Accuracy of the Non-destructive Surface Nanostructure QuantiÐcation Technique Based on Analysis of the XPS or AES Peak Shape

Transcription

1 SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) Accuracy of the Non-destructve Surface Nanostructure QuantÐcaton Technque Based on Analyss of the XPS or AES Peak Shape S. Tougaard Physcs Department, Odense Unversty, DK-5230 Odense, Denmark The accuracy of XPS and AES quantðcaton by peak shape analyss was establshed from a detaled analyss of a range of model spectra and three sets of experments. It was found that nformaton on the concentraton depth proðle n the surface regon up to depths of 5k (where k s the nelastc electron mean free path) s prmarly contaned n the spectral energy regon up to 100 ev below the peak energy and s essentally completely contaned by the energy regon up to 200 ev below the peak. Analyss of a larger energy range than 100 ev does not add much to the nformaton on the detals of the structure n the outermost 5k but gves the possblty to determne addtonal structural parameters that descrbe the composton at larger depths. The structural parameters that descrbe the chemcal composton of the outermost 5 10k of the sold were dvded nto prmary and secondary parameters: the prmary parameters are the three most mportant parameters needed to descrbe the man characterstcs of the dstrbuton of atoms; the secondary parameters are parameters other than the three prmary parameters that descrbe the Ðner detals of the depth dstrbuton of atoms n the outermost 5 10k of the surface regon. The uncertanty n the determned three prmary parameters s typcally 5 10%. The uncertanty n the determned secondary parameters s typcally Z35%. D erent models of depth proðles can be dstngushed when they d er sgnðcantly over a wdth of more than 1/3k at any depth [5k. The uncertanty n the total determned amounts of atoms wthn the surface regon s 5 10% as long as the depths are wthn the prmary probng depth of the method (.e. Æ5k ). The absolute quantðcaton of a set of samples where the n-depth dstrbu- ton vares consderably gves a root-mean-square scatter of 15%. Ths s reduced to 10% when elastc scatterng e ects are modelled by a smple analytcal expresson. ( 1998 John Wley & Sons, Ltd. Surf. Interface Anal. 26, 249È269 (1998) KEYWORDS: quanttatve surface analyss; XPS; AES; nano-structures; peak shape analyss INTRODUCTION In the past 30 years there has been a contnuous e ort to mprove the accuracy of x-ray photoelectron spectroscopy and Auger electron spectroscopy (XPS and AES).1h14 The motvaton for ths s the hgh and contnuously growng technologcal mportance n ths tme perod of the propertes of sold surfaces on the nanometre depth scale. Algorthms for quantðcaton rely on several factors, such as relable values of the nelastc electron mean free path (j ), good descrptons of elastc scatterng e ects and accurate procedures to determne the analyser transmsson as a functon of energy. To secure a cost-e ectve output from ths e ort, t s mportant to stress the obvous fact that when several factors contrbute roughly equally to the error, even a consderable mprovement n the uncertanty from a sngle factor has essentally no nñuence on the total error. When we want, n the future, to mprove procedures for quanttatve surface analyss by XPS and AES, t s then necessary Ðrst to establsh those leadng factors that contrbute most to the error. It s these factors that should be the focus of research, because f ther accuracy s not mproved, any mprovement n the * Correspondence to: S. Tougaard, Physcs Department, Odense Unversty, DK-5230, Odense, Denmark. less mportant factors wll have essentally no nñuence on the accuracy of the quantðcaton procedure. In a prevous paper14 we compared the errors assocated wth d erent factors. It was shown that the factor that by far contrbutes most to the naccuracy n quantðcaton s the lack of knowledge of the n-depth composton of the sold. For a meanngful quantðcaton, assumptons on the n-depth dstrbuton of atoms must be made because the measured peak ntensty depends strongly on ths. In practcal analyss, however, the ndepth atomc dstrbuton s never known, because f t were t would be a waste of tme and e ort to do the analyss. Usually the sold composton s, for convenence, qute arbtrarly assumed to be homogeneous up to a depth of several nanometres and then the surface concentraton wll be proportonal to the measured peak ntensty. Ths assumpton may result n enormous errors n quantðcaton.9,14 Thus, the n-depth composton of solds that are analysed by XPS and AES s hardly ever known to be homogeneous up to a depth of several nanometres. It s precsely because samples are nhomogeneous on the nanometre depth scale that analyss s done wth XPS or AES, rather than wth other well-establshed15 but less surface-senstve technques. To llustrate the fundamental problem wth the assumpton of homogeneous composton wth depth, we consder an example of model spectra calculated for CCC 0142È2421/98/040249È21 $17.50 Receved 19 September 1997 ( 1998 John Wley & Sons, Ltd. Accepted 3 December 1997

2 250 S. TOUGAARD d erent depth dstrbutons. Thus, Fg. 1 shows spectra of the Cu 2p peaks correspondng to four d erent surface morphologes.14 The XPS peak ntensty from all four solds s exactly dentcal, although the surface compostons are wdely d erent. Analyss of these spectra under the assumpton that the surface concentraton s proportonal to the peak ntensty would result n a quantðcaton where the naccuracy s such that the true concentraton at the surface could be anywhere from 0% [as n (d)] to 100% [as n (a)] and the true total amount of copper materal wthn the surface regon could be anywhere from the equvalent of 1.1 Ó [as n (a)] or 10 Ó [as n (c)] or even hgher [as n (d)]. The uncertanty n quantðcaton s consequently several hundred per cent. QuantÐcaton based on peak ntenstes alone s thus subject to ncredbly large uncertantes. From the above, the clear concluson s that our focus of research should be to mprove spectral analyss of XPS and AES such that t provdes nformaton on the n-depth dstrbuton of atoms on the nanometre depth scale. Wthout ths, we can expect essentally no mprovements n the accuracy of XPS and AES even f we Ðnd substantally better descrptons of other parameters n the quantðcaton procedure. From Fg. 1, t s clear that the peak shape n a wder energy range below the peak depends crtcally on the n-depth dstrbuton of the element. It would thus be very easy to dstngush expermentally between the peak shape of the four spectra n a 100 ev energy regon. Much more accurate quantðcaton can therefore be acheved f the dependence of peak shape on surface morphology s taken nto account n the analyss. Ths s the dea behnd the formalsm devel- Fgure 1. Four wdely dfferent surface structures of copper n gold that gve dentcal peak ntenstes. