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A self-modeling approach to the resolution of XPS spectra into surface and bulk components
Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197–210 www.elsevier.nl / locate / elspec
A self-modeling approach to the resolution of XPS spectra into surface and bulk components a, b a Gary W. Simmons *, David L. Angst , Kamil Klier a
Department of Chemistry and Zettlemoyer Center of Surface Studies, Lehigh University, Bethlehem, PA 18015, USA b Lucent Technologies Research Laboratory, Breinigsville, PA 18031, USA Received 10 March 1999; accepted 26 June 1999
Abstract A method of resolving XPS spectra into surface and bulk component spectra for cases of moderate energy resolution is designed based on an analysis of a family of spectra acquired at different polar angles from flat specimens. Assumptions about line shapes are not required, but an analytical model of the angular dependence of the bulk and surface XPS signals is needed when the component spectra overlap in the range of binding energies of interest. Recommendations are made of the error limits of experimental variables for the successful application of the self-modeling method. The method was used successfully in the separation and quantitative analysis of the O(1s) XPS spectra of surface silanols from bulk silicon dioxide of a fully hydrated silicon dioxide surface. The surface (silanol) and bulk (oxide) components were found to be separated by 0.30 eV, and the surface component was found to be broader (1.58 eV) than the bulk component (1.15 eV). 1999 Elsevier Science B.V. All rights reserved.
1. Introduction X-ray photoelectron spectroscopy (XPS) has been employed widely for studies of the chemical and physical properties of surfaces. Chemical information is limited, however, for cases when photoemission spectra of different chemical species are not clearly energy resolved. A typical example of this limitation is shown in Fig. 1a of the O(1s) spectrum from a hydrated silicon oxide surface. These spectra, normalized to unit areas, were taken at different photoelectron detector angles. The relative increase in intensity on the high binding energy side of the O(1s) line with increasing angle can be attributed to oxygen such as surface silanol groups, ‘strained’ *Corresponding author.
oxygen in the oxide surface and oxygen in residual organic compound(s). Since steps were taken to fully hydrate and clean the surface of this specimen, it is assumed that the spectra shown in Fig. 1a represents mostly bulk silicon oxide and surface silanols. Quantitation of the silanol coverage requires curveresolving methods to determine the relative areas of the silanol and oxide O(1s) photoemission lines. Because of the relatively small chemical shift between silanol and oxide species, regression methods for resolving these two components require assumptions about line shapes and fitting constraints. Analysis of the spectra shown in Fig. 1a using such methods have not given unequivocal results. More satisfactory curve resolution, however, can be obtained using a self-modeling method that is based on principal component analysis. Although restricted to
0368-2048 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 99 )00066-3
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Fig. 1. (a) Oxygen (1s) XPS spectra of hydrated silicon oxide surface as a function of detector angle: 08, 458, 658, 758, 808, and 858. (b) Simulated spectra comprised of the sum of two arbitrary functions representing bulk and surface components.
two component systems, surface and bulk components in this case, self-modeling can be used to obtain the spectrum of each component without assumptions about their line shapes. The developments of principal component analysis is well documented [1–14]. The method described in this paper follows closely that given by Sylvestre et al. [7,8] and presents an analysis of angle-resolved spectra by this method for the first time. It is assumed that the energy (E) spectrum of a two component system has the form: Y(E) 5 a f1 (E) 1 b f2 (E)
(1)
where the spectrum has been normalized to unit area, f1 (E) and f2 (E) are unknown, nonnegative, linearly independent functions also normalized to unit area, and a and b are nonnegative variables independent of E. The normalization is taken over the binding energy range of L1 to L2 . L2
E Y(E) dE 5 1;
L1
L2
and
E f (E) dE 5 1 i
L1
(2)
This normalization requires that a 1 b 5 1 and makes Y(E) a convex combination of f1 (E) and f2 (E). In the discussion that follows, f1 (E) is the photoelectron spectrum from the bulk and f2 (E) the spectrum from the surface. It is important to realize that fi (E) may represent one or more chemical species. When more than one chemical species is present, other resolution methods may be employed after the surface and bulk spectra have been separated. The variables a and b in Eq. (1) represent the composition of a two component system, and the resolution of the spectral line for each of the pure components is subject to the restriction that there is no region in the binding energy range L1 to L2 for which f1 (E) and f2 (E) are simultaneously identically zero: f1 (Ej ) 5 0 and f1 (Ek ) . 0 and
f2 (Ej ) . 0 f2 (Ek ) 5 0.
