Estimation of the uncertainties associated with XPS peak intensity determination

SURFACE A N D INTERFACE ANALYSIS, VOL. 18, 323-332 (1992)

Estimation of the Uncertainties Associated with XPS Peak Intensity Determination

Stephen Evans The Chemistry Centre, Department of Biochemistry, University College of Wales, Penglais, Aberystwyth, Dyfed SY23 3DD, UK

Formulae are presented for estimation of the statistical uncertainties (standard deviations) associated with the peak heights and integrated intensities (areas) required for elemental quantification in XPS in situations where peak overlap is not significant. The situation arising when curve-resolution procedures are used to separate discrete XPS signals for quantification is discussed, and a simple, rapid, but approximate procedure to estimate standard deviations for both the parameters characterizing the fit and the overall component intensities is developed. Both minimum-Z’ and unweighted least-squares procedures are considered. Some difficulties in relation to background subtraction and the inevitable presence of systematic errors in most XPS curve-fitting procedures are identified and discussed.

INTRODUCTION

Although XPS peak intensities are routinely interpreted to yield ‘quantitative’ elemental analyses, little attention has generally been given to the statistical precision or uncertainty that should properly be associated with each determination. Possibly this is because the approximations necessarily involved in the conversion of XPS relative intensities to relative atomic abundances are, for data with good counting statistics, generally much greater than the uncertainty intrinsically associated with the measured peak height or area. However, this is not necessarily the case for low concentrations near to the XPS detection limit, and in general there is often con- siderable advantage to be gained by knowing the statistical significance or otherwise of changes in the measured intensity, i.e. in the precision of the quantification as opposed to its accuracy.

The present paper, that by Harrison and Hazell’ and a National Physical Laboratory (NPL) report’ have all arisen from a workshop session of the UK ESCA Users Group in January 1991, our aim being the estab- lishment of recognized procedures for uncertainty estimation in XPS quantification.

In the first part of this paper, we address the estimation of uncertainty for both peak heights and integrated peak intensities (loosely, peak areas) determined directly from the data, both with and without prior smoothing. Although the formulae derived follow straightforwardly from first principles, they have not, to the author’s knowledge, previously been reported and they are cer- tainly not widely applied. They provide a firm basis for estimating the statistical uncertainty in any XPS elemental analysis not involving the use of curve-fitting (curve-resolving) techniques. This section of the paper is essentially as presented at the Workshop.

In the latter part, the more complex problem of the uncertainties to be associated with the results of curve- fitting procedures3 is discussed. The NPL report’ pro- vides an essential grounding in the principles governing

0142-2421/92/050323-10 $05.00 0 1992 by John Wiley & Sons, Ltd.

random uncertainty estimation, but concentrates on procedures that can be applied using standard commercial datasystem software. The methods recom- mended (section 2.9)’ take roughly as long to compute as the entire original fit, for each peak considered. This can become impractical for complex fits when computing power is severely limited.3 In addition, little guid- ance is offered for the practically important situation in which both random and systematic uncertainties coexist. Although the systematic errors may be very dif- ficult to quantify, the magnitudes of the random uncertainties are still required as a monitor of data quality. Indeed, without random uncertainty estimates- however approximate-for every component in an XPS analysis, no truly quantitative significance can be associated with the resultant derived stoichiometry. We shall therefore show, first, that random uncertainties can still be estimated in the presence of systematic effects, and, secondly, develop a new route to rapid, but approximate, random uncertainty estimation that requires only minimal computation. Complementary associated papers’.’ describe how uncertainty limits may be propagated through typical XPS quantification procedures.

~~~

STATISTICAL UNCERTAINTIES ASSOCIATED WITH DIRECT MEASUREMENTS OF PEAK HEIGHTS A N D INTEGRATED PEAK INTENSITIES

Peak heights

If the intensity of a peak is monitored by recording the number of electrons detected, Ni, during a predeter- mined time interval at the energy of the peak maximum (assuming the uncertainty in this energy to be negligibly small), the uncertainty (one standard deviation (SD), represented by a) in the number recorded is simply

Received 23 August 1991 Accepted 30 November 1991

324 S. EVANS

1 = channel no.

A. Usually, however, a 'background' signal N b is also measured, at an energy adequately removed from the peak maximum, and subtracted from N . The uncertainty (6) in the background is then f i b , and the uncertainty in the net intensity (height) is given by

O(Ni - Nb) = ,/", (1) If the height is determined not by direct measurement but by fitting, say, the five or nine highest points of a fully recorded peak to a q ~ a d r a t i c ~ , ~ and taking the coordinates of the turning point to represent the peak maximum, the corresponding uncertainty is reduced significantly. This process is closely related to convolutional ~moothing,4,~-~ and the effective improvement factor, f, in the signal-to-noise ratio (SIN) is similarly given by

f=&E (2) where c,, is the normalizing coefficient for the convolution and ci the central (largest) coefficient of the set. For the widely used nine-point Savitsky-Golay (SG) procedure: c, = 231 and ci = 59, and the SIN ratio of the maximum reading is improved by a factor of 1.9fL6r7 The uncertainty in a peak height, net of background, is then given by

Integrated peak intensities

We define the integrated intensity of a peak as the number of detected electrons (net of background) contributing to the peak in question. The intensity then consists simply of a number of counts, and does not have energy or time dimensions. In any subsequent comparisons between spectra recorded using differing channel widths and/or differing acquisition times, it is straightforward to normalize the intensity for these variables. Alternatively, identical results can be obtained by

m +J

c 3 0 u \

> I- H cn Z W I- Z H

considering intensity as peak area per unit time of data collection; see Harrison and Hazell.'