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) oped by Tougaard et al.,7h9,14,16h18 whch provdes quanttatve nformaton on the surface nanostructure of the sold by analyss of XPS or AES peak shapes. The technque reles on the phenomenon seen n Fg. 1: that the energy loss structure that accompanes an XPS or AES peak carres nformaton on the depth of orgn of the detected electrons. The method s non-destructve and therefore allows the change n surface morphology f a gven surface atomc structure durng surface treatment (e.g. chemcal reactons and gradual annealng) to be studed. It has been appled n the study of a wde range of systems and physcal phenomena,19h41 ncludng growth mechansms and nanostructures of metal/ metal,20,28 slcon/metal,32h34 slcon/germanum41 and polymer/metal systems,29,31 carbon segregaton on N,35 metal oxde growth19 and the depth exctaton functon n electron-stmulated AES.38h40 Several tests on the valdty of the method have also been done. It s the purpose of the present paper to gve a crtcal evaluaton of the potental of ths technque. We wll determne the accuracy of quantðcaton and also the accuracy wth whch the technque can determne detals of the n-depth dstrbuton of atoms on the nanometre scale. THEORY To obtan a practcal and useful model, we treat electron exctaton and electron transport as separate events. It s well known that ths assumpton s not strctly vald,42h46 but at typcal XPS and AES energes t s vald to a good approxmaton, as we wll now argue. It was shown many years ago that at low energes, where the outgong photoelectron or Auger electron has an energy of a few tmes the plasmon energy, the plasmon ntensty s dmnshed consderably due to a quantum nterference term between the extrnsc and the ntrnsc plasmon exctatons.42h45 The e ect may be sgnðcant when the energy of the excted electron s only a few tmes the typcal energy loss n a sngle scatterng event. The most probable energy loss for electrons movng n solds s, n general, D10È30 ev.47,48 In ultravolet photoelectron spectroscopy (UPS) where the energy of the outgong electron s typcally \50 ev, the nterference terms are therefore mportant and t s then a questonable approxmaton to evaluate such energy spectra wthn a model where the exctaton and transport processes are treated as separate e ects. The quantum nterference term becomes less mportant at hgh energes because the ntrnsc and extrnsc plasmon exctatons become decoupled by manly occurrng n spatally separated regons. For the spectroscopes of nterest here, where the typcal electron energy exceeds 200 ev, the nterference term s therefore of mnor mportance and t s a reasonable approxmaton to treat the processes as separate e ects and dvde the e ects nto an exctaton process, whch ncludes any ntrnsc exctatons (see below), and a transport process where changes n peak shape result from electron scatterng durng transport to the surface. The peak shape of a photon-excted core level s nherently Lorentzan, the wdth of whch s the nverse of the core-hole lfetme. Photon broadenng wth essentally a Gaussan character wll, partcularly for polar ( 1998 John Wley & Sons, Ltd.

3 NON-DESTRUCTIVE SURFACE NANOSTRUCTURE QUANTIFICATION TECHNIQUE 251 materals, add consderably to the lne wdth determned by lfetme broadenng n both XPS and AES.42 Measured peak shapes wll generally be mxed, wth addtonal Gaussan broadenng orgnatng from the response functon of the spectrometer. The photoelectron peak shape s also greatly nñuenced by what s known as the many-body phenomenon,42h45,49h52 whch results n so-called ntrnsc exctatons: after the core electron has been photon excted, the potental from the resultng core hole leads to electronèhole par exctatons near the Ferm level50 and to plasmon exctatons of the valence sea of electrons.51,52 For metals, where the densty of electron states (DOS) s hgh n the regon of the Ferm level, the former process leads to a skewed peak shape toward lower bndng energy and the e ect n the mmedate vcnty of the peak s larger for solds wth a large DOS at or near the Ferm energy. For energes of only a few electron-volts from the peak energy, where the valence band DOS may be consdered roughly constant, the combned e ect of these electronè hole par exctatons and the lfetme broadenng s descrbed by a formula due to Donach and Sunjc.49 Ths formula s, however, not a vald approxmaton for the full extenson of the ntrnsc peak. In fact, the DonachÈSunjc formula Ðts the expermentally observed XPS peak shapes n an energy range of only a couple of electron-volts from the peak energy. More than a few electron-volts from the peak energy, the structure n the DOS and the contrbuton from ntrnsc plasmon exctatons51,52 wll cause the exctaton probablty of the varous electronèhole par processes to vary wth energy and the ntrnsc peak shape devates consderably from the smple DonachÈSunjc form.52 In the energy regon around 10È30 ev below the peak energy, correspondng to the regon of plasmon exctatons, ntrnsc exctatons have been found to account for about one-thrd of the measured ntensty n XPS peaks from both smple metals51 and transton metals.52 These Ðndngs have recently been conðrmed drectly by Auger photoelectron concdence spectroscopy.53 The ntrnsc exctatons then cause XPS peaks to extend typcally D50 ev below the peak energy.52 Durng transport out of the sold, the electrons experence nelastc scatterng events. Ths leads to a dstorton of the energy dstrbuton compared wth the orgnal dstrbuton at the pont of exctaton n the sold. It s thus the nelastc scatterng processes that are responsble for the transport-nduced changes n peak shape. However, elastc scatterng6 as well as d racton and forward focusng e ects54,55 also occur and these wll ndrectly nñuence the peak shape. The energetc electrons are emtted from the atom wth an angular dstrbuton determned by the exctaton process. Elastc scatterng leads to a change n the angular dstrbuton of electrons and wll also cause the electron trajectores to devate from straght lnes. The measured energy spectrum wll then have contrbutons from electrons that have followed d erent paths wth a relatve weght determned by the elastc scatterng crosssecton. It turns out that for energes of nterest n XPS and AES (.e. Z100 ev), small-angle scatterng s far most probable56 and ths tends to favour trajectores where most of the emtted electrons have followed nearly straght lnes. In the next secton we breñy dscuss models to descrbe the measured peak shapes and then we go on to dscuss models that descrbe the connecton between atomc n-depth composton