated for 200–1000 eV elec-trons crossing the surfaces of Au, Cu, Ag, Fe, Si, Ni, Pd, MgO and SiO2with various crossing angles. Such calculations were performed based on the dielectric response theory for both incident (from vacuum to solid) and escaping (from solid to vacuum) electrons. Calcu-lated results showed that the surface excitation parameter increased with crossing angle but decreased with electron energy. This was due to the longer time for electron–surface interaction by glancing incident or escaping electrons and by slow moving electrons. The results were fitted very well to a simple formula, i.e. Ps¼ aE
b
ðcos aÞc, where Psis the surface excitation parameter, E is the electron energy, a is the angle between the electron trajectory and the surface normal, and a, b and c are material dependent constants.
Ó 2006 Elsevier B.V. All rights reserved.
Keywords: Electron; Surface excitation parameter; Glancing angle
1. Introduction
Quantitative information on inelastic interactions be-tween electron and solid plays an important role in the surface sensitive spectroscopies such as Auger electron spectroscopy (AES), X-ray photoelectron spectroscopy (XPS) and reflection electron energy-loss spectroscopy (REELS), etc. Previous studies showed that surface excita-tions contributed significantly to the spectra of electrons backscattered from solid surfaces [1–3]. Therefore, an ac-count of surface excitations in the analyses of electron spec-troscopies should be established theoretically.
Many authors suggested that surface loss function could be used to estimate the contribution from surface excita-tions to electron inelastic cross-section [4–7]. This ap-proach, however, was oversimplified and provided limited quantitative information on the characteristic energy loss
spectra [8]. Later, the specular reflection model of the
dielectric response theory was applied[1,9,10] to calculate the inelastic cross section due to surface excitations for an electron moving (in vacuum or solid) close to the sur-face. Recently, a surface excitation parameter (SEP) was introduced to characterize the average number of surface plasmons generated by an electron moving across the solid surface [2,11,12]. In the article of Tung et al. [2], they calculated the SEP by assuming that surface excitations occurred right on the surface. But actually surface excita-tions were possible for an electron moving at the position extending to a few angstroms on both sides of the surface.
Chen and Kwei[11]derived the SEP by an integration of
the inverse inelastic mean free path (IMFP) for surface excitations over electron path length across the surface on both sides. In their approach, plasmon excitations inside the solid were separated into individual contribu-tions from surface and volume excitacontribu-tions. Because the compensation of surface and volume excitations led to a
roughly depth-independent IMFP[12], it was more
conve-nient to deal with surface and volume excitations together rather than separately inside the solid. Kwei et al.[12]then
0039-6028/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2006.01.071
*
Corresponding author. Tel./fax: +886 3 5727300. E-mail address:[email protected](C.J. Tung).
calculated the SEP by integrating the inverse IMFP over electron path length in vacuum for normally incident and escaping electrons. For tilted crossing electrons, the SEP was approximated by multiplying the SEP for a = 0° (the crossing angle) by (cosa)1. In their work, however, conser-vations of energy and momentum were not completely satisfied due to the treatment of momentum transfer in cylindrical coordinates that carried no restriction on the normal component. In order to satisfy conservations of
en-ergy and momentum, spherical coordinates[13]should be
adopted. The newly developed model by the present authors applied spherical coordinates in the derivation of position-dependent inelastic cross-section for an electron with arbitrary crossing angle[14].
In the present work, the newly developed model was ap-plied to calculate crossing-angle-dependent and energy-dependent SEPs for 200–1000 eV electrons crossing several solid surfaces. The calculated results were compared with
corresponding data of other works [12,15,16]. The
pres-ently calculated SEPs were fitted to a simple formula for the convenience of applications. The best-fitted parameters of various materials were listed.
2. Methods
Using the dielectric response theory, a modified inelas-tic-scattering model was developed for an electron moving normally or obliquely across the solid surface[14]. In this model, the SEP was determined by an integration of the
inverse IMFP over electron path length in vacuum [12].