It is necessary first to subtract some assumed background from the gross channel intensities, and initially it will be assumed that this background is obtained by the simplest possible procedure-linear interpolation between left and right limits (L and R), chosen arbitrarily as the 'ends' of the peak. The SD on such an integrated peak intensity then comprises three independent components : (1) The uncertainty op in the total counts in all channels

contributing to the peak

(Tp = JX. i = R - 1 (4)

(2) The uncertainty ( T ~ resulting from the uncertainty in the left-hand background limit count, fiL

oL = JN,(R - L - 1)/2 (5 ) (3) The analogous uncertainty consequent on the 'noise'

in the right-hand background limit count

The origins of Eqns (4)-(6) can be visualized more easily with the aid of Fig. 1: essentially, oL and oR are representations of 'triangles of uncertainty' at the base of the peak area. Note particularly that the L and R limit channels contribute nothing to op, which is thus independent of ( T ~ and oL and so may be combined with them by quadratic summation to yield the total uncertainty in the integrated peak intensity, (T,

For intense peaks on a very low background the first term in Eqn (7) can dominate, but usually the second term is the more important, a point emphasized by Har- rison and Hazell.' Using unsmoothed data, it can be reduced significantly by averaging the data over several

high-energy integration limit

low-energy i = R integration limit

I i = L ,d'

1 I f 'J /!!

/ Y I bi components o f uncertainty

in integrated intensity due to 'noise. in L H limit

components o f uncertainty in integrated intensity due to 'noise' in RH limit I / '

Figure 1. Schematic diagram to show how the uncertainties in the intensities at the left- and right-hand limits of a peak affect the net integrated intensity. Note that & and &are shown grossly exaggerated.

UNCERTAINTY ESTIMATION IN XPS 325

channels to set the limits: using n channels, this is equivalent to smoothing such that f= & (cf. Eqn (2)). In such circumstances, or if the data have been smoothed by a convolutional procedure before intensity determination, both oL and CT, are reduced by the factor f, and (NL + NR) in Eqn (7) should be replaced by (NL + N,)/f2. This generally results in a significant and

worthwhile reduction in the uncertainty. The first term, essentially the total gross intensity of the peak, is at most only marginally affected by the smoothing process; thus, although the two terms in Eqn (7) are, strictly, no longer totally independent, the accuracy of the estimate of oi is not significantly reduced. When only the popular SG nine-point smooth is available, it may be useful to remember6.8 that the efficiency of n passes of an SG quadratic convolutional smooth routine will be such as to yieldf, = f i ~ O . ~ ’ .

Care must, however, always be taken in using SG convolutional smoothing that the lineshape is not significantly distorted by oversmoothing;’ negative-going wings are introduced, which can easily lead to the selection of inappropriate background limits with consequent gross errors in derived intensities. If it is essential to obtain integrated intensities from unavoidably poor- quality data, it is safer to apply Gaussian convolution to smooth the data.’ The peak distortion accompanying this process is merely a broadening, and it remains obvious where to set the integration limits. Note, however, that if the induced broadening is sufficiently great to cause the background limits to be moved further apart, there is an adverse effect on the uncertainty estimate through the increase in (R - L - 1) in

If a more sophisticated background subtraction routine such as that introduced by Shirley” is pre- ferred, the above analysis is not fundamentally affected. The changes in net integrated intensit resulting from the limit uncertainties ffiL and &R can be calculated numerically by recomputing the background in each case: but in the great majority of cases at least this seems unnecessary. The slightly different upper and lower error limits that result are, for most practical pur- poses, an undesirable complication. When one term in the uncertainty is increased by the use of the Shirley background, another is normally decreased, and the overall mean uncertainty estimate should not in general be seriously altered.

Eqn (7).

STATISTICAL UNCERTAINTIES FOR PARAMETERS OBTAINED BY LEAST-SQUARES FITTING PROCEDURES

Choice of function for least-squares minimization

A procedure for estimating the standard deviations of fitted parameters exists,’ ‘*12*2 to the author’s knowledge, only for the case of fits obtained by minimizing x2, defined in the present terminology by

in which Ci is the calculated number of counts in channel i. This procedure is appropriate when the residuals are normally di~tr ibuted,~.~.’~ a condition that is close to being satisfied when fitting ‘noisy’ data with the usual approximate lineshape and background func- t i o n ~ , ~ ’ ~ but which, unfortunately, is frequently not satisfied when the data being fitted show a good SIN ratio. The residuals in many representative fits reproduced el~ewhere,~ for example, show systematic deviations from the data far greater than the random errors and the residuals are, in such circumstances, most unlikely to be normally distributed. Those fits3 were obtained by minimizing the unweighted mean square deviation, 1 ( N i - Ci)2, the correct procedure to follow when no assumption regarding the distribution of the residuals is to be The effect of minimizing x2 in such circumstances is shown in Fig. 2: because the residuals near the peak maximum have much lower weights, the effect is marginally to improve the fit in the ‘wings’ of the peak-where the lineshape function is least accurate3-at the expense of the quality of fit near the peak maximum. This is not desirable when both the lineshape and the background function are approximations. We therefore prefer the unweighted minimization procedure for general use : error estimation procedures for minimum-X2 fits may be adapted in this situation, as we shall show. However, meanwhile, we shall retain the minimum-x2 criterion to maximize com- patibility with the NPL report2 and current commercial datasystems.

