Curve-fitting in XPS using extrinsic and intrinsic background structure

Journal of Electron Spectroscopy and Related Phenomena 106 (2000) 65–80 www.elsevier.nl / locate / elspec

Curve-fitting in XPS using extrinsic and intrinsic background structure a, a b c J.E. Castle *, H. Chapman-Kpodo , A. Proctor , A.M. Salvi a

School of Mechanical and Materials Engineering, University of Surrey, Guildford, GU2 5 XH, UK b Quantegy Inc., 2230 Marvyin Parkway, Opelika AL 36803, USA c Dipartimento di Chimica, Facolta’ di Scienze, Universita’ della Basilicata, Potenza, 85100, Italy Received 15 February 1999; received in revised form 7 June 1999; accepted 14 September 1999

Abstract It has been shown by Tougaard that the extrinsic background to a photoelectron peak is very small at the position of the peak. Most of the background step generated by the peak arises from other forms of energy loss which we have suggested is intrinsic to the photo-excitation process. We have defined this intensity, using a specific modified Shirley background in the immediate vicinity of the peak, and hence fixed it for a given element and chemical state by means of a shape parameter. In this paper we describe an examination of the passive film on stainless steel by ARXPS in which the fixed intrinsic backgrounds are used in fitting oxide and metal for iron and chromium and also for oxygen and carbon. By using peaks of identical shape, position and background for each angle it becomes easy to spot the behaviour of the residual extrinsic background, which as expected depends on angle. In this work we demonstrate the use of a polynomial function, that approximates the Tougaard background, included in the fitting process. This inclusion allows to define a parameter of the polynomial which represents the background slope and use this to assign different background tails to each component in a multiple peak. The behaviour of the extrinsic background can, by this means, be compared with the changing ratio of peak areas as a function of the take-off angle. This work on the behaviour of the extrinsic component of the background within one or two peaks widths of the peak enables surface structures to be placed in depth-wise order with only a small extension to the peak-fitting routines that are normally employed in X-ray photo-electron spectroscopy. We conclude that within the background close to the peak there lies information on the layer thickness and that this might be developed into a reliable method for the estimation of overlayer thickness.  2000 Elsevier Science B.V. All rights reserved. Keywords: XPS; Curve fitting; Extrinsic and intrinsic background; Angle resolved XPS (ARXPS); Thin films

1. Introduction In Fig. 1a we show, using the copper 3s peak as a typical example, the two types of background which are commonly used in data analysis for XPS. Back*Corresponding author. Tel.: 144-1483-259-150; fax: 144-1483259-508. E-mail address: [email protected] (J.E. Castle)

ground (i) is the Shirley [1] background, here extended into a linear tail. Removal of a Shirley background leaves a nearly symmetrical Gaussian / Lorentzian (G / L) peak which is ideal for use in quantitative analysis. Background (ii) is the function developed by Tougaard [2] from a priori principles to represent the extrinsic energy losses associated with the transport of photoelectrons through the solid. We might note that this extrinsic background is

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Fig. 1. Illustration of fitting procedure used in this paper: (a) data from a typical region of an XPS spectrum; (b) construction of peak and background using a fixed value of k in order to define the background rise, h; (c) the choice of different values for the linear term in the polynomial creates a set of tails with a common origin at the peak centre and height, h.

very small at the position of the peak centre which is where the Shirley function has a major influence. In recent papers [3–5] we have suggested that the difference in intensity between the two backgrounds can be attributed to intrinsic losses associated with photoelectron excitation of the electronic structure. We have defined this intensity by means of a shape parameter, k, which represents the intrinsic

background intensity normalized by the peak area, i.e.,

k 5 h /A

(1)

where h is the maximum value (on the high BE side of the peak) of the linear background projected to the peak centre, as shown in Fig. 1b, and A is the Voigt peak area.

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In our work so far we have fitted data using a background function plus a peak function. The peak is a Voigt function (G / L) defined by binding energy, FWHM, and G / L mixing ratio. The background consists of a Shirley function (essentially the integral of a Gaussian approximating the Voigt peak) which has been multiplied with a polynomial, P, so as to produce a background tail, where P is given by:

fit the background using a linear tail and recording its slope. The procedure to do this is carried out within the latest version of GOOGLY [8] which allows the Shirley type of background to each peak to be fixed using a predetermined value of k. In order to handle more simply the fitting of tails, in addition to peaks, the polynomial tail, P, has been recast in the following form:

P 5 B 09 1 B 91 D/W 1 B 29 (D/W )2 1 B 39 (D/W )3.