proðle and peak shape. Photoelectron peak shapes After the photoexctaton process, some of the electrons are transported to the surface and enter the spectrometer. QuantÐcaton reles on an accurate descrpton of how ths transport nñuences the energy spectrum. For nhomogeneous samples, the e ect s substantal (see Fg. 1) and quantðcaton requres a detaled and accurate descrpton of nelastc electron scatterng. The total energy loss of an electron movng n a sold s determned by the nelastc scatterng cross-secton. Multple scatterng events are mportant, because n typcal cases the energy spectrum ncludes electrons that have travelled a dstance of several nelastc electron mean free paths. Let F(E, ), x)d2) de dx be the Ñux of electrons excted at depth 0 0 x, dxn 0 an 0 energy nterval E,dE nto the sold angle ), d) and let Q(E, 0 ), x; 0 R, ))dr d2) be the probablty 0 0 that an electron 0 0 excted wth energy E at depth x n drecton ) wll arrve at 0 0 the surface n the drecton ), d2)after havng travelled the path length R, dr. Then the number of electrons emtted per second, per unt energy and sold angle s7 J(E, )) \ P de 0 P d2)0 P dxf(e0,) 0,x) ] P Q(E 0,) 0,x;R,))G(E 0,R;E)dR (1) where G(E, R; E)dE s the probablty that an electron 0 wth ntal energy E has energy n the nterval E, 0 E ] de after havng travelled the path length R. It s possble n prncple that the energy dstrbuton at the pont of exctaton may vary wth depth. Ths may arse as a result of peak shape dependence on the local chemcal composton, whch n a typcal sample wll vary wth depth. Ths e ect may also occur even n a homogeneous sold, because the d erence n the chemcal envronment of an atom present n the surface layer to that of an atom stuated a few layers underneath the surface may lead to d erences n electron energy levels and n the local densty of electron states (DOS). Ths n turn wll a ect the response of the surroundng electrons to the exctaton process and thus a ect also the shake-up processes and consequently the energy dstrbuton of emtted electrons. Because these depth-dependent e ects are usually small and a complete quanttatve descrpton s not practcally possble because of lack of detaled models of general valdty, t s usually a vald and reasonable approxmaton to assume that the concentraton of electron emtters f (x) may vary wth depth x but that the energy dstrbuton s ndependent of depth,.e. F(E, ), x) \ f (x)f(e, ) ) (2) where F(E, ) )d2) de s the number of electrons per second, per 0 atom 0 and 0 0 per unt energy excted n an energy nterval E,dE nto the sold angle ),d2) and f (x) s the number 0 0 of atoms per unt volume 0 at 0 depth x. ( 1998 John Wley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998)

4 252 S. TOUGAARD Elastc electron scatterng, whch we dscuss n the followng secton, enters as a path-length-ncreasng e ect and s descrbed by the functon Q n Eqn (1). Elastc electron scatterng. The nelastc processes are clearly the domnatng factor n nterpretaton of measured peak ntenstes and peak shapes. QuantÐcaton therefore reles on accurate values for the nelastc scatterng cross-secton and the nelastc electron mean free path j. In ths secton we wll dscuss some e ects of elastc electron scatterng n quanttatve nterpretaton of XPS. Elastc scatterng e ects have often been neglected n practcal applcatons of the peak shape analyss method. We wll therefore dscuss the valdty of ths approxmaton Ðrst. In a recent systematc study of Monte Carlo-smulated peak ntenstes57 t was found that for typcal geometres appled n practcal XPS analyss the nñuence of elastc scatterng s small and amounts to [10È15% for electrons from depths [1.5j, whle for electrons emtted at larger depths, the elastc electron scatterng gves a substantal reducton n the measured peak ntensty. The e ect of elastc electron scatterng on quanttatve XPS of homogeneous solds, where most of the peak ntensty comes from depths [j, wll consequent- ly also be small. Ths was studed for a partcular XPS geometry n Ref. 58, where ntensty ratos of expermental XPS peaks from several one-element solds were compared to two Ðrst-prncple theores correspondng to neglectng and ncludng the e ects of elastc electron scatterng. Elastc electron scatterng was smulated by a Monte Carlo calculaton. The theoretcal peak ntenstes were found to change by an average of 14% as a result of elastc scatterng. Surprsngly, however, the standard devaton between theory and experment was practcally unchanged, namely D15% n both cases (.e. whether neglectng or ncludng elastc scatterng e ects). Ths ponts to the concluson that for practcal analyss of homogeneous solds t makes only a very small mprovement n the accuracy of quantðcaton to nclude elastc scatterng e ects. Ths result can be understood from the followng consderaton. The error on quantðcaton for the peaks from the pure elemental solds depends manly on the accuracy of sx factors: the rato of nelastc mean free paths, the rato of photoonzaton cross-sectons, the procedure for peak ntensty determnaton (.e. the method used for nelastc background correcton), the nñuence of elastc electron scatterng, the stablty of the nstrument and the energy dependence of the electron spectrometer transmsson functon. (Other factors wll also contrbute to the error, such as the role of surface roughness and surface plasmon exctatons, but for smplcty we assume that the error comes from the above-mentoned sx factors.) Let us assume that all factors contrbute wth the same amount to the error and let us assume that ths s 6% for each factor (ths number s chosen just to llustrate a pont and s qute arbtrary but probably not too far from realty). The total relatve error due to these sx factors s 14.7% whch s close to what was observed n the comparson of XPS peak ntensty ratos to theory58 (see above). Let us then assume that, somehow, we are able to completely elmnate the uncertanty from one of these sx SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) factors. Then we have Ðve factors each contrbutng 6% to the error and ths results n a relatve error of 13.4%. Ths s only slghtly smaller than 14.7% and llustrates that when several factors contrbute roughly equally to the error, even a consderable mprovement n the uncertanty from a sngle factor has only lttle nñuence on the total error. D erent analytcal formulae to descrbe the e ects of elastc electron scatterng have been proposed n the past.59h63 As the example dscussed above shows, the e ects of elastc electron scatterng are often small. It therefore seems natural to treat them by a correcton factor to a calculaton where elastc scatterng e ects have been neglected. Ths was done recently by Jablonsk and Tougaard,63 and they found a smple yet rather accurate