Influence of the direction of motion on the inelastic interaction between electrons and solid surfaces

(1)

Inﬂuence of the direction of motion on the inelastic

interaction between electrons and solid surfaces

Y.C. Li

a

, Y.H. Tu

a

, C.M. Kwei

a,*

, C.J. Tung

b a

Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan

b

Department of Nuclear Science, National Tsing Hua University, Hsinchu 300, Taiwan Received 19 February 2005; accepted for publication 25 May 2005

Available online 20 June 2005

Abstract

Theoretical derivations of the inelastic differential inverse mean free path (DIMFP) and inelastic mean free path (IMFP) for electrons crossing solid surfaces were made for different crossing angles and electron distances relative to the crossing point at the surface. Individual contributions from volume and surface excitations were separated and analyzed for electrons traveling inside and outside the solid. Extended Drude dielectric functions were employed to calculate the DIMFP and inverse IMFP for electrons incident into and escaping from Cu. It was found that the DIMFP and inverse IMFP for electrons moving inside the solid were approximately independent of crossing angle and position of electrons due to the compensation of volume and surface excitations. For electrons moving deep inside the solid, the DIMFP and inverse IMFP reduced to the values for electrons moving in an infinite solid. As electrons traveling in the vacuum, the DIMFP and inverse IMFP became greater for glancing incident and escaping angles since surface excitations were more probable. The surface excitation parameter (SEP) for electrons traveling in vacuum showed an angular dependence. The SEP of escaping electrons was found larger than that of incident electrons due to the attractive force exerted by the induced surface charges. The calculated SEP was found to follow a simple expression.

Keywords: Electron; Inelastic mean free path; Surface excitation; Volume excitation

1. Introduction

Quantitative information on inelastic interac-tion cross-secinterac-tions of low-energy electrons crossing solid surfaces is important in surface sensitive spec-troscopies such as Auger electron spectroscopy

*

Corresponding author. Tel.: +886 3 5712121x54136; fax: +886 3 5727300.

E-mail address:cmkwei@cc.nctu.edu.tw(C.M. Kwei).

(2)

(AES), X-ray photoelectron spectroscopy (XPS) and reﬂection electron energy-loss spectroscopy (REELS), etc. This information can be extracted from experimentally measured REELS spectra by

the deconvolution method [1]. The extracted

inelastic cross-sections contain the combined effects arisen from volume and surface plasmon excitations. Volume excitations occur for electrons traveling inside the solid. Surface excitations orig-inate from electrons, either inside or outside the solid, moving at a distance in the order of ang-stroms from the surface. Within the solid, the decrease in volume excitations as electrons move close to the surface is compensated by the increase in surface excitations. This makes the total inelas-tic cross-section at any position inside the solid nearly independent of depth [2,3]. For electrons traveling outside the solid, only surface excitations are attainable over an effective region close to the surface [3]. In such a case, the surface excitations are usually characterized by the so-called surface excitation parameter (SEP), defined as the average number of surface plasmons excited by electrons

outside the solid [3]. Tougaard and Chorkendorﬀ

[1] extracted the inelastic cross-sections from the REELS spectra without separating out the contri-bution of surface excitations. Chen [4]singled out this contribution by incorporating a surface eﬀect into the Landau formula in the deconvolution method and applying an approximate relation of

1_ﬃﬃ E

p _{for the energy dependence to the surface} excita-tion probability.

Theoretical derivations of inelastic cross-sec-tions for low-energy electrons crossing solid

sur-faces have been made using the dielectric

response theory[2,3,5]. Yubero et al.[6–8]derived the effective (or lumped) inelastic cross-sections for backscattered electrons assuming a single-elastic backward-scattering in the trajectory of the elec-tron. In reality, the electron might experience plu-ral or multiple elastic scatterings in its trajectory before being scattered out of the solid. In spite of this, the effective inelastic cross-sections of Yubero et al. were compared well with experimentally determined inelastic cross-sections. The effective inelastic cross-sections contain information on reflected electrons that cross the surface twice. Be-cause it is difficult to separate out the contribution

of surface excitations from the lumped cross-sections, the eﬀective cross-sections should not be applied directly to the quantitative analysis of XPS and AES for which electrons cross the surface

only once. Simonsen et al. [9] later proposed a

model which included the eﬀect of static core hole for XPS and AES. In this model, the trajectory of the electron was assumed to follow a straight line due to its neglect of elastic scatterings. Elastic scat-terings, however, are important in the REELS analyses. In the present work, attempts were made to develop position-dependent inelastic cross-sections for obliquely incident and escaping electrons that were applicable to the REELS analyses.

