SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

Elastic Scattering Corrections in AES and XPS.III. Behaviour of Electron Transport Mean FreePath in Solids for Kinetic Energies in the Range100 eV Æ E Æ 400 eVs

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

Quantitative Auger electron spectroscopy (AES) and x-ray photoelectron spectroscopy (XPS) depend on an accu-rate knowledge of the correct depth scale of emission of the signal electrons. This depends on both inelastic andelastic scattering processes occurring in the specimen under analysis.

A previous paper showed that the depth scale in AES and XPS is signiÐcantly inÑuenced by unexpected structurein the transport mean free path for electrons in high-atomic-number elements over the approximate energy range100–400 eV. This behaviour is implicit in Mott cross-sections, which are known to be reliable and have been used inlow-energy electron di†raction (for example) for decades. This paper examines the features of electron–atomelastic scattering over the range of energy and atomic number important in XPS and AES analysis. In particular,the third and fourth partial waves, together with LevinsonÏs theorem are shown to give rise to the structure in thetransport mean free path. This is a transport cross-section analogue of the Ramsauer–Townsend e†ect exhibited atlow energies in total elastic scattering cross-sections. 1997 by John Wiley & Sons, Ltd.(

Surf. Interface Anal. 25, 447È453 (1997)No. of Figures : 9 No. of Tables : 0 No. of Refs : 15

KEYWORDS: XPS; AES; elastic scattering ; IMFP; transport mean free path

INTRODUCTION

In the preceding paper1 we simulated electron transportin a classical overlayer/substrate structure using anaccurate Monte Carlo method2 based on elastic scat-tering cross-sections as calculated by Czyzewski et al.3We examined how far the e†ects of elastic scattering canbe taken into account by retaining the usual quantiÐca-tion equations, but replacing the inelastic mean freepath (IMFP) value that appears in them with an attenu-ation length (AL) with a di†erent numerical value. Themost unexpected feature of the results presented was thelarge AL/IMFP ratio in the 100È400 eV kinetic energyregion for the Period 6 elements considered (Hf, Ta, W,Re, Os, Ir, Pt, Au and Bi). This e†ect already has a Ðrmquantitative basis because the Monte Carlo calculationsand the relativistic scattering cross-sections on whichthey are based are reliable at these energies. We nowexamine qualitatively the physical origin of this feature.

* 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.

MODELLING TRANSPORT CROSS-SECTIONSFOR XPS AND AES

The most important single value (as opposed to theentire di†erential scattering cross-section) in this elec-tron transport problem is the transport mean free path(TrMFP) This is sometimes also known as thejtr .4,5momentum-transfer mean free path, because it is ameasure of how far an electron must travel before itsinitial momentum is transferred to the matrix atoms byelastic scattering

jtr\ (Nptr)~1 (1)

where N is the atomic number density and is theptrtransport cross-section

ptr\ 2nP0

n dpd)

(h)(1[ cos h) sin h dh (2)

Figure 1 shows three separate mean free paths forelectrons in gold : the IMFP from Tanuma et al.6 andthe elastic scattering mean free path (EMFP) and theTrMFP (both calculated by numerical integration fromthe di†erential cross-sections of Czyzewski et al.3).

Clearly the TrMFP does not increase monotonicallyas one might expect, but has a maximum at D200 eV

CCC 0142È2421/97/060447È07 $17.50 Received 15 November 1996( 1997 by John Wiley & Sons, Ltd. Accepted 18 February 1997

448 P. J. CUMPSON

Figure 1. The three important mean free paths for electrons in gold. Note the peak in the transport mean free path at a kinetic energy ofÁ200 eV. These accurate theoretical calculations of elastic scattering properties are computationally intensive, and therefore have beenconducted at only a small number of discrete energies indicated by the points in this figure. These are joined by straight lines to guide theeye.

that gives rise to an AL value very close to the IMFPvalue, unlike those at other energies. Calculations forelements from Z\ 47 to Z\ 83 show a gradualincrease in TrMFP in the region of D200 eV as atomicnumber increases. Monotonicity is lost at aroundZ\ 64 (gadolinium). The TrMFP values tabulated byLiljequist et al.7 also show this feature.

Figure 2 shows TilininÏs simple approximation for theTrMFP,8 derived for atoms of any atomic number, forscattering of electrons of kinetic energy greater thanD27 Z2@3 eV (i.e. valid for electron energies [150 eV inthe case of Al, [350 eV for Ag or [500 eV for Au).

