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Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51

Contents lists available at ScienceDirect

Journal of Electron Spectroscopy andRelated Phenomena

journa l homepage: www.e lsev ier .com/ locate /e lspec

omparison of regularization methods for the inversion of ARXPS data

.W. Payntera,∗, M. Rondeaub

INRS Énergie Matériaux Télécommunications, 1650 boul. Lionel-Boulet, Varennes, Québec, CanadaDépartement de physique, Université de Sherbrooke, 2500 boul. de l’Université, Sherbrooke, Québec J1K 2R1, Canada

r t i c l e i n f o

rticle history:eceived 13 October 2010eceived in revised form5 December 2010ccepted 16 December 2010

a b s t r a c t

Starting from a posited input depth profile of oxygen in a hydrocarbon polymer surface, 100 sets of noisysimulated ARXPS data were created at each of 5 noise levels, 0.3%, 1%, 3%, 6% and 9%. Oxygen depth profileswere then recovered from the noisy simulated data using nine regularized methods, including MaximumEntropy and Tikhonov regularization. The various regularization schemes evaluated were ranked withrespect to their ability to reproduce the input profile, as a function of the level of added noise.

vailable online 25 December 2010

eywords:ngle-resolved X-ray photoelectronpectroscopyoncentration gradient

© 2011 Elsevier B.V. All rights reserved.

aximum Entropy methodikhonov regularization

. Introduction

The adoption of nanometer-scale technologies in industriesuch as those related to information technology has resulted inneed for new metrology tools with a depth resolution of a few

tomic layers. Angle-resolved XPS [1], a technique for the non-estructive depth-profiling of the topmost few nanometers of aaterial surface, has provided a solution to this need [2–4]. InRXPS, the sample is irradiated with a soft X-ray beam and pho-

oelectrons are collected at several photoemission angles. Becausehe photoelectrons generated by the X-ray beam are attenuated bynteractions with the sample as they make their way to the sur-ace, varying the photoemission angle has the effect of varying thehotoelectron path length in the condensed material and hencehe effective sampling depth of the analysis, and allows the sur-ace structure (composition depth profile) to be reconstructed bymathematical manipulation of the data equivalent to an inver-

ion of the Laplace transform. This inversion is an “ill-conditioned”roblem, the extracted depth profile being unstable with respecto random counting noise in the data [1]. A major challenge inhe interpretation of ARXPS data is therefore the development and

haracterization of methods for dealing with the ill-conditioning ofhe inversion problem.

The destabilizing effect of counting noise in the ARXPS data is soevere that, for routine measurements using a laboratory spectrom-

∗ Corresponding author. Fax: +1 450 929 8102.E-mail address: Royston Paynter@emt.inrs.ca (R.W. Paynter).

368-2048/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.elspec.2010.12.025

eter, a quantification of the amount of information available, the“number of degrees of freedom”, is just three [1]. In depth profilesmodeled as a series of line segments joining ten to twenty depth-concentration coordinates, the number of free parameters exceedsthree by a considerable amount. In ARXPS a procedure known as“regularization” [5–7] is generally adopted to deal with this issue,and we investigate this approach in some detail in this paper.

In a “regularized” procedure the optimization of the model pro-file shape amounts to a minimization of a quantity calculated fromthe “joint function”, which has the form {residual norm + ˛ solu-tion norm}. The residual norm is a measure of the goodness of fitto the experimental data given by a calculation of what the pro-posed depth profile would give, such as the �2 statistic. A smallerresidual norm corresponds to a better fit to the data. The solutionnorm is some measure of the complexity of the proposed depth pro-file, the most widely used example being the profile cross-entropy(the so-called Maximum Entropy method [8]). A smaller solutionnorm corresponds to a simpler profile, which generally translatesinto smoothness. The “regularization parameter” ˛ balances thetwo considerations in the joint function: achieving a good fit to thedata, and avoiding unphysical spikes and steps in the depth profilearising from over-fitting the (noise in the) data. In essence, then,regularization can be thought of as a way to limit the complexityof a multi-parameter profile to the extent that it could ideally have

been constructed from a three-parameter model.

In a recent paper [9] we interpreted a single set of ARXPS data,taken on a polystyrene film exposed to an oxygen/helium plasma,comparing the profiles obtained using a variety of mathematicalformulae, or “regulators”, for the calculation of the solution norm

44 R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51

Table 1Input oxygen profile (balance carbon) used in this paper.