Recently, the present authors developed a dielectric response function model[10]to calculate the depth-dependent differential inverse mean free path (DIMFP), the inelastic mean free path (IMFP) and surface excitation parameter (SEP) for electrons normally crossing through solid sur-faces[3]. For other crossing angle a, the SEP val-ues were found by multiplying the results on SEP for normally crossing angle with (cos a)1 [3,11]. However, momentum transfer in cylindrical coor-dinates with no restriction on its normal compo-nent was adopted for performing the momentum integration in the model. The conservations of en-ergy and momentum were not completely satisfied, and the derived DIMFP for volume excitations in the limit of large depths did not match the DIMFP in an infinite solid.

In the present work, a similar approach as that in the previous model[2,3]was followed to deter-mine the angular dependence of inelastic cross-sec-tions for electrons incident into or escaping from the solid. Spherical coordinates were employed in the momentum integration to satisfy the energy

and momentum conservations [12]. Further, the

dependences of DIMFP and inverse IMFP on the crossing angle and the position of electrons were established from ﬁrst principles based on the extended Drude dielectric functions [10]. The angular distribution of SEP was also investigated. Calculated SEPs for copper were ﬁtted to an ana-lytical formula. A comparison was made between presently calculated results and corresponding data of other models[11,13,14].

(3)

2. Methods

As illustrated in Fig. 1, an electron of charge e and velocity ~vtravels across an interface at time t = 0 from medium 1 of dielectric function e1ð~q;xÞ to medium 2 of dielectric function e2ð~q;xÞ, where ~q is the momentum transfer and x is the energy transfer. Note that atomic units are used through-out this paper unless otherwise speciﬁed. The crossing angle a is deﬁned as the angle between the interface normal and the electron moving direction. The instant position of the electron is ~r¼ ~vt, relative to the crossing point at the inter-face. As the electron crosses the interface, surface and volume excitations are probable due to elec-tron–solid interactions. Surface excitations occur when the electron travels near the interface, while volume excitations arise as the electron moves inside the media. These two excitations can be described using the dielectric response theory

[2,3]. By solving the PoissonÕs equation, the Fou-rier components of the scalar potentials in media 1 and 2 are given by[3,15]

Uð1Þ_ð~_q;_{xÞ ¼} 8p2 q2_e 1ð~q;xÞ d xð qv cos bÞ þ qsð~Q;xÞ h i for t < 0 ð1Þ and Uð2Þ_ð~_q;_{xÞ ¼} 8p2 q2_e 2ð~q;xÞ d xð qv cos bÞ qsð~Q;xÞ h i for t > 0; ð2Þ

where ~q¼ ð~Q; q_zÞ; ~Q and qz are the parallel and normal components of ~q with respect to the sur-face, b is the angle between ~q and ~v, and qsð~Q;xÞ is the induced surface charge density. The signs accompanying the induced surface charge density are opposite for t < 0 and for t > 0, which is due to the requirement for the continuity of the nor-mal component of the electric displacement at the interface. The other boundary condition, i.e. the continuity of the tangential components of the electric ﬁeld at the interface, requires that the induced surface charge density follows:

qsð~Q;xÞ ¼ Rþ1 1 dðx~Q~vkqzv?Þ q2 _e 1 2ð~q;xÞ 1 e1ð~q;xÞ h i dq_z Rþ1 1 1 q2 1 e2ð~q;xÞþ 1 e1ð~q;xÞ h i dq_z ; ð3Þ where v?and vkare the normal and parallel com-ponents of ~vwith respect to the surface.

Now the Fourier components of the scalar potentials, Uð1Þð~q;xÞ and Uð2Þ_ð~_q;_{xÞ on either side} of the interface, can be obtained after substituting Eq. (3) into Eqs. (1) and (2). The induced poten-tials in real space can be derived by the inverse Fourier transforms of Uð1Þð~q;xÞ and Uð2Þ_ð~_q;_xÞ after removing the potential of the electron in vac-uum. Adopting spherical coordinates in the inte-gration of momentum transfer, the induced potentials can be written as

Uð1Þ_indð~r; tÞ ¼ 1 p Z Z Z d xð qvcosbÞ 1 e1ð~q;xÞ 1 ei qvt cos bxtð Þ_{sin bdbdqdx} 1 2p2 Z Z Z Z _q sð~Q;xÞ e1ð~q;xÞ eið~Q~RxtÞ eiqzz_{sin hd/dhdqdx for t < 0} _ð4Þ and

Fig. 1. A sketch of the problem studied in this work. An electron of velocity ~vmoves across the interface at time t = 0 from medium 1 of dielectric function e1ð~q;xÞ to medium 2 of

dielectric function e2ð~q;xÞ with crossing angle a. The instant

position of the electron is~r¼ ~vt, relative to the crossing point at interface.