(3)
This restriction is not met for the spectra shown in Fig. 1a, where the surface and bulk spectra overlap over most of the O(1s) photoelectron range of
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
199
binding energies. The restriction given by Eq. (3), however, can be lifted when a and b are known functions [8]. Since the relative intensities of the surface and bulk signals change with detector angle, the variables a and b are functions of this angle. The form of the a(u) and b(u) functions used in lifting the restrictions [Eq. (3)] are, however, dependent on the model chosen to describe the angular dependence. Although this model dependence may be considered a limiting feature, it does offer a more rational approach to spectrum resolution than guessing at line shapes for the O(1s) photoemission from the surface and bulk components; an even more unpleasant task if the surface and bulk components each have several overlapping photoemission lines. The objective of this paper is to demonstrate a method for separating the surface and bulk components of angle resolved photoelectron spectra without any assumption about the line shapes of either component. Firstly, various surface layer models are considered to arrive at possible angular dependent functions, a(u) and b(u). Secondly, a model considered to be applicable to a monolayer of silanols on silicon dioxide is used to simulate spectra for evaluating the self-modeling resolution technique. Thirdly, an assessment of the effects of experimental errors is made, and finally, the technique is used to separate the surface and bulk components of the experimental spectra shown in Fig. 1a.
Fig. 2. Geometry used to model the angular dependence of the surface and bulk photoelectron intensities.
2. Surface layer models
1. Photoemission from an infinitely thick substrate (x 0 → `) through a layer of finite thickness, x 1 :
The angular dependence of photoemission from surface layers has been detailed in the literature [15–19]. The essential features of these treatments are given to serve as reference for the discussion of the surface / bulk resolution technique that follows. The specimen surface is assumed to be flat and photoelectron spectra are acquired over a large range of photoelectron detector angles, u, as defined in Fig. 2. The photoemission intensity Iw in the direction u from a point located at x is Iw 5 I0 e
2w / l e 2w 1 / l s
e
(4)
where le is the mean escape depth in the material of the emitting layer and ls is the mean escape depth in
the screening layer. In the simplest approximation, le 5 ls 5 l, and backscattering and refraction are neglected. Integration of Eq. (4) for a given u over all w, w5x / cos u, between w5 0 and w0 5x 0 / cos u gives: I(u ) 5 I0 e 2x 1 / l cos u l[1 2 e 2x 0 / l cos u ]
(5)
which represents a master equation for two layers of arbitrary thicknesses x 1 and x 0 . It has all the features to describe the angular dependence of photoemission from uniformly thick layers.
I(u ) 5 I0 le 2 x 1 /lcos u
(6)
2. Photoemission from a surface monolayer of thickness t 0 (x 0 5 t 0 , x 1 5 0): I(u ) 5 I0 t 0 / cos u
(7)
3. Photoemission from an infinitely thick substrate with no overlayer (x 0 →`, x 1 50): I(u ) 5 I0 l
(8)
4. Photoemission from a thin uniform elemental layer (x 0 → t 0 ) screened by an outer layer of thickness x 1 :
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I0 t 0 2x / I(u ) 5 ]] e 1 l cos u cos u
(9)
The photoelectron emission is also a function of photoelectron diffraction effects and instrumental factors. Diffraction effects are not likely to be significant for a monolayer coverage of silanols on a substrate of amorphous silicon oxide. An angular instrument response function, F(u ), is incorporated into the model equations to account for changes in photoelectron signal specific to the geometry defined by the x-ray source, specimen and detector. The experimental photoemission is then given by I(u ) experimental 5 F(u )I(u )
(10)
where I(u ) is the photoemission from one of the specific models described above.
3. The surface silanol model The spectra shown in Fig. 1a were taken from a specimen prepared in the following manner. A silicon wafer of highly polished integrated circuit substrate material, was cleaned by a procedure used in integrated circuit manufacturing [20]. The substrate was etched for 5 min in 10% HF / water to remove oxide. The substrate was then immersed for 15 min each in two separate baths of ammonium hydroxide, hydrogen peroxide, and water (5:1:1 ratio, 758C), followed by immersion in a bath of hydrochloric acid, hydrogen peroxide and water (6:1:1 ratio, 758C) for an additional 15 min. The cleaned substrates were oxidized at 11508C in 100% oxygen using a Tamarack 100-M rapid thermal annealing unit. The thickness of the oxide formed during a 5-min oxidation was on the order of 130– ˚ The substrate was then immersed in high 150 A. purity deionized water and held at 908C for 20 days to completely hydrate the surface oxide. Residual organic carbon was reduced by a five min exposure of the substrate to UV-ozone just prior to insertion into the spectrometer. X-ray photoelectron spectra 1 1
XPS spectra were taken with a SCIENTA ESCA 300 spectrometer using monochromatic Ka X-rays from a rotating anode source and a computer controlled goniometer.
