Resolution enhancement of x-ray photoelectron spectra by maximum entropy deconvolution

SURFACE AND INTERFACE ANALYSIS, VOL. 26, 195È203 (1998)

Resolution Enhancement of X-ray PhotoelectronSpectra by Maximum Entropy Deconvolution

S. J. Splinter and N. S. McIntyre*Surface Science Western, The University of Western Ontario, London, Ontario N6A 5B7, Canada

The maximum entropy method (MEM) is applied to the deconvolution of x-ray photoelectron spectra. This methodprovides the least-biased estimate of the unbroadened spectrum by using the Shannon information content as theregularizing functional. The large-scale, non-linear optimization problem is solved using a robust variable metricsequential-quadratic programming (SQP) algorithm implemented on a personal computer (PC). The program istested on simulated spectra and then shown to provide reliable resolution enhancement of measured spectra byunfolding a measured instrumental resolution function. Typical resolution enhancements of 50% are achievable inÆ15 min of computer time. 1998 John Wiley & Sons, Ltd.(

Surf. Interface Anal. 26, 195È203 (1998)

KEYWORDS: XPS, x-ray photoelectron spectroscopy, deconvolution, resolution

INTRODUCTION

X-ray photoelectron spectroscopy (XPS) is widely usedto provide chemical composition and chemical stateinformation of surface constituents. The main advan-tage of XPS over other surface analytical techniques liesin its ability to detect subtle di†erences in surfacebonding states via well-deÐned variations in elementalbinding energies (the well-known “chemical shiftÏ). Themagnitude of these shifts can, however, be quite small,often of the order of the exciting x-ray linewidth or theenergy analyser broadening. In many cases, therefore,detailed analysis is limited by the resolution available inthe measurements.

Energy resolution improvement can be achieved inpractice by monochromatization of the exciting x-raysand/or by very high energy resolution analysis of theemitted photoelectrons through improved instrumen-tation. Both approaches, however, necessarily result ingreatly decreased count rates and associated signal-to-noise ratios. As a result, the use of expensive high-intensity photon sources is required. This added costtherefore provides an acute motivation for the develop-ment of a reliable computational means for resolutionenhancement. In this paper, we describe a method forenhancing the energy resolution of photoelectronspectra by deconvolution based on the maximumentropy formalism.

All real data are distorted by the Ðnite resolution ofthe measuring device. In spectroscopy, for example, wewant to measure the quantity s(E) (the spectrum) bymeans of an instrument with a resolution r(EÈE@). Thisresolution function (or apparatus function) gives the

* Correspondence to : N. S. McIntyre, Surface Science Western, TheUniversity of Western Ontario, London, Ontario N6A 5B7, Canada.

response of the instrument at E to an impulse at E@. Themeasured signal is then given by

d(E) \P

s(E) É r(E[ E@)dE\ s(E) ? r(E) (1)

where the symbol ? represents circular convolution.This convolution integral occurs frequently in experi-mental physics and chemistry and is a special case ofthe general Fredholm integral equation of the Ðrst kind.The recovery of the underlying signal by deconvoluting,or unfolding, Eqn. (1) is an extremely important yet alsovery difficult problem. This is because the problem isill-posed, i.e. small perturbations in d(E) due to randomnoise can lead to large perturbations in s(E) because ofthe smoothing e†ect of integration. The deconvolutionmethod must therefore apply some form of regulariza-tion or restriction on the class of estimating functionsfor s(E).1 Deconvolution methods employing variousregularization schemes have been subject of a vast liter-ature. An excellent review of the subject, particularly asit pertains to spectroscopy, can be found in the mono-graph edited by Jansson.2

This paper describes how the maximum entropymethod (MEM) can be applied reliably to the deconvol-ution of photoelectron spectra. We Ðrst review thetheory of deconvolution as applied to XPS and discussprevious work in this area. We then present the mathe-matical description of MEM, explaining why this rep-resents a better approach to take, and describe thealgorithmic and computational details used to solve thenon-linear constrained optimization problem. Finally,we present several representative results obtained withsimulated and experimental data.