formula vald for typcal geometres appled n practcal XPS and AES. They formulated the e ect of elastc scatterng on peak ntensty as a smple correcton factor (CF) to the result of a descrpton that neglects elastc scatterng. The CF s the rato of emtted peak ntensty from a layer of atoms located at a gven depth z n a sold calculated from theores that take nto account (Iel) and neglect (Inel) elastc electron scatterng. They made extensve Monte Carlo calculatons of CF under varaton of the full relevant parameter range of electron energy, matrx atomc number, depth x of orgn of emtted electrons and angular emsson ansotropy. By expressng these results n unts of the dmensonless varable x/j they found the followng general et analytcal expresson63 CF(x/j )\Iel/Inel\exp[[0.1578(x/j )[1.251] et et ]exp[[ (x/j )2 et ] (x/j )[0.2020] (3) et where j \ (j j )/(j ] j ) and j s the transport et tr tr tr mean free path for elastc electron scatterng. Three assumptons were made n the calculatons, namely that the geometry s close to normal emsson, that the angle between x-ray source and analyser axs s close to the magc angle (54 ) and that the rato j /j s approx- tr mately constant over the analysed depth. However, the result devates only slghtly from Eqn (3) when these assumptons are not strctly fulðlled.64 Equaton (3) provdes a very smple way to correct peak ntenstes for elastc scatterng e ects. Let the n-depth concentraton proðle be gven by the functon f (x) descrbng the number of atoms per unt volume at depth x. When elastc scatterng s neglected, the measured peak ntensty from ths sold s P = Inel \ I f (x)exp A x [ 0 j cos 0 hb dx (4) where h s the emsson angle wth respect to the surface normal and I j cos h s the ntensty from a sold wth f (x) 4 1. The 0 measured ntensty when elastc scatterng s accounted for s P A = Iel \ I CF(x/jet ) f (x)exp [ x 0 j cos 0 hb dx (4a) where we have assumed that j s approxmately con- stant n the surface regon. When et j vares consderably over the analysed depth, Eqn (4a) et s no longer strctly vald. ( 1998 John Wley & Sons, Ltd.

5 NON-DESTRUCTIVE SURFACE NANOSTRUCTURE QUANTIFICATION TECHNIQUE 253 It was shown n Ref. 64 that the general expresson for the functon CF [Eqn (3)] s to a good approxmaton vald for depths up to 50 Ó when the angle between axs of the analyser and the x-rays s n the range 45È65, the emsson angle s \30 and the transport mean free path j s n the range 1.3j Oj O 7j. The latter condton s tr satsðed for essentally tr all XPS peaks. For depths O2j, the root-mean-square (RMS) devaton between CF [Eqn et (3)] and the ratos calculated by Monte Carlo smulatons amounts to D5%.64 Except where explctly mentoned, n the calculatons presented n ths paper we wll neglect elastc electron scatterng. Because of elastc electron scatterng, the depth of orgn s d erent from the path length travelled by the electron. The mportance of ths e ect can be estmated from the results of Ref. 63. From Ðgure 4 of that paper, the path-length-ncreasng e ect of elastc scatterng s small for depths smaller than D1.5j. For larger depths the e ect ncreases, and for depths of D3È5j the ntensty s seen (Fg. 4 of Ref. 63) to be decreased by an addtonal factor of D3 due to elastc scatterng e ects. A decrease n ntensty of a factor of 3 corresponds to a path length ncrease of D1j. An electron excted at depth D4j has consequently moved an average dstance of D5j before beng emtted. As a rough estmate, the actual dstance that emtted electrons have travelled n the sold s thus, on average, D15È30% larger than the straght-lne dstance from the pont of exctaton to the surface. Forward focusng and d racton e ects. Because elastc scatterng of electrons on atoms s much more probable n the forward drecton,56 structures n measured spectra as a functon of emsson angle are frequently observed wth maxma occurrng n drectons correspondng to emtted electrons beng scattered on neghbourng atoms.54,55 Varatons as hgh as 30È50% n measured peak ntenstes as a functon of take-o angle have been observed.54,55 The enhanced ntensty s caused by focusng n the forward drecton of the emtted electrons by the attractve Coulomb potental on neghbourng atoms. In a smple qualtatve pcture, ths leads to enhanced ntensty n drectons that drectly correspond to the near neghbours of the electron-emttng atom. In XPS of Al(001)65 and of NO, MnO and CoO,66 ths forward-focusng e ect was nvestgated n the peak energy regon as well as n the energy loss regon below the peak energy. By observng the ntensty varaton as a functon of both the emsson angle and the energy dstance to the peak, t was found that the further the energy dstance to the man peak, the less structure s observed n the measured ntensty. Ths was nterpreted as beng due to a substantal reducton n the forward focusng for the electrons that orgnate from deeper layers. Thus, electrons below a peak energy have travelled a typcal dstance roughly n proporton to ther energy loss. The mean energy lost per nelastc mean free path travelled s D15È30 ev.47,48 Features due to forward focusng observed expermentally are then concluded to orgnate predomnantly from electron emtters wthn the outermost two to four layers of atoms.55 Several examples of the forward-focusng e ect have been reported.54,55,65h68 To account quanttatvely for the e ect as well as the Ðner detals n the ntensty varatons wth take-o angle, detaled models are beng used wth consderable success.54,55,67 Analyss of the preferred angular drectons of Auger or photoelectrons therefore gves very drect nformaton on the geometrc arrangement of the outermost atoms and has led to the development of a powerful technque for nvestgatons of the atomc surface structure.54,55 Although an advantage for nvestgatons of the geometrcal structure of surface atoms, forward-focusng e ects have severe negatve mplcatons for quanttatve surface composton analyss by AES and XPS of snglecrystallne substrates. It can lead to errors as hgh as 50% n determned stochometres.54,55,65,66 Averagng over several drectons wll reduce the e ect.54,66,69 Ths s, however, often mpractcal because the total data collecton tme s severely ncreased or t may even be mpossble because many nstruments do not allow for varatons of both azmuthal and polar angles of the electron energy analyser. The e ects are largely avoded n polycrystallne and amorphous materals as long as the polycrystallne materal s free of preferental crystal orentaton.69 In any case, on bombardment used for sample cleanng, for example, wll to some extent destroy the crystal structure n the outermost two to four atomc layers and wll thus tend to reduce forwardfocusng and d ractons e ects. Inelastc electron scatterng. The detaled energy