Elastic Scattering Corrections in AES and XPS. II. Estimating Attenuation Lengths and Conditions...

SURFACE AND INTERFACE ANALYSIS, VOL. 25, 430È446 (1997)

Elastic Scattering Corrections in AES and XPS. II.Estimating Attenuation Lengths and ConditionsRequired for their Valid Use inOverlayer/Substrate Experimentss

P. J. Cumpson* and M. P. SeahCentre for Materials Measurement and Technology, National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK

We examine substrate/overlayer experiments and the equations commonly used to quantify overlayer thicknesses.Comparisons with accurate Monte-Carlo simulations show that using attenuation lengths (rather than inelasticmean free paths) eliminates most of the error due to elastic scattering without increasing the complexity of thequantiÐcation.

We give attenutation lengths for 27 elements, calculated by the criterion that systematic errors in such quantiÐ-cations should be minimized. These are therefore the best attenuation length values to use in layerwise quantiÐca-tion. We show that, provided these attenuation length values are used, the error in estimation of the thickness of anoverlayer due to elastic scattering can be limited to »(5% + 1 for an emission angle O58Ä from the surfaceA� )normal, and »(10% + 1 for an emission angle O63Ä from the surface normal. This accuracy is acceptable forA� )most analytical work. Other methods (such as analytical transport theory) are much more complicated, andachieve a high precision that is often unnecessary in view of other uncertainties typically present in these experi-ments (such as errors due to surface morphology and di†raction e†ects). The results presented here, using the fulltheory, show that the analystÏs simple straight-line approximation is in fact of adequate accuracy, provided that thecorrect values of attenuation length are used.

Simple semi-empirical equations are presented, which allow the analyst to estimate the attenuation length forelectrons of kinetic energy between 50 and 2000 eV, to a standard uncertainty of 6% . 1997 by John Wiley &(

Sons, Ltd.

Surf. Interface Anal. 25, 430È446 (1997)No. of Figures : 16 No. of Tables : 1 No of Refs : 42

KEYWORDS: XPS; AES; attenuation length ; IMFP; elastic scattering

INTRODUCTION

In XPS or AES, in order to obtain an accurate quanti-tative analysis of a surface whose composition in inho-mogeneous with depth, one needs to know the depth oforigin of the measured electrons. The standard quantiÐ-cation equations1 used by analysts involve the assump-tion that signal electrons travel in straight lines, andthat the specimen is atomically Ñat, as shown in Fig. 1.When AES and XPS were Ðrst developed, it wasthought that the electron inelastic mean free path(IMFP) was the only natural unit of depth in thesespectroscopies. The IMFP was substituted in the simplequantiÐcation equations to allow, for example, over-layer thicknesses to be calculated and matrix factorsevaluated. This quantiÐcation method allowed rapidprogress in many areas of application, such that both

* Correspondence to : P. J. Cumpson, Centre for Materials Mea-surement and Technology, National Physical Laboratory, Tedding-ton, Middlesex TW11 0LW, UK.

¤ Paper presented at the 9th International Conference on Quanti-tative Surface Analysis (QSA-9), Guildford, UK, 15È19 July 1996.

Contract grant sponsor : UK Department of Trade and Industry Figure 1. Geometry of AES and XPS.

CCC 0142È2421/97/060430È17 $17.50 Received 15 April 1996( 1997 by John Wiley & Sons, Ltd. Accepted 18 February 1997

ATTENUATION LENGTHS FOR OVERLAYER/SUBSTRATE EXPERIMENTS 431

AES and XPS are now common materials analysis tech-niques. The IMFP is an intrinsic property of thematerial under analysis, so it can be measured usingcompletely di†erent techniques, such as from back-scattering intensity measurements.

Around the mid-1980s, it was suggested that theappropriate length scale to substitute into the quantiÐ-cation equation was not the IMFP but the “attenuationlengthÏ (AL). Powell2,3 deÐned the AL as “a valueresulting from overlayer-Ðlm experiments on the basisof a model in which elastic electron scattering isassumed to be insigniÐcantÏ. This deÐnition, with minormodiÐcations, is that adopted by the American Societyfor the Testing of Materials (ASTM).4 Other possibledeÐnitions have recently been reviewed by Jablonskiand Powell ;5 most are motivated by the uncomfortablefact that the AL, as deÐned by Powell and ASTM, is notan intrinsic property of the specimen material, but alsodepends to some extent on experimental geometry.Therefore, deÐnitions have been proposed that arebased on microscopic transport theory or deÐne the ALfrom the depth distribution function (DDF). Forexample, to place results in the context of otherpublished work, in related paper I6 we calculatedattenuation length from the gradient of a DDF plot,along the lines suggested by Gries and Werner.7However, the most important point behind using an ALis the fact that we wish, Ðrst and foremost, to achieve amore accurate quantiÐcation. The whole motivationbehind deÐning an AL is that, when substituted into thestandard quantiÐcation equations, it gives a more accu-rate result than does the IMFP. This is due to the factthat the standard equations neglect a number of sec-ondary (but nevertheless signiÐcant) e†ects8 apart fromsimple inelastic losses, notably elastic scattering as indi-cated in the pioneering Monto-Carlo simulations ofBaschenko and Nefedov.9h11 By using a value of ALdi†ering slightly from the IMFP, it was hoped thaterrors introduced by neglecting this e†ect could bereduced or eliminated.

A considerable literature has developed dealing withthe di†erences between the straight-line approximation(SLA) model and these fuller calculations, and also theinadequacy and major shortcomings of the SLA model.However, we should also note that experimentalanalysts Ðnd in their studies that the SLA model worksvery well if the correct value of AL is used.1,12 Intu-itively, it is clear that the SLA model must be a reason-able approximation for some range of :(1) atomic number Z for the overlayers ;(2) energy E of the signal electrons ;(3) angle of emission h.

If experimental analysts, whose major interests maybe in materials science and not aspects of electron trans-port theory, are to use electron spectroscopies e†ec-tively, they need to have and use transparent simplerelationships to solve their problems. They also need anintuitive feel for the behaviour in order to design experi-ments. Of course, if the behaviour is intrinsicallycomplex it cannot be altered. What we can do, however,is search Z, E and h space for the optimal volume inwhich a simple expression is an adequate description ofthe best theoretical calculations we can make. We shalluse tabulated values for IMFP from Tanuma, Powelland PennÏs13 detailed analysis of optical data, as PennÏs

method is estimated to be accurate to D10%. Further-more, in providing a generic description of IMFPsusing an empirical equation, Tanuma et al. work to afurther 10% scatter. In the present work, therefore, weseek to keep uncertainties signiÐcantly below theselimits.

Powell originally deÐned the AL as a value calculatedfrom substrate/overlayer measurements when applyingthe conventional SLA equations for Ðnding layer thick-ness. In this paper we shall turn this deÐnition around,and calculate AL values that minimize the errorinvolved in applying the SLA model to the routineanalysis of specimens. Therefore, we shall use the AL inthe sense of the ASTM deÐnition.4 In this paper wesimulate electron transport in a classical overlayer/substrate structure using an accurate Monte-Carlomethod.6 We consider how AL values may be chosenspeciÐcally to minimize errors in substrate/overlayerquantiÐcation. We provide AL values for the 27 ele-ments considered by Tanuma et al.14 for the energyrange 50È2000 V and an assessment of which experi-ments can be quantiÐed by this route with negligiblesystematic error. Based on these Monte-Carlo resultswe then develop semi-empirical equations that Ðt theseAL values for any electron energy between 50 and 2000eV, with a standard uncertainty of 6%. These equationsare simple enough to be applied to any matrix : the useof AL values, though deprecated by some, is perfectlylegitimate provided that one knows the systematic errorthat results and that such an error is acceptable.