Tilinin analysed the ThomasÈFermi potential in aquasi-classical scattering approximation9 in which the

di†erence between successive partial wave phase shifts isassumed to be small. Although there are a number ofproblems in applying the ThomasÈFermi potential tocalculate di†erential cross-sections at these energies(notably its unrealistically large Ðeld far from thenucleus), these a†ect the small-angle part of the cross-section most strongly and transport cross-sections aremuch less sensitive to this. The continuous line in Fig. 2shows TilininÏs predicted transport cross-section plottedagainst a reduced energy, giving a “universal curveÏ validfor all atomic numbers. Also shown are transport cross-sections from the di†erential cross-section data of Czy-zewski et al.3 for Ag and Au, calculated by numericalintegration of Eqn. (2). One can see that TilininÏs

Figure 2. Transport mean free path (TrMFP) for electrons in gold and silver. Points represent accurate numerical calculations based onintegration of the Mott cross-sections published by Czyzewski et al .,3 which used partial wave expansion in the relativistic Hartree–Fock–Slater potentials of Au and Ag. The continuous line is Tilinin’s ‘Universal Function’ SÉ, which approximates the TrMFP values well for allelements at medium energies. However, at energies eV the transport cross-section is substantially less than that predicted by the low[400energy limit of Tilinin’s SÉ function.

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

UNEXPECTED BEHAVIOUR OF TRANSPORT MFP 449

expression is excellent at medium reduced energies andonly D10% too high for larger energies. Even at verysmall energies for low Z elements the approximationperforms well, but for gold it fails to predict theminimum in the transport cross-section at D200 eV,which we have seen gives rise to a maximum in theTrMFP and hence also in the AL/IMFP ratio at aboutthis energy.

E†ects such as exchange, polarization and the ratherpoor asymptotic behaviour of the ThomasÈFermipotential are well known to cause difficulties in calcu-lating the total elastic cross-section (or the di†erentialcross-section) below D100 eV.10 However, as oneexamines the behaviour of electrons of progressivelylower energy, below TilininÏs chosen energy re� gime, thepeak in the TrMFP at D200 eV is the Ðrst deviationfrom his expression, and this is caused by none of theproblems listed above. Finding such a structure in theTrMFP is a surprise, because it might be expected to bemore robust with respect to these e†ects than the totalcross-section, which shows no such feature.

APPROXIMATE PHASE-SHIFT ANALYSIS BYWKB METHOD

The explanation lies in the fact that at energies less thanabout 27 Z2@3 eV the di†erences between successivepartial wave phase shifts are not small. In the kineticenergy region of interest for XPS and AES, two “reso-nancesÏ turn out to be important.

Although Mott cross-sections have been carefully cal-culated by a number of authors, and should be used forall numerical work in this Ðeld, it is nevertheless instruc-tive to look at the simplest model that will exhibit thisphenomenon, so as to gain a semi-quantitative under-standing of what is happening. We can use the WKBmethod to Ðnd approximate phase shifts in simplescalar wave mechanics scattering from an atomic poten-tial. In the low energy region where TilininÏs quasi-classical approximation is breaking down, it seemssensible, for the sake of clarity, to avoid using aThomasÈFermi potential because this will exaggeratethe low-l phase shifts, due to its r~3 asymptotic behav-iour. Instead we can use an equally simple potential, inatomic units, based on a sum of three exponentials thatapproximate the results of accurate DiracÈHartreeÈFockÈSlater calculations by Salvat et al.11

V (r)\ [ Zr

(A1 e~a1r] A2 e~a2r ] A3 e~a3r) (3)

Because the values of the parameters are derived from arelativistic calculation, they will give a good representa-tion of the true screening function, especially for high Zelements. Salvat et al.11 tabulated values for parameters

and for most elements. BuildingA1 , A2 , A3 , a1 , a2 a3on the isotropy of this potential, we can ensure the cal-culation is only one-dimensional by adding a centrifugalterm to form an e†ective radial potential InVeff(r).atomic units this is

Veff(r)\ V (r) ] l(l ] 1)r2 (4)

The WKB method involves integrating a “local wave-numberÏ of the incident electron over distance to obtainthe phase shift of each partial wave12

dlWKB\ lim

r?=

CPa

rJ'(r@) dr@[ kr ] (l ] 12)

n2D

(5)

where '(r) represents the “local wavenumberÏ once thelocal potential is considered (the e†ective potential thatincludes a centrifugal term)