Estimation of random error in the presence of systematic error

As the data acquisition time for a spectrum is increased, a point is reached at which systematic errors in any sim- plified model used to fit it can unambiguously be shown to be present.2 The ‘reduced’ x2 (x2/v, the number of degrees of freedom) approaches unity for a fit domi- nated by purely random error. More precisely, the quantity Q(x& I v), which can readily be calculated, lies in the range 0.84 > Q > 0.16 for 68% of fits with purely random errors and within the range 0.995 > Q > 0.005 for 99% of such fits.2 Both parameters will be quoted for the fits discussed.

For data with high SIN, random uncertainties become negligibly small, and the systematic errors then dominate,2 as in Fig. 2. Often, however, although the value of Q shows beyond doubt that systematic effects are present, the random errors are still too large to be neglected. This is especially important for multicomponent fits, as shown below : uncertainties arising from the limited S/N of the data can still dominate, even when Q = 0, because of the effects of inter- parameter correlation. It is therefore necessary to establish that the magnitude of the random uncertainties can still be assessed in the presence of substantial systematic error.

Two examples will suffice to demonstrate that this is indeed possible. First, a series of Cu 2p3,, spectra was acquired with data collection times of 1, 2, 4, 8, 16 and 32 units, thus covering a wide range of SIN values, and each spectrum was fitted by a single symmetrical peak. The first fit lay well within the 1 - SD range for purely

326 S. EVANS

m u n C 3

1 L m L c) n n L m \

>- + H ffl z W + z H

minimum c h i - s q u a r e d f i t

1 u n w e i g h t e d l e a s t - s q u a r e s f i t

962 963 964 965 966 967 96E 969 970 962 963 964 965 966 967 968 969 970

K I N E T I C ENERGY / eV K I N E T I C ENERGY / e V

Figure 2. C Is XPS from single-crystal diamond fitted with a single Lorentzian-Gaussian sum function: left, by minimization of xz k’ = 451 3; reduced x2 = 51 ; RMS deviation = 0.78%. cf. ideal’ of 0.16%; right, by minimization of the unweighted mean square deviation k’ = 4743, RMS = 0.95%).

random error (Q = 0.51, reduced x2 = 0.99), while the later fits showed unmistakeable systematic deviations from random error. The last, reproduced in Fig. 3, had Q = 0 and reduced x’ = 11. Plots of the estimated random parameter uncertainties (obtained as described below) us. the reciprocal square root of the peak intensity nevertheless showed virtually perfect straight-line fits ( R = 0.9999). Thus the random uncertainties, which must follow such a square-root law, can be estimated reliably even in the presence of relatively large systematic differences between the assumed lineshape and the data. The same conclusion was also evident from con- sideration of a single Cu 3s spectrum (0.1 eV per channel, with one-quarter of the data acquisition time of that shown in Fig. 3). Using well-chosen background limits, this spectrum yielded Q = 0.61, well within the 1 - SD limit for random error. By displacing the background limits, unmistakeable systematic error could be introduced, reducing Q to as low as 7 x but the random uncertainty in the net intensity, estimated as described below, changed through no more than a neg- ligible 12%.

The presence of systematic errors in the lineshape and background in the fits to be discussed thus has no significant impact on the following assessment of the random uncertainties involved in these fits.

Uncertainties in peak parameters and their relationship to integrated-intensity uncertainty

The established criterion2T’1.12 for a 1 - SD uncertainty limit in a single parameter is that the minimized x’ should be increased by unity, after reoptimization of the fit with respect to all other parameters. The direct implementation of this criterion for all parameters in a complex fit requires extensive computation, although with modern commercial software the uncertainties for farily simple fits can be obtained acceptably quickly.’

However, a problem arises in connection with uncertainties in integrated intensities, the quantity most frequently carried forward for quantification in terms of elemental composition. The uncertainties in area associated with each parameter uncertainty cannot simply be added quadratically, because they are highly correlated.’ An increase in height, for example, can always be partly compensated by a decrease in width, leading to a much smaller uncertainty in the integrated intensity than might initially be supposed. The NPL report’ (section 2.9) discusses three possible solutions. The first, and most sophisticated, assumes both that the inverse Hessian matrixI4 is available and that the required uncertainty can be derived from it by ‘the appropriate matrix transformation’. This may not be possible when no straightforward analytical expression for the peak area exists, such as some representations3 of asymmetric XPS peaks; in the present author’s work at least areas are normally determined by numerical integration of the fitted curve^.^ The NPL report itself admits that its second proposal is in practice ‘not feasible’, while the third requires that only height and width parameters (and not shape parameters, such as Lorentzian- Gaussian (LG) mix, asymmetry and tail) have been fitted.

Here, therefore, an unconventional alternative approach is explored. Each correlated uncertainty is factorized into an intrinsic (statistical) component and components attributable to correlation. For a single- peak fit, the intrinsic parameter uncertainties are then by definition uncorrelated and independent, and quadratic summation of the associated integrated-intensity uncertainties immediately yields the required overall uncertainty in integrated intensity. For multiple- component fits, the correlation between the parameters characterizing each discrete peak is formally distin- guished from correlation between the peaks, so that the former can again be excluded when evaluating the uncertainties in the integrated intensities. A very rapid


m u - c 3

route also emerges for approximate parameter uncertainty estimation, useful when the available computing power is severely limited.