QUANTIFYING SUBSTRATE/OVERLAYEREXPERIMENTS

A large proportion of all quantiÐcation in AES andXPS consist of the following four steps :(1) The specimen is assumed to consist of one or more

layers on a substrate.(2) Spectra are acquired at one or more emission angles

(shown as h in Fig. 1).(3) The peak intensities of two or more chemical species

are measured (e.g. by peak synthesis15,16) and theratios of these peak intensities are calculated. Theuse of intensity ratios rather than absolute inten-sities removes common factors such as the incidentx-ray intensity from the subsequent quantiÐcation.

(4) The thickness and/or composition of the layers arequantiÐed, usually taking into account a good dealof prior knowledge about the chemistry of the speci-men and details of how it was prepared.

We take the simple case of an overlayer of element Aon a substrate of element B, each being amorphous orÐnely polycrystalline. In the SLA, in which it is assumedthat electrons follow straight-line paths from creation toemission (i.e. no elastic scattering), one can write theintensity of electrons that we expect to observe fromIAan overlayer of thickness d as

IA \ IA=M1 [ exp [[d/jimfpA (EA) cos h]N (1)

where is the intensity that we would observe from aIA=specimen of pure A and is the IMFP of electronsjimfpAin element A, which is a function of the kinetic energy

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 430È436 (1997)

432 P. J. CUMPSON AND M. P. SEAH

of the Auger or photoelectrons from the overlayer.EANote that kinetic energy is vacuum level-referencedthroughout this work.

The intensity of signal electrons that we expect toobserve from the substrate composed of element B is

IB\ IB= exp [[d/jimfpA (EB) cos h] (2)

where is the kinetic energy of electrons originatingEBin the substrate. and are usually unavailable ;IA= IB=however, the analyst will often have sensitivity factorsand which are proportional to andsA sB , IA= IB=.

These can be used provided that we deal with only theratio so thatIA/IB ,

(IA/sA)(IB/sB)

\ 1 [ exp [[d/jimfpA (EA) cos h]exp [[d/jimfpA (EB) cos h]

(3)

Usually, of course, we are interested in calculating thethickness d for a given measured intensity ratio. Inmany cases the energy of the peak from element A willbe very close to that of element B (e.g. A may consist ofan oxide of the substrate element B), in which case wecan make the assumption that the IMFPs of signal elec-trons in the overlayer are identical17 whether they orig-inate in the overlayer itself or in the substrate, i.e.

say, and obtainjimfpA (EA) \ jimfpA (EB)\ jimfp ,

d \ jimfp cos h lnC1 ] (IA/sA)

(IB/sB)D

(4)

This result in used to plot Fig. 2. We can see that themeasurement of the thickness of an overlayer consists inÐnding the x-axis value corresponding to the measuredintensity ratio and, then reading o† from the graph they-axis value corresponding to the overlayer thickness(here plotted in reduced form).

Two regions of Fig. 2 have been shaded ; these rep-resent specimens and conditions that would not nor-mally be quantiÐed using this popular layerwise model.The shading at low overlayer thickness represents thefact that this analysis breaks down for d ^ 0.2 nmbecause in such cases we have fractional monolayer

coverage and quantiÐcation is better carried out byanother route that explicitly describes this.1 The shadedarea at low substrate intensity (shown at the right-handside of Fig. 2) indicates that, given the inelastic back-ground present in all AES and XPS spectra, one canseldom measure two peaks with widely di†ering inten-sities sufficiently accurately for their intensity ratio to bevery meaningful. Here, we have assumed that for thesubstrate peak to be measured it must have at least 5%of the intensity of the overlayer peak (this boundarymay be placed slightly di†erently if A and B have verydi†erent sensitivity factors, but the results derived inthis paper are not sensitively dependent on it).

Equation (4) does not account for elastic scattering ofsignal electrons. However, Monte-Carlo methods can beused to trace the trajectories18 of a large number ofAuger or photoelectrons through the solid from atomicemission to escape from the surface. This allows theintensities of particular spectral peaks to be calculatedfor any arbitrary specimen composition.19,20 In relatedpaper I,6 one of us presented new Monte-Carlomethods that give very rapid and accurate results insuch problems. Using the “Berger-DoggettÏ algorithmdescribed in paper I, and using published elastic scat-tering cross-sections21 calculated from relativisticHartreeÈFockÈSlater atomic potentials, we now look athow elastic scattering changes the curve shown in Fig.2. Figure 3(a) shows the DDF from a Monte-Carlosimulation of an overlayer of amorphous carbon on asubstrate of amorphous carbon for 1000 eV electrons.Figure 4(a) is a similar plot for 700 eV electrons in iron.Figures 3(b) and 4(b) are rather more interestingbecause they show how elastic scattering modiÐes Fig.2. Here we are imagining the analysis of an overlayer ofeach element on a substrate of the same element.Finally, Figs 3(c) and 4(c) show the ratios of the Monte-Carlo results of Figs 3(b) and 4(b) to the Fig. 2 curve.

Several important features can be seen immediatelyfrom Figs 3 and 4, as follows :(1) Figures 3(b) and 4(b) show that the relationship

between d and is more complex than the SLAIA/IB

Figure 2. Calibration curve for overlayer/substrate determination in the absence of any elastic scattering.



Figure 3. (a) Depth distribution function (DDF), (b) overlayer/substrate intensity ratios and (c) those ratios divided by the curve in Fig. 2for 1 keV signal electrons in matrix of glassy carbon.

(shown in Fig. 2) would predict. In particular, thecurves now depend on the angle of emission, espe-cially for greater than aboutIA/sA 2IB/sB .

(2) Generally the Monte-Carlo results in Figs 3(b) and4(b) lie below the SLA curve plotted in Fig. 2,showing that if we quantify overlayer thicknessusing the SLA model with the IMFP, value, we willgenerally overestimate the thickness of the overlayerby neglecting the e†ect of elastic scattering. The

exception to this rule occurs at large depths foremission angles greater than D70¡ ; in these cases,the SLA model with IMFP values can actuallyunderestimate the overlayer thickness. This is due toelectron trajectories such as that shown in Fig. 5,where elastic scattering can clearly enhance the sub-strate intensity for large emission angles.

(3) Those curves in Figs 3(b) and 4(b) representing elec-trons emitted at 45¡ with respect to the surface most



Figure 4. (a) Depth distribution function (DDF), (b) overlayer/substrate intensity ratios and (c) those ratios divided by the curve in Fig. 2for 700 eV signal electrons in an iron matrix.

closely resemble the SLA plot shown in Fig. 2, atleast when this SLA plot is scaled vertically by anappropriate factor. This is conÐrmed by the plots inFigs 3(c) and 4(c), which show the curves in Figs 3(b)and 4(b) divided by the SLA curve plotted in Fig. 2.Figures 3(c) and 4(c) show the h \ 45¡ curves to bealmost Ñat.

(4) Comparison of Figs 3(b) and 4(b) with Fig. 2 indi-cates that the amount of vertical rescaling requiredto make the SLA curve Ðt the Monte-Carlo results

reasonably well increases with atomic number. Herewe have plotted data for only two elements, but thisis a general trend. An exception to this rule thatapplies at low kinetic energies will be discussedseparately.22 Corrections for elastic scattering e†ectstherefore usually become more important for largeratomic number.