Profile point index i 1 2 3 4 5 6 7 8 9 10

304

iafirttrtlsiasht

tdiafhthttt

2

mtmb

I

aotaXgtattaepalkp

Depth zi (A) 0 10 20Oxygen concentration ci (at.%) 32 16 8

n the joint function. Using a common criterion for the selection ofvalue for ˛, we found that although some regulators gave pro-les that were quite similar, other regulators gave rather differentesults. Because we did not know the form of the real profile inhe sample under evaluation, we discussed the profiles obtained inerms of plausibility with respect to our expectations of what theeal depth profile might look like. According to our a priori criteria,he most plausible results were obtained using Tikhonov-c′2 regu-arization, in which the solution norm is calculated as the sum of thequares of the slopes in the profile (see below). However, due to thell-conditioning of the inversion problem, any particular result fromsingle ARXPS experiment is possibly just anecdotal. Had the same

ample been measured a second time, the counting noise wouldave been slightly different, different enough, perhaps, to result inhe calculation of a significantly different depth profile.

In this paper, using Mathematica software (Wolfram), we syn-hesize multiple sets of noisy ARXPS data from a common, knownepth profile, and invert them using various regularization schemes

n an attempt to recover the original profile. The input profilepproximates an exponential decay, characteristic perhaps of a dif-usion process [10], rather than the type of discrete layer profile thatas been a popular test for the inversion of ARXPS data [11–15]. Ashe input profile is known, we can rank the recovered profiles, andence the regularization schemes used to generate them, in a quan-itative manner. The results obtained suggest a new paradigm forhe classification of the various regularization schemes available tohe interpretation of ARXPS data.

. Theoretical

As we are interested mainly in ARXPS data taken on low-Zaterials, i.e. oxygen-treated hydrocarbon polymers, we choose

o ignore the elastic scattering of photoelectrons in our simplifiedodel [16,17]. The starting point for our calculations is therefore

ased on the familiar Beer–Lambert equation

(�) = sK(�)F�[

1 + 12

ˇ(

32

sin2 � − 1)]∫ ∞

0

c(z) exp( −z

� cos �

)dz

(1)

I(�) is the peak intensity at the photoemission angle �, defineds the angle between the normal to the sample surface and the axisf the photoelectron collection optics. s is a scale factor, equivalento the total counting time in a real measurement. K(�) representscombination of instrumental and geometric factors, such as the-ray flux and the analysis area; K is the same for all peaks at aiven photoemission angle. F is the analyzer transmission func-ion, whose value is particular to the photoelectron kinetic energy,nd the value of �, the photoionization cross section, is particularo a given atomic orbital. c(z) is the composition depth profile ofhe element giving rise to the peak; in this work, we will assumeconstant total atom density of 100% as a function of depth and

xpress the concentration of each constituent element in atomic

ercent. z is the depth into the sample perpendicular to the surface,nd � is the photoelectron inelastic mean free path in this ‘straightine’ approximation, whose value is particular to the photoelectroninetic energy. The value of � will also be specific to the sam-le composition; in this paper simulating data taken on oxidized

40 50 60 70 80 902 1 0 0 0 0

polymers, however, we make the simplifying assumption that thevalue of � is negligibly sensitive to variations in the atom concen-trations. ˇ is the photoemission asymmetry parameter and � theangle between the X-ray source and the direction of photoemission[18].

We make several simplifying assumptions in our simulationof a spectrometer with a “parallel collection” geometry [19]. Thesolid angle of collection of photoelectrons is taken to be negligi-bly small, and 16 � values evenly distributed between 24.875◦ and81.125◦ were used. The transmission function F is assumed to beflat with respect to energy and equal to unity. The photoemissioncross-section � was 1 for carbon (C1s peak) and 2.93 for oxygen(O1s peak). The value of ˇ was 2 for both these peaks, and � wasset to � + 15◦. The photoelectron inelastic mean free paths weretaken to be 30.78 A (O1s) and 36.96 A (C1s) calculated from the NISTIMFP database software [20]. As K will be eliminated by cancellationwhen we combine the simulated peak intensities to obtain appar-ent concentrations in at.%, we do not need to consider it further. Weemploy a ten-point linear segment model for the input depth pro-file, which is modeled in terms of ten pairs of depth-concentrationcoordinates (zi,ci), with the profile continuing to infinite depthat the concentration value of the tenth profile point beyond thetenth profile point. The total atom density (carbon plus oxygen)is assumed to be constant, so that ci(carbon) = 100% − ci(oxygen).The zi were uniformly spaced at 10 A intervals with z1 = 0 being thesample surface. The synthetic peak intensity is calculated as [21]

I(�)synth = s�32

sin2(� + 15)