A vlrrv

Uncertainties in single-component fits.

r

Parameter uncertainties: theory. When the value of one parameter, p, characterizing an optimized fit is displaced by Ap, the value of x’ rises to Ax’ above the minimum. Theorem 4 (section 2.6) of the NPL report,’ valid for ‘small’ perturbations, then indicates that a2 = ( A P ) ~ / A X ~ for the parameter in question, so long as it is totally uncorrelated with any other parameter(s) optimized in the fit. Reoptimization then results in no change to the fit, and Ax2 remains constant. However, if the parameter is correlated with others, a’ is increased by a factor that we shall term the correlation factor, F, for that parameter. Since the initial Ax’ and the final (reduced) reoptimized Ax’ are both independently pro- portional to (Ap)’ (with different proportionality constants) according to Theorem 4, F should be constant for any specified parameter and equal to the ratio of initial Ax2 to final Ax’. The value of F is thus directly indicative of the extent of correlation and represents the initial displacement of x’ required for a final, reopti-

m U

C 3

> L m L U

.A

- n m L

\

> + +I Ln z w + z b 4

F 2 s ( L i F ) S i 2p (Li-mica)

K I N E T I C ENERGY / eV K I N E T I C ENERGY / eV

Cu 3s (Cu/Au)

2. L ID L Y - n L ID

\

> c

H 1 1 1 1 1 1 1 1 1 1

1124 1126 1120 I t 3 0 1132 1134

K I N E T I C ENERGY / eV

mized, value of unity-the 1 - SD limit for the parameter in question:

where

Parameter uncertainties: practical application. The validity of Eqn (9) was investigated using a wide variety of spectra, chosen to cover as evenly as possible the whole range of symmetric peak shapes, from nearly pure Gaussian to Lorentzian, as well as a range of asymmetric shapes progressively approaching the maximum asymmetry normally encountered. A representative selection is shown in Figs 2 and 3. All the fits were optimized as previously described3 (but minimizing x’), using compiled Prosper0 FORTRAN 77 on a Research Machines Nimbus PC-186 microcomputer with 8087 coprocessor. LG sum functions3 were used throughout for symmetric peaks : asymmetric peaks were represented as b e f ~ r e . ~ Each parameter in each fit was then repeatedly displaced from its optimum value, the

S i 2s (Li-mica) Cu 2 p (Cu/Au)

Cu 3s (Cu/Au)

1124 1126 1128 1130 1t32 1134


1 , 1 , , , , , , , , , , , , ,

318 317 310 319 320 321 322


Fe 2p ( i r o n f o i l )

- 110 am azz a24 azo a21 aao a s as,


Figure 3. Representative single-component fits optimized by minimization of x2 and used to obtain the data for Table 1. The second Cu 3s spectrum illustrated has rn = 0.92; the other two 0.4 eV per channel Cu 3s fits were visually similar. The graphite and platinum spectra are reproduced (but with 0.1 eV channels) in Ref. 3 (fig. 2).

328 S . EVANS

change in xz noted in relation to the displacement and the fit reoptimized with the parameter in question fixed. The results are summarized in Table 1.

The expected quadratic relationship between Axtitial and Ap was closely followed: a single trial displacement usually allowed any desired Ax2 to be achieved to within 5%. However, when the trial perturbations were vastly greater than those for the desired change in x2, the necessary parameter changes were overestimated by up to 25%. The repeated displacements and reoptimiza- tions showed that F was generally constant to within 5-15%, even for very substantial initial displacements, but varied markedly from one parameter to another.

However, the value of F for each parameter describ- ing a symmetric peak was found not to depend strongly on the peak shape (characterized by the LG mix parameter, m). Moreover, no dependence on peak width was found (or expected): the degree of correlation does not depend on the experimental channel width. Approx- imation to the mean values (Table 1) of F, (2.3), F, (3.6) and F , (2.1) for all symmetric peaks would thus lead to errors in estimated uncertainties derived via Eqn (9) of no more than N 12%. However, the data tabulated are suficient to establish closer approximations

F, = 1.9 + 0.7rn; F, = 2.8 + 1.5~1; F, = 1.6 + 0.8m

(R > 0.71; significant at >95% level)

Adequate estimates of F are thus available for all parameters characterizing symmetric XPS peaks represented by LG sum functions.

A rather greater dependence of F on peak shape is apparent for asymmetric peaks. Although constant values for F , (2.4), F, (1.6) and Ft (4.2) are acceptable, the degree of correlation between width and asymmetry depends on the value of the asymmetry parameter, pa :

F, = 5 + 13p, and Fa = 11 + 21p,

(0 < pa 5 0.6); R = 0.82

from the four fits in the table. Even here, however, use of the average values would lead to errors in the derived uncertainties of no more than 25%. The lower value of F, for asymmetric peaks was expected, since m now only governs the peak shape on one side of the maximum3; F, is close to that for symmetric peaks because height is only weakly correlated with asymmetry.