(4)

Uð2Þ_indð~r; tÞ ¼ 1 p Z Z Z d xð qv cos bÞ 1 e2ð~q;xÞ 1 ei qvt cos bxtð Þ_{sin bdbdqdx} þ 1 2p2 Z Z Z Z _q sð~Q;xÞ e2ð~q;xÞ eið~Q~RxtÞ eiqzz_{sin hd/dhdqdx for t > 0;} ð5Þ where ~r¼ ð~R; zÞ, ~Rand z are the parallel and nor-mal components of ~r with respect to the surface, and q_sð~Q;xÞ ejð~q;xÞ ¼1 p Qv_? ðx ~Q~v_kÞ2þ Q2_v2 ? e1ð~Q;xÞ e2ð~Q;xÞ ejð~Q;xÞ e1ð~Q;xÞ þ e2ð~Q;xÞ h i for j¼ 1 and 2. ð6Þ

Eq. (6) was derived under the assumption that

eð~q;xÞ eð~Q;xÞ[6,7]. The integrations over x in the second integrals of Eqs.(4) and (5)can be per-formed by closing the contour in the upper and lower half planes for t < 0 and t > 0, respectively. To carry out the contour integration in the lower half plane, it is convenient to convert it into the upper half plane by replacing eið~Q~RxtÞ _{in Eq.} ₍₅₎ with eið~Q~RxtÞ_{¼ 2 cosðxt ~}_Q_~_{RÞ e}ið~Q~RxtÞ_{. The}

stopping power, dW

ds, can be related to the in-duced potential, Uindð~r; tÞ, by[16]

dW ds ¼ 1 v oUindð~r; tÞ ot ~r¼~vt ; ð7Þ

where the derivative of the induced potential is evaluated at the position of the electron. And the stopping power can be expressed in terms of the position-dependent DIMFP, l(a, E, x, r), accord-ing to dW ds ¼ Z E 0 xl a; E; x; rð Þdx. ð8Þ

In the case of an electron traveling from solid to vacuum, i.e. s! v, e1ð~q;xÞ and e2ð~q;xÞ may be re-placed by eð~q;xÞ and 1, respectively. The DIMFP is therefore given by ls!v_ð_{a; E; x; r}_Þ ¼ 2 pv2 Z qþ q dq1 qIm 1 eð~q;xÞ HðrÞ 2 cos a p3 Z qþ q dq Z p=2 0 dh Z 2p 0 d/ qsin 2

hcos qð _zrcos aÞ exp jrjQ cos að Þ ~ x2þ Q2_v2 ? Im 1 eð~Q;xÞ " # HðrÞ þ4 cos a p3 Z qþ q dq Z p=2 0 dh Z 2p 0 d/ qsin 2

hcos qð zrcos aÞ exp jrjQ cos að Þ ~ x2þ Q2_v2 ? Im 1 eð~Q;xÞ þ 1 " # HðrÞ þ4 cos a p3 Z qþ q dq Z p=2 0 dh Z 2p 0 d/ qsin 2 hexpðjrjQ cos aÞ ~ x2þ Q2_v2 ? Im 1 eð~Q;xÞ þ 1 " # 2 cos xr~ v exp jrjQ cos að Þ HðrÞ; ð9Þ where ~x¼ x qv sin h cos / sin a, Q = q sin h, qz= q cos h, v?= v cos a, E¼v

2

2, and H(r) is the Heavi-side step function. Applying the energy–momen-tum conservation relations, the upper and lower limits of q are q ¼pffiffiffiffiffiffi2Epffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðE xÞ. The terms involving Imð1

eþ1Þ are due to the contribution from surface excitations, whereas those involving Imð1

eÞ

are contributed from volume excitations. Eq. (9)

reveals that only surface excitations are possible for the electron traveling outside the solid. How-ever, both volume and surface excitations may oc-cur for the electron moving inside the solid. The term exp(jrjQ cos a) in Eq. (9) indicates that the contribution from surface excitations decreases exponentially with the increase in distance from the surface. On the other hand, the reduction in the contribution from volume excitations increases rapidly as the electron moves near the surface. When electron moves deep inside the solid, i.e.