Consider a semi-infinite solid (r > 0) of dielectric function eðq*;xÞ, where q*is the momentum transfer, x is the energy
transfer, and r is the radial distance from the crossing point on the surface. The SEPs for escaping (from solid to
vac-uum: s! v) and incident (from vacuum to solid: v ! s)
electrons are given by Ps!vs ða; EÞ ¼4 cos a
p3 Z 0 1 dr Z E 0 dx Z qþ q dq Z p=2 0 dh Z 2p 0 d/ 2 cos xr~ v
expðjrjQ cos aÞ
qsin 2
hexpðjrjQ cos aÞ ~ x2þ Q2v2 ? Im 1 eðQ*;xÞ þ 1 2 4 3 5; ð1Þ and
Pv!ss ða; EÞ ¼4 cos a p3 Z 0 1 dr Z E 0 dx Z qþ q dq Z p=2 0 dh Z 2p 0 d/qsin 2
hcosðqzrcos aÞ ~
x2þ Q2v2 ?
expðjrjQ cos aÞIm 1
eðQ*;xÞ þ 1 2
4
3
5; ð2Þ
where ~x¼ x qv sin h cos / sin a, Q = q sinh, qz= q cosh, q¼pffiffiffiffiffiffi2Epffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðE xÞ, v?= vcosa, v is electron velocity, and E¼v2
2 is electron kinetic energy. The crossing angle a is defined as the angle between surface normal and electron direction. Note that atomic units (a.u.) are used through-out this paper unless otherwise specified. The calculated
Fig. 1. Crossing-angle-dependent SEPs calculated using Eq.(1)for 800 eV electrons escaping from Au to vacuum (solid circles). The solid curve is a fit of the calculated results using Eq.(3). The dashed, dash–dot and dotted curves are, respectively, the results of Oswald[15], Werner et al.[16]and our previous work[12].
SEPs using Eqs.(1) and (2)were found to follow a simple formula Ps!vs ða; EÞ or Pv!s s ða; EÞ ¼ aEb cosca; ð3Þ
for both escaping and incident electrons. 3. Results and discussion
In the present work, the extended Drude dielectric func-tion was applied to calculate the SEP for an electron inci-dent into or escaping from the solid. The fitting parameters of the dielectric function were taken from our
previous work [13,17]. Fig. 1 shows the
crossing-angle-dependent SEPs calculated using Eq.(1)for 800 eV
escap-ing electrons movescap-ing from Au to vacuum (solid circles). It can be seen that the SEP increases with increasing crossing angle a. For a > 70°, the SEP increases rapidly, indicating a high probability of surface excitations for glancing emer-gence electrons. The results of Oswald[15](dashed curve),
Werner et al.[16](dash–dot curve) and our previous work
[12](dotted curve) are included in this figure for compari-sons. These results correspond to calculations based on the simple assumption of a cosine dependence on the cross-ing angle. The present calculations, however, are based on the derived formulas of the angular dependence using the dielectric response theory. It reveals from the comparisons that the cosine assumption works only approximately. The presently calculated SEPs are fitted to Eq. (3) with fitting results plotted as solid curves. With E in electron-volts, we list best-fitted values of parameters a, b and c inTable 1 for all solids studied. These fitting values are different from the corresponding values of our previous work for
a= 0°[12]where conservations of energy and momentum
are not completely satisfied by the adoption of cylindrical
coordinates there.Fig. 2shows the energy-dependent SEPs
calculated using Eq.(1) for electrons escaping from Ni to vacuum at a = 60° (solid circles). Again, solid, dashed, dash–dot and dotted curves are, respectively, the results of fittings, Oswald, Werner et al., and our previous work. It shows that the SEP decreases with increasing electron energy because of the less interacting time for surface excitations.
Similarly, the SEPs calculated using Eq.(2)as a function of crossing angle for 800 eV electrons incident from
vac-uum to Fe are plotted inFig. 3. The SEPs as a function
of energy for electrons incident from vacuum to Pd for
a= 60° are plotted in Fig. 4. In both figures, symbols
and curves have the same meanings as those described above. Note that the SEPs for incident electrons exhibit similar energy and angular dependences as for escaping electrons. However, the SEPs for incident electrons have
Table 1
Fitted values of parameter a, b and c in Eq.(3)
Ps!vs Pv!ss a b c a b c Au 1.8695 0.4052 0.80 1.8848 0.5060 1.06 Cu 1.9994 0.4166 0.82 1.2999 0.4664 1.13 Ag 2.1203 0.4260 0.74 1.4260 0.4821 1.05 Fe 1.9489 0.4321 0.80 1.2562 0.4827 1.05 Si 1.7295 0.4201 0.83 1.1313 0.4756 1.12 Ni 2.1712 0.4283 0.79 1.4244 0.4786 1.03 Pd 2.0564 0.4177 0.79 1.4934 0.4843 1.03 MgO 0.7535 0.3771 0.85 0.6413 0.4634 1.09 SiO2 0.5707 0.3764 0.85 0.5095 0.4687 1.08
Fig. 2. Energy-dependent SEPs calculated using Eq.(1)for electrons escaping from Ni to vacuum at a = 60° (solid circles). The solid curve is a fit of the calculated results using Eq.(3). The dashed, dash–dot and dotted curves are, respectively, the results of Oswald[15], Werner et al.[16]and our previous work[12].