All parameter uncertainties for single-peak fits using these lineshape functions can therefore be closely estimated by using one trial displacement each, without reoptimizing the fit, and applying Eqn (9) with the values of F indicated above. A similar treatment should be possible for other lineshape functions, following experimental redetermination of F for each parameter; one might reasonably expect similar values to apply for modified LG product function^.^

Integrated-intensity (area) uncertainties. Having now specifi- cally identified the correlation-induced components of the uncertainties, we can exclude them when combining parameter uncertainties to obtain the uncertainty in

Table 1. Experimental correlation factors for representative single-peak fits

Symmetric peaks

Channel Mean (initial A,f)/(final A,$) No. of width Reduced LG mix, (No. of determinations in parentheses)b

Peak (sample) channels (CW) (eV) FWHMjCW ;i! 0 np Height FWH M LG mix

F 2s (LiF) Si 2p (Li-mica) Si 2s (Li-mica) C 1 s (diamond)

Cu 3s (Cu/Au) Cu 3s (Cu/Au)' Cu 3s (CU/AU)~ Cu 3s (Cu/Au)"

CU 2 ~ q Z (cu/Au)

32 0.25 8 32 0.25 8 40 0.25 11 92 0.1 10 81 0.1 12

125 0.1 28 29 0.4 7 30 0.4 7 27 0.4 7

2.4 2.0 1 .o

51 11

1.5 1.5 1.7 1.4

0 0.001 3 0.43 0 0 0.0002 0.054 0.01 9 0.090

0.06 0.22 0.40 0.51 0.59 0.64 0.77 0.92 1 .o Mean f ,

1.8 (4) 2.8 (4) 1.7 (3) 2.0 (2) 3.0 (4) 1.7 (3) 2.1 (4) 2.9 (4) 1.7 (4) 2.6 (3) 4.1 (3) 2.3 (3) 2.6 (8) 4.2 (8) 2.2 (5) 2.4 (4) 3.8 (4) 2.1 (4) 2.8 (4) 4.2 (4) 2.6 (3) 2.5 (4) 3.8 (4) 2.5 (3) 2.3 (4) 4.0 (3)d 2.1 (4) = 2.3 f 0.3 f, = 3.6 * 0.6

For parameterization of these factors, see text

f , = 2.1 + 0.3

Asymmetric peaks

Channel Asym. Mean (initial A,f)/(final @) No. of width Reduced index, (No. of determinations in parentheses)

Peak (sample) channels (CW) (eV) FWHMjCW ;i! 0 p. Height FWH M LG mix' Asymmetry TailD

C 1s (graphite)= 46 0.4 2.5 295 0 0.03 2.6 (2) 5.5 (4) 1.8 (2) 10 (2) 3.8 (4) Pt 4f,,, (Pt)" 45 0.4 3.8 50 0 0.18 2.2 (4) 5.5 (4) 1.3 (2) 13 (4) 3.6 (3) Pt 4f,,, Pt)e 45 0.4 3.8 50 0 0.29 2.5 (4) 12 (4) 1.9 (4) 23 (2) 5.3 (4) Fe ZP,,, (Fe) 47 0.4 6.5 2.5 6 x lo- ' 0.61 2.5 (4) 12 (5) 1.6 (4) 22 (4) 4.1 (4)

Mean f ,=2.4+0.2 f,.,=9*4 f ,=1.6*0.3 f , = 1 7 * 6 f ,=4.2+0.8 (For parameterization of f , and f,, which increase sharply

with asymmetry, see text)

a 0 = Gaussian, 1 = Lorentzian-see Ref. 3. SD for these entries range from 0.1 to 0.3. Optimum LG mix varied by changing peak limits. For increase only: decreases cannot be compensated by increase in m.

'The second component of these fits was held constant throughout. ' LG mix parameter values of 0.28,0.002.0.28 and 0.26. respectively. Higher values would attract slightly more correlation, as for symmetric peaks. gTail parameter values of 0.99. 0.94, 0.76 and 0.69, respectively; no significant correlation with f .


integrated intensity. In confirmation of this, observation of the area changes accompanying both the initial displacement and the subsequent reoptimization of the fits showed that the integrated intensity change after reoptimization was always close to that produced by a Axz = 1 parameter change without reoptimization. For- mally, if each parameter displacement Ap results in an area change Aa, then

where the summation is carried out over all parameters affecting the peak area, and provided that Aa and Ap are linearly related. This is the case for all parameters of the peak shapes used, other than the tail parameters of asymmetric peaks. Here, linearity may be assumed over the limited range required. Although the integrated intensity is not as sharply dependent on the LG mix as it is on the height and width, the contribution of the former to the summation is significant because of the lower precision generally associated with this parameter. The peak area may, however, often still be more closely defined (in percentage terms) than some of the individual parameters that characterize the peak shape.