dstrbuton G of electrons as a functon of the path length travelled n the sold can be calculated provded that the energy loss probablty per unt path length travelled s known. If elastc scatterng e ects are neglected, the energy spectrum of emtted electrons s J(E, )) \ P de 0 F(E 0, )) P f (x)g(e 0, x/cos h; E)dx (5) where h s the emsson angle wth respect to the surface normal. The functon G, whch essentally gves the energy dstrbuton of an electron as a functon of path length x/cos h travelled n the sold, s thus of central mportance n any quanttatve analyss of energy spectra of emtted electrons. It s determned by the nelastc scatterng cross-secton. We denote by K(E, T ) the d erental nelastc scatterng cross-secton,.e. K(E, T )dr dt s the probablty that an electron of energy E wll lose energy n the nterval T, T ] dt after havng travelled a path length dr n the sold. In general K(E, T ) depends strongly on T, moderately on the specðc sold and only weakly on E.47,48 For energy spectra where the total energy loss s small compared wth the prmary electron energy, K(E, T ) + K(T ),.e. ndependent of E. Then the e ect of multple scatterng has a rgorous soluton and the spectrum of emtted electrons s wth J(E, )) \ P de 0 F(E 0, )) P ds e~2ns(e~e0) ] P dxf(x)e~x&(s)@cos h (6) &(s) \ 1 j [ P 0= K(T )e~st dt (7) ( 1998 John Wley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998)

6 254 S. TOUGAARD Inelastc electron scatterng s clearly the domnatng Ðrst-order e ect n quanttatve understandng of peak ntenstes and peak shapes, and ths has been the subject of several papers.19h41 Inelastc electron scatterng cross-secton. For practcal applcaton of the above formalsm, t s essental to have methods for fast calculatons of the energy loss of electrons at any energy as they move n solds of general and non-unform composton. Ths s possble wth the Unversal cross-secton that was ntroduced a decade ago.47 Recently, a crtcal revew of the Unversal crosssecton was gven48 n the lght of the results of research carred out snce t was Ðrst ntroduced. In partcular, ts valdty was compared to the valdty of expermental cross-sectons determned by analyss of reñected electron energy-loss spectra (REELS). It was shown that for applcatons n quanttatve surface analyss by XPS and AES, the solds can be dvded nto classes accordng to the full wdth at half-maxmum of the domnatng shape of the nelastc scatterng cross-secton. Dependng on the class of materals, a functon wth ether two or three parameters s needed to descrbe the most mportant general characterstcs common to the crosssectons of that class.48 For most metals, ther oxdes and alloys, the Unversal cross-secton47,48 j (E)K(E, T ) \ BT (8) (C ] T 2)2 wth C \ 1643 ev2 and B B 3000 ev2 apples wth reasonable accuracy. The cross-secton s normalzed to unt area for B \ 2C \ 3286 ev2. Ths cross-secton was successfully appled n studes of many systems (ncludng pure metals, alloys, metal oxdes and thn Ðlms of metals and metal oxdes). For solds wth a narrow plasmon structure, the cross-sectons cannot be well descrbed by a functon wth two parameters. For these, however, t was shown48 that the man characterstcs of the crosssecton can be descrbed by the T hree-parameter Unversal cross-secton BT j(e)k(e, T ) \ (9) (C [ T 2)2]DT 2 where the three parameters B, C and D have been determned for each class of materals (e.g. polymers, semconductors, free-electron-lke solds48). It was shown48 that the Unversal cross-secton (Eqn (8)) s qute accurate for solds when ther cross-secton has a wdth at half-maxmum Z20 ev. For solds wth a cross-secton wdth of 10È15 ev, the Unversal crosssecton s stll farly good for the descrpton of the far peak regon (Z30 ev from the peak energy) but t s less accurate to account for the near peak regon ([10 ev from the peak energy). For the solds wth a crosssecton wdth [5 ev, the Three-parameter Unversal cross-secton [Eqn (9)] s always more accurate than the Unversal cross-secton. Quanttatve XPS by peak shape analyss There are two d erent approaches to the applcaton of the new formalsm: ether algorthms are used to SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) remove the nelastc background from the measured spectrum; or the peak shape of the spectrum of emtted electrons s calculated. In both cases spectral evaluaton s done by formulae that depend on the n-depth concentraton proðle f (x). The deas for mproved quantðcaton breñy descrbed below have been made avalable n the form of a software package QUASESTM (Quanttatve Analyss of Surfaces by Electron Spectroscopy),18 whch was used n all spectral analyss n the present work. QuantÐcaton by peak shape calculaton. In ths approach, the spectrum J(E, )) s calculated by Eqn (6). The functon F(E, )) may convenently be determned by the procedure descrbed n the next secton from a measured spectrum of a pure elemental sample. The ndepth concentraton proðle f (x) s then vared untl a good agreement wth the measured spectrum s obtaned. In ths way, the detaled n-depth concentraton proðle f (x) s determned. QuantÐcaton by background removal. Formulae to determne the atomc exctaton functon F(E, )) from a measured spectrum were developed for d erent types of n-depth proðles.7h9,14,16h18 It was shown that the ntegral equaton [Eqn (6)] may be solved rgorously for the prmary exctaton spectrum F(E, )). where and F(E, )) \ 1 P 1 G J(E, )) [ P de@ J(E, )) P ds A ] exp[2ns(e@ [ E)] 1 [ P 1 (10) P(s)BH P(s) \ P dxf(x)exp C [ x &(s)d (11) cos h P A P \ dxf(x)exp [ x (12) 1 j cos hb Equaton (10) may be used to determne ether F(E, )) f f(x) s known (e.g. for a one-element sample) or to determne f (x) f F(E, )) s known. The exact peak shape n the energy regon close to the peak energy up to D20 ev below the peak energy s not known because t s largely determned by the chemcal bond, lfetme broadenng and ntrnsc exctatons n the photoemsson process, all of whch may depend on the local chemcal envronment. However, the spectrum after background correcton must be of zero ntensty n an energy regon beyond D30 ev below the prmary peak energy. Furthermore, t was ponted out that the spectral ntensty must stay at zero ntensty for all energes below the peak energy untl the energy of another peak n the energy spectrum s reached. Ths puts a strong constrant on the functon F(E, )) and ths may be appled as a crteron to determne f (x) n the sense that f (x) s vared untl the constrant s fulðlled. As another crteron, one can use knowledge on F(E, )) determned from the analyss of spectra from samples wth a wellcharacterzed n-depth concentraton proðle, such as a sngle-element sold. One should be aware of the possble peak shape changes caused by the d erence n ( 1998 John Wley & Sons, Ltd.