'(r) \ k2[ 2Veff(r) (6)

or

'(r) \ 2E0[ 2V (r) [ (l ] 12)2r2 (7)

where r \ a is the largest zero of '(r) (the classicalturning point) and k is the incident electron wavenum-ber in atomic units). The term l(l] 1) has(k2\ 2E0been replaced with to improve the accuracy of(l ] 12)2the WKB results, as is discussed in most texts. Theabove expressions allow us to calculate the phase shiftof each partial wave to reasonable accuracy with nomore than a single numerical integration for eachenergy, a calculation simple enough to be performed ona programmable calculator.

Figure 3 shows the results of such calculations forscattering from an isolated gold atom. The atomicpotential V (r) is sufficient to overcome the centrifugalcontribution to the e†ective potential for l O 3 so that

is attractive for some r but not (in the case of an AuVeffatom) for l [ 3. The result is that forlimE?0 d

l\ 0

l[ 3.Therefore, those partial waves that vanish at zero

incident energy, i.e. l P 4, do not penetrate at low ener-gies due to the centrifugal barrier. This is made formalin LevinsonÏs theorem (Ref. 9, p. 553). LevinsonÏstheorem states that will vanish at zero energy if thered

lare no atomic bound states of orbital angular momen-tum l. If, however, there are a total of m states(regardless of principal quantum number) with thatangular momentum, then

limE?0

dl\ mn (8)

Therefore, if our WKB calculation was accurate downto the lowest energies, we would see the E\ 0 inter-cepts at the left-hand side of Fig. 3, representing the

Figure 3. Phase shifts for elastic scattering of electrons from goldatoms.

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

450 P. J. CUMPSON

total multiplicities of l \ 0, 1, 2 and 3 states within thegold atom.

The total elastic cross-section and the transportpelcross-section can both be calculated easily fromptrthese phase shifts ; most quantum mechanics texts derivethe relation

pel\4nk2 ;

l/0

=(2l ] 1) sin2 d

l(9)

for the total elastic cross-section, and the analogousexpression for the transport cross-section is13

ptr\4nk2 ;

l/0

=(l] 1) sin2 *

l(10)

where Both equations arise from simple*l\ d

l`1 [ dl.

orthogonality properties of Legendre polynomials. Thevital point here is that the transport cross-section is sen-sitive to the di†erence in phase between successivepartial waves : we can see from Fig. 3 that and ared3 d4particularly important and that varies*3\ d4[ d3quite rapidly over 100È400 eV. We can deÐne partialtransport cross-sections

ptr,n \ 4nk2 (n ] 1) sin2 *

n(11)

so that the transport cross-section is the sum of thesepartial transport cross-sections

ptr\ ptr,0 ] ptr,1 ] ptr,2 ] É É É (12)

At energies of a few hundred electron-volts, only thelowest partial transport cross-sections contribute, sothat if one or two of them disappear at a particularenergy due to the corresponding values passing*

nthrough some multiple of n, this can cause a signiÐcantdrop in transport cross-section (and hence a peak inTrMFP) around that energy. This is a “transport ana-logueÏ of the RamsauerÈTownsend e†ect for total elasticcross-section, which is well known in low-atomic-number elements at low energies. Analogues of the

RamsauerÈTownsend e†ect have been found before :14minima in the di†erential cross-sections of certain ele-ments at particular scattering angles. However, for XPSand AES it is surprising and signiÐcant that such ane†ect can be seen in the transport cross-section, an inte-grated measure of scattering that might have beenexpected to behave more like the total elastic cross-section that shows no structure at these energies.

Figure 4 shows calculated values for and from*2 *3our WKB calculation for a progression of elementsbetween Z\ 74 (tungsten) and Z\ 83 (bismuth). As wemove further up in atomic number to platinum andgold, zeros of both partial transport cross-sectionsalmost coincide at D200 eV. This leads to a fall inoverall transport cross-section around Z\ 78 (E\ 200eV), which is exactly the behaviour seen in morecomplex relativistic calculations of the type used inplotting Fig. 1.