Uncertainties in multiple-component fits

Theory. In multiple-peak fits the internal (intra-) peak correlation still exists, but in addition there is strong interpeak correlation. As for single peaks, we wish to separate out the effects of intrapeak correlation in order to justify the use of quadratic summation to obtain uncertainties in integrated intensity. Following Eqn (9), for each parameter we may write

where the values of F for intrapeak correlation remain as before; F for the peak position parameter is unity, because for single peaks the position is not significantly correlated with the other parameters : Finterpeak can be determined for each parameter from Eqn (11) by the displacement/reoptimization procedure, since c2 is still given by Eqn (9), or via the inverse Hessian The uncertainty in each integrated intensity can then be obtained by quadratic summation, again excluding the intrapeak correlation (cf. Eqn (10)):

(Aa)2Finterpeak AXiZnitial

az(area) = C

the summation running over all parameters affecting the area of the peak in question. Alternatively, and more conveniently when the time taken for multiple reoptimization is significant, an empirical parametrization may be sought for Finterpeak such that a representative set of experimentally-determined error bars is acceptably reproduced. We demonstrate the possibilities of this approach below.

Practical application. A PTFE C 1s spectrum fitted by six peaks was selected as a representative multipeak fit (Fig. 4). The peaks were constrained to remain symmetric

C I s region from contaminated PTFE

- A A v w u- W’

m U

C 3

> L (0 L JJ

.d

.d

n L m \

> I- H v) z W I- z H

95 1 955 959 963 967


Figure 4. Example six-component fit optimized by minimization of X2. The peak numbers indicate the order of addition to the fit: the need for components 1, 2 and 3 was immediately obvious, and peaks 4-6 were then added sequentially until a satisfactory fit could be achieved. or2 = 149; reduced xz = 2.4; RMS deviation = 1.64%. cf. ideal3 of 1.47%.)

except for one asymmetric function3 modelling gra- phitic carbon on the probe tip. The reduced x 2 is 2.4 and Q = 0, indicating that systematic effects are highly statistically significant; yet there is no indication from the residual to suggest that the introduction of any additional peaks is physically justified. Moreover, the counting statistics are relatively poor, and it is obvious that many of the parameters characterizing the fit must have uncertainties related to the noise level (i.e. random uncertainties traceable to the counting statistics) that are large relative to any demonstrable magnitude of systematic error.

Eleven trial displacements (each) of peak height and width, and six (each) of LG mix and peak position were followed by reoptimization, as for the single-peak fits discussed above. Five were duplicates using increased initial displacements, as a check on consistency, thus yielding a total database of 29 discrete parameter uncertainty limits via Eqn (9), as shown in Table 2. The uncertainties relating to the smaller components are surprisingly large, confirming that correlation effects are easily underestimated. There may, however, also be physical constraints on peak parameters : thus the very large widths for peaks 4 and 5 which would be statistically possible, are clearly not physically acceptable.

In five instances, displacements were carried out in both of the two possible directions; note that in three of these the size of the resultant uncertainty differed sub- stantially with direction. Clearly, it would be unwise to rely too critically on any single-valued uncertainty estimate, however obtained : applied in one specified direction, it may easily be in error by a factor of two or more. This should be borne in mind when assessing the results from the approximation offered below.

We then sought the identity of the principal variables governing the extent of interpeak correlation. One

3 30 S . EVANS

Table 2. Uncertainty limits (SD) for Fig. 4, computed via reoptimization, compared with estimates via Eqns (12) and (13y

Mean Peak Fractional Position (eV) Height (%) FWHM (%) LG mix ratio, Area (%) No.b area Corn.= Estd Est./Com. Corn." Est.d Est./Com. CormC Estd Est./Com. Corn.' EsLd Est./Com. Est./Com. Corn.= Est./Com.

1 0.48 0.020 0.018 0.90 1.8 2.2 1.2 2.8+/2.2 - 2.6 1.04 0.05 0.06 1.1 1.07 2.3 2.5 1.09 2 0.23 0.046 0.042 0.91 2.2+/11- ' 4.5 0.7 8.5+/8.0- 5.3 0.64 0.15 0.13 0.9 0.78 8 6 0.75 3O 0.11 0.051 0.040 0.78 4.5 7.0 1.6 10+ 15 1.55 0.18 0.25 1.4 1.32 8 10 1.25 4 0.064 0.22 0.21 0.94 16 20 1.3 64+/35- 30 0.60 0.66 0.60 0.9 0.93 37 29 0.78 5 0.052 0.29 0.21 0.72 16 22 1.4 100+/25- 33 0.52 0.39 0.58 1.5 1.02 41 30 0.73 6 0.060 0.13 0.19 1.45 15 22 1.5 17+ 33 1.95 1.0 0.89 0.9 1.44 34 37 1.09

Mean 0.95 50.3 Mean 1.26 5 0.3 Mean 1.05 5 0.6 Means 1.1 0 5 0.3 1.09 0.2 Mean 0.92 t 0.2

aMinus signs denote results from reoptimization following parameter decreases (where investigated); estimates via Eqn (1 3) are all means from positive and negative trial displacements of 5% on height and width, 0.2 on LG mix and 0.1 eV in position. The ratios between the estimated and computed values show that the approximation yields results within a factor of two (0.5 <ratio < 2.0) for all parameters. The means reflect the reliability of the approximation averaged over each parameter and each peak. Because of the successive-approximation nature of the optimization technique used,3 it is possible that further small reductions in 2 could be achieved. The true SD may thus be marginally higher than reported below. bAs labelled in Fig. 4. "Computed by Eqn (9) via reoptimization.

Estimated via Eqn (1 3). Estimated via Eqns (11) and (1 2) using the parameter uncertainties computed via reoptimization (Eqn (9)).

'Accompanied by physically excessive broadening (250%) of peaks 4 and 5. SAsymmetric peak: estimated uncertainties (SD) for asymmetry and tail parameters of 103% and 0.45, respectively.