P 5 k 1 B1 (E 2 E0 ) 1 B2 (E 2 E0 )2

(2)

where W5full width half maximum (FWHM) in eV, D 5 E 2 E0 , where E0 is the peak centre and E 2 E0 expresses the distance in eV from the peak centre. When B 09 51 and B 92 and B 93 are set to zero a family of tails results, taking the form of straight lines with a slope which depends on the value of B 19 . As reported in our previous work [3], these linear tails have a common origin at the peak centre E0 at a height specified by the value of the shape parameter, k. By using a background which is fixed in this manner we fix the whole shape of the peak. In previous papers we have explored the behaviour of the shape parameter, k, using horizontal backgrounds (B 19 50). This was justified because we were only interested in the peak region and chose to ignore the tail itself. However, the steady change in the balance of extrinsic and intrinsic contributions to the tail, as shown separately and clearly in the studies of loss features made by Tougaard [6], means that the tail, even close to the peak, does contain useful information. Tougaard has described the general shape of the extrinsic structure and shown how this relates to the presence of an overlayer and it is this which gives the theoretical basis for the interpretation of the extrinsic background. A full analysis can be undertaken using his program QUASES [7]. However, examination of the shape of the Tougaard function in Fig. 1a shows that the initial part can be approximated by a straight line. The use of a straight line tail as an extension to the Shirley function, i.e., combining together the two background curves of Fig. 1a has the advantage of yielding the simple G / L peak used in quantitative analysis whilst retaining some of the information on overlayer thickness available from Tougaard’s background. The simplest way of generating a parameter is to

(3)

It will be seen that this modification yields the polynomial directly in terms of the x-scale units of electron volts and not in the dimensionless quantity of peak width used in Eq. (2). This allows the tails from peaks of differing widths to be mixed in a logical manner. Further, the value of k, related to the origin of the tail, has been introduced directly into the polynomial which again facilitates the generation of appropriate tails for each peak. The tails obtained when B2 is set to zero and B1 allowed to take a range of different values is illustrated in Fig. 1c, based on the chromium 2p 3 / 2 peak. The program iterates (a) the peak parameters to obtain a best fit in the peak region and (b) the value of B1 in order to obtain a good fit in the tail region. The background region of Fig. 1c contains contributions from each peak in the cluster but for simplicity these other contributions are not shown. In this paper we have the limited objective, relative to Tougaard’s more complete approach, of establishing whether including a small part of the background in the peak fitting routine, using a linear tail, is of assistance in assigning a depth-wise sequence to the materials from which the individual peaks originate. Curve fitting is normally used to separate two or more chemical states and it is probable that these will normally lie at separate depths, as with a metal and its native oxide, and each tail will have a distinctive value of B1 . However the data merge in the post-peak region to give a single total background, subsuming the individual contributions from each peak. In order to obtain sample spectra with which to see whether this situation can be resolved we use an examination of the passive film on stainless steel by angle resolved X-ray photoelectron spectroscopy (ARXPS). Passivation

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generates a thin oxide layer of approximately 2–3 nm in thickness which is ideal for XPS. This oxide is expected to be covered with a contamination layer which will have an increasing influence as the takeoff angle is decreased (towards glancing emission).

2. Experimental

2.1. Material and specimen preparation The chemical composition of the of 316 stainless steels is given in Table 1. The procedure chosen for preparing the samples was similar to that employed by a round-robin exercise involving ten European laboratories [9] to develop a standard method for the use of surface analytical techniques for corrosion studies. This method has been shown to give the least possible contamination with organic compounds. A specimen of 1 cm 2 was polished with 1 mm alumina polish on a felt cloth at 250 rpm. It was then degreased with acetone and ultrasonically cleaned in Milli Q water for 2 min followed by acetone for 2 min. To minimize hydrocarbon contamination from the polishing process and other possible sources, the specimen was ultrasonically cleaned again in isopropanol for 2 min.