7 NON-DESTRUCTIVE SURFACE NANOSTRUCTURE QUANTIFICATION TECHNIQUE 255 chemcal envronment of the atoms n the reference sample and the sample beng nvestgated. To the extent that these d erences can be neglected, the spectrum may be appled as a reference and f (x) s vared untl analyss yelds a spectrum wth the same absolute ntensty and peak shape as the reference spectrum. If the peak shape analyss ncludes peaks from all the elements n a sample, then the constrant that the sum of the concentraton of the ndvdual elements at any depth must add up to 100% may also be appled. OBJECTIVES OF QUANTIFICATION The man purpose of ths paper s to study the accuracy of quantðcaton wth XPS and AES based on the present knowledge about the physcs behnd the technques. To do ths, we must Ðrst deðne the objectves of XPS and AES quantðcaton. Ths s necessary because the accuracy depends on what one attempts to quantfy. For example, XPS may gve detaled and rather accurate nformaton on the composton of the outermost few atomc layers but the quantðcaton of depths more than say 100 atomc layers s naturally subject to a much larger uncertanty. Lkewse, parameters that descrbe the approxmate depth dstrbuton of atoms n the surface regon can be quantðed wth much better precson than parameters that descrbe the Ðner detal of n-depth dstrbuton. The queston of accuracy of quantðcaton s therefore not well deðned unless we dvde t nto several more specðc questons. Ths dvson must be based on a study of the physcal processes because t s these that wll determne the accuracy. Ths s done here. There s a strong correlaton between the ntensty of a gven spectral energy regon and the depth where these electrons were emtted. As an example, the peak ntensty depends strongly on the atomc concentraton at depths [1j but s rather ndependent of the concen- traton at depths exceedng 3j below the surface. On the other hand, the spectral ntensty 100 ev below the peak energy depends strongly on electrons excted at depths of 3È4j but depends less on the concentraton for depths [1j. D erent energy regons of the spec- trum thus carry nformaton on the atomc concentraton at d erent depths. We must therefore dvde the surface regon nto d erent regons of depth and, lkewse, we must also dvde the spectrum nto d erent energy regons below the peak energy. Ths dvson wll be made n the next secton. Havng determned the connecton between depths and spectral energy regons we wll then go on to determne how much nformaton about the detaled concentratonèdepth proðle can be determned wth reasonable accuracy. As we shall see, the certanty wth whch a gven structural parameter s determned depends on the number of parameters that are used to descrbe the detals of the depth proðle. Thus, f we consder only a few parameters that descrbe the rough structure of the sample, the uncertanty on each parameter s smaller than the uncertanty f we use more parameters n an attempt to determne more detals of the chemcal structure of the surface regon. We wll dvde the structural parameters nto prmary and secondary parameters: prmary parameters deðne the rough nanostructure of the depth composton; secondary parameters deðne the Ðner detals of the depth composton. As expected, we wll Ðnd that the uncertanty of the prmary structural parameters s smaller than that of the secondary parameters. Depth of orgn of electrons n a spectrum D erent energy regons of a spectrum have ntensty contrbutons that orgnate from electrons excted at d erent depths. In ths secton we estmate the depth of orgn of electrons that contrbute to the varous energy regons of a spectrum. Ths knowledge s then used to dvde the sold nto two regons of depths. Before we consder detaled quanttatve model calculatons, we wll Ðrst make a rough estmate based on a smple model. Let us denote the typcal energy loss n a sngle scatterng event by de. Then the ntensty n an energy range *E below the peak energy s prmarly determned by the dstrbuton of electron emtters wthn the outermost depth range R \ *E de ] j (13) *E s typcally D15È30 ev dependng on the sold.47,48 As an example, *E \ 100 ev yelds R D 4È7j. The spectrum n an energy range up to D100 ev from the peak energy s then prmarly determned by the composton wthn the outermost 5j. The contrbuton from electrons excted at larger depth wll be small. We wll now consder detaled model calculatons of spectra for d erent n-depth dstrbutons of electron emtters and use these results to establsh the connecton between path length travelled and the energy regon of the spectrum. The model spectra are calculated (wth an exstng software package18) for electron emsson normal to the surface and usng the normalzed Unversal cross-secton (.e. B \ 2C \ 3286 ev2).48 The correlaton between energy loss and depth wll scale wth the nelastc electron mean free path j. All depths are therefore gven n unts of j. The am s to Ðnd the correlaton between the depth of orgn of emtted electrons and ther energy loss *E,.e. ther energy *E below the peak energy n the measured spectrum of emtted electrons. To ths end we deðne two measures for the connecton between spectral ntensty contrbuton and depth of exctaton. For the Ðrst measure we take a thn layer of materal, place t at varyng depths z below a surface and montor the ntensty n the emtted spectrum at energy *E below the peak. We deðne three depths: z0 s the depth where the layer gves the hghest contrbuton to the ntensty and z1 and z2 are the two depths where the same layer gves half of that ntensty contrbuton. Fgure 2 shows spectra of a 1 Ó thck layer of atoms placed n gold at the three depths z0, z1 and z2 vald for *E \ 100 ev [Fg. 2(a)] and *E \ 200 ev [Fg. 2(b)]. The values determned for z0, z1 and z2 are plotted n Fg. 4 for energes *E up to 500 ev below the peak energy. It s seen that the contrbuton to the spectral ntensty at a gven energy comes from a range of depths. For example, at 100 ev below the peak energy ( 1998 John Wley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998)

8 256 S. TOUGAARD Fgure 2. Model spectra from a 1 A thck layer of materal that emts electrons of 1000 ev energy. The layer s placed at varyng depths z below a surface: z0 s the depth where the layer gves the hghest contrbuton to the ntensty DE below the peak energy; z1 and z2 are the two depths where the same layer gves half of that ntensty contrbuton. The sold has nelastc mean free path l and the Unversal nelastc scatterng cross-secton ÍEqn (8)Ë. the spectral ntensty contrbuton from electrons that are excted at all depths between 0.73j and 5.1j vares by only a factor of two. For the second measure we take a sold wth constant atomc concentraton extendng from the surface down to depth R. When R s nðnty we have a homogeneous sold. We montor the ntensty n the emtted electron SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) Fgure 3. Model spectra from a sold wth constant concentraton that extends from the surface to the depth R : R1 and R2 are the values for R for whch the ntensty DE below the peak energy s respectvely 67% and 90% of the ntensty from a homogeneous sold (R ¼ Æ ). The sold has nelastc mean free path l and the Unversal nelastc scatterng cross-secton ÍEqn (8)Ë. spectrum at energy *E below the peak. For a gven energy *E we now Ðnd the values R1 and R2 ofrfor whch we get respectvely 67% and 90% of the ntensty that we get from a homogeneous sold. Fgure 3 shows spectra for *E \ 100 ev and *E \ 200 ev. The values of R1 and R2 are plotted n Fg. 4 as a functon of *E. From Fgs 2È4, and from further extensve nvestgatons not presented here, we conclude that the ntensty and shape of the spectrum n the energy range up to D100 ev from the peak are prmarly determned by the concentraton of electron emtters wthn the outermost ( 1998 John Wley & Sons, Ltd.