major factor was clearly' the energy separation between the peaks, i.e. the degree of peak overlap. The ratio between the sum of the peak widths and the energy separation between them was thus an essential parameter. However, this variable alone could not account for an evident trend for the uncertainties to increase as the peak intensity decreased ; relative intensity was therefore introduced as a second parameter. Empirically, the simple formula

where wi, are the half-widths at half-maximum, Ei, the peak energies and ai, the peak intensities, was found to reproduce all 24 experimental uncertainty limits, ranging over more than an order of magnitude, to within a factor of two (see Table 2), an accuracy comparable with the intrinsic limitations of the concept of the single-valued uncertainty limit. Peak-wise contributions to the summation are shown in Table 3; comparison with Fig. 4 confirms that these accord with qualitative expectations.

On the evidence of Table 2, Eqn (13) might reasonably be applied to any fit comparable with Fig. 4, i.e.

Table 3. Interpeak correlation factor contributions for the data of Fig. 4'

Peak Contribution to correlation factor summation by peak no. no 1 2 3 4 5 6 F,",,,,*,k

1 - 0.06 0.01 0.09 0.02 0.29 2.6 2 0.14 - 0.13 0.55 0.42 0.12 5.8 3 0.04 0.22 - 0.06 0.45 0.03 3.8 4 0.62 1.77 0.11 - 0.20 0.95 13.8 5 0.16 1.90 1.16 0.29 - 0.11 13.7 6 2.15 0.40 0.05 0.99 0.08 - 13.8

a Each entry represents one term in the summation shown in Eqn (1 3). Major component(s) are shown highlighted by bold face.

one with relative peak intensities in the range -1-10, and peak maxima separated by energies comparable with or greater than their FWHM. These conditions cover many XPS peak fits, and this approximate procedure requires only minimal computation. Outside this range, however, or where greater accuracy is required, the displacement-reoptimization (and Hessian- rna t r i~ '~ ) routes to Finterpeak via Eqn (1 1) remain available.

Further work using a much larger database could provide a better parameterization of Finterpeak . The present data do not justify a more complex formulation; in respect of any specified peak or parameter, the SD of the mean ratios in Table 2 are generally greater than the deviation of the means from their target values of unity.

Background errors

Integrated-intensity uncertainties estimated by the above methods do not, however, yet take into account the full extent of the possible random error. The background assumed for each fit is also associated with a statistical uncertainty in the total area, via Eqns (5 ) and (6). The fitting procedure in effect partitions a given area amongst the optimized peaks as closely as the systematic errors discussed below will permit. What is exam- ined is, therefore, the probability that the combination of peaks with the calculated parameters plus the assumed background adequately represents the data. For single-peak fits, with the assumed background calculated directly from L and R channel intensities, this additional intensity uncertainty is larger than the uncertainty in intensity derived above from the fit parameters. Every effort should thus be made to minimize the background error, either by averaging the signal level over several channels when setting the background limits (see discussion of Eqn (7) above) or by using improved methods of data collection.' Even after averaging over as many as nine channels, this uncer-


tainty may still be comparable with the uncertainty derived from the fit parameters, and it might seem advantageous to include the L and R background intensities as additional parameters to be optimized in the fit, as indeed is often done. The introduction of adjustable parameters not directly associated with any specific component of the fit may, however, cause difficulties for the present approach to uncertainty estimation.

Moreover, in multicomponent fits this background uncertainty is far less dominant. For the data of Fig. 4, for example, the overall intensity uncertainty due to the background limits is 2.9% without channel averaging (only 1 % with nine-channel averaging), compared with a 2.5% uncertainty from the fit for the largest component. For the smaller components, the uncertainties derived from the fit are magnified by interpeak correlation, and completely dominate the overall uncertainties in component intensity, as shown in Table 2.

Whenever an assumed background is used, as here and previ~usly,~ one may allow for the additional uncertainty (where significant) by partitioning (arbitrarily) the uncertainty in total area calculated via Eqns (5) and (6) between the components of the fit, and adding the increments to the above integrated-intensity uncertainties quadratically on the assumption that they are essentially independent of the fitted parameter uncertainties. This is not entirely satisfactory; not only may the background error affect some components in the fit more than other but also the random parameter uncertainties calculated above cannot fully reflect the uncertainty introduced by the statistical character of data acquisition.

Systematic errors

Systematic differences between the optimized fitted peaks and the data frequently occur as a result of the lineshape and/or background approximations. The consequent differences between the fitted and the directly calculated total areas are often comparable with the statistical uncertainty on the latter (Eqn (7)), and can amount to as much as several per cent. These discrep- ancies may sometimes be reduced by increasing the sophistication of the fit+e.g. by adding peaks to represent very small components neglected in simpler interpretations-but the intrinsic limitations of the lineshape and background limit the possible improvement. When the SIN ratio is very high, these systematic effects may be much larger than the random uncertainties ; it is then especially important that these errors are not simply neglected. As with the background uncertainties discussed above, however, the effect in multicomponent fits may be significant only for the major components, especially when the SIN ratio is relatively low. Other sources of systematic error in XPS quantification have been discussed by Powell and Seah.”