2.2. Passivation conditions The specimen was exposed at the free corrosion potential in a solution of 0.5 M NaCl10.5 M H 2 SO 4 10.08% H 2 O 2 for 10 s. All experiments were carried out at room temperature between 22 and 268C. After exposure to the electrolyte, washing was done by dipping into very pure Milli Q water for 1 s and dried by blotting with tissue on a lower corner. The specimen was then mounted and entered into the UHV system.

2.3. XPS measurement Angle-resolved-XP spectra were recorded using a VG Scientific ESCALAB MkII spectrometer with unmonochromated Al Ka excitation (hv51486.6 eV) operating at 10 kV and 34 mA. The base vacuum during acquisition was better than 10 29 mbar. The analyser was operated in the constant analyser energy (CAE) mode, with a pass energy of 50 eV for survey spectra and 20 eV for narrow scans and channel width of 2 and 0.2 eV, respectively. Four take-off angles of 75, 35, 20 and 158 respectively, (relative to the sample surface), a series which is approximately linear in 1 / sin u, were used throughout the experiment. No argon ion etching was done prior to exposure. Spectra acquisition was done using the datasystem attached to the spectrometer and were then transferred to an IBM compatible PC computer where the peak-fitting program, GOOGLY, was run after data files had been converted to a standard format required by the program.

3. Results and curve fitting. The values of k for iron, chromium and nickel have been published elsewhere [3,5] and are given with other fitting parameters in Table 2. Suitable values for oxygen in the passive layer and contamination carbon were determined by trial and error and then held constant throughout the angular series: the parameter B2 was set to zero. Thus the area of the peak and the slope parameter of the tail (B1 ) are the only variables determined by the peak fitting. By using peaks of identical shape (including the intrinsic background) and position, the angular variation is confined to variations in the oxide / metal peak areas and in the value of B1 . Spectra from the 2p regions of Fe and Cr together with the 1s of oxygen and

Table 1 The chemical composition of 316 stainless steels Element wt%

Cr 17.05

Ni 11.14

Mo 2.39

Mn 1.69

C 0.051

Si 0.27

P 0.015

S 0.009

Cu 0.045

Ti 0.03

Fe balance

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69

Table 2 FWHM (eV), and B1 coefficient parameters used for fitting Fe2p, Cr2p, O1s and C1s peaks Element

FWHM (eV)

Sensitivity factor

k

B1 (u 5758)

B1 (u 5358)

B1 (u 5208)

B1 (u 5158)

Fe2p 3 / 2 (Met) Fe2p 3 / 2 (Ox) Fe2p 3 / 2 (Sat) Cr2p 3 / 2 (Met) Cr2p 3 / 2 (Ox) Cr2p 3 / 2 (Sat) O1s (Peaks 1 and 2) O1s (Peak 3) C1s

1.7 4.35 4.35 1.8 3.35 3.35 1.84 1.84 1.84

2.0 2.0 2.0 1.5 1.5 1.5 0.66 0.66 0.25

0.100 0.095 – 0.07 0.06 – 0.005 0.004 0.004

0.0040 0.0006 0.0006 0.0070 0.00026 0.00026 0.0026 0.0018 0.0018

0.0160 0.0008 0.0008 0.0300 0.0009 0.0009 0.0033 0.0016 0.0016

0.0400 0.0009 0.0009 0.0700 0.0011 0.0011 0.0037 0.0022 0.0022

0.0650 0.0012 0.0012 0.1800 0.0017 0.0017 0.0043 0.0018 0.0018

carbon were fitted using the modified version of GOOGLY. In GOOGLY the contribution, T(E), by the tail region of a given peak, to the signal in the background, N(E), at any point (E 2 E0 ), is given by:

mation. To obtain the condition for a linear overall slope as a sum of partial slopes we need to separate E and E0 . Rearranging Eq. (5) for the tail region, in which Q(E) 5 1, we obtain:

T(E) 5 AQ(E)P

T(E) 5 A.B1 .E 1 A(k 2 B1 E0 ) 5 A.B1 E 1 C

(4)

in which A is the Voigt area of the peak, P is the value of the polynomial at the given position and Q(E) is the integral of a Gaussian, scaled to have a value between 0 and 1 across the width of the peak and hence to define the Shirley background shape. Thus:

where

T(E) 5 AQ(E)[k 1 B1 (E 2 E0 )]

and the slope of the tail for this peak is

(5)

and the contribution made by a given peak to the approximately linear slope of the background after the peak, d(T(E)) / d(E 2 E0 ), is given by: d(T(E)) / d(E 2 E0 ) 5 AB1