Point (3) stated above reveals the essence of the argu-ment for replacing IMFP by AL in routine quantiÐca-tion ; by replotting the SLA curve shown in Fig. 2, this



Figure 5. At very large emission angles the intensity observedfrom the substrate is strongly enhanced by elastic scattering eventsin the overlayer. A signal electron emerging at the large emissionangle illustrated here is very unlikely to have followed the longdirect path. Such an electron is much more likely to have beenemitted at a relatively small angle with respect to the surfacenormal and then elastically scattered close to the surface. It is thisprocess that prevents the use of the straight-line approximation(SLA) to quantify overlayer thickness at very large emissionangles. However, for a typical emission angle of 45¡, the SLA canperform extremely well, as discussed in the text.

time with a y-axis plotted in units of ratherd/(jAL cos h)than the SLA curve approximates thed/(jimfp cos h),Monte-Carlo results extremely well. This means thatthe form of Eqn. (4) is still good for quantiÐcation in thepresence of elastic scattering, but that it can be mademore accurate by rescaling the characteristic length thatappears in the equation. Therefore, we can replace Eqn.(4) with an equation that is no more complex but ismore accurate because it takes most of the e†ect ofelastic scattering into account.

d \ jAL cos h lnC1 ] (IA/sA)

(IB/sB)D

(5)

where is the attenuation length.jALIt is clear from the ratio plots shown in Figs 3(c) and4(c) that, provided the appropriate value of AL ischosen, the SLA model can give an extremely accurateestimate of overlayer thickness when h ^ 45¡. Clearly, ifone intends to use the simple SLA model in quantiÐca-tion, then a “take-o† Ï angle of 45¡ will minimize the sys-tematic error due to elastic scattering ; indeed, using thisemission angle this error is minimized to the extent thatit is almost certain to be small compared to otherexperimental uncertainties (especially the e†ect ofsurface roughness, di†raction and lateral inhomogeneity).Of course, by choosing an emission angle of 45¡, onelimits oneself to analysing only D70% of the depthaccessible using normal emission, and this may notalways be desirable, especially when dealing with thickoverlayers where the substrate signal is small.

These considerations give us the following deÐnition.

DEFINITION

In this section we show how, after careful considerationof Figs 2 and 3(b), a very good general deÐnition of ALsuggests itself. We then show how this general deÐnitioncan be applied to the speciÐc task of calculating theseALs from Monte-Carlo data. Other deÐnitions may beuseful in special circumstances ; however, as we shall see,

this deÐnition leads to a value that, with Eqn. (5), pro-vides the widest range of d and h values within whicherrors are minimized.

Geometrical interpretation

Comparing Fig. 2 with the h \ 45¡ curve shown in Fig.3(b), we can see that the AL we need is simply theIMFP multiplied by a factor equal to the area underthe curve in Fig. 3(b) divided by the area under thecurve in Fig. 2, over the interval of peak intensities com-monly encountered in XPS or AES. Mathematically, thisis exactly equivalent to performing a least-squares Ðt ofEqn. (5) to the Monte-Carlo curve in Fig. 3(b) to Ðndthe single unknown parameter The areas under thejAL .two curves are easy to express mathematically in termsof two integrals, but to give the result the greatest prac-tical validity, we need to consider carefully the limits ofintegration and the “spaceÏ in which the integration isperformed.

Practical integration limits

The experimentally important interval over which theÐt should be performed is from ln (1 ] IA sB/IB sA) \ 0.1,which represents an overlayer thickness of about onemonolayer for normal emission, to ln (1]IA sB/IB sA)\3,which represents a peak intensity ratio of D20 beyondwhich, as we have discussed already, spectrum back-grounds usually make it very difficult to measure peakratios with any accuracy. These are therefore the practi-cal integration limits that cover the entire interval overwhich data can be sensibly quantiÐed, as indicated inFig. 2.

Integration space

Figure 2 provides a “calibration curveÏ, allowing one tostart with a peak intensity ratio measured by XPS andthen read o† from the graph the thickness of the over-layer. There are Ðrm practical reasons for the choice ofsemi-logarithmic scale in Fig. 2, which lead to a particu-lar choice of integration space.

The intensity ratio is plotted on a logarithmic scale,because from the whole range of specimens that onecould be given to analyse (whether from electronicmaterials, industrial catalysts or anything else), anyvalue of this ratio is about equally likely along this scalebetween the two limits we have just chosen.

The overlayer thickness is plotted on a linear ordinatescale because one usually wants to quote overlayerthickness in the format “d \ x ^ y nanometresÏ, inwhich it is the absolute uncertainty (not the fractionaluncertainty) in the thickness that is important. If we hadplotted Fig. 2 as a log-log graph, small errors in valuesfrom near the top of the vertical axis would lead toquite large absolute errors in the overlayer thicknessvalue obtained.

This means that to calculate the AL value for 1 keVelectrons in carbon, for example, we should ratio thearea under the h \ 45¡ plot in Fig. 3(b) to the areaunder Fig. 2 in exactly the semi-logarithmic space used inthese plots. This, in turn, means the appearance of expo-nentials in the integrals used to represent the areas



Table 1. Inelastic mean free paths (IMFPs) and calculated attenuation lengths (ALs) for the 27 elements studied by Tanuma, Powelland Penn14 for electrons of kinetic energy 50–2000 eV (Note the discussion in the text regarding the emission angle at whichthese have been evaluated, and that sum rule errors suggest that equation CS2 provides a more accurate estimate of AL forthe four elements Al, Nb, Re and Hf)

Carbon Magnesium Aluminium Silicon Titanium Vanadium Chromium Iron Nickel

IMFP AL IMFP AL IMFP AL IMFP AL IMFP AL IMFP AL IMFP AL IMFP AL IMFP AL

50 eV 0.59 0.34 0.40 0.31 0.32 0.24 0.41 0.30 0.45 0.28 0.42 0.26 0.44 0.26 0.43 0.26 0.49 0.29

75 eV 0.61 0.39 0.47 0.37 0.37 0.28 0.47 0.35 0.47 0.31 0.45 0.29 0.42 0.27 0.42 0.28 0.46 0.30

100 eV 0.64 0.45 0.54 0.43 0.42 0.32 0.53 0.40 0.51 0.35 0.49 0.32 0.43 0.29 0.44 0.29 0.46 0.31

200 eV 0.88 0.69 0.82 0.68 0.63 0.51 0.78 0.62 0.73 0.52 0.68 0.47 0.57 0.40 0.58 0.40 0.57 0.39

300 eV 1.12 0.93 1.07 0.92 0.83 0.69 1.03 0.84 0.95 0.70 0.88 0.62 0.72 0.52 0.72 0.51 0.69 0.49

400 eV 1.37 1.17 1.30 1.14 1.00 0.85 1.25 1.05 1.16 0.88 1.07 0.78 0.86 0.63 0.85 0.62 0.81 0.59

500 eV 1.60 1.40 1.53 1.35 1.17 1.01 1.46 1.24 1.37 1.05 1.25 0.92 1.00 0.75 0.98 0.73 0.92 0.68

600 eV 1.84 1.63 1.75 1.57 1.33 1.17 1.66 1.43 1.56 1.21 1.43 1.07 1.14 0.87 1.12 0.84 1.04 0.78

700 eV 2.06 1.85 1.96 1.77 1.49 1.32 1.86 1.62 1.76 1.38 1.60 1.21 1.27 0.98 1.24 0.95 1.15 0.88

800 eV 2.28 2.07 2.17 1.97 1.65 1.48 2.06 1.81 1.95 1.55 1.77 1.35 1.40 1.09 1.37 1.06 1.27 0.98

900 eV 2.49 2.28 2.38 2.18 1.81 1.63 2.25 2.00 2.13 1.71 1.94 1.50 1.53 1.21 1.49 1.16 1.38 1.07

1000 eV 2.70 2.49 2.59 2.38 1.96 1.77 2.44 2.18 2.32 1.87 2.10 1.63 1.66 1.32 1.62 1.27 1.49 1.17

2000 eV 4.68 4.46 4.53 4.28 3.41 3.19 4.25 3.93 4.02 3.37 3.64 2.95 2.86 2.39 2.77 2.28 2.53 2.09

Copper Yttrium Zirconium Niobium Molybdenum Ruthenium Rhodium Palladium Silver


50 eV 0.50 0.29 0.50 0.37 0.44 0.30 0.60 0.33 0.51 0.28 0.49 0.26 0.48 0.26 0.48 0.28 0.62 0.34