Figure 5 illustrates, schematically, the progressionshown in the calculated results plotted in Fig. 4. TheWKB method is imperfect in that it integrates only upto the classical turning point of the incident particle, sothat the values (and hence the values) all tendd

l*

lunphysically to zero at very low energies. Nevertheless,Fig. 5(c) shows how particularly low transport cross-sections can arise. For any two successive partial wavedi†erences and as Z is increased there will be a*

k*

k`1 ,particular energy at which both are close to [n invalue, and hence the corresponding partial transportcross-sections almost vanish. This is not to say that thisbehaviour is observable for any pair of partial wavephase di†erences, because this may occur for an atomicnumber larger than that of any stable element, or atbelow D100 eV where our scattering approximationsare invalid.

To delineate the regions of ZÈE space where wemight expect small transport cross-sections, we can plotcontours representing zeros of each successive partialtransport cross-section that occur for successive *

lvalues passing through a value of [n. Figure 6 showsschematically what we should expect this contour plotto look like.

Figure 4. Contributions to the transport cross-section resulting from the difference between the second and third shown as a(¼D2,

continuous line) and the third and fourth shown as a dashed line) partial waves. At around Z ¼78 and E ¼200 eV both of the phase(¼D3,

differences pass through Én, causing these two partial transport cross-sections to disappear and giving a peak in the transport cross-sectionin these elements at an electron kinetic energy of Á200 eV.

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

UNEXPECTED BEHAVIOUR OF TRANSPORT MFP 451

Figure 5. This figure shows schematically what we should expectfrom WKB calculations of phase shift differences as atomic numberZ increases from (a) to (b) to (c). For two successive partial wavephase differences and we should expect that at a particu-D

kD

k½1lar atomic number shown in (c) both can pass through Én atalmost the same energy. Cases (a), (b) and (c) correspondapproximately to Z ¼74 (tungsten), Z ¼77 (iridium) and Z ¼78(platinum), as shown in Fig. 4.

Very low transport cross-sections might be expectedwhere two of these contours cross. Therefore, it is thepoints of intersection of these contours that are of prac-tical signiÐcance, and here the WKB method can be

Figure 7. Zeros in the partial transport cross-sections of the ele-ments as a function of Z calculated by the WKB method using theatomic potentials of Salvat et al .11

expected to be reasonably accurate. Figure 7 shows theWKB results and is a quantitative conÐrmation of ourschematic picture.

Note that the abscissa is not linear in Z ; instead, it isplotted in units derived from a ThomasÈFermi model,which should ensure that atoms with new angularmomentum states (i.e. l \ 1, 2, 3 . . .) appear at approx-imately equal intervals. Using this abscissa scale, thefamily of contours of might be expected to be*

k\ [n

approximately self-similar, and even though the Z rangeis necessarily small in this Ðgure, one can see that this isat least roughly the case. When examining Fig. 7, bearin mind that below D100 eV this method of calculatingtransport cross-section is unreliable because a numberof approximations (e.g. neglect of polarization,exchange, etc.) break down both simultaneously andrapidly below about this energy. This, together with thefact that the largest atomic number of naturallyoccurring elements is Z\ 92, leaves the simultaneousdisappearance of the second and third partial wave con-tributions at about Z\ 78 and E\ 200 eV as the mosteasily observable occurrence of this e†ect.

Electron spectroscopy may be the only Ðeld in whichthis is currently important ; whilst low-energy electron

Figure 6. Schematic diagram showing zeros in the partial transport cross-sections of the elements as a function of Z. The dashed branchesof these contours should be expected for a WKB calculation because of its use of a classical turning point, but they are not real. Therectangular region outlined in dots is the region for which we have performed WKB calculations, the results of which are shown in Fig. 7.

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

452 P. J. CUMPSON

di†raction calculations are often performed for theseelements and energies, the entire Mott cross-section isused in the calculation without explicitly having to con-sider structure in the TrMFP. Other disciplines, such asastrophysics for example, regularly deal with complexradiative transfer calculations using TrMFPs, some-times at energies in the range 50È400 eV but typicallyonly for light elements where other e†ects are dominant.

APPROXIMATE ENERGY DEPENDENCE OFATTENUATION LENGTH

Consider two elements as examples : aluminium as aarchetypal example of a free-electron-like metal ; andgold as an example of a transition metal of fairly highatomic number. Both are of great practical signiÐcanceto many users of AES and XPS. In Fig. 8 we Ðrstcompare their IMFPs.