Deconvolution of XPS data

The observed core-level XPS spectrum of a solid is aconvolution of the natural line-shape with various dis-

CCC 0142È2421/98/030195È09 $17.50 Received 10 March 1997( 1998 John Wiley & Sons, Ltd. Accepted 28 October 1997

196 S. J. SPLINTER AND N. S. McINTYRE

torting functions. Madden and Houston3 have classiÐedthese distortions as being either intrinsic (inherentresponses of the material being studied) or extrinsic(responses due to the instrument). The intrinsicbroadening mechanisms derive from inelastic scatteringprocesses, including phonon scattering, and a variety ofother electronÈelectron interactions. The extrinsicbroadening mechanisms, on the other hand, derivesolely from the Ðnite width of the excitation source andfrom the imperfect measuring device. It is these latterdistortions that are presented by the resolution func-tion, r(E), in Eqn. (1) and are the principal emphasis ofthis work. The source and analyser broadening aremore important than the energy-loss tail when examin-ing close-lying core levels, whereas the intrinsic losse†ects tend to have a greater e†ect on the shape ofbroader spectra such as valence bands. Deconvolutionof intrinsic loss mechanisms from XPS spectra has beenexamined in detail by other authors.2h6

Our goal, then, is to recover the spectrum that wouldhave been obtained using a hypothetical, perfectlyresolving instrument. To do so, we Ðrst require ameasure or an estimate of the resolution function, r(E).(An approach for estimating r(E) in XPS is outlinedbelow.) The true spectrum free from instrumentalbroadening can then be recovered by deconvoluting orunfolding the estimated resolution function from the as-measured spectrum. As mentioned above, however, thisis a very difficult problem in reality. The reason is thatfor real spectra we must replace Eqn. (1) with

d(E) \ s(E) ? r(E) ] n(E) (2)

where n(E) is the inevitable random noise present in themeasurements. Any solution to Eqn. (2) is thereforenon-unique because any function, n@(E), which satisÐes

r(E)? n@(E) \ 0 (3)

can be added to s(E) without greatly a†ecting its con-volution with r(E). Random, high-frequency noise isknown to satisfy Eqn. (3).7 The result is that even smallnoise perturbations in the measured d(E) can introducelarge, unphysical oscillations in the deconvoluted spec-trum. In other words, the uncertainty in the measure-ments plays a fundamental role in our ability to recoverthe true, unbroadened spectrum.

In order to alleviate this noise ampliÐcation problem,deconvolution algorithms must apply some mechanismof regularization. In general, regularization methods canbe categorized as being either linear or non-linear.Linear deconvolution algorithms have output elements(the deconvoluted spectrum) that can be expressed as alinear combination of the input elements (the measuredspectrum). Non-linear algorithms, on the other hand,involve implementing non-linear regularizing function-als and (usually) non-linear constraints comprising apriori information. Whereas the linear methods arecomputationally much simpler, they generally performmuch more poorly, particularly in the case of band-limited data.2 Nevertheless, the linear methods haveattracted the most attention for deconvolution of XPSspectra.

One of the simplest linear forms of regularization isthe iterative algorithm originally proposed by VanCittert.8 In this method and its variants,9 successive

reÐnements of an approximation to the true func-sü (E)tion s(E) are made according to the iterative formula

sü 1(E) \ sü 0(E) ] [d(E) [ sü 0(E)? r(E)] (4)

This approach is essentially the Jacobi method forsolving simultaneous linear systems as applied to signalprocessing. The number of iterations then acts as aparameter of regularization. This method is computa-tionally very simple and has the advantage that thenoise accumulates with each iteration, so the number ofiterations can be terminated manually when the noiseampliÐcation is judged to be excessive. The method is,however, obviously arbitrary in that the number of iter-ations is arbitrary. In addition, because it is a linearmethod, its performance is highly dependent on the fre-quency content of the data, often leading to unphysicalsolutions (e.g. solutions having negative values) and it isvery sensitive to errors in the estimate of the resolutionfunction. Nevertheless, iterative deconvolution methodsstill have utility in cases where only modest correctionis required and they have, in fact, been applied exten-sively to XPS data, especially for the removal of intrin-sic broadening e†ects. 3h6,10h13

Another simple linear method or regularization is toapply a low-pass Ðlter in Fourier space.14 According tothe convolution theorem, Eqn. (1) can be represented bythe simple form

d \ s ? r 7 Dk\ Sk É Rk (5)

where and represent the (discrete) FourierDk , Sk Rktransform (FT) of the measured spectrum, theunbroadened spectrum and the resolution function,respectively. For perfect, noise-free analytial functions,it is possible to recover s(E) exactly by simply comput-ing the inverse FT of the ratio In real life,Dk/Rk .however, data contain noise that extends to high fre-quencies. The noise ampliÐcation problem inherent todeconvolution is thus particularly easy to understand inthis case. Because the noise-free components of fallDkto zero as k ] O, and because is also a decreasingRkfunction of frequency, the high-frequency noise contentof is preferentially ampliÐed when we divide byD

kRk .