{c1� cos � + �2 cos2 �

i=9∑i=1

(ci+1 − ci

zi+1 − zi

)

×[

exp( −zi

� cos �

)− exp

( −zi+1

� cos �

)]}(2)

using the appropriate values for carbon or oxygen for �, � and the(zi,ci). The input profile used in this paper, expressed as ten (zi,ci)pairs, is given in Table 1. To this “noiseless” synthetic data, I(�)synth

carbon

and I(�)synthoxygen, random noise is added, to give 100 sets of noisy syn-

thetic data. The noise was made to obey Poisson statistics [22] withrespect to the value of I by means of a special command in Mathe-matica. In this way, the noisiness of the data is controlled by varyingthe magnitude of the scale factor s, with the noise being relativelygreater for smaller s values. This emulates the relationship betweenthe counting time and the signal-to-noise ratio in a real spectrome-ter. The noisy synthetic data I′ is then combined to give a simulatedset of ARXPS data in terms of apparent concentration as a functionof the photoemission angle, at . % (�)synth, for example, for oxygen

at.%(�)synthoxygen = 100I′(�)synth

oxygen/�oxygen

I′(�)synthoxygen/�oxygen + I′(�)synth

oxygen/�carbon

(3)

The Poisson noise, in terms of the standard deviations on the

I′(�)synth, i.e.√

I′(�)synth, was propagated through this calculation

following Harrison and Hazell [22], to give the standard devia-tion on at . % (�)synth. Because of the inclusion of the asymmetryparameter, the standard deviation on at . % (�)synth, expressed as apercentage of at . % (�)synth, the “% noise”, varies with �, being largestfor the smallest and greatest values of �. This emulates the way that

R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51 45

Table 2Regularization schemes.

Regularization scheme Joint function

No regularization “No Reg” �2 = 2∑

j

(%Ojsynth−%Ocalc

j)2

�2j

Tikhonov-c2 (oxygen only) “c-sq ox” �2 + ˛ t∑

i

c2i

Tikhonov-c2 “c-sq O + C” �2 + ˛t∑

O,C

∑i

c2i

(c-bulk)2 “c-bulk-sq” �2 + 2˛t∑

i

(ci − c10)2

Maximum Entropy (m = 50) “MaxEnt-50” �2 − ˛t∑

O,C

∑i

{ci − 50 − ci log

(ci50

)}Maximum Entropy (m = bulk) “MaxEnt-bulk” �2 − ˛t

∑O,C

∑i

{ci − c10 − ci log

(ci

c10

)}

Tikhonov-c’2 “Slopes-sq” �2 + 2˛t

i=9∑i=1

(ci+1−ci

t

)2

Tikhonov-c”2 “Curves-sq” �2 + 2˛t

i=8∑i=1

{[(ci+2−ci+1)/t−(ci+1−ci )/t]

t

}2

c′ entropy “Entropy(Slopes)” �2 + ˛t∑

O,C

i=9∑i=1

∣∣ ci+1−cit

∣∣ log{

1 +∣∣ ci+1−ci

t

∣∣}

c′′ entropy “Entropy(Curves)” �2 + ˛t∑

O,C

i=8∑i=1

∣∣ (ci+2−ci+1)/t−(ci+1−ci )/t

t

∣∣ log{

1 +∣∣ (ci+2−ci+1)/t−(ci+1−ci )/t

t

∣∣}W profil

f er.

timugtcfir

sea(�tcds

a

dad

a

f

here indicated by∑

O,C the solution norm is summed over the oxygen and carbon

or the carbon profile. t is the point spacing in the profile model, i.e. 10 A in this pap

he uncertainties on the measured concentrations are distributedn real ARXPS data that we have obtained on plasma-treated poly-

ers using a parallel acquisition spectrometer. The “% noise” figuresed in this paper refers to the average over � and the 100 data setsenerated for each noise level (0.3%, 1%, 3%, 6% and 9% noise). Forhe purposes of comparison, the uncertainties on the oxygen con-entrations in our recent paper [9], relative to the concentrationgures themselves, averaged 7% for a single set of data over theange of photoemission angles employed.