9 NON-DESTRUCTIVE SURFACE NANOSTRUCTURE QUANTIFICATION TECHNIQUE 257 Fgure 4. Values of z0, z1, z2, R1 and R2 n unts of l as a func- ton of DE. 5j and are essentally completely determned by the electron emtters wthn 8j. For *E [ 100 ev, the cor- respondng depths are larger (see Fg. 4). For example, t was prevously found that nformaton on the depth exctaton functon n electron-stmulated AES can be found from analyss of the expermental AES spectrum over an energy range of several hundred electronvolts.38h40 Guded by these results, we now dvde the surface nto two depth regons: the surface regon, whch s the outermost 5È10j of the sold; and the subsurface regon, whch corresponds to depths Z5È10j. The results of ths secton regardng nformaton on n-depth dstrbuton of atoms contaned n the detaled spectral peak shape can then be summarzed as follows. Informaton on the atomc concentratonèdepth proðle n the surface regon up to depths D5j s prmarly determned by the spectral energy regon up to D100 ev below the peak energy and s essentally completely determned by the energy regon up to D200 ev below the peak. The spectral ntensty for energes more than 200 ev below the peak energy has a sgnðcant contrbuton from electrons orgnally excted at depths larger than 5È10j. Prmary and secondary structural parameters In the prevous secton we establshed the connecton between the spectral energy range and the depth of orgn of electrons that contrbute to the spectrum. In ths secton we wll estmate the number of structural parameters that can be determned from analyss of the spectrum. The cross-secton for electron energy loss n solds s n general a wde functon of energy loss. There s therefore consderable overlap n the ntensty contrbuton at a gven spectral energy from electrons excted at dfferent depths. Ths s the reason why there s not a drect one-to-one connecton between depth of orgn and energy loss (see prevous secton). To estmate the amount of nformaton that can be extracted from an energy spectrum, we Ðrst note that most solds have cross-sectons that are at a maxmum for 15È30 ev energy loss and that have a full wdth at half-maxmum of 15È25 ev.48 As a rule of thumb, each 15È30 ev of the spectrum therefore gves nformaton from whch one structural parameter can be determned. For example, the ntensty and shape of the spectrum n the energy range up to D100 ev from the peak s prmarly determned by three and essentally completely determned by sx structural parameters. We therefore dvde the structural parameters that descrbe the chemcal composton of the outermost 5È10j of the sold nto two groups: prmary and secondary parameters. The prmary parameters are the three most mportant parameters needed to descrbe the man characterstcs of the dstrbuton of atoms wthn depths up to 5È10j. Two of these parameters descrbe the structure and the thrd gves the number of atoms wthn the structure. The two parameters that characterze the structure are not suffcent to determne the number of atoms because the structure may contan several types of atoms and ther concentraton may vary. For example, the three prmary parameters for a rectangular structure are the start and end depths and the concentraton of analysed atoms wthn that depth range. Lkewse, for an sland structure, the three prmary parameters are the surface coverage, the sland heght and the concentraton of analysed atoms wthn the sland. The secondary parameters are addtonal parameters n excess of the three prmary parameters that characterze the Ðner detals of the depth dstrbuton of atoms n the outermost 5È10j of the surface regon. Normally there wll be only three secondary parameters because more than sx parameters that characterze the structure of ths depth range can seldom be determned wth any degree of accuracy from analyss of the energy spectrum. The spectrum does, however, contan nformaton on more than sx structural parameters f the spectrum s analyzed over an energy range consderably larger than 100 ev. The extra parameters descrbe the structure of the deeper layers ([5È10j ) because t s prmarly electrons excted at these larger depths that determne the shape of the spectrum beyond D100 ev from the peak energy (see prevous secton). So analyss of a larger energy range than 100 ev does not add much to the nformaton on the detals of the structure n the outermost 5j but gves the possblty to determne addtonal structural parameters that descrbe the structure for larger depths. EXPERIMENTAL STUDIES In the foregong sectons t was argued that the level of accuracy of quantðcaton n surface analyss cannot be ( 1998 John Wley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998)

10 258 S. TOUGAARD found wthout specfyng n more detal what s quant- Ðed. Ths s related to the surface senstvty of the technques appled n quanttatve surface analyss. From analyss of theoretcal model spectra we then formulated specðc objectves of quantðcaton, namely two regons of depths (.e. the outermost 5È10j and larger depths) and two sets of parameters (the prmary and secondary parameters) to deðne the concentratonè depth proðle of the outermost 5È10j. In ths secton we shall Ðnd the level of accuracy n quantðcaton of these parameters. We wll base ths nvestgaton on analyss of three experments. We have chosen here three of the Ðrst experments where peak shape analyss was appled.19,20,28 Snce then, several other expermental systems have been studed wth the technque,19h41 but the three early experments serve well to study the uncertanty n the quantðcaton. In addton, the present work wll also gve a new and mproved analyss of these experments because the prevous analyss was based on less accurate algorthms than n the present analyss. Here we use a formalsm that takes nto account multple electron scatterng and we use a software package that allows a wde varety of n-depth dstrbutons to be studed and provdes a smpler and more drect quantðcaton procedure.18 Ths results n a more detaled, accurate and complete