(5)

r! 1, Eq.(9)reduces to the same expression as that for an electron moving in the inﬁnite solid. The inverse IMFP for an escaping electron can be calculated using

ls!v_ð_{a; E; r}_{Þ ¼} Z E

0

ls!v_ð_{a; E; x; r}_Þdx. _ð10Þ

For an electron moving outside the solid, the SEP may be obtained by integrating the inverse IMFP over the whole path length of the electron. Thus the SEP for escaping electrons is given by

Ps!v_s ða; EÞ ¼

Z 1

0

ls!vða; E; rÞdr. ð11Þ

Similar derivations can be performed for an elec-tron entering the solid by taking e1ð~q;xÞ ¼ 1 and e2ð~q;xÞ ¼ eð~q;xÞ. In this case, i.e. v ! s, the DIMFP for the incident electron is given by

lv!sða;E;x;rÞ ¼4cosa p3 Z qþ q dq Z p=2 0 dh Z 2p 0 d/qsin 2 hcos qð zrcosaÞ ~ x2þ Q2 v2 ?

exp jrjQcosað ÞIm 1 eð~Q;xÞ þ 1 " # HðrÞ þ 2 pv2 Z q_þ q dq1 qIm 1 eð~q;xÞ HðrÞ 2cosa p3 Z qþ q dq Z p=2 0 dh Z 2p 0 d/qsin 2 hexpðjrjQcosaÞ ~ x2 þ Q2_v2 ? Im 1 eð~Q;xÞ " # 2cos xr~ v exp jrjQcosað Þ HðrÞ þ4cosa p3 Z q_þ q dq Z p=2 0 dh Z 2p 0 d/qsin 2_hexp jrjQcosa ð Þ ~ x2 þ Q2 v2 ? Im 1 eð~Q;xÞþ1 " # 2cos xr~ v exp jrjQcosað Þ HðrÞ. ð12Þ

Corresponding inverse IMFP and SEP can be ob-tained from lv!sða; E; rÞ ¼ Z E 0 lv!sða; E; x; rÞdx ð13Þ and Pv!s_s ða; EÞ ¼ Z 0 1 lv!sða; E; rÞdr. ð14Þ

Letting a = 0° in Eqs. (9) and (12), formulas for normally escaping and incident electrons exhibit similarities between present and previous works

[2,3]. The diﬀerence between these works is attrib-uted to the application of momentum–energy con-servations. In the previous work, the integration over the normal component of the momentum transfer, qz, was not restricted by the conservation relations. In the present work, however, these rela-tions are completely satisﬁed by electron–solid inelastic interaction cross-sections using spherical coordinates.

3. Results and discussion

Using Eqs. (9)–(14) and the extended Drude

dielectric functions, we have calculated the

DIMFP, inverse IMFP and SEP for an electron entering or escaping from the solid. Fitting param-eters of the dielectric functions were taken from our previous work[10].Figs. 2 and 3show the re-sults of calculations on the DIMFP for a 500 eV electron escaping from Cu to vacuum with various electron distances from the surface crossing point, r, either outside (Fig. 2) or inside (Fig. 3) Cu for diﬀerent crossing angles, a. When the electron is in the vacuum,Fig. 2shows that the DIMFP is en-tirely contributed from surface excitations. As the electron moves away from the surface, corre-sponding to larger r values, the DIMFP becomes smaller due to the decrease in surface excitations. The DIMFP becomes larger for greater a values at a ﬁxed r value because of the shorter distance to the surface. The characteristic surface plasmon energy, or the peak energy loss, is nearly indepen-dent of the crossing angle and the electron distance from the crossing point. For the electron inside the

(6)

Fig. 2. Calculated results of the DIMFP in vacuum for a 500 eV escaping electron from Cu to vacuum with diﬀerent crossing angles and distances from the crossing point at the surface.

Fig. 3. Calculated results of the DIMFP in Cu for a 500 eV escaping electron from Cu to vacuum with diﬀerent crossing angles and distances from the crossing point at the surface.