The curves in Fig. 8(a) have been plotted from thevalues tabulated by Tanuma et al.6 but this Ðgureshould really be treated as schematic because we haveextended them (dashed lines) to below 50 eV, the lowestenergy for which Tanuma et al.6 performed calculations.We plot this low-energy schematic extension to showthat the IMFP curve of Al does have a minimum atD20 eV. The important point in comparing Al and Auin this Ðgure is that for the free-electron metal, theminimum in the IMFP occurs at a relatively lowenergy, one that is comparable to the bulk plasmonenergy in Al. Loss of kinetic energy in the excitation ofa bulk plasmon is the dominant loss mechanism for asignal electron escaping from Al metal, and electronenergy-loss spectra of Al always show nice sharp peaksat 16 eV multiples of the plasmon energy. As Zincreases through a particular group of the Periodic

Figure 8. Comparison of (a) the energy dependence of inelasticmean free paths and (b) the transport mean free paths of electronsin aluminium and gold.

Table, the number of metallic electrons per atomincreases. In the Debye model this means that theplasmon energy will increase ; hence, one sees anupward trend in the minimum in the IMFP to [100 eVfor gold. Simple though it is, this Debye model suggestsa trend for the minimum in the IMFP plots to move tohigher energy as Z increases,15 even though the collec-tive excitations are really much more complex.

A cursory look at an electron energy-loss spectrum ofgold will easily show that Au has an extended lossstructure ; signal electrons traversing a gold specimencan lose kinetic energy in larger and less well-deÐnedsteps to collective excitations of Au electrons, particu-larly the Au 4f shell. The minimum in the IMFP curvefor Au is rather Ñattened, and occurs at a higher kineticenergy than for Al. If one simply looks at the energyrange important in AES and XPS (say 50 eV upwards),then the IMFP behaviour of Au appears rather morecomplex than that of Al because the IMFP minimumoccurs in this range for Au.

However, when considering attenuation lengths thiscomplexity is cancelled, to a large extent, by elastic scat-tering behaviour. Figure 8(b) shows TrMFP for elec-trons in Al and Au, calculated by integration of thedi†erential cross-sections published by Czyzewski etal.,3 although again we have extended these curves withdashed lines to indicate qualitative behaviour in thelow-energy region where calculations are rather uncer-tain. This plot too, therefore, is schematic. As with theIMFP data, Al behaves relatively simply. The impor-tant point here is that both IMFP and TrMFP inÑu-ence the attenuation length, and the e†ect of thelow-energy complexity of the TrMFP behaviour of goldroughly cancels that of the low-energy IMFP complex-ity. This gives attenuation length values that are veryclose to linear with kinetic energy, as shown in Fig. 9.This suggests that between about E\ 50 eV andE\ 500 eV we may successfully model the energydependence of AL as having a non-zero E\ 0 intercept.

Figure 9. Attenuation lengths for aluminium and gold, showingthat the effects of inelastic and elastic scattering combine to giveapproximately linear energy dependence as a function of energybetween 50 eV and 2 keV. The straight lines are plotted to guidethe eye, and represent no particular least-squares fit.

( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)

UNEXPECTED BEHAVIOUR OF TRANSPORT MFP 453

This is the motivation for including the parameter d inthe semi-empirical equations for estimating AL thatwere developed in Part II.1

CONCLUSIONS

The IMFP of high-atomic-number metals shows con-siderable structure in the low energy range of interest toAES and XPS. This is due to inelastic loss structuresthat result from collective excitations of metallic elec-trons as the signal electron escapes. Elastic scattering,rather surprisingly, gives rise to a compensating e†ect.The TrMFP for elastic scattering passes through amaximum for high-Z elements, due to a combination ofLevinsonÏs theorem (specifying which partial waves canhave Ðnite shifts at zero energy) and the “chanceÏ coin-cidence of phase shift di†erences between second and

third and third and fourth partial waves at aroundZ\ 78. This resonant behaviour is reminiscent of theRamsauerÈTownsend e†ect, but occurs only in thetransport cross-section (the elastic cross-section showsno signiÐcant structure) and at a much higher energy ofD200 eV. This compensation gives rise to a simplerenergy dependence of attenuation length than that ofeither the IMFP or TrMFP separately.

Acknowledgements

The author wishes to thank Dr M. P. Seah for many useful dis-cussions and a critical reading of the manuscript, Dr M. M. ElGomati for useful comments and the authors of Ref. 3 for a copy oftheir database of di†erential scattering cross-sections.

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.

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( 1997 by John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 25, 447È453 (1997)