By applying a low-pass Ðlter, therefore, the high-frequency components of are removed and the noiseDkampliÐcation is alleviated. The regularization parameterin this case is the cut-o† beyond which the frequenciesare suppressed. A particularly good Ðlter to use is theWiener inverse Ðlter,15,16 which minimizes the meansquare deviation between the true uncorrupted signaland a calculated estimate averaged over the randomparameters. It thus represents the best linear Ðlter pos-sible, assuming that the noise is additive and Gaussiandistributed.17 Fourier transform-based methods havebeen applied to XPS spectra4,10 with moderate success.The problem is that such Ðlters are again arbitrary inthat the degree of spectral sharpening is dictated by thechosen cut-o† frequency. In addition, oscillations of“ringingÏ e†ects are often introduced due to the sharpcut-o†s sometimes employed. Finally, the use of low-pass Ðlters assumes that all or most of the informationresides in the low-frequency portion of the data ; theiruse therefore leads to a permanent loss of all high-frequency information and can result in distortions inthe deconvoluted spectrum.

Linear deconvolution methods, as a consequence of

SURFACE AND INTERFACE ANALYSIS, VOL. 26, 195È203 (1998) ( 1998 John Wiley & Sons, Ltd.

XPS RESOLUTION BY MAXIMUM ENTROPY 197

their linearity, often lead to physically impossibleresults. The reason is that, if noise predominates in thefrequency range required to achieve a restoration, linearmethods tend to “restoreÏ these noise values to physi-cally meaningless results. Because of the ill-posed natureof the deconvolution problem, a unique solution doesnot exist. Therefore, in the absence of any constraints,any solution is possible so long as it satisÐes the data.Linear deconvolution techniques are not well suited toconstraints ; for example, positivity is a function of all ofthe Fourier coefficients together, and not of eachseparately. It is the noise in the data that thus deter-mines which solution is favoured over all the rest. Muchimproved results can therefore be obtained using non-linear methods that introduce a compromise betweenÐdelity to the data and Ðdelity to some prior knowledgeabout the solution.

Most prior knowledge built into deconvolution algo-rithms is deterministic in nature, such as through theuse of positivity or boundedless constraints. Forexample, MacNeil and Dixon18 and Delwicke et al.19used a modiÐed Van Cittert algorithm that employsmultiplicative corrections instead of additive correctionsto deconvolve photoelectron spectra. This algorithm,initially developed by Gold20 eliminates negative solu-tion values. Similarly, Beatham and Orchard21 appliedpositivity constraints by employing BiraudÏs22 Fourier-series squared method. These authors experienced onlylimited success. Jansson,2 however, obtained excellentresults on XPS spectra using his constrained iterativemethod and employing a variable relaxation factor.

All of the non-linear methods mentioned above,although somewhat successful, only reduce the numberof possible solutions to the deconvolution problem. Weare still left with the possibility that more than one solu-tion may satisfy the data. It is, however, possible toobtain a unique solution using MEM. This method pro-vides the least-biased estimate of the underlying spec-trum possible for the given information.23h25 It hasbeen applied successfully to numerous deconvolutionproblems, including image enhancement,26h28 energyresolution enhancement of EELS,29 Raman30,31 andSIMS32 spectra and depth resolution enhancement ofXPS33,34 and SIMS35,36 depth proÐles. There havebeen very few attempts to use MEM to enhance XPSspectral resolution. Vasquez et al.37 used MEM tosharpen simulated spectra. They did, however, haveproblems with stability, and reported small spuriouspeaks of unknown origin. Mahl et al.38 used MEM as aprelude to curve Ðtting ; they did not, however, reportalgorithmic or resolution function details. In this paper,we show that reliable energy resolution enhancementsof XPS spectra can be achieved using a robust algo-rithm based on a variable-metric sequential-quadraticprogramming (SQP) method for solving the non-linearoptimization problem.