75 eV 0.49 0.33 0.52 0.41 0.45 0.34 0.59 0.38 0.46 0.30 0.43 0.28 0.42 0.27 0.47 0.30 0.53 0.34

100 eV 0.50 0.35 0.55 0.44 0.48 0.37 0.60 0.41 0.45 0.32 0.42 0.29 0.41 0.29 0.48 0.33 0.49 0.34

200 eV 0.62 0.44 0.75 0.60 0.66 0.50 0.77 0.53 0.56 0.40 0.52 0.37 0.50 0.36 0.62 0.44 0.55 0.41

300 eV 0.77 0.55 0.98 0.78 0.86 0.65 0.97 0.67 0.71 0.51 0.65 0.46 0.61 0.44 0.78 0.54 0.64 0.48

400 eV 0.90 0.65 1.19 0.96 1.05 0.80 1.17 0.81 0.85 0.62 0.78 0.56 0.73 0.53 0.94 0.65 0.76 0.58

500 eV 1.02 0.75 1.40 1.13 1.23 0.95 1.37 0.96 1.00 0.73 0.91 0.65 0.85 0.62 1.10 0.77 0.87 0.67

600 eV 1.14 0.85 1.60 1.30 1.41 1.09 1.56 1.10 1.13 0.83 1.04 0.75 0.97 0.72 1.25 0.88 0.99 0.76

700 eV 1.27 0.96 1.79 1.47 1.58 1.23 1.74 1.23 1.27 0.94 1.16 0.85 1.08 0.80 1.40 0.99 1.10 0.85

800 eV 1.39 1.07 1.98 1.63 1.75 1.37 1.92 1.37 1.40 1.05 1.28 0.94 1.20 0.90 1.54 1.10 1.21 0.94

900 eV 1.51 1.17 2.16 1.79 1.91 1.50 2.09 1.50 1.52 1.14 1.40 1.03 1.30 0.98 1.68 1.20 1.32 1.04

1000 eV 1.63 1.27 2.34 1.95 2.07 1.63 2.26 1.63 1.65 1.25 1.51 1.12 1.41 1.07 1.82 1.31 1.42 1.12

2000 eV 2.77 2.28 4.04 3.47 3.57 2.91 3.84 2.85 2.80 2.19 2.57 1.97 2.39 1.88 3.09 2.31 2.40 1.97

Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Bismuth


50 eV 0.58 0.33 0.48 0.29 0.50 0.30 0.52 0.30 0.55 0.30 0.53 0.29 0.50 0.29 0.67 0.36 0.49 0.35

75 eV 0.59 0.42 0.45 0.33 0.43 0.31 0.42 0.30 0.46 0.32 0.45 0.31 0.44 0.31 0.54 0.37 0.52 0.39

100 eV 0.62 0.49 0.45 0.36 0.41 0.33 0.38 0.31 0.43 0.33 0.43 0.33 0.42 0.33 0.48 0.37 0.55 0.44

200 eV 0.80 0.65 0.55 0.46 0.50 0.42 0.43 0.36 0.50 0.41 0.52 0.42 0.49 0.41 0.51 0.44 0.72 0.62

300 eV 1.02 0.79 0.68 0.55 0.61 0.49 0.51 0.42 0.60 0.47 0.64 0.49 0.60 0.48 0.60 0.49 0.88 0.75

400 eV 1.20 0.89 0.80 0.62 0.73 0.56 0.60 0.47 0.71 0.53 0.75 0.56 0.71 0.55 0.71 0.56 1.06 0.88

500 eV 1.38 1.01 0.92 0.70 0.84 0.63 0.69 0.54 0.81 0.60 0.87 0.63 0.82 0.62 0.81 0.63 1.23 1.00

600 eV 1.56 1.12 1.04 0.78 0.94 0.70 0.77 0.59 0.91 0.67 0.97 0.70 0.92 0.69 0.91 0.70 1.40 1.13

700 eV 1.73 1.24 1.15 0.86 1.04 0.77 0.85 0.65 1.01 0.73 1.08 0.77 1.02 0.76 1.01 0.77 1.56 1.25

800 eV 1.90 1.36 1.27 0.95 1.14 0.85 0.94 0.72 1.11 0.80 1.18 0.84 1.12 0.83 1.11 0.84 1.72 1.38

900 eV 2.06 1.47 1.37 1.03 1.24 0.92 1.01 0.77 1.20 0.87 1.28 0.91 1.22 0.90 1.20 0.91 1.87 1.50

1000 eV 2.22 1.59 1.48 1.11 1.34 1.00 1.09 0.84 1.29 0.94 1.38 0.98 1.31 0.97 1.29 0.98 2.02 1.62

2000 eV 3.73 2.70 2.49 1.90 2.24 1.70 1.83 1.43 2.15 1.59 2.29 1.66 2.19 1.65 2.16 1.68 3.44 2.79

under the two plots, and which form the following deÐ-nition.

General deÐnition

Note that both the choice of limits and the choice ofintegration space are dictated by practical XPS andAES considerations. This leads to a general deÐnition ofAL that is based on the practical requirements of quan-tiÐcation. If overlayer thickness is regarded as a func-tion of the peak intensity ratio and the emission angleat which that ratio was measured, i.e. h),d\d(IA sB/IB sA ,

then

jAL\

Pa

bd(es, h) ds

Pa

bln (1 ] es) ds cos h

(6)

where s is the logarithm of the normalized peak inten-sity ratio

s \ lnAIA/sAIB/sB

B(7)



Figure 6. Maximum emission angles for which layer thickness can be quantified to accuracies of À5%½1 (½) and À10%½1A� A� (L)using the attenuation lengths in Table 1 derived from Monte-Carlo calculations at h¼45¡. Note the three breaks in the abscissa scale, forpresentation purposes.

with our practical integration limits ofa \ ln [exp (0.1)[ 1] and b \ ln [exp (3)[ 1]. Here his a typical analytical emission angle, the exact value ofwhich is chosen so that is not strongly sensitive tojALthat value. Figures 3(c) and 4(c) suggest that h \ 45¡ isbest. The ratio in Eqn. (6) ensures that the resulting jALvalue, when substituted into Eqn. (5), gives the bestapproximation in the least-squares sense to the truethickness d over the experimentally signiÐcant interval.

One does not normally think of the overlayer thick-ness d as being a function of the intensity ratio becausethe measured intensities are a consequence of the over-layer thickness for any particular specimen. Here,however, the integrals provide our averaging over thewhole ensemble of overlayer thicknesses that one couldencounter in practical quantiÐcation, so that we obtainan AL value that is acceptable accurate for all of them.For this deÐnition we need the thicknesses d for regularincrements in s, and hence d is e†ectively a function of

and h.IA , IB

Applying this deÐnition to Monte-Carlo calculations

We could evaluate Eqn. (6) from experimental data.One could make an independent measurement of thick-ness d (e.g. by ellipsometry or quartz crystalmicrobalance) for an overlayer that has a measuredintensity ratio and emission angle. One would thenrepeat these measurements for a few di†erent specimenswith di†erent overlayer thicknesses and numericallyintegrate the nominator integral in Eqn. (6) to calculateAL.

In this paper, however, we aim to provide valuesapplicable to a wide range of materials, so we shallapply this deÐnition to obtain the AL by making thefollowing choices :(1) Use a Monte-Carlo method to relate d and the peak

intensity ratio. The Monte-Carlo calculation givesus for any given peak intensity ratio and emis-dMCsion angle. This is the most accurate theoretical esti-mate of d currently available that is applicable to awide range of materials.

(2) Put h \ 45¡. (The validity of using ALs calculatedfor this emission angle is conÐrmed in the nextsection. They will be shown to be acceptably accu-rate over a wide range of emission angle.)