(6)

since in the linear region Q(E) 5 1. When the program is used to fit an overlapping cluster of peaks, it does not yield unique values of tail slope B1 : the only requirement is that the sum of all linear components from the individual tails is equal to the best straight line fit to the tail region of the raw data, i.e., the slope of the line fitting the data in the tail region is given by the sum of each partial contribution. Since each individual tail commences at a unique individual value of E0 this sum can only be made in a meaningful way in an overlap region in which each partial tail has reached its linear approxi-

(7)

C 5 A(k 2 B1 E0 ) 5 constant For any peak i, we have: T(E) i 5 A i B1,i E 1 Ci

(8)

dT(E) i / dE 5 A i .B1,i

(9)

These tails can be summed for all n peaks for the region in which the contribution of all values of Ci have been incorporated (i.e., beyond the peak of highest binding energy). dN(E) / dE 5

O (A .B i

1,i

).

(10)

i 51 to j

where A i . and B1,i are the individual values for each component peak. To obtain the fits shown in Figs. 2–5 the peakfitting parameters given in Table 2 were used, together with linear tails having a slope determined by varying the parameter B1 . In recording the parameters used in Table 2 for oxygen and carbon the peaks were assigned numbers, peak 1, peak 2, etc. starting from that of the lowest binding energy.

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Fig. 2. The Iron 2p spectra at four take off angles, fitted using linear tails and the published value of k.

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Fig. 3. The chromium 2p spectra at four take off angles, fitted using linear tails and a published value of k.

Satellite peaks were inserted in their published positions but, unlike the primary photo-peaks were given a k value of zero, since they themselves represent discrete final states within the general intrinsic loss tail of the primary peak. Examination of the figures shows that relatively good fits can be

obtained using the predetermined k values and a tail of constant slope. As expected, equally good fits are obtained for sets of tails ranging from positive to negative values of B1 as long as the tails on differing peaks behave in a complementary manner. Notwithstanding the behaviour of the tail slope,

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Fig. 4. The oxygen 1s spectra at four take off angles, fitted using linear tails and a fixed value of k.

the peak areas are reliably derived from the fitting procedure and do not depend on the manner in which the tail slopes are chosen. As we explain below, the

peak areas were first used to derive a structure and then the structure was used to guide the choice of the particular values of B1 used in assessing the contri-

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Fig. 5. The carbon 1s spectra at four take off angles, fitted using linear tails and a fixed value of k.

73

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bution to the tail slopes due to each peak component. The results can be seen in Figs. 2–5.

4. Discussion

4.1. Composition and intensity ratios The contamination and oxide layer thickness were modelled using the program ANGULAR [10] and are as shown in Fig. 6. ANGULAR is typical of the programs which predict an angular variation of intensity from a specified structure and is most reliable when rather simple structures are used. In this case we use a three layer model with a small blurring of the interfaces between the layers. On this basis the results should be reliable. The examination of chemically passivated steel by ARXPS is relatively unusual. However, the thickness of the passive layer is 2.7 nm which is slightly thicker than the minimum found on electrochemically passivated steel [9]. The thickness of the contamination layer, at 0.4 nm, is not at all unusual. Nickel does not oxidize [11] during passivation but concentrates slightly at the metal / oxide interface. This model structure gives a basis from which to consider the behaviour of the background for each individual element.

4.2. Background structure The above structure has two layers, contamination of thickness d 1 and oxide of thickness d 2 . When this is examined at four angles, u1 2u4 , we obtain a set of four effective values, d 1 / sin u1 – 4 , for the contamination layer thickness. This will determine the background slopes, B1,oxide , arising from the oxide component of the element. There will also be four effective values for the contamination plus oxide thickness, (d 1 1 d 2 ) / sin u1 – 4 which will influence the background slope, B1,metal , arising from the metallic component of the element. These two sets of B1 values are correlated by the fact that pairs of values, for each value of u must obey the sum given in Eq. (10), i.e., A oxide B1,oxide 1 A metal B1,metal must give the derivative of the best line through the data points in the tail region. In order to provide sets of B1 values that match this criterion for each angle we first fitted the data

with the slope parameters for oxide and metal equal. Following this the value for the oxide slope was varied in approximately 10% steps and the program used to generate an appropriate value for the metal slope. This ensured that for all pairs the criterion represented by Eq. (10) was maintained. As an illustration of the data set thus produced, four pairs of values of B1,oxide , and B1,metal have been plotted for each angle against d /lsin u, in Fig. 7. In this plot l is the effective attenuation length for the given photoelectron and d is the overlayer thickness, i.e. either d 1 or (d 1 1 d 2 ). In order to interpret the plots in Fig. 7 we need to recall that B1 5 hd(T(E)) / dEj /A