The maximum entropy method

In 1948, Shannon39 used an axiomatic approach toderive a measure of information conveyed by specifyingone of several alternatives, and thus quantiÐed theconcept of uncertainty of a probability distribution,

. . . , He called this measure the “informa-p \ (p1 , p2 , pn).

tional entropyÏ

H \ [ ;i/1

Npilog p

i(6)

ShannonÏs entropy can be viewed as being an inversemeasure of information : a shape with greater entropy isless informative than one with less entropy. Forexample, a delta function has the minimum possibleentropy (greatest information), whereas a smoothsurface has the maximum possible entropy (leastinformation). Informational entropy appeared innumerous problems in various scientiÐc areas, but itwas not until 1957 that Jaynes23 explicitly stated the“principle of maximum entropyÏ, proposing that when-ever we make inferences based on incomplete informa-tion, we should draw them from the probabilitydistribution that has the maximum uncertainty(entropy) permitted by the information that we do have.In other words, we should only use the informationavailable to us and we should strictly avoid the use ofany additional information. Because maximum entropyimplies minimum information (or maximumuncertainty), when making inferences we should maxi-mise the uncertainty in that which we do not know!

In terms of the deconvolution problem, we wish toÐnd the “bestÏ spectrum s(E) in Eqn. (1) that does notviolate any prior knowledge about the true solution.According to the principle of maximum entropy, weshould therefore search for the deconvoluted structurethat has maximum entropy (i.e. maximum uncertainty)subject to the known information. The informationalentropy, H, thus becomes our regularizing functional.The “known informationÏ can be quantiÐed by requiringthat our calculated spectrum convolved with theresolution function agrees with the measured spectrum,within the noise. This condition is imposed by ensuringthat the weighted sum of squared errors, /2, is consis-tent with the uncertainty within the data

/2\ ;i/1

N [di[ (s ? r)

i]2

p2 O N (7)

where p is the noise variance in the data and N is thenumber of data points in the spectrum. This conditionensures that the calculated spectrum does not departfrom the experimental mean by more than one standarddeviation. (When the error distribution is Gaussian, /2becomes the chi-squared (s2) statistic.) It should benoted that this condition is valid only when N is suffi-ciently large. For example, Seah and Cumpson40 haveshown that /2 will di†er from N by [10% whenN \ 135 channels. This variable drops to 7.3% forN \ 256 channels, to 5.1% for N \ 512 channels and to3.6% for N \ 1024 channels. It is recommended, there-fore, that N be kept as large as possible (more than atleast 256 channels) to ensure reliable results.

To obtain the maximum entropy solution in practice,we extend the entropy functional, Eqn. (6), to include allpositive additive distributions, as pointed out by Skil-ling and Gull41

H \ [ ;i/1

N Csi[ m

i[ s

ilogA s

im

i

BD(8)

where is a “default modelÏ to which the estimate col-mi

( 1998 John Wiley & Sons, Ltd. SURFACE AND INTERFACE ANALYSIS, VOL. 26, 195È203 (1998)


lapses in the absence of constraints. This form ofentropy is invariant under a change of coordinatesystem and the distribution does not need to be normal-ized. The deconvolution problem can now be formu-lated mathematically as

maximize H \ [;Csi[ m

i[ s

ilogA s

im

i

BDs.t. /2D N

(9)

It should be noted that the maximum entropy decon-volution problem as described by Eqn. (9) represents the“classicalÏ or “historicalÏ approach to MEM. It is alsopossible to adopt a “statisticalÏ approach, which is basedon Bayesian inference.42 An advantage of the lattermethod is that it also provides vertical “error barsÏ forthe deconvoluted spectrum. It is, however, computa-tionally more difficult. A good recent set of reviews con-cerning the latter approach can be found in themonograph edited by Buck.43