were taken of the O(1s) binding energy region for detector angles of 08, 458 658, 758 808 and 858. A zero base-line was established for each spectrum by subtracting a straight line defined by the average of the first and last ten data points. Each spectrum was then normalized to unit area, Fig. 1a. The silicon dioxide is assumed to be covered with a saturated layer of silanols (|4.6 OH groups per nm 2 ) [21], the angular dependent O(1s) photoemission from the silanol layer is assumed to be given by Eq. (7) (x 1 5 t 0 ) and from the oxide substrate by Eq. (6). Eq. (1) is now written for silicon oxide covered with a uniform monolayer of silanols as: Yi (E) 5 F(ui )I0sn O le 2t 0 / l cos ui f1 (E) 1 n OH t 0 f2 (E) / cos uid
(11)
where n O is the volume concentration of oxygen in the oxide and n OH is the volume concentration of silanols in the surface layer. Since the silanol layer has a thickness on the order of the diameter of an -OH group, then to a first approximation t 0 < l. Y(E) 5 F(u1 )I0 (n O l f1 (E) 1 n OH t 0 f2 (E) / cos ui )
(12)
The integral used to find the normalized form of Eq. (12) is written as follows: L2
L2
E Y (E) dE 5 F(u )I 1n l E f (E) dE i
L1
i
0
O
1
L1
L2
1 n OH t 0 / cos ui
E f (E) dE2 2
(13)
L1
Since eLL12 5 1 and eLL12 Yi (E) dE 5 1, the normalized form of Eq. (12) is given by Eq. (14). n O l f1 (E) 1 n OH t 0 f2 (E) / cos ui Y1 (E) 5 ]]]]]]]]] n O l 1 n OH t 0 / cos ui
(14)
Since the quantity n 0 l is a constant over the energy range of the O(1s) emission, the concentration of surface silanols can be referenced to the oxide substrate, k 5 n OH t 0 /n O l. The model O(1s) angular dependent photoemission for a monolayer of silanols on silicon oxide is given in the final form as follows:
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S
1 Yi (E) 5 ]]]] 1 1 k / cos ui
DS
k f1 (E) 1 ]] f2 (E) cos ui
([V ] 21 [M][V ]) jm 5 l jdjm
D
201
(19)
where l j are eigenvalues of the set of equations (15)
Arbitrary functions for f1 (E) and f2 (E) (mixed Lorenzian / Gaussian) were used in Eq. (15) to simulate a set of spectra for a value of k50.150 and for angles of 08, 458, 658, 758, 808 and 858. These spectra, shown in Fig. 1b, exhibit features similar to the experimental data shown in Fig. 1a. The next section describes the procedure for retrieving the value of k and the functions f1 (E) and f2 (E) from the simulated spectra.
4. Self-modeling curve resolution of simulated spectra In the procedures that follow, assumptions are not made about the value of k nor the functions f1 (E) and f2 (E) that were used to produce the simulated data, and for this reason the resolution method can be thought of as self-modeling [7,8]. The mathematical steps are presented in sufficient detail such that they can be readily implemented using a high-performance, high-level platform such as MATLAB [22] 2 . In vector notation, Eq. (1) is written as follows:
[M]Vj 5 l jVj ,
(20)
Vj is the jth column of the matrix [V ] and djm is the Kronecker delta. The relative magnitude of the eigenvalue l j corresponding to an eigenvector Vj indicates the relative contribution of that eigenvector to the variance of the data. For the two component system under consideration, all of the variance is represented by the eigenvectors associated with two largest eigenvalues. Any remaining non-zero eigenvalues are attributed to system noise and experimental error, and the corresponding eigenvectors are therefore not included in the analysis. If Eq. (16) holds exactly, then all the spectra, Yi (E) can be expressed as linear combinations of the two eigenvectors, V1 (E) and V2 (E) associated with the two largest eigenvalues of M: Yi (E) 5 j 1iV1 (E) 1 j 2iV2 (E).
(21)
Once the eigenvectors V1 (E) and V2 (E) have been determined, the model spectra Yi (E) are found by the best least squares fit for each spectrum given by Eq. (21), where j 1i , and j 2i (scores or weights) are the values which minimize the sum of squares: p
Yi (E) 5 ai f1 (E) 1 bi f2 (E)
(16)
The experimental data Yi (E) are expressed as an n3p matrix, where n is the number of spectra and p is the number of points in each spectra, Y1 (E1 ) Y5 ? Yn (E1 )
*
? ? ?
? Y1 (Ep ) ? ? ? Yn (Ep )
*
(17)
and the covariance matrix, M, of this data matrix is formed by multiplying the data matrix, Y, by its transpose, Y9 M 5 [Y9Y] /n
(18)
The covariance matrix is then diagonalized by finding a matrix [V ] such that 2
A file of the MATLAB commands and output for the analysis of the simulated spectra can be obtained by contacting the first author at [email protected].
O [Y (E ) 2 Y (E ) ] i
, s
i
2
(22)
, m
, 51
where Yi (E) s is the observed spectra and Yi (E) m the model spectra. Because of the orthogonality of the eigenvectors, the scores can be found from the following dot products:
j ij 5Vj ? Yi
(23)
Yi (E) s are then compared with the predicted Yi (E) m , and if they agree within experimental error, the form given by (Eq. 16) is verified. If the errors are large, or show lack of fit then self-modeling of the spectra is not applicable. The error can be estimated as follows [8]. p
OO n
2
[Yi (E, ) s 2 Yi