The 100 sets of noisy simulated data, all derived from theame set of noiseless data, are then each interpreted using sev-ral candidate regularization schemes (Table 2). First, 100 profilesre extracted from the 100 noisy data sets without regularization“No Reg”), by optimizing just the fit to the data (minimizing the2 statistic) as the ci in the profile are varied without regard to

he shape of the profile obtained. This should provide the “worstase” scenario with the unbridled potential to fit the noise in theata, the “over-fitting” phenomenon that the various regularizationchemes are intended to mitigate. In Table 2, %Osynth

jis the value of

t.%(�)synthoxygen at the jth photoemission angle in the noisy synthetic

ata set used in the calculation. %Ocalcj

is the value of at.%(�)calcoxygen

t the jth photoemission angle obtained from the extracted profileuring the optimization of the ci, calculated as

t.%(�)calcoxygen = 100I(�)calc

oxygen

calc calc(4)

I(�)oxygen + I(�)carbon

I(�)calcoxygen and I(�)calc

carbon are obtained from the extracted profilesollowing the optimization of the ci to minimize the joint function,

es, otherwise, it is summed over the oxygen profile only, the prefactor 2 accounting

via

I(�)calc = c1� cos � + �2 cos2 �

i=9∑i=1

(ci+1 − ci

zi+1 − zi

)

×[

exp( −zi

� cos �

)− exp

( −zi+1

� cos �

)](5)

using values for � and ci appropriate to either oxygen or carbon. �j

is the standard deviation on %Osynthj

.The same 100 data sets are then evaluated using the nine can-

didate regularization schemes, in sequence. Each of the schemesemploys the �2 statistic (Table 2) to calculate the residual norm,and adds a regulator as a second term in joint function, which is thequantity to be minimized by the variation of the ci in the profile.The value of the regulator is the solution norm, which quantifies thecomplexity of the profile. By simultaneously maximizing the fit tothe data (minimizing the residual norm) and minimizing the com-plexity of the profile (expressed in the solution norm), the intentis to find the simplest profile that provides a reasonable fit to thedata. In the minimization of the joint function via the variation ofthe ci, the ci are constrained so that 10−10 at.% ≤ ci ≤ 100–10−10 at.%.This is necessary for the completion of calculations using regulatorsthat contain a logarithm, and was applied to the other regulatorsfor consistency.

The weighting of the two aforementioned considerations in theoptimization of the profile shape, i.e. achieving a good fit to the dataand limiting the complexity of the depth profile, is controlled by the

regularization parameter ˛. Using an excessively large value for ˛will result in a profile that is too simplified and which gives a poorfit to the data; using an excessively small value for ˛ will result ina profile that contains unphysical spikes, steps, etc. in order to fitsmall excursions in the data that are due to counting noise. A critical

46 R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51

F(p

ch

apgocioatgFh1

ott[tnvci

tibwtdfi

l

extracted profiles. This way of comparing the extracted profile withthe input profile does not however penalise errors in c10, the bulkconcentration, in any special way.

ig. 1. Oxygen depth profiles obtained from the minimization of the joint functionTikhonov-c′′2 regularization) as a function of the logarithm of the regularizationarameter ˛ used in the calculation, for one case of 1% added noise.

hoice in any regularized interpretation of ARXPS data is therefore:ow to decide on a value for ˛.

For the purposes of this paper, in which we wish to compare thebilities of the nine regularization schemes to recover a given inputrofile shape, as a function of the added noise, we employ a “surro-ate” L-curve method [5,23] as a common criterion for the choicef the value for ˛. The joint function is first minimized, varying thei in the oxygen profile, using a wide range of values for ˛ (numer-cally 10−8, 10−7, 10−6, . . ., 106, 107, 108). The range of solutionsbtained can be plotted as a 3-D graph; an example for 1% noisend Tikhonov-c′′2 regularization is shown in Fig. 1. It can be seenhat for values of ˛ smaller than 10−1 A3 at.%−2, the extracted oxy-en profile contains a subsurface spike, a symptom of overfitting.or ˛ values greater than 106 A3 at.%−2, the profile is over-regulated,aving zero curvature. An acceptable value for ˛ must lie between0−1 and 106 A3 at.%−2, therefore.

To construct the L-curve we take these solutions and for eachf the minimized joint functions plot the logarithm of the solu-ion norm against the logarithm of the residual norm, Fig. 2. Joininghese points, the resulting curve has the form of a truncated letter L24]. The L-curve method is to identify the value of ˛ correspondingo the corner on the L-curve where the solution norm and residualorm are minimized simultaneously. It turns out that deriving thisalue for ˛ from the L-curve directly is not a straightforward pro-edure, so we have developed a surrogate method that better lendstself to automation.