Equation (9) represents a non-linear optimizationproblem with non-linear constraints. It is this non-linearity that greatly increases the computationalburden of MEM compared to linear deconvolutionmethods. In addition, because the length of the datavector, N, can be large, standard solution algorithms(such as modiÐed NewtonÏs methods) that require thestorage and inversion of N ] N matrices cannot beimplemented on a personal computer. Some authorshave therefore attempted iterative solutions.44,45 Thesealgorithms are known to su†er from instability prob-lems and, in addition, they lack clear termination cri-teria, thus again making them somewhat arbitrary. Forthese reasons, we have chosen to implement an algo-rithm based on WilsonÏs variable-metric SQPmethod.46

This algorithm was originally proposed to solvelarge-scale astronomical image-enhancement problemsby Skilling and Bryan,26 and was subsequentlyextended by Reiter.27 One of us (S.J.S.) recently imple-mented a modiÐed version running on a personal com-puter for deconvolution of electron energy-lossspectra.29 The computational details are discussed indetail in the above references and will not be repeatedhere. In brief, the algorithm breaks the deconvolutionproblem down into the sequential solution of quadraticsubproblems subject to linearized constraints. At eachiteration a step vector is computed, which sequentiallysteps towards the solution. To reduce the dimension-ality of the problem, the step vector is represented as alinear combination of a small number of suitablychosen search directions. The length of the step vector isconstrained at each iteration using the Hessian of theentropy functional as the “variable metricÏ. It is this lastpoint that has been found to be the key to the develop-ment of a robust algorithm.26

The algorithm has been coded in the MATLABTMlanguage running on a PentiumTM-based personal com-puter. The program does not require any N ] N matrixinversions of multiplications ; most of the central pro-cessing unit (CPU) time is spent in computing 16 data-space to frequency-space transforms (via the FFTalgorithm) per iteration. Typical spectra can be sharp-ened in \15 min of computer time.

RESULTS AND DISCUSSION

simulated spectra

In order to demonstrate the validity of the MEMdeconvolution technique in general and the presentprogram in particular, we Ðrst present results obtainedon synthetic spectra. To facilitate comparison withother methods, the present algorithm was applied totwo tests taken from the literature.46,47 In both of thesetests, a known function, s(x), is folded with a resolutionfunction, r(x), and pseudo-random noise is added togive the “measuredÏ function, d(x). The deconvolutedresult, is then compared to the original knownsü (x),function.

In the Ðrst example, we use the test data given byJohnson47 and described by Verkerk.46 Johnson devel-oped a deconvolution method based on the Taylorseries expansion of s(x). His test data consist of a Gauss-ian s(x) and a Gaussian r(x)

s(x) \ 1

psJ2nexpA[ x2

2pr2B

r(x) \ 1

prJ2nexpA[ x2

2pr2B

(10)

so that d(x) is also Gaussian with pd\ (ps2] pr2)[email protected] uses and d(x) was sampled atps\ 1.0 pr \ 0.7 ;43 points with *x \ 0.2, and normally distributed addi-tive zero-mean (white) noise with a 2% standard devi-ation was added to Figure 1 shows the results of ad

i.

deconvolution of JohnsonÏs data using the presentMEM program. If we deÐne the error between the truespectrum, s(x), and the computed estimate, as46sü (x),

v\S 1

N;i

*i2 É

1s'

(11)

where then we obtain v\ 1.27% using the*i\ (s

i[ sü

i),

MEM program. This indicates that there is excellent

Figure 1. Test of the MEM deconvolution program using thesimulated data of Johnson,47 represented by the dashed line, d(x).The error between the true spectrum, s(x), and the computed esti-mate, MEM, is Á1.3%.



agreement between the intrinsic and deconvolutedspectra. For comparison, Johnson obtained 2.7% usinghis Taylor series method, Verkerk 46 obtained 1.3%using a modiÐed FT-based algorithm and Dierckx48obtained 1.36% using a deconvolution algorithm basedon spline functions.