Unweighted least-squares fits

Finally, we return to the question of uncertainty estimation for fits in which the unweighted mean square deviation is minimized. No rigorous procedure exists, but it

may reasonably be assumed that the statistical uncertainties are of the same order as those applicable to the x2 minimization. Compare, for example, the two fits in Fig. 2, remembering that it has been shown above that the presence of residual systematic error in a minimum- x z fit does not significantly affect the estimation of the statistical uncertainties. Consequently, we introduce a ‘pseudo-xz’, defined by

i R

Y = ( N i - Ci)’ K i = L

where K is chosen such that Y = x2 at the mean-square minimum. Y is then close to, but slightly larger than, the minimum x2 for the fit in question. Y is minimized by the unweighted least-squares procedure, and changes in it resulting from parameter perturbation can be interpreted analogously to the changes in x z discussed above.

The effect of the change in weighting from Eqn (8) to Eqn (14) depends strongly on the peak-to-background (P/B) ratio in the spectrum being fitted. Where the background is at least comparable with the. signal, as for the Cu 3s spectra in Fig. 3 (P/B - 0.3), the estimated uncertainties are almost identical for the two procedures. For the typical Cu 2p,,, fit in Fig. 3, with P/ B - 2.4, the uncertainty estimates for the unwejghted fit range from 70 to 100% of those for the minimum-x2 fit, while in an extreme case such as the diamond C 1s fits in Fig. 2, with P/B - 57, the unweighted-fit uncertainty in height is as low as 32% of that for the minimum-x2 fit, and that in area as low as -50%. These reductions are most pronounced for uncertainties in height and position, the parameters most heavily dependent on the highest data points, which are much lower weighted in x2 than in Y. Uncertainties estimated via Y may thus be (predictably) too small, but they are still valuable because of their objectivity and direct link with data quality.

CONCLUSIONS

The formulae presented in the first part of this paper permit the rapid calculation of statistical uncertainties for any XPS quantification not requiring the resolution of overlapping components.

When a curve synthesis procedure has to be used to distinguish discrete XPS signals, the assignment of statistical uncertainty is only possible if a least-squares optimization has been carried through to refine the fit. It is possible to obtain uncertainty estimates both for minimum-x2 fits and unweighted least-squares fits, although in the latter case their validity has not been rigorously established.

It has been shown that by factorizing each uncertainty into components dependent on peak shape, peak size and overlap, and a statistical component obtained by trial displacement of the optimized fit, acceptable random uncertainty estimates can be obtained both for individual parameters and integrated intensities without having subsequently to reoptimize the fit for each uncertainty required. The results are necessarily approximate, but when more sophisticated procedures are not

332 S. EVANS

available they offer a rapid route to semi-quantitative uncertainty estimates acceptable for routine XPS analysis.',2

Although the resultant uncertainties in composition reflect only the random error in the experiment (while systematic errors in XPS quantification are leg i~n) , '~ the routine adoption of such procedures for random error estimation in XPS quantification will permit immediate direct assessment of the quality of the data used. This is of increasing significance, as the impor-

tance of quality assurance in surface analysis generally is becoming more widely recognized.

Acknowledgements

I wish to thank Dr L. B. Hazell and Dr K. Harrison (BP Research Centre) and Dr P. Cumpson and Dr M. P. Seah (NPL) for draft copies of their papers on these topics;'.' and Dr Cumpson for drawing my attention to Avni's paper," an essential stimulus for this work.

REFERENCES

1. K. Harrison and L. 8. Hazell, Surf. Interface Anal. 18, 368 (1 992).

2. P. J. Cumpson and M. P. Seah, Random Uncertainties in AES and XPS: Peak Energies, Areas and Quantification. NPL Report DMM(A) 26', May 1991 ; Surf. Interface Anal. 18, 345 (1992) and 18,361 (1992).

3. S. Evans, Surf. InterfaceAnal. 17, 85 (1 991). 4. A. Savitsky and M. J. E. Golay,Anal. Chem. 36, 1627 (1964);

corrections, Anal. Chem. 44,1906 (1 972). 5. S. Evans and D. A. Elliott, Surf. Interface Anal. 4, 267 (1 982). 6. M. P. Seah, W. A. Dench, B. Gale and T. E. Groves, J. Phys. E:

7. M. P. Seah and W. A. Dench, J. Electron Spectrosc. Relat.

8. S. Evans and A. G. Hiorns, Surf. lnterface Anal. 8,71 (1 986).

21,351 (1988).

Phenom. 48,43 (1 989).

9. P. M. A. Sherwood, in Practical Surface Analysis by Auger and X-Ray Photoelectron Spectroscopy, ed. by D. Briggs and M. P. Seah, p. 459. Wiley, Chichester (1983).

10. D. A. Shirley, Phys. Rev. B 5,4709 (1 972). 11. Y. Avni, Astrophys. J. 21 0,642 (1 976). 12. P. R. Bevington, Data Reduction and Error Analysis for the

Physical Sciences. pp. 242-246. McGraw-Hill, New York (1969).

13. See, for example, P. R. Mason, J. B. Hasted and L. Moore, Adv. Mol. Relaxation Processes 6, 21 7 (1 974).

14. A. F. Carley and P. H. Morgan, Computational Methods in the Chemical Sciences. Ellis Horwood, Chichester (1 989).

15. C. J. Powell and M. P. Seah, J. Vac. Sci. Technol. A 8, 735 (1 990).

Estimation of the uncertainties associated with XPS peak intensity determination

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