Our method is to take the solutions shown in Fig. 1 and to plothe logarithm of the residual norm as a function of log ˛. The result-ng sequence of points, which we call the “S-curve”, can generallye well-fit with an error function having the form a + b erf(cx + d),here x = log ˛, Fig. 3. To calculate the final depth profile, we used

he value of ˛ corresponding to the first maximum in the fourtherivative of the fitted error function (Fig. 4), which in terms of the

tted parameters can be obtained from

og ˛ = −√

2(3 +√

6)c2 − 2cd

2c2(6)

Fig. 2. L-curve for the solutions shown in Fig. 1. The point on the figure correspondsto the solution found using a value for ˛ given by the S-curve method (see text).

When employed in the minimization of the joint function, thisvalue of ˛ usually corresponds to a point close to the subjective“corner” on the L-curve (Fig. 2). This “S-curve” method also has theadvantage of giving a result even in cases where the L-curve cannotbe properly constructed due to technical issues with the calculation.It should nonetheless be born in mind that whereas our intent wasto employ the well-known L-curve criterion for the choice of theoptimal value of ˛, we used a different, more tractable, mechanism,the S-curve, rather than the L-curve directly.

Each of the regulators quantifies the complexity of the profile ina different way, and in order to compare them with each other andwith the “No Reg” calculation, we need to be able to quantify theirability to reproduce the profile used to generate the synthetic data.For this purpose we calculate the “goodness of profile” as

i=10∑i=1

(cinputi

− ccalci )

2(7)

where cinputi

is the oxygen concentration at the ith profile pointin the input profile (Table 1) and ccalc

iis the oxygen concentra-

tion at the ith profile point in the profile extracted using the valuefor ˛ given by the S-curve criterion. A lower goodness of profilenumber denotes a better correspondence between the input and

Fig. 3. S-curve for the solutions shown in Fig. 1 (points) and fit error function (line).

R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51 47

F firstt

3

swlmdd

pcdopfitiroe

tFlTttpafi

TG

A

ig. 4. Fourth derivative of the fit error function shown in Fig. 3, with a point at thehe optimized depth profile.

. Results and discussion

As mentioned above, we extracted 100 profiles from 100 noisyynthetic data sets for each of 5 levels of noise. For each noise level,e summarize the profiles obtained in two ways. First, we calcu-

ate the average and standard deviation of the goodness of profileetric for the 100 profiles. Second, we plot the average and stan-

ard deviations of the extracted ci values for oxygen in an “averageepth profile”.

The data for the goodness of profile are given in Table 3 andlotted without the standard deviations in Figs. 5 and 6. The ill-onditioning of the inversion problem is reflected in the standardeviations on the average goodness of profile numbers, which areften as large as, or larger than, the means themselves. Whenlotted as a function of the % noise, the average goodness of pro-le numbers appear to tend towards straight lines, the fidelity ofhe extracted profiles degrading in a linear fashion as the noisencreases. All of the regularization schemes perform better than noegularization whatever the level of noise (although in the absencef added noise, the No Reg calculation recovers the input profilexactly).

Turning now to the averaged extracted profiles, we will illus-rate our findings with the results obtained with 1% added noise.ig. 7 shows the average of the 100 profiles extracted with no regu-arization, compared with the input profile shown as a broken line.he relatively close correspondence between the input profile andhe mean ci values in the extracted profiles illustrates the principle

hat if 100 ARXPS measurements were to be made on the same sam-le, the noise in each individual measurement would average out,nd an averaged recovered profile would resemble the true pro-le closely. The ill-conditioning of the inversion problem is seen

able 3oodness of profile numbers (means and standard deviations) for the various regularizat

Goodness-of-profile (at.%2)

Noise (%) 0.30 ± 0.04 1.00 ± 0.13No Reg 84 ± 52 213 ± 185c-sq ox 17 ± 12 49 ± 26c-sq O + C 85 ± 57 216 ± 159c-bulk-sq 28 ± 12 99 ± 41MaxEnt-50 60 ± 45 134 ± 109MaxEnt-bulk 36 ± 22 98 ± 47Slopes-sq 9 ± 7 31 ± 21Curves-sq 11 ± 7 29 ± 22Entropy(Slopes) 8 ± 6 27 ± 19Entropy(Curves) 12 ± 7 28 ± 21

lower goodness of profile score is indicative of a better correspondence between the in

maximum, indicating the value of ˛ used in the joint function for the extraction of

therefore in the relatively large error bars (±1 standard deviation)on the mean ci values, indicating relatively large variations in theindividual ci values from extracted profile to extracted profile. It isthis variability that translates into the relatively high slope of theNo Reg line in Figs. 5 and 6.