As a second example, we use the standard spectro-metric problem proposed by Crilly.49 Here, s(x) is thesum of two Gaussians

s(x)\ 2 expC[ (x [ 10)2

2(2.25)2D

] 6 expC[ (x [ 30)2

2(2.25)2D

(12)

and the resolution function, r(x), is a unit area Gaussianwith We have given the folded d(x) pseudo-pr \ 8.0.random white noise with 1% standard deviation. Figure2 shows the results of deconvolution using our MEMprogram. In this case, an error analysis yields v\ 1.6%,again suggesting that excellent agreement exists betweenthe true spectrum and the computed estimate.Marawski et al.50 used the same test to comparevarious iterative deconvolution algorithms. Theseauthors used the relative root-mean-square error toassess the accuracy of the compared algorithms

d2\ p s [ s p2p s p2

(13)

where the symbol represents the 2-norm. Table 1p p2shows the results obtained by Marawski et al. usingd2the Van Cittert,8 Gold,20 Tikhonov1 and Jansson2 algo-rithms compared with the present MEM program.From Table 1 it is evident that the present programoutperforms the iterative deconvolution methods onsimulated spectra.

Measured spectra

All of the measured spectra were obtained with aSurface Science Laboratories SSX-100 spectrometer.

Figure 2. Test of MEM deconvolution program using the stan-dard spectrometric problem proposed by Crilly.49 The dashed line,d(x), represents the true underlying spectrum. The error betweenthe true and computed spectrum is 1.6%.

Table 1. Error of reconstructionfor the standardd

2spectrometric problemobtained by Morawskiet al.45 for several iter-ative deconvolutionalgorithms comparedto the present MEMprogram

Algorithm d2

Van Cittert8 0.179

Gold20 0.134

Tikhonov1 0.507

Jansson2 0.090

MEM (presemt study) 0.078

This instrument makes use of a concentric hemispheri-cal analyser (CHA) for measuring photoelectron kineticenergies. The incoming electrons are retarded to a Ðxedpass energy before entering the focal plane of theanalyser ; the spectroscopist is a†orded a choice of passenergies ranging from 150 down to 10 eV, providingprogressively increasing energy resolution at theexpense of reduced count rates. The monochromaticx-ray source (Al Ka or Mg Ka) can be focused to spotsizes ranging from 600 to 50 lm, again providingincreasing resolution and decreasing count rates. Thebinding energy scale is referenced to an Au line4f7@2position of 83.93 ^ 0.05 eV for metallic gold.

Resolution function. In order to recover the unbroadenedspectrum by deconvolution, we Ðrst require an estimateof the spectrometer resolution function, r(E). In general,the broadening function is dependent upon the meanradius of the analyser, which is Ðxed, and the passenergy and source spot size mentioned above, which arevariable. (For some twin anode instruments theresolution width is also dependent upon the input slit.)Wertheim and Dicenzo51 and Barrie and Christensen52describe a method for estimating r(E) using the shape ofthe Fermi cut-o† in a metal such as silver, which has afeatureless occupied conduction band (CB) with essen-tially constant DOS over a range of almost 4 eV.According to these authors, if the thermal width of theFermi edge is small compared to the analyserresolution, then the CB cut-o† can be considered to bea step function. The resolution function is then esti-mated by simply computing the derivative of the mea-sured Fermi edge and (to Ðrst order) Ðtting the result toa Gaussian. Figure 3 shows the results of such a mea-surement on a clean, pure polycrystalline Ag surfaceusing an x-ray spot size of 300 lm and a Ðxed analyserpass energy of 25 eV. The dotted line in Fig. 3 showsthe calculated Fermi-Dirac distribution at room tem-perature, indicating that the width of the Fermi edge isindeed small compared to the resolution. The resolutionfunction for these spectrometer conditions can thereforebe represented as a Gaussian with a full-width at half-maximum (FWHM) of 0.53 eV. These measurementswere repeated for various conditions of x-ray spot sizeand analyser pass energy for our spectrometer ; theresults are presented in Table 2. It should be mentionedthat if the resolution is comparable to the thermal



Figure 3. The Ag Fermi edge measured on a clean, polycrystallineAg surface using an x-ray spot size of 300 lm and a fixed analyserpass energy of 25 eV. The inset shows the computed derivative.The resolution function for these spectrometer conditions is there-fore assumed to be a Gaussian, and an FWHM equal to the widthof this derivative, 0.53 eV.

width, or if the counting statistics are too poor for reli-able di†erentiation, then it is possible to estimate theresolution function by either Ðtting the spectrum with aconvolution of a Fermi function with a parametrizedrepresentation of r(E), or by Ðtting a spectrum of a core-level whose Lorentzian width and asymmetry are wellknown.51