analyss of these three experments n comparson to the prevously publshed analyss.19,20,28 Copper oxde growth on copper Ths system was studed n Ref. 19. The copper fol was exposed n stu to oxygen at 2 ] 10~3 Torr whle beng held at 650 C. Ths combnaton of temperature and oxygen pressure les n the range where only Cu O and 2 no CuO s formed.70 That no CuO was formed was also checked after each exposure to oxygen by analyss of the Cu 2p spectrum.19 The O 1s peak overlaps n energy wth the Cu LMM Auger peak. When peaks from two d erent atoms overlap n energy, the peak shape analyss technque s more nvolved. Ths s because F(E) s d erent for the two atoms. Therefore, the spectra for each of the two atoms must be calculated separately and then added before t s compared to the measured spectrum. Rather than usng the O 1s peak n the analyss of ths partcular system, t was therefore more convenent to apply the O KVV Auger peak excted wth Mg Ka radaton. Fgure 5. Oxygen KVV Auger spectra from a copper sample after dfferent oxygen exposures. The Auger transton s excted wth Mg Ka radaton. Fgure 5 shows O KVV Auger spectra measured for d erent oxygen exposures. The exposures are gven n Table 1 and the curves n Fg. 5 show spectra correspondng to sx of these oxygen exposures. The spectra were obtaned by subtractng from the measured O KVV spectra the spectrum from pure copper n the same energy range after normalzng to the ntensty level n the background on the hgh energy sde of the O KVV peak. Ths procedure was found necessary for the lowest two exposures where the O KVV sgnal s qute small. For the remanng seven exposures there was no d erence n the analyss between ths procedure and the usual procedure, where a straght lne s Ðtted on the hgh energy sde of the O KVV peak regon and then subtracted from the entre spectrum. All spectra had Ðrst been corrected for the energy dependence of the analyser transmsson, whch was Table 1. Structures of grown Cu O determned from analyss of O KVV (Fgs 5 8) 2 Oxygen Island Total amount of Cu 2 O exposure (L)a Coverage Heght (A ) Á4 Ã À5% 25 À20% 7À15% Ã À5% 60 À20% 20 À15% Ã À5% 90 À30% 54 À20% 1.37 Ã À5% Heght 100 A (Á5l ) 4.07 Ã À3% Heght 100 A (Á5l ) 9.74 Ã À3% Heght 100 A (Á5l ) 21.3 Ã À2% Heght 100 A (Á5l ) 42.9 Ã À2% Heght 100 A (Á5l ) 59.0 Ã Æ a 1 Langmur ¼10É6 Torr s. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998) ( 1998 John Wley & Sons, Ltd.

11 NON-DESTRUCTIVE SURFACE NANOSTRUCTURE QUANTIFICATION TECHNIQUE 259 taken to be nversely proportonal to the square root of the electron energy. The spectra have been smoothed by an 11-pont SavtzkyÈGolay smooth wth second-degree polynomal Ðt. Ths s not essental for the analyss procedure and the only reason for ths smoothng s that when, as here, multple colours are not used n Ðgures t s smply mpossble on a plot to dstngush two nosy spectra that are close together. Now the peak shape analyss formalsm s appled to these spectra; j \ 19.0 Ó s used as estmated from the Seah and Dench formula.13 The sample correspondng to the hghest exposure s assumed to have oxygen wth constant concentraton to all relevant depths (.e. at least D10j ). Then the optmum value of B n the Unversal cross-secton s found for ths system by analysng the spectrum n Fg. 5 correspondng to 59 ] 106 L exposure wth an assumed proðle that s homogeneous to all depths. The best value of B s the one that gves approxmately zero ntensty over a wde energy range below the peak. Ths results n B \ 2900 ev2 and the spectrum s the lowest one n Fg. 6. Ths spectrum s used as a reference F(E) spectrum n all the followng analyss. The spectra n Fg. 5 were analysed under varaton of the prmary parameters of the surface structure untl the resultng spectrum n each case matches the reference F(E) spectrum wth respect to both absolute peak area and peak shape. It was found that a satsfactory analyss of the spectra can only be acheved when the Cu O s dstrbuted n the form of slands (see 2 below). The parameters are shown n Table 1 and the resultng F(E) spectra are shown n Fg. 6. The uncertanty on the structural parameters n Table 1 s now estmated. In Fg. 7, the spectrum corre- Fgure 7. The spectrum of the sample exposed to Ã106 L oxygen (Fg. 5) after analyss, assumng that the oxde covers the complete surface. From ths t s concluded that the oxde does not cover the complete surface. Fgure 6. The spectra n Fg. 5 after analyss wth the parameters gven n Table 1. spondng to ] 106 L oxygen exposure s analyzed under the assumpton that the small amount of Cu O s 2 dstrbuted as a unform layer over the surface (.e. coverage \ 1.0). Fgure 7 shows the analyss as the thckness d of the oxde layer s vared untl t matches the reference spectrum. Wth d \ 7 Ó the peak ntensty s matched, but the ntensty further away from the peak s clearly wrong. Wth d \ 60 Ó, the ntensty for energes far below the peak energy s zero, as t should be, but the peak ntensty s wrong by a factor of three. It s therefore obvous that there s no soluton wth coverage \ 1.0. The behavour of F(E) for a wde parameter range can easly be found and by comparson to the F(E) reference the uncertanty n parameter values can readly be estmated. The resultng values are: coverage \ 0.34 ^ 5%; sland heght \ 60 Ó^ 20% (see Table 1 and Fg. 6). As another example, the spectrum correspondng to 9.74 ] 106 L oxygen exposure s analysed n Fg. 8. The optmum values, whch are gven n Table 1, are found by varaton of the prmary structural parameters (.e. thckness and surface coverage of the copper oxde) and the resultng spectrum s shown n Fg. 6. Fgure 8(a) llustrates that the agreement wth the reference spectrum s sgnðcantly worse when the coverage s 0.72 and 0.92 rather than the optmum value of Fgure 8(b) shows that wth 0.82 coverage, the sland heght must be at least 100 Ó. After consderng further sets of parameter values, the concluson s quckly reached that ( 1998 John Wley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 249È269 (1998)