In the case of Tikhonov-c2 regularization, the solution norm canbe calculated on the basis of the oxygen profile alone (“c-sq ox”)or as a sum over the oxygen and carbon profiles (“c-sq O + C”).The average profile, over 100 synthetic data sets with 1% addednoise, for the sum over the oxygen profile alone is shown in Fig. 8,and the average profile for the sum over both the oxygen and car-bon profiles is shown in Fig. 9. Although the regulator summedover both oxygen and carbon gives an average result that bet-ter reproduces the input profile in the 0–30 A region (Fig. 9), theprofile is less stable (larger error bars) and shows a tendency toproduce an artifactual subsurface peak at 80 A depth, where thedepth interval modeled (0–90 A in this study) exceeds the depthinterval occupied by oxygen in the input profile (0–50 A, Table 1).As indicated by the relatively large error bar on the profile pointat 80 A depth, this artifactual peak can vary quite considerably inheight.

The Tikhonov-c2 regularization scheme has the disadvantagethat the bulk concentrations should contribute to the solutionnorm, a shortcoming that the (c-bulk)2 regulator is intended to rec-tify. In the case of this regulator, because of the complimentarityof the oxygen and carbon profiles, it is sufficient to calculate thesolution norm on the basis of the oxygen profile alone. The average

profile produced using the (c-bulk)2 regulator (Fig. 10) resemblesthat produced using the “c-sq ox” regulator (Fig. 8) in the 0–40 Aregion, but with a tendency to produce a higher, anomalous bulkconcentration of oxygen (profile point at 90 A depth).

ion schemes as a function of the level of added noise.

3.00 ± 0.39 6.00 ± 0.80 9.00 ± 1.24494 ± 627 938 ± 1676 1336 ± 1921

93 ± 48 118 ± 46 133 ± 70530 ± 381 894 ± 671 1377 ± 882174 ± 72 237 ± 173 334 ± 312322 ± 227 571 ± 486 922 ± 682174 ± 77 238 ± 172 358 ± 315

71 ± 82 135 ± 267 224 ± 52669 ± 103 152 ± 320 303 ± 65968 ± 76 115 ± 225 230 ± 53669 ± 101 158 ± 339 312 ± 653

put and extracted profiles.

48 R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51

0 2 4 6 8 10

0

200

400

600

800

1000

1200

1400

Goo

dnes

s of

pro

file

(at.%

2 )

% noise

c-sq ox c-bulk-sq Slopes-sq Curves-sq No Reg

Fig. 5. Goodness of profile score as a function of the level of added noise for thenon-entropy regulators, compared with no regularization. A lower goodness of pro-file score is indicative of a better correspondence between the input and extractedprofiles.

0 2 4 6 8 10

0

200

400

600

800

1000

1200

1400

Goo

dnes

s of

pro

file

(at.%

2 )

% noise

MaxEnt-50 MaxEnt-bulk Entropy(Slopes) Entropy(Curves) No Reg

Fig. 6. Goodness of profile score as a function of the level of added noise for theentropy regulators, compared with no regularization. A lower goodness of pro-file score is indicative of a better correspondence between the input and extractedprofiles.

0 20 40 60 80 100-5

0

5

10

15

20

25

30

35

40

O c

once

ntra

tion

(at.%

)

Depth (Å)

input No Reg

Fig. 7. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, with no regularization. The input profile isshown as a broken line.

0 20 40 60 80 100-5

0

5

10

15

20

25

30

35

40

O c

once

ntra

tion

(at.%

)

Depth (Å)

input c-sq ox

Fig. 8. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, using Tikhonov-c2 regularization based onthe oxygen profile only. The input profile is shown as a broken line.

0 20 40 60 80 100-5

0

5

10

15

20

25

30

35

40O

con

cent

ratio

n (a

t.%)

Depth (Å)

input c-sq O+C

Fig. 9. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, using Tikhonov-c2 regularization based onthe oxygen and carbon profiles. The input profile is shown as a broken line.

The Tikhonov-c′2 and Tikhonov-c′′2 regulators would not receivea contribution from the constant concentrations in the bulk andso do not require correction. The average profile produced by theTikhonov-c′2 regulator (“Slopes-sq”) is shown in Fig. 11. Althoughon average the profiles tend to undercut the input profile in the20–50 A depth region and to overestimate the bulk concentra-tion, the error bars are relatively small, meaning that the extractedprofile is relatively more stable. The Tikhonov-c′′2 (“Curves-sq”) reg-ulator produces quite similar results, Fig. 12.