Finally, the denominator integral of Eqn. (6) can beevaluated, giving the equation that we used to estimateAL

jAL\ 0.2422Pa

bdMC(es, 45¡)ds (8)

where is the result of a Monte-CarlodMC(IA sB/IB sA , h)calculation.

Strictly, AL should be restricted to behaviourdescribed by an exponential decay with distance. Thepresent close approximation therefore yields an “e†ec-tiveÏ or “practical averageÏ AL. Rather than createanother term, this parameter should be refered to as theCumpson-Seah AL if any potential for confusion exists.For brevity here we shall use AL and the symbol tojALrepresent values calculated using this equation.

Attenuation length values calculated in this way areapplicable to the major part of quantiÐcations currentlyperformed, and allow the traditional SLA-based quanti-Ðcation to be performed with a known uncertainty dueto elastic scattering. In Table 1 we list ALs calculatedfrom Eqn. (8) for the 27 elements for which MFPs areavailable from the work of Tanuma et al.,14 evaluatedfor h \ 45¡. It should be stressed here that we use thevalues calculated by Tanuma et al.14 from optical data,rather than values from their empirical Ðtting equation.These ALs, when used with the SLA model, predict thee†ects of electron transport for 45¡ emission extremelywell. The next point to examine is over what range ofemission angle around this value can these values beused to speciÐed accuracy.

REGION OF VALIDITY

Our Ðrst temptation may be to quote an accuracy ofoverlayer thickness wholely in percentage terms,perhaps ^5% (or ^10%). However, such percentages



become meaningless for small depths where they mayrepresent a small fraction of a monolayer. Instead weshall look for the maximum emission angles for whichan SLA-based quantiÐcation of overlayer thickness isvalid to 5% (or 10%) plus or minus 1 The small con-A� .stant part of this quoted uncertainty means that it isonly the systematic error occurring at large depths thatneeds to be borne in mind, yet it still gives a very usefuland practical error estimate.

We shall, in particular, estimate the “maximum usableemission angleÏ for the case of a kinetic energy of1000 eV (a typical kinetic energy for XPS) and a ratio ofoverlayer-reduced intensity to substrate-reduced inten-sity of 10, which is about the largest value that could beused reliably in practice. Figure 6 shows thesemaximum usable emission angles. What this means isthat if one uses the ALs given in Table 1 and

(IA/sA)(IB/sB)

O 10

then to ensure an accuracy of ^(10%] 1 in lay-A� )erwise quantiÐcation of these intensities, one must notuse an emission angle larger than and to ensureh10†MAX ,an accuracy of ^(5%] 1 one must not use an emis-A� )sion angle larger than A cursory examination ofh5†MAX.Fig. 6 will show that for the 27 Tanuma elements,14

in all cases but As one mighth5†MAX[ 58¡ h10†MAX[ 63¡.expect, larger emission angles can be used for the (moreweakly scattering) low Z elements while maintaining thesame accuracies.

Figure 6 gives conÐdence in the use of the values inTable 1 for accurate quantiÐcation of a wide range oflayered specimens under nornal analysis conditions.The reason for the failure at high emission angles isshown schematically in Fig. 5. Larger emission angles(perhaps 80¡ or 85¡) can be very valuable in providingthe greatest possible surface sensitivity for qualitativework. However, quantitative analysis of layer thicknessusing the AL values in Table 1 (or, indeed, the equa-

tions CS1 and CS2 that we shall introduce later) shouldbe performed within the above emission angle limits.

SEMI-EMPIRICAL EQUATION FORATTENUATION LENGTH

It is extremely useful to be able to estimate AL using asimple equation, analogous to those proposed for esti-mating IMFP values by Tanuma et al.14 and Gries.23Our present purpose is rather di†erent from theirs,because the AL is a di†erent quantity to the IMFP and,as the calculated AL values in Table 1 show, can have asimpler energy dependence than either the IMFP or theelastic scattering properties.

In Fig. 7 we can see that for gold, the highest Zelement plotted, the IMFP has a very non-linear energydependence over the low energy range of 50 eV toD500 eV. Yet this non-linearity is largely cancelled inthe AL curve by the e†ect of elastic scattering, givingan AL that is nearly linear with energy. Therefore, thefunctional forms of the AL curves for the elementsplotted in Fig. 7 are simpler than those of their IMFPcurves, particularly in the low energy regime, andindeed simpler than the (AL/IMFP) ratio plots of Fig. 8,at least over the energy range of interest in AES andXPS. It is unlikely that this near-cancellation persistsmuch below 50 eV. The physical origin of this near-cancellation will be examined in detail in paper III22 inthis series, but for our present purposes the conclusionis that it is more straightforward to look for a semi-empirical equation for AL directly, rather than attemptto use, say, the Ðtted relation TPP-2M13 for IMFP, anda further Ðtted relation for the ratio (AL/IMFP),because each would have to be more complex than isnecessary for our present purposes. Nevertheless, wemust be guided by theory rather than proposing apurely empirical Ðt, due to the limited range of elementsfor which Tanuma et al.14 were able to calculate IMFP

Figure 7. Attenuation length (AL) and inelastic mean free path (IMFP) as a function of electron kinetic energy for matrices of silicon, ironand gold. Note the near-linearity of AL at low energies above 50 eV, even for a large atomic number element such as gold. In contrast, theIMFP of gold has a finite minimum above 100 eV.



Figure 8. Attenuation length divided by inelastic mean free path(IMFP) for the 27 elements for which Monte-Carlo calculationswere performed, grouped by period of the Periodic Table. Notethat at high energies this ratio typically decreases with increasing Zand that monotonicity with energy is lost between Period 5 and 6.Period 6 elements in particular show interesting behaviour below500 eV, which is discussed in detail in paper III.22

values. These were to some extent self-selected,23because elements that are difficult to prepare (due torapid oxidation, for example) have no published opticaldata and are therefore not represented in the IMFPtables of Tanuma et al.14 TilininÏs comparison24 ofasymptotic formulae for elastic and inelastic scatteringis particularly valuable here.

In the high kinetic energy limit, Gryzinski25 obtainedan approximation for inelastic scattering cross-section(in atomic units)

pinel\ n ;t/1

N ni

UiE

fAEU

i

B(9)

where is the number of atomic electrons contributingnito the ith inelastic channel, is the binding energy ofU

ithose electrons and forf (E/Ui) P [1 ] 13 ln (E/U

i)]

large E. At energies typical of AES or XPS, weaklybound electrons contribute most to this sum and have

in atomic units or in SI units. This isUiD 1 U

iD 27 eV

larger than the mean binding energy of valance elec-trons because the set of weakly bound electrons alsoincludes some of the outermost atomic orbitals, appro-priately weighted in Eqn. (9). The number of weaklybound electrons is DZ2@3 in the statistical ThomasÈ

Fermi model.24 Therefore, one might expect a reason-able approximation to be

jAL PE

[1] 13 ln (E/27)]Z2@3 (10)

where E is the electron kinetic energy (in eV) and Z isthe atomic number of the element. This high energylimit is similar to the semi-empirical modiÐed Betheequation for IMFP of Tanuma et al.13 and the semi-empirical equation for IMFP proposed by Gries.23 TheThomasÈFermi model, which gives us the Z2@3 divisor,“smoothsÏ over the periodic electronic structure so thatit is only a rough estimate of the number of electronstaking part in inelastic scattering processes. Some suchaveraging is surely valuable because Eqn. (9) does notsum over valence electrons alone. More importantly,the ThomasÈFermi model typically overestimates theelectron density of weakly bound electrons far from thenucleus, especially for large Z atoms. Hence, Gries23found an exponent of atomic number less than 2/3 to bein better agreement with IMFP data. Gries suggestedZ0.5. We shall choose an exponent of 0.45, which gives aslightly better Ðt.