(11)

i.e., B1 is the relative importance of the slope to the area of the peak and matches our subjective feel for the importance of the sloping background. In particular the effect of extrinsic scattering is to give a steady increase in the value of B1 . When the sets of possible values for B1 for the iron plus iron oxide spectrum, shown in Fig. 7, are examined it can be seen that there is one set which passes from oxide to metal without a discontinuity in the curve. The cross-over point is shown more clearly in the inset. The physical significant of this smooth behaviour is that it satisfies the condition that the background slope should steadily increase with the increase in overlayer thickness, whatever the composition of the superficial scattering layers. It is this set which was chosen for the fitted spectra shown in Fig. 2. A similar set was adopted from the chromium spectra of Fig. 3. In the case of the carbon spectra it was assumed that the value of B1 for each component peak should take the same value since they are all located essentially at the same depth within the contamination layer. The peaks for oxygen were treated differently. Here it was recognised that the high binding energy component was located with carbon in the contamination layer and was given the corresponding value of B1 : fixing this enabled the value appropriate to the oxide (oxygen, peak 1) and hydroxide (oxygen, peak 2) components to be derived from the program. The actual values for all elements and take-off angles are given in Table 2 and are plotted in Fig. 8a and b). The parallel behaviour of the oxygen and chromium

J.E. Castle et al. / Journal of Electron Spectroscopy and Related Phenomena 106 (2000) 65 – 80

Fig. 6. The model depth profile and resultant match between calculated and experimental compositions as a function of angle.

75

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Fig. 7. The variation of the relative slope, B1 , with take-off angles. The cross-over region is shown enlarged in the inset figure.

values for the oxide layer, derived by differing routes, is especially gratifying since these elements are both overwhelmingly found in this single layer. Having obtained a set of valid values for B1 , the values of B1 A can be obtained. These values are

plotted in Fig. 9a–c, where again, for clarity, they are separated into the separate regions of contamination, passive film and substrate. Contrary to the values for B1 , the values for B1 A, the absolute slope in counts per eV, decreases with increased overlayer

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77

Fig. 8. The variation of the slope parameter, B1 with angle: (a) for the oxide and (b) for the metallic substrate.

thickness. The parallel behaviour shown by O, Cr and Fe in both oxide and metal is immediately apparent from both Figs. 8 and 9.

4.3. A normalised background parameter The fact that the values of B1 A for the key elements are similar in behaviour suggests that a normalisation based on this product would give a

more general representation of the slope as a function of the overlayer thickness. Using standard relationships, the signal derived from the metallic substrate, A metal , can be expressed as: A metal 5 JnS exp(2(d 1 1 d 2 ) /lsin u )

(12)

the exponential term allowing for the attenuation of the signal by the total overlayer thickness, d, in terms

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determining peak area means that it is impossible to obtain a ‘universal’ curve by manipulation of these parameters. In fact, it is preferable to base the normalisation on Eq. (11), since we can express this as: B1 5 hd(T(E)) / dEj /JnS exp(2d /l sin u )

(13)

and hd(T(E)) / dEj /J should be independent of flux. Hence: B1 nS 5 hd(T(E)) / dEj /J exp(2d /l sin u ).

(14)

The right hand side is now independent of the sensitivity factor and concentration of a given element and varies only with the depth distribution. Thus, for a given photon flux and spectrometer condition the left hand side represents a normalisation of the slope parameters and for convenience it will be referred to as the function B(d). By means of Eq. (14) we can readily explore the behaviour of B(d) as a function of overlayer thickness. In order to create a unified depth scale we have replaced ‘d /l’ with the ‘Z’ function, the characteristic depth parameter, proposed by Seah et al. [13], where Zi 5 Di 1 [1 /(cosec u 2 1)]ln h[cosec u (1 2 exp(2T i ))] / [1 2 exp(2T i cosec u )]j.