smaller values than for escaping electrons. This is because the attractive force acting on the incident electron (in vacuum) by the surface induced charges accelerates the electron. On the other hand, the attractive force on the escaping electron (in vacuum) decelerates the electron. Therefore, the time spent near the surface for incident elec-tron is less than for escaping elecelec-tron, thus leading to less surface excitations for incident electron.
4. Conclusions
Applying the recently constructed dielectric model for inelastic cross sections, we calculated the crossing-angle-dependent and energy-crossing-angle-dependent SEPs for electrons cross-ing solid surfaces, such as Au, Cu, Ag, Fe, Si, Ni, Pd, MgO and SiO2. Calculations were performed using the extended Drude dielectric functions derived from optical data. The
Fig. 3. Crossing-angle-dependent SEPs calculated using Eq.(2)for 800 eV electrons incident from vacuum to Fe (solid circles). The solid curve is a fit of the calculated results using Eq.(3). The dashed, dash–dot and dotted curves are, respectively, the results of Oswald[15], Werner et al.[16]and our previous work[12].
Fig. 4. Energy-dependent SEPs calculated using Eq.(2)for electrons incident from vacuum to Pd at a = 60° (solid circles). The solid curve is a fit of the calculated results using Eq.(3). The dashed, dash–dot and dotted curves are, respectively, the results of Oswald[15], Werner et al.[16]and our previous work[12].
calculated results showed that the SEP increased with crossing angle due to the longer time for electron–surface interaction by glancing incident or escaping electrons. Also, the SEP decreased with electron energy due to the shorter time for electron–surface interaction by fast mov-ing electrons. The presently calculated SEPs were fitted very well to a simple formula. This formula may be applied to the analyses of the intensity reduction for reflected or emitted electrons in surface and interface analyses. Acknowledgement
This research was supported by the National Science Council of the Republic of China under Contract No. NSC94–2215–E–009–080.
References
[1] F. Yubero, S. Tougaard, Phys. Rev. B 46 (1992) 2486.
[2] C.J. Tung, Y.F. Chen, C.M. Kwei, T.L. Chou, Phys. Rev. B 49 (1994) 16684.
[3] Y.F. Chen, P. Su, C.M. Kwei, Phys. Rev. B 50 (1994) 17547. [4] G. Chiarello, E. Colavita, M. De Cresenzi, S. Nannarone, Phys. Rev.
B 29 (1984) 4878.
[5] Y. Ohno, Phys. Rev. B 39 (1989) 8209.
[6] J.C. Ingham, K.W. Nebesny, J.E. Peemberton, Appl. Surf. Sci. 44 (1990) 279.
[7] F. Yubero, J.M. Sanz, E. Elizalde, L. Galan, Surf. Sci. 237 (1990) 173.
[8] F. Yubero, S. Tougaard, Surf. Interf. Anal. 19 (1992) 269.
[9] F. Yubero, J.M. Sanz, B. Ramskov, S. Tougaard, Phys. Rev. B 53 (1996) 9719.
[10] A.C. Simonsen, F. Yubero, S. Tougaard, Phys. Rev. B 56 (1997) 1612.
[11] Y.F. Chen, C.M. Kwei, Surf. Sci. 364 (1996) 131.
[12] C.M. Kwei, C.Y. Wang, C.J. Tung, Surf. Interf. Anal. 26 (1998) 682.
[13] C.M. Kwei, S.J. Hwang, Y.C. Li, C.J. Tung, J. Appl. Phys. 93 (2003) 9130.
[14] Y.C. Li, Y.H. Tu, C.M. Kwei, C.J. Tung, Surf. Sci. 589 (2005) 67. [15] R. Oswald, Ph.D. thesis, University of Tu¨bingen, Tu¨bingen, 1997. [16] W.S.M. Werner, W. Smekal, C. Tomastik, H. Sto¨ri, Surf. Sci. 486
(2001) L461.
[17] C.M. Kwei, Y.F. Chen, C.J. Tung, J.P. Wang, Surf. Sci. 293 (1993) 202.