Figure 7 shows that even for gold, the highest Zelement plotted, the minimum in AL lies below E\ 50eV, whereas the minimum in the IMFP curve lies above100 eV, as discussed by Tanuma et al.14 The TPP-2Mequation of Tanuma et al.13 includes Ðtting parametersC and D speciÐcally to model the minimum in theIMFP at this energy. Our present purpose is to modelAL, and for this the partial linearization resulting fromelastic scattering e†ects described in part III22 suggestsan additional constant d.

We can deÐne two equations, CS1 and CS2, each ofwhich may prove useful in estimating AL, depending onhow much one knows about the specimen matrixbeforehand. Equation CS2 is likely to be the most accu-rate, whereas CS1 requires the least independent infor-mation (e.g. no knowledge of material properties suchas density is required). Both equations are very simpleand easy to use. Equation CS1 in particular shouldallow many types of layered structure to be quantiÐedin units of monolayer thickness without having to con-sider absolute length units such as nanometres orA� ngstroms.

Therefore, our Ðrst proposed approximation equationis

jAL/a \ kG EZ0.45[ln (E/27) ] 3]

] dH

(11)

where, again, E is the energy (in eV) and Z is the atomicnumber of the matrix. Performing a least-squares Ðt toall our Monte-Carlo results in Table 1, we obtain valuesfor the two parameters k and d, giving equation CS1

jAL\ 0.160G EZ0.45[ln (E/27) ] 3]

] 4H

monolayers

(12)

which predicts the values in Table 1 with a standarddeviation of 11%.

So far we have taken no account of electronic struc-ture, instead using only a fractional power of Z toapproximate the number of electrons available for



inelastic processes. We can begin to include electronicstructure dependence in a simple way by modifyingEqn. (11) to allow a non-unity power in another quan-tity that is dependent on electronic structure, the latticeparameter a. Lattice parameter a can be estimated (innanometres) from

a \ 108A koNAv

B1@3(13)

where mol~1 is the Avogadro con-NAv \ 6.02 ] 1023stant, o is the density of the matrix (in kg m~3) and k isthe average atomic mass of the matrix (in g).

Modifying Eqn. (11) to allow a non-unity power r forlattice parameter gives

jAL\ karG EZ0.45[ln (E/27) ] 3]

] dH

nanometres (14)

and performing a least-squares Ðt to all our Monte-Carlo results we obtain values for r, k and d, givingequation CS2

jAL\ 0.316a3@2G EZ0.45[ln (E/27) ] 3]

] 4H

nanometres

(15)

which, for a typical XPS kinetic energy of 1 keV, i.e.E\ 1000, predicts the values in Table 1 with a standarddeviation of 9%. In fact, the best least-squares Ðt wasachieved using r \ 1.52, but the simpler value of r \ 3/2was chosen for convenience and does not add signiÐ-cantly to the residuals. The Ðt was then repeated todetermine the optimum values of the remaining threeparameters, giving the values shown in Eqn. (15). Figure9 compares the AL predicted by Eqn. (15) with theMonte-Carlo results listed in Table 1 for a typical signalelectron energy of 1000 eV. Note that the constant 27that appears is the logarithm of both CS1 and CS2 isnot arbitrary, but is simply one hartree in units ofelectron-volts, rounded to the nearest integer for con-venience.

If one knows the density of the matrix, one can useEqn. (13) to estimate a and use CS2 to obtain a valuefor the AL (in nm) that is a little more precise than thatpossible using CS1. Both CS1 and CS2 are extremelyeasy to use, and due to the fortunate cancellation ofelastic and inelastic scattering e†ects (described ingreater detail in part III22) they have fewer parametersthan Tanuma et al.Ïs semi-empirical equation forIMFP.13

Neither of the above equations should be usedoutside the energy range 50È2000 eV. Below 50 eV inparticular, a number of the inherent approximationsrapidly become invalid. Furthermore, because it isextremely difficult to obtain reproducible measurementsof XPS and AES peak intensity measurement belowD100 eV, the usefulness to the analyst of any predictiveequations much below this energy is itself rather ques-tionable.

UNCERTAINTY BUDGETS

We need to estimate :(1) The uncertainty in AL that results from using an

approximate equation such as CS1 or CS2 to esti-mate it.

(2) The accuracy of the Monte-Carlo results themselves,from which CS1 and CS2 were developed.

Accuracy of equation CS2 for estimating AL

Figure 10 compares the residuals obtained when onesubtracts the Monte-Carlo AL from that estimatedusing CS2. Each point represents one of the 27 elementsfor an energy of 1000 eV. On the vertical axis we haveplotted the sum of the ps- and f-sum rule errors for theoptical data used by Tanuma et al.,26 which they usedto estimate the likely sign and magnitude of the error in

Figure 9. Attenuation lengths for 27 elements predicted using a simple semi-empirical equation CS2 ÍEqn. (15)Ë and from the detailedMonte-Carlo results listed in Table 1. Equation (15) is simple to use and of sufficient accuracy for most layer thickness measurements.



Figure 10. There is a correlation between the apparent residualerrors in using the simple semi-empirical equation CS2 ÍEqn.(15)Ë and the estimated error in the original optical data used byTanuma et al . (TPP). This suggests that, for the four elements Al,Nb, Re and Hf, the semi-empirical equation CS2 provides a moreaccurate estimate of attenuation length than the Monte-Carlo cal-culations themselves.

their tabulations of IMFP for both elements and com-pounds. The f-sum error gives a rough estimate of thelikely error in the IMFP at higher energies, while theps-sum error gives a rough estimate of the likely errorat low energies. Here we have simply added the two togive a rough estimate of the likely error in the IMFPvalue used as part of the Monto-Carlo AL calculation.The important point to note is that the residualsobtained in using CS2 are correlated with the likelyerrors in the IMFP input data, at least as far as the fourlabelled “outridersÏ Al, Nb, Re and Hf are concerned.We can conclude that, for these four elements at least,the equation CS2 is likely to give a more accurate esti-mate of AL than the Monte-Carlo results in Table 1,due to uncertainties in the optical data on which thosefour sets of IMFP values are based.

Figure 11 shows the residuals obtained when onesubtracts the Monte-Carlo AL from the estimated usingCS2, and in the same way subtracting the optical IMFPof Tanuma et al. from the estimate given by theirTPP-2M equation.13 The four open circles represent Al,Nb, Re and Hf, which the independent indication pro-vided by the sum rule indicates are less reliable than therest. Each Ðlled circle represents one of the 23 remain-ing elements. Equation CS2 is able to predict the ALvalues of these 23 elements to a standard deviation of6% at both 200 eV and 1 keV. This is therefore a rea-sonable guide to the uncertainty resulting from the useof CS2 to estimate AL, provided that 6 O ZO 83 and100 eVO EO 2000 eV.

Uncertainty in the Monte-Carlo calculations

Stochastic uncertainty from Monte-Carlo calculation. After2000 iterations, the remaining stochastic variance in theAL values of Table 1 is insigniÐcant compared to theother, systematic, uncertainties involved.

Figure 11. Residuals from the equations CS2 and TPP-2M13 for27 elements for electron energies of 200 eV (a) and 1 keV (b). Thefour elements Al, Nb, Re and Hf are denoted by open circles (thelarge uncertainty in these four values, judged from sum ruleresiduals in the underlying optical data, means that they are muchless reliable than the other 23 values). This scatter of the CS2residuals has a standard deviation of only 0.9 at 1000 eV forA�these 23 elements.