Fig. 9. The variation of the absolute slope, B1 A, with angle: (a) for the contamination layer, (b) for the oxide and (c) for the metallic substrate.

of the effective attenuation length of the given element / compound [12]. J is the photon flux, n the concentration of the given element and S its sensitivity factor. Whilst this gives an apparent basis for normalisation the fact that the photon flux is instrumental in

(15)

Zi is a weighted depth, relative to the effective attenuation length, and corresponds to a virtual depth from which the photoelectrons of a given species ‘i’ can be thought to originate. The dimensionless values, D and T, are obtained as d /l and t /l where d is the depth of the surface of a given layer and t is the thickness of that layer as derived from the modelled depth profile reported in Fig. 6. Thus: B1 nS 5 hd(T 2) / dEj /J exp(2Zi / sin u ).

(16)

Fig. 10 shows all the values of B(d) plotted against the appropriate values of Zi li / sin u. The logarithmic plot used to encompass that wide range of values also minimises very real departures from a truly universal curve. However, it does demonstrate the manner in which the values of B(d) show a broad difference of a factor of ten for each layer. This

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79

Fig. 10. The normalised slope parameter, B(d), as a function of thickness.

gives clear indication as to the benefit which can be obtained from using this parameter to obtain a first indication of layer sequence. All the oxygen values fall at the higher extreme of the scattered values along the curve for B(d). It is possible that this arises because of the intrusion of a discrete energy loss into the fitted region. The rise of data points into this loss peak has the effect of putting an offset into the slope determined by the programme. Where this occurs it will be important to acquire a wider window of data in order that a line can be fitted across the whole of the loss feature. Graat, Somers and Bottger [14] show in their detailed study of the near peak background that each scattering event requires an additional self-convolution of the background signal. Thus, a simple approximation to the magnitude of the extrinsic losses using slope alone can never succeed in giving anything but an empirical relationship. The tendency of B(d) to become constant at large overlayer thickness is in accord with their view that repetitive scattering moves background intensity steadily away

from the peak so that the near-peak region effectively saturates with regard to slope. The importance of the present interpretation of the background is that within a single peak fitting routine for often encountered structures such as an oxide on metal it will be possible to assess overlayer thickness and composition using a single take-off angle. This will be done by selecting values of B1 which satisfy both Eqs. (10) and (16), using Fig. 10 as a guide. It might be argued that this is no more than can be done already by use of metal / oxide peak intensity ratios but this is suitable for pure metals alone. However when there is a significant composition change in the oxide the ratio method is invalid. Overlayer thickness estimated by use of the background does not suffer from this deficiency. As we indicated in the Section 1, the new fitting procedure separates the general change, with angle, of the background from the changing ratios of the component peaks. This work has shown that the background on iron is different from that on chromium because of the very different intensities of

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background derived from the substrate spectra. The backgrounds arising from the influence of contamination on the oxide are almost equal. The background feature in fact shows at a glance that the concentration of iron in the oxide is depleted relative to the metal whilst that of chromium is enhanced The ability to reach this conclusion in absolute confidence does, we believe, provide a worthwhile justification for this new form of data analysis.

Ricerca Scientifica e Tecnologica in collaboration with the Conferenza dei Rettori delle Universita’ Italiane and the British Council (UK) under the ‘British–Italian Collaboration in Research and Higher Education’ 1998–1999 project. H.C.-K. acknowledges the University of Surrey for provision of his research scholarship.

References 5. Conclusion This work builds on our earlier definition of the intrinsic background shape in order to define a slope parameter which varies with extrinsic background. The programme GOOGLY has been implemented to allow the use of a specific modified Shirley background associated with each component of the mixture (i.e., this background is compound specific and independent of take-off angle) and the use of approximate Tougaard background (in form of peak tail) to take into account the expected differences in background slope with take-off angle. We conclude that, as shown by the Tougaard model, the slope of the extrinsic background close to the peak can be used for layer sequencing and can provide a first estimation of overlayer thickness during routine curve fitting. In the examination of a passive film on stainless steel we have shown the very large extrinsic background associated with the iron peak, relative to that on chromium peaks, stems entirely from the negative concentration gradient of iron: high in the metal and low in the oxide.

Acknowledgements This work was carried out with the financial assistance of the Ministero della Universita’ e della

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