(7)

solid, Fig. 3 shows that the DIMFP also varies with electron distance from the crossing point. Here the DIMFP exhibits broad peaks due to the overlapping of the contributions from surface and volume excitations. Inside the solid, the DIMFP is either weakly dependent or independent of the crossing angle. This DIMFP quickly reduces to the asymptotic value for an electron moving in the inﬁnite solid. This reveals that surface excita-tions are essentially possible only in a limited region near the surface. Similar results on the DIMFP for a 500 eV incident electron moving

from vacuum to Cu are plotted in Figs. 4 and 5

for the electron moving outside and inside Cu, respectively. Again, surface excitations are impor-tant when the electron moves near the surface. In the case of the electron moving in vacuum, the DIMFP is enhanced for glancing incident angles. For the electron moving in solid, this angular dependence becomes relatively weak.

Using Eq.(10), we have calculated the inverse IMFP for a 500 eV electron escaping from Cu to vacuum as a function of electron distance relative to the surface crossing point. Results are plotted

in Fig. 6for different crossing angles. As the elec-tron travels outside the solid, i.e. r > 0, the inverse IMFP falls off rapidly, especially for smaller cross-ing angles. This indicates that the glanccross-ing escap-ing electron is more likely to induce surface excitations than the normally escaping electron be-cause the former electron spends longer time near the surface. For the electron traveling inside the solid, i.e. r < 0, the inverse IMFP is roughly inde-pendent of electron distance and crossing angle due to the approximate compensation of volume and surface excitations. Therefore, the total in-verse IMFP of the electron inside the solid can be approximated by a constant value equal to the inverse IMFP for the infinite solid. A similar plot of the inverse IMFP for a 500 eV incident electron moving from vacuum to Cu is shown in

Fig. 7for several crossing angles. Here the inverse IMFP of the incident electron in vacuum, i.e. r < 0, is smaller than that of the escaping electron

in vacuum shown in Fig. 6. This is because the

attractive force acting on the incident electron in vacuum by the induced surface charges is parallel to electron moving direction and thus accelerates

Fig. 4. Calculated results of the DIMFP in vacuum for a 500 eV incident electron from vacuum to Cu with diﬀerent crossing angles and distances from the crossing point at the surface.

(8)

the electron. On the other hand, the attractive force on the escaping electron in vacuum is anti-parallel to electron moving direction and hence decelerates the electron. Therefore, the time spent near the surface for incident electron is less than

that for escaping electron, leading to less surface excitations for incident electron.

Fig. 8 shows the results (solid circles) of the

angular dependent SEP calculated using Eq. (11)

for a 500 eV electron escaping from Cu to vacuum.

Fig. 5. Calculated results of the DIMFP in Cu for a 500 eV incident electron from vacuum to Cu with diﬀerent crossing angles and distances from the crossing point at the surface.

Fig. 6. A plot of the inverse IMFP for a 500 eV escaping electron from Cu to vacuum with diﬀerent crossing angles as a function of electron distance from the crossing point at the surface.

Fig. 7. A plot of the inverse IMFP for a 500 eV incident electron from vacuum to Cu with diﬀerent crossing angles as a function of electron distance from the crossing point at the surface.

(9)

Similar results (solid circles) calculated using Eq.

(14)are plotted inFig. 9for a corresponding inci-dent electron from vacuum to Cu. It can be seen that the SEP rises slowly with increasing crossing angle until about a = 70°, above which such a rise becomes rapidly. Examining the calculated results, the SEP is found to follow an equation:

Ps!v_s ða; EÞ or Pv!s

s ða; EÞ ¼ aðEÞ

cosb_a. ð15Þ

The best-fitted values of the parameters are a = 0.1516 and b = 0.85 for 500 eV escaping elec-trons and a = 0.0720 and b = 1.09 for 500 eV inci-dent electrons. The fitting results are also plotted inFigs. 8 and 9as solid curves. A comparison be-tweenFigs. 8 and 9indicates that the SEP is larger for escaping electrons than for incident electrons, an effect mentioned above. Also, results of Oswald

[13](dotted curve), Werner et al.[14,17,18]

(dash-dot curve) and Chen [11] (dashed curve) are

included in these ﬁgures for comparisons. Some discrepancies among all theoretical results are found. In spite of the diﬀerences in magnitude for the SEP, the angular dependences are all simi-lar for a less than 60°. For a more than 60°, the angular dependences are also similar for the pres-ent results of escaping electrons and the results

of Werner et al. The ﬁtted formula by Chen [11]

is available only for escaping electrons. Our pres-ent ﬁttings are made for both incidpres-ent and escap-ing electrons. Here we ﬁnd that the angular dependence of the SEP does not follow the (cos a)1law but follows Eq. (15).