Uncertainty in screening potential. The Coulomb potentialof a given atomic nucleus is always the same, but thescattering of low-to-medium energy electrons by theatom depends on the screening provided by atomic elec-trons, which in turn depends on electronic conÐgurationand hence on how the atom is bonded to its neighbours.We should therefore expect, for example, di†erentelastic scattering behaviour in di†erent allotropes of thesame element. Fortunately this a†ects the screening ofthe nuclear charge only far from the nucleus, and there-fore has a large e†ect only on the likelihood of electronsbeing scattered through small angles, to which AL cal-culations are rather insensitive. However, this hasnevertheless led to previous results by other authors



being questioned,27 so it is worthwhile attempting toquantify the magnitude of this uncertainty.

Table 1 comprises ALs calculated from relativisticHartreeÈFockÈSlater calculations of scattering cross-sections for isolated atoms. Czyzewski et al.21 alsocalculated cross-sections using a ThomasÈFermi/MuffinÈTin28 model of the type often used very suc-cessfully in dynamic low-energy electron di†raction(LEED) calculations for metal surfaces. In terms ofpotential far from the nucleus, these two potentials use-fully bracket all those that would expect to encounter ina real specimen, and we can compare them to judge thereliability of applying the ALs of Table 1 to a specimenin which the bonding is unknown.

Czyzewski et al.21 present ThomasÈFermi/MuffinÈTin cross-section sufficient to calculate ALs for 18 of the27 elements given in Table 1. Rather than tabulate these18 elements separately, in Fig. 12 we plot the ALs calcu-lated from these cross-sections divided by those ofTable 1. Despite having total elastic scattering cross-sections that di†er substantially (typically by a factor ofD2), the ALs calculated from them agree very well, withthe mean of the plotted ratios for the 18 elements beingclose to unity at all energies between 50 eV and 2 keV.The standard deviation of the ratio evaluated for the 18elements at each energy gives us a means to judge theuncertainty in the ALs in Table 1. These standard devi-ations have been used to produce the inset plot in Fig.12. One can see that, in quantifying the thickness ofoverlayer/substrate structures, this source of uncertaintyis only about ^1.5% for 1 keV signal electrons, risingto about ^2.5% at 200 eV at the one standard devi-ation (68%) conÐdence limit. This is perfectly acceptablecompared to the other uncertainties involved.

Typical overall uncertainty

Let us compose an “uncertainty budgetÏ for AL in atypical analytical problem:(1) Uncertainty in the AL estimated by CS2 : ^6%.

(2) Uncertainty due to atomic potential : ^1.5%.(3) Uncertainty due to use of a single AL value to

model elastic scattering (assuming emission angle\58¡) : ^5%.

Adding in quadrature we obtain a standard uncer-tainty of D8%, so that conservative 95% conÐdencelimits might be ^20%. This percentage uncertaintyapplies also to thickness measurements performed usingan AL value estimated in this way.

This uncertainty is acceptable for most analyticalwork. It is likely that an attempt to achieve signiÐcantlygreater accuracy would require very signiÐcant addi-tional e†ort.

COMPARISON WITH PUBLISHEDEXPERIMENTAL AND THEORETICALATTENUATION LENGTHS

Experimental data

Fabrication of substrate/overlayer reference materialsfrom which AL values can be measured is often difficult.The overlayer needs to be laterally homogeneous,nearly atomically Ñat and have a uniform thicknesscomparable to the AL. Uniform thickness is particularlydifficult to achieve, island growth being the most ener-getically favourable mode for vacuum or vapour deposi-tion of most materials on most substrates. Therefore,the most reliable AL measurements result from studiesof thin passivating oxide layers. Figures 13 and 14 showexperimental values for AL in Si and Al, respectively,taken from the measurements of Tracy,29 Zaporozh-chenko et al.,30 Flitsch and Raider,31 and the inter-laboratory comparison conducted by Marcus et al.12Agreement with the Monte-Carlo results is clearlyextremely good, and our empirical approximation CS2[Eqn. (15)] performs rather well given the scatter of theexperimental data, especially in the case of silicon.

Figure 12. Ratios of attenuation length (AL) calculated from two feasible models of atomic potential : Thomas–Fermi/Muffin–Tin andrelativistic Hartree–Fock–Slater (RHFS). Although the total elastic scattering cross-sections of these two potentials differ considerably,particularly at low energies, this difference is largely due to the small angle scattering that has little effect on AL. Inset : plot of the standarddeviation of these ratios as a function of energy. This represents a good estimate of the uncertainty in the results in Table 1, arising from thedifficulty in choosing an accurate but simple atomic potential.



Figure 13. Literature measurements of attenuation length byZaporozhchenko et al .30 and Flitsch and Raider31 New(…) (=).Monte-Carlo results are plotted as a continuous curve, and anempirical approximation to those results ÍEqn. (15)Ë as a brokenline. Agreement between all three is extremely good. Note thatthere are no adjustable parameters in this plot.

Figure 14. Literature measurements of attenuation length byTracey29 and Marcus et al .12 New Monte-Carlo results(…) (=).are plotted as a continuous curve, and an empirical approximationto those results ÍEqn. (15)Ë as a broken line. Note that there are noadjustable parameters in this plot.

Figure 15. Literature measurements of attenuation length byFlitsch and Raider31 and Fulghum et al .34,35 The(…) (=, >).empirical approximation developed here ÍEqn. (15)Ë is plotted as abroken line. Note that there are no adjustable parameters in thisplot.

Figure 16. Literature measurements of attenuation length byBattye et al .33 Olefjord et al .32 and Marcus et al .12(…), (=) (>).Our empirical approximation, Eq. (15), based on a fit to Monte-Carlo calculations is shown as a broken line. Note that there are noadjustable parameters in this plot.

Figures 15 and 16 show experimental data for theAL values of electrons in the corresponding oxide over-layers, from measurements tabulated by Flitsch andRaider,31 Olefjord et al.,32 Marcus et al.,12 Battye,33Fulghum et al.34 and Fulghum.35 Although no Monte-Carlo calculations for compounds were used indeveloping CS2 (and detailed discussion of how toapply CS1 and CS2 to compounds must be describedelsewhere36), the empirical equation nevertheless esti-mates AL in these oxides very well, given the scatter inavailable measurements.

Theoretical data

Jablonski has recently published extensive tables37 ofparameters for analysis of homogeneous specimens byXPS, derived from careful Monte-Carlo calculations.These tables do not concern AL in substrate/overlayersystems, however, and they are therefore not directlycomparable with the values tabulated here.

The most extensive tabulated data are those of Ebelet al.38 They gave values of the escape depth (assumingemission normal to the surface) for 32 elements at 200eV intervals between 200 and 2400 eV. These valueswere calculated using the Monte-Carlo simulation pro-gramme described earlier by Jablonski and Ebel,39 theIMFP values of Penn40 and elastic scattering cross-sections calculated from a ThomasÈFermi potential. Allof these input data are drawn from di†erent sources tothose of the present study, and the escape depth (ED)itself is a di†erent quantity to the ALs that we calculate ;however, they should be examined because their deÐni-tions are similar and their values should be identical inthe limit of weak elastic scattering. The ED values tabu-lated by Ebel et al., when plotted as ratios to the PennIMFP values on which they are based, show a largescatter of approximately ^10% due to the Monte-Carlo technique alone. This is a much greater uncer-tainty than for the AL values we report here, due to theuse of the Berger-Doggett technique. Nevertheless,trends in the value of ED/IMFP reported by Ebel etal.38 are in qualitative agreement in two essential pointswith the AL/IMFP value that we present in Fig. 8 :there is a general trend to smaller ED/IMFP values



with increasing atomic number ; and for low atomicnumber elements there is a trend to larger ED/IMFPratio as kinetic energy increases in the range 200È2400eV.

Seah proposed1 the following empirical equation forAL based on Monte-Carlo results published byJablonski

jALjimfp

\ (1[ 0.028JZ)[0.501] 0.0681 ln (E)] (16)

This simple equation models the Monte-Carlo results ofTable 1 extremely well for carbon but performs ratherpoorly for large Z elements (even when one allows theÐtted parameter values of 0.028, 0.501 and 0.0681 to beadjusted) because it assumes the ratio of IMFP to ALto be monotonic for all elements, which as one can seefrom Fig. 8 is not the case for Period 6 and some Period5 elements.