Now we turn to the regulators based on the profile entropy.The so-called “Maximum Entropy” regulators have the formci − mi − ci log(ci/mi). The interpretation given to mi by Smith andLivesey [8] is 100%/(number of atomic species modeled), and using

mi = 50 at.% we obtain the average profile shown in Fig. 13. It can beseen to resemble the profile obtained using the “c-sq O + C” regula-tor, Fig. 9, in that an artifactual subsurface peak is often producedclose to the deepest profile point. In this interpretation of mi thebulk concentrations should contribute to the solution norm and

R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51 49

0 20 40 60 80 100-5

0

5

10

15

20

25

30

35

40O

con

cent

ratio

n (a

t.%)

Depth (Å)

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ig. 10. Mean ± standard deviation of the ci from the profiles extracted from 100ynthetic data sets with 1% added noise, using (c-bulk)2 regularization. The inputrofile is shown as a broken line.

e have argued [25] that it would be better to set mi = c10, so thathe constant bulk concentrations make no contribution. Doing so,e obtain the average profile shown in Fig. 14, which can be seen to

esemble the result obtained using the (c-bulk)2 regulator, Fig. 10.The close similarities in the behaviour of the Tikhonov-c2

nd MaxEnt (m = 50) regulators, and the (c-bulk)2 and MaxEntm = bulk) regulators, noted above, begs the question: in practice,s it the operand that determines the result, rather than the formatentropy-based or not) of the regulator? We can explore this ques-ion by trying two new “entropy-like” regulators, one based on thelopes in the profile, the other on the curvatures. The c′ entropy reg-lator does in fact give a result (Fig. 15) very similar to that obtainedsing the Tikhonov-c′2 regulator, Fig. 11, and the c′′ entropy regulatorives a result closely resembling that obtained using Tikhonov-c′′2

egularization, Figs. 16 and 12 respectively. The dependence of theesult on the operand (concentrations, slopes or curvatures) ratherhan whether or not the regularization is based on entropy, can alsoe seen in comparing the data graphed in Figs. 5 and 6, and suggests

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ig. 11. Mean ± standard deviation of the ci from the profiles extracted from 100ynthetic data sets with 1% added noise, using Tikhonov-c′2 regularization. The inputrofile is shown as a broken line.

Fig. 12. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, using Tikhonov-c′′2 regularization. The inputprofile is shown as a broken line.

a new paradigm for the classification of the most popular methodsfor the inversion of ARXPS data.

With the particular combination of input profile and the num-ber and spacing of points in the profile model used in this paper,the methods based on slopes or curvatures gave results that canbe judged superior overall to the results obtained with meth-ods based on concentrations, including our implementation of thewidely used Maximum Entropy method. We do not, however, claimuniversality for this result at this stage in our investigations. Prelim-inary calculations, for example, show that the artifactual subsurfacepeaking seen with the MaxEnt (m = 50) method can be mitigated bybetter scaling the model (number and spacing of the profile points)to the input profile. The mis-match in scaling that characterizes thisstudy, however, perhaps better represents a typical real-life case in

which, in the absence of knowledge about the depth regime cov-ered by the real profile, the depth regime covered by the model isset to roughly 3 times the photoelectron inelastic mean free path.On the plus side for the entropy methods, we noted that in Math-

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Fig. 13. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, using Maximum Entropy regularizationwith the mi set to 50 at.%. The input profile is shown as a broken line.

50 R.W. Paynter, M. Rondeau / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 43–51

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ig. 14. Mean ± standard deviation of the ci from the profiles extracted from 100ynthetic data sets with 1% added noise, using Maximum Entropy regularizationith the mi set to c10. The input profile is shown as a broken line.

matica, the entropy-based calculations completed in roughly halfhe time it took for the equivalent calculations in the non-entropyormalism.

The unexpected exception to the overall trends discussed aboves the anomalously good performance of Tikhonov-c2 regularizationased on the oxygen profile alone. We hypothesize at this stagehat it might be related to the particular form of the input pro-le, in which the oxygen concentration falls to zero in the bulk.ikhonov-c2 regularization will naturally tend to minimize the oxy-en concentration throughout the extracted profile, and comparingigs. 8 and 10, for example, one can see that the better goodnessf profile scores obtained by Tikhonov-c2 regularization are mainlyue to the much lower oxygen concentrations it produces at depthsreater than 50 A.

We would like to underscore the point that the relative perfor-ances of these regulators have been compared using a common

riterion for the choice of the regularization parameter ˛ that isntended to conform to the L-curve method. It may be that the var-

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ig. 15. Mean ± standard deviation of the ci from the profiles extracted from 100ynthetic data sets with 1% added noise, using c′ entropy regularization. The inputrofile is shown as a broken line.