Pyrite S 2p. As a Ðrst example of the ability of the MEMprogram to sharpen real spectra, we show resultsobtained for the S 2p spectrum from natural pyrite

Figure 4(a) shows the measured core-level XPS(FeS2).spectrum obtained from a freshly vacuum-fracturedpyrite crystal. The fracture surface was conchoidal sothat no crystallographic face could be assigned to theanalysed surface. The spectrum was measured using a25 eV pass energy and 300 lm x-ray spot size, thusgiving a constant absolute energy resolution of 0.53 eV.The main S line located at D162.45 eV binding2p3@2energy had a measured width of 0.77 eV. The 2p1@2spin-orbit contribution is located D1.2 eV to higherbinding energy. Several authors have used peak-Ðttingmethods to postulate the existence of a surface state atD161.8 eV binding energy, although some controversyexists.53,54 Figure 4(b) shows the sharpened pyrite S 2pspectrum obtained by unfolding the 0.53 eV wide

Table 2. Measured resolution function widthfor the SSX-100 spectrometer

Pass energy (eV)

Spot size (lm) 25 50 150

150 0.46 eV 0.57 eV 1.22 eV

300 0.53 eV 0.64 eV 1.44 eV

600 – 0.74 eV 1.55 eV

Figure 4. Maximum entropy method (MEM) deconvolution ofthe S 2p spectrum of natural pyrite, (a) Measured spectrumFeS

2.

obtained using a 300 lm x-ray spot size and a 25 eV analyser passenergy. (b) Sharpened spectrum obtained by unfolding the 0.53eV wide resolution function using the MEM program. The MEM-sharpened spectrum shows the presence of a low-binding-energycontribution at 161.7 eV, indicative of non-ideal surface latticestates.

Gaussian resolution function from the measured datausing the present MEM deconvolution program. Theresolution of the sharpened spectrum, sh(E), was esti-mated from the width of a modiÐcation function, g(E),given by55

g(E) \ i†tASH

kÉ R

kD

k

B(14)

where is the FT of the sharpened spectrum and i†tSHkrepresents the inverse FT operator. Using this estimate,

we Ðnd that the resolution of the sharpened spectrum isimproved by almost 50% from 0.53 eV to 0.27 eV usingthe MEM program. This is conÐrmed by the [50%decrease in the width of the line.2p3@2From Fig. 4 it is evident that the present MEMprogram is capable of providing signiÐcant resolutionimprovement in measured XPS spectra. The MEM-sharpened spectrum, Fig. 4(b), clearly shows the pres-ence of a low-binding-energy contribution at 161.7 eV,indicative of non-ideal surface lattice states as proposedby Nesbitt and Muir.53 Further evidence for the pres-ence of these surfaces states was given by the variable-energy synchrotron radiation photoelectronspectroscopy work of Bronold et al.,56 lending addi-tional credence to the reliability of the MEM methodfor spectral resolution enhancement. We have usedMEM to examine the chemistry of pyrite, arsenopyriteand marcasite surfaces in detail ; the results will bepublished elsewhere.57



Figure 5. The influence of the signal-to-noise ratio of the mea-sured spectrum on the MEM deconvolution procedure. Becausedeconvolution necessarily degrades the signal-to-noise ratio, it isimportant to start with high-quality data. Signal-to-noise ratiosgreater than Á100–150 are required to obtain reliable results.

E†ect of signal-to-noise ratio. Because resolution enhance-ment is the opposite of smoothing, high signal-to-noiseratios are required of the measured data to ensure thesuccess of the deconvolution procedure. We have exam-ined the inÑuence of the initial signal-to-noise ratio on

Figure 6. Maximum entropy method (MEM) deconvolution ofthe Si 2p XPS spectrum measured from a slightly oxidized p-Si(100) surface. The sharpened result clearly shows the 2p

1@2 ,spin-orbit split, and the FWHM of the Si peak is found2p

3@2 2p3@2

to be 0.25 eV. The MEM does not, however, greatly affect theshape of the broad peak centred at Á103.8 eV. This peak is due tothe presence of oxidized silicon atoms present on the surface andis intrinsically wide.