SURFACE ROUGHNESS AND ANALYSERACCEPTANCE

In the present work, most of the e†ect of elastic scat-tering in AES and XPS has been incorporated bysimply adjusting the value of the characteristic depthparameter that occurs in SLA quantiÐcation. Some pre-vious authors have gone further, and used this param-eter to account for other e†ects, such as surfaceroughness and Ðnite analyser angular acceptance. Infact there is good reason for not trying to lump all thecomplexity of an analysis into one (however wellchosen) parameter. Including the e†ect of Ðnite analyseracceptance, for example, leads to the use of di†erentALs for a hemispherical analyser as opposed to a cylin-drical mirror analyser for the same specimen ; the ALbecomes an uncomfortable mixture of material andinstrument properties, and one has to compile largetables to take account of possible roughnesses andanalysis geometries.

The simplest approach conceptually is to use ALs astabulated here, to account for most of the e†ect ofelastic scattering in the specimen, while dealing withessentially geometrical e†ects such as roughness andanalyser acceptance separately in the analysis.

FUTURE THEORETICAL DEVELOPMENTS

It should be remembered that the AL is a useful deviceonly in the context of the traditional, greatly simpliÐed,quantiÐcation equations ; Powell41 has pointed out thatif one develops quantiÐcation equations that take intoaccount elastic scattering or Ðnite analyser acceptance(for example) by modelling these processes explicitly,then AL values become irrelevent and one returns tousing intrinsic material properties such as IMFP, trans-port mean free path (TrMFP),42 di†erential elastic scat-tering cross-section, etc. This is the more intellectuallysatisfying approach, but such a route has yet to be setout in the literature, especially for the type of inhomo-geneous specimens that are most commonly encoun-

tered by AES and XPS analysts. Therefore, whileMonte-Carlo or transport theory calculations are best,it is clear from the uptake of earlier ideas that practicalanalytical work requires simple equations to extract asmuch information as rapidly as possible. Most analystswill generally pass over any algorithm that they do nothave time to apply rigorously in favour of a robust,simple equation that allows them to control theirexperiment conceptually. Provided that it has adequateaccuracy, there is no reason why such a simple equationshould not be used.

Topics that justify future study include :(1) A wider range of AL measurements for Ðlms whose

morphology has been independently assessed, sayby scanning tunnelling microscopy or atomic forcemicroscopy.

(2) ConÐrmation of the usefulness of CS1 and CS2 forstructures in which the layers have very di†erentelastic scattering strengths (as measured by trans-port mean free path, for example).

(3) ConÐrmation of the usefulness of CS1 and CS2 fornon-isotropic emission in XPS for common instru-ment geometries.

CONCLUSIONS

(1) An emission angle of 45¡ is a “magic angleÏ for elasticscattering at which errors in the measurement ofoverlayer thickness are minimized.

(2) Attenuation lengths have been deÐned as thoselength parameters that give the most accurateresults when used in the traditional SLA quantiÐca-tion equations applied to overlay/substrate struc-tures.

(3) Elastic scattering causes ALs to be typically between10% and 25% shorter than IMFPs.

(4) Attenuation length values have been calculated for27 elements using a rapid and accurate Monte-Carlo algorithm.

(5) Under normal analysis conditions, and provided nodata are acquired at emission angles greater than58¡, these ALs will lead to a quantiÐcation that isaccurate to on the depth scale.^(5% ] 1 A� )

(6) For accuracy, the corresponding^(10% ] 1 A� )upper limit of emission angle is 63¡.

(7) Semi-empirical equations CS1 and CS2 have beendeveloped to allow easy estimation of ALs in anysolid over the energy range 100È2000 eV. Theseequations take account of the correct low-energyelastic scattering behaviour of large Z elements forthe Ðrst time.

(8) Equation CS1 is

jAL\0.160G EZ0.45[ln (E/27)]3]

]4H

monolayers

where E is electron kinetic energy (in eV) and Z isthe average atomic number of the matrix. EquationCS1 should be used when one has no knowledge ofthe lattice parameter or density of the matrix. Onecan easily quantify layer thicknesses in units ofmonolayers without having to express these innanometres or A� ngstroms.



(9) Equation CS2 is

jAL\0.316a3@2G EZ0.45[ln (E/27)]3]

]4H

nanometres

and does require knowledge of the lattice parametera in nanometres or an estimate of it from Eqn. (13)using a known value for the density of the matrix.Equation CS2 is more accurate for estimating ALs,involving an additional uncertainty of only about^6% (at a typical kinetic energy of 1000 eV) com-pared to the detailed Monte-Carlo calculations.

This is acceptable in virtually all routine analyticalwork.

Acknowledgements

We thank the authors of Ref. 21 for a copy of their database of di†er-ential scattering cross-sections, and Dr Hugh Bishop for useful com-ments and a critical reading of the manuscript.

The work described in this paper was supported under contractwith the UK Department of Trade and Industry as part of theNational Measurement System Valid Analytical Measurement Pro-gramme.

REFERENCES

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APPENDIX: NUMERICAL PROCEDURES

A more detailed description of the Monte-Carlo methodused to calculate Table 1 may be useful.

The Monte-Carlo calculations were conductedaccording to the “Berger-DoggettÏ method set out inrelated paper I6 using programs written in MATLABon a personal computer. These routines were validatedin paper I comparison with analytical results for the

two special cases of negligible and isotropic elastic scat-tering.

The intensities of overlayer and substrate wereobtained by integration of equation (25) of paper I.6Therefore, a single Monte-Carlo calculation gave us theratio of overlayer to substrate intensities for any over-layer thickness viadA ,

(IA/sA)(IB/sB)

\;i/1N ;

n/0q ;m/0n #(g

m(i))A

nm(i) gm(i)[1[ exp ([dA/jimfp g

m(i))]

;i/1N ;

n/0q ;m/0n #(g

m(i))A

nm(i) g

m(i) exp ([dA/jimfp g

m(i)) (A1)



where successive random electron paths in the simula-tion are labelled i\1, 2, 3, . . . . Equation (A1) is analo-gous to Eqn. (3), but taking elastic scattering intoaccount by the use of the Berger-Doggett method. Thecalculations involved Monte-Carlo simulations usingthe Berger-Doggett method for a large number of iter-ations. These calculations were performed to 16 digits ofprecision by the use of the MATLAB program.Occasional combinations of random scattering anglescan lead to an almost singular A matrix, which canamplify rounding errors. Therefore an “error budgetÏwas maintained for each A matrix calculated, and if suf-Ðciently singular to render these 16 digits inadequate,the scattering angles were passed to a separate routinethat allows calculation of that particular iteration to berepeated to over 1000 digits of precision. The “round-o†error budgetÏ ensures that this was sufficient to avoidany error in our AL values, to well beyond the numberof digits of precision quoted in Table 1. Other tests were

performed to conÐrm the reliability of the Monte-Carlocalculations, such as running the program Ðve timeswith di†erent initial “seedsÏ for the random number gen-erator, to ensure that the generator was sufficientlyrandom to give us conÐdence in the ALs to the numberof decimal places that we quote.

Note that since this work was begun, several softwarepackages have become available that allow arbitraryprecision arithmetic to be performed much morequickly. The Berger-Doggett simulation methodappears particularly suited to such systems.

Our AL values in Table 1 are tabulated for fewerkinetic energies than the Tanuma et al. tabulatedIMFPs, because our Monte-Carlo calculation alsorequired the relativistic HartreeÈFockÈSlater di†erentialelastic scattering cross-sections calculated by Czyzewskiet al.,21 which were not calculated for kinetic energies of1000È2000 eV.


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