4. Conclusions

Based on the dielectric response theory, a theo-retical derivation was made to calculate the

posi-tion-dependent inelastic cross-sections for

electrons moving across a solid surface at different crossing angles. Formulas were derived for both incident and escaping electrons using spherical coordinates that completely satisfied the conserva-tion constraints of energy and momentum. Calcu-lations of these cross-sections were performed using the extended Drude dielectric functions de-rived from optical data. It was found that the DIMFP and inverse IMFP for electrons moving inside the solid were approximately independent of the crossing angle and position of electrons ow-ing to the approximate compensation of volume and surface excitations. For electrons moving deep inside the solid, the calculated DIMFP and inverse IMFP reduced to the values for electrons moving in an infinite solid. As electrons traveling in the vacuum, the DIMFP and inverse IMFP became larger for glancing incident and escaping angles

Fig. 8. A plot of the SEP for a 500 eV escaping electron from Cu to vacuum as a function of crossing angle. Symbols are the calculated results using Eq.(11). Solid curve is the ﬁtting results using Eq. (15). Corresponding data of Chen (dashed curve), Oswald (dotted curve) and Werner et al. (dash-dot curve) are plotted for comparisons.

Fig. 9. A plot of the SEP for a 500 eV incident electron from vacuum to Cu as a function of crossing angle. Symbols are the calculated results using Eq.(14). Solid curve is the ﬁtting results using Eq.(15). Corresponding data of Oswald (dotted curve) and Werner et al. (dash-dot curve) are plotted for comparisons.

(10)

since surface excitations were more probable than for normally incident and escaping angles. The SEP of escaping electrons was found larger than that of incident electrons due to the attractive force exerted by the induced surface charges. The presently calculated SEP was found to follow a simple expression valid for practical applications. The consideration of SEP in the intensity reduc-tion for reﬂected and emitted electrons is impor-tant in experimental and theoretical surface and interface analyses.

Acknowledgement

This research was supported by the National Science Council of the Republic of China under contract no. NSC92-2215-E-009-063.

References

[1] S. Tougaard, I. Chorkendorﬀ, Phys. Rev. B 35 (1987) 6570. [2] Y.F. Chen, C.M. Kwei, Surf. Sci. 364 (1996) 131.

[3] C.M. Kwei, C.Y. Wang, C.J. Tung, Surf. Interface Anal. 26 (1998) 682.

[4] Y.F. Chen, Phys. Rev. B 58 (1998) 8087.

[5] C.J. Tung, Y.F. Chen, C.M. Kwei, T.L. Chou, Phys. Rev. B 49 (1994) 16684.

[6] F. Yubero, S. Tougaard, Phys. Rev. B 46 (1992) 2486. [7] F. Yubero, J.M. Sanz, B. Ramskov, S. Tougaard, Phys.

Rev. B 53 (1996) 9719.

[8] F. Yubero, D. Fujita, B. Ramskov, S. Tougaard, Phys. Rev. B 53 (1996) 9728.

[9] A.C. Simonsen, F. Yubero, S. Tougaard, Phys. Rev. B 56 (1997) 1612.

[10] C.M. Kwei, Y.F. Chen, C.J. Tung, J.P. Wang, Surf. Sci. 293 (1993) 202.

[11] Y.F. Chen, Surf. Sci. 519 (2002) 115.

[12] C.M. Kwei, S.J. Hwang, Y.C. Li, C.J. Tung, J. Appl. Phys. 93 (2003) 9130.

[13] R. Oswald, Ph.D. Thesis, University of Tu¨bingen, Tu¨bin-gen, 1997.

[14] W.S.M. Werner, W. Smekal, C. Tomastik, H. Sto¨ri, Surf. Sci. 486 (2001) L461.

[15] C.M. Kwei, S.Y. Chiou, Y.C. Li, J. Appl. Phys. 85 (1999) 8247.

[16] F. Flores, F. Garcia-Moliner, J. Phys. C 12 (1979) 907. [17] W.S.M. Werner, W. Smekal, H. Sto¨ri, Surf. Interface Anal.

31 (2001) 475.

[18] W.S.M. Werner, C. Eisenmenger-Sittner, J. Zemek, P. Jiricek, Phys. Rev. B 67 (2003) 155412.