Figure 7. Maximum entropy method (MEM) deconvolution ofthe C 1s XPS spectrum measured from a clean polyethylenesurface. The sharpened spectrum clearly shows a vibrational asym-metry on the high-binding-energy side of the C 1s line. This asym-metry has been reported by Beamson et al .,60 who used a ScientaESCA 300 rotating-anode XPS instrument.

the deconvoluted result for the pyrite S 2p XPS spec-trum. The results are shown in Fig. 5. The signal-to-noise ratios of the original spectra were calculated usingthe method described by Koenig and Grant.58 FromFig. 5, it is evident that signal-to-noise ratios aboveD100È150 are required to achieve resolution enhance-ments of a factor of 2. The impact of noise on deconvol-ution of XPS spectra is currently being studied in detailin this laboratory and the results will be presented else-where.

Silicon Si 2p. As a second example of the utility of theMEM technique, we show results obtained for the Si 2pcore-level XPS spectrum measured from a lightly oxi-dized p-type Si(100) surface. Figure 6 shows the mea-sured, background-subtracted spectrum compared withthe MEM-sharpened spectrum. The measured data inthis case were obtained using a spectrometer passenergy of 50 eV and x-ray spot size of 150 lm, giving aconstant absolute resolution of 0.57 eV. This resolutionis just sufficient to observe the and spin-2p1@2 2p3@2orbit split separation of 0.61 eV. The MEM-sharpenedspectrum, on the other hand, shows both componentsclearly ; the resolution after deconvolution was againimproved by almost 50% to 0.29 eV. The FWHM of thesharpened Si peak was found to be 0.25 eV,2p3@2approaching the theoretical value of 0.20 eV.59 We notealso that the intensity ratio of two peaks in the sharp-ened spectrum is very close to the statistical branchingratio of 2 : 1, thus lending further conÐdence to thesuccess of our method. It is also interesting to note thatdeconvolution does not greatly a†ect the shape of thebroad peak centred at D103.8 eV. This peak is due tothe presence of oxidised silicon atoms present on thesurface and is inherently wide because of the distribu-tion of oxidized chemical states with only slightly di†er-ent binding energies. It is not surprising, therefore, thatdeconvolution of the instrumental broadening did nota†ect this intrinsically wide signal.



Polyethylene C 1s. Finally, we show that the MEMprogram can be applied to the deconvolution of XPSstructures having a marked asymmetry. Fig. 7 showsthe C 1s XPS spectrum measured from polyethylene.Recently Beamson et al.60 used a Scienta ESCA 300rotating-anode XPS instrument to report vibrationalasymmetry on the high-binding-energy side of poly-ethylene and other solid-state organic polymers. Thisasymmetry, which is due to changes in equilibriumbond length and force constant during core level ioniza-tion, had not been observed previously for solid-statepolymers. Beamson and co-workers state that the asym-metry only becomes apparent for experimental line-widths \1.0 eV. The high x-ray intensity of the Scientainstrument permits high-resolution studies on lowcross-section materials. The FWHM of the polyethyleneC 1s line was thus observed to be 0.84 eV and a clearhigh-binding-energy asymmetry was reported. In Fig. 7,the polyethylene C 1s line measured with the SSX-100spectrometer was 1.10 eV wide and displayed onlyminor asymmetry. The data were obtained using aspectrometer pass energy of 25 eV and x-ray spot size of300 lm. The resolution-enhanced spectrum obtained byMEM deconvolution, on the other hand, had a width of0.71 eV, which is [15% narrower than that reported byBeamson et al. In addition, the deconvolution pro-

cedure clearly emphasized the asymmetric nature of theline, again lending credibility to the method.

CONCLUSIONS

The maximum entropy method has been applied toresolution enhancement of XPS spectra. The algorithmdescribed avoids the subjective nature of many decon-volution methods by Ðnding the least-biased estimate ofthe unbroadened spectrum available for the given infor-mation. Excellent agreement between simulated spectraand their MEM-deconvoluted counterparts wasachieved. It appears that the program described herecan provide reliable resolution enhancements of mea-sured spectra by unfolding a measured instrumentalresolution function.

Acknowledgements

The authors wish to thank Dr Allen Pratt for his help in collectingand interpreting the pyrite S 2p data, Dr Wayne Chang for collectingthe Si 2p spectra and Mary Jane Walzak for her help in collecting andinterpreting the polyethylene C 1s spectra.

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