[

[[

Fig. 16. Mean ± standard deviation of the ci from the profiles extracted from 100synthetic data sets with 1% added noise, using c” entropy regularization. The inputprofile is shown as a broken line.

ious regulators evaluated in this study will perform differently withrespect to one another when a different criterion is employed forthe choice of the regularization parameter. Neither does this studyevaluate the suitability of the S-curve method in choosing an opti-mal value for ˛. These simulations were furthermore limited toone particular input profile, and other types of profile have yet tobe investigated. These matters, along with other questions raisedabove, are topics for further enquiry.

4. Conclusion

Using synthesized data and a simple physical model, we havecompared and quantified the abilities of various regularizationschemes to recover an input depth profile in the face of added noise.The results obtained suggest that the operand used in the regulator(concentrations, slopes or curvatures in the depth profile) is moreimportant to the result than whether or not the regulator calcu-lates the profile entropy. With the particular combination of inputprofile, model and regularization parameter choice method usedin this study, the regulators based on slopes or curvatures did bet-ter than those based on concentrations, with one exception, theTikhonov-c2 calculation based on the oxygen profile alone.

Acknowledgement

This work was supported by the Natural Sciences and Engineer-ing Research Council of Canada.

References

[1] P.J. Cumpson, J. Electron Spectrosc. Relat. Phenom. 73 (1995) 25.[2] C.R. Brundle, G. Contia, P. Mack, J. Electron Spectrosc. Relat. Phenom. 178–179

(2010) 433.[3] C.R. Brundle, Surf. Interface Anal. 42 (2010) 770.[4] R.L. Opila, J. Eng Jr., Prog. Surf. Sci. 69 (2002) 125.[5] P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadel-

phia, 1998, p. 84.[6] P.J. Cumpson, Appl. Surf. Sci. 144–145 (1999) 16.[7] B.J. Tyler, D.G. Castner, B.D. Ratner, Surf. Interface Anal. 14 (1989) 443.[8] G.C. Smith, A.K. Livesey, Surf. Interface Anal. 19 (1992) 175.

[9] R.W. Paynter, J. Electron Spectrosc. Relat. Phenom. 177 (2010) 5.10] R.W. Paynter, D. Roy-Guay, G. Parent, M. Ménard, Surf. Interface Anal. 39 (2007)

445.11] S. Oswald, F. Oswald, J. Appl. Phys. 100 (2006) 104504.12] M.A. Scorciapino, G. Navarra, B. Elsener, A. Rossi, J. Phys. Chem. C 113 (2009)

21328.

ectros

[[[

[

[[

[[

R.W. Paynter, M. Rondeau / Journal of Electron Sp

13] C.J. Powell, W.S.M. Werner, W. Smekal, Appl. Phys. Lett. 89 (2006) 172101.14] J.E. Fulghum, Surf. Interface Anal. 20 (1993) 161.15] J. Polcák, J. Cechal, P. Bábor, M. Urbánek, S. Prusa, T. Sikola, Surf. Interface Anal.

42 (2010) 649.

16] N. Suzuki, K. Iimura, S. Satoh, Y. Saito, T. Kato, A. Tanaka, Surf. Interface Anal.

25 (1997) 650.17] N. Suzuki, T. Kato, S. Tougaard, Surf. Interface Anal. 31 (2001) 862.18] M.P. Seah, in: D. Briggs, J.T. Grant (Eds.), Surface Analysis by Auger and X-Ray

Photoelectron Spectroscopy, IM Publications and Surface Spectra Ltd., Chich-ester, 2003, p. 360.

[[[

[[

copy and Related Phenomena 184 (2011) 43–51 51

19] M.S. Vinodh, L.P.H. Jeurgens, Surf. Interface Anal. 36 (2004) 1629.20] C.J. Powell, A. Jablonski, NIST Electron Inelastic-Mean-Free-Path Database, SRD

71, US Department of Commerce, National Institute of Standards and Technol-ogy, Gaithersburg, MD, 2000.

21] R.W. Paynter, Surf. Interface Anal. 3 (1981) 186.22] K. Harrison, L.B. Hazell, Surf. Interface Anal. 18 (1992) 368.23] Y.A. Babanov, O.M. Nemtsova, T. Reich, L.N. Romashev, M.A. Milyaev, V.V. Usti-

nov, J. Struct. Chem. 49 (2008) S165.24] R.W. Paynter, Z. Chanbi, Appl. Surf. Sci. 255 (2008) 2746.25] R.W. Paynter, J. Electron Spectrosc. Relat. Phenom. 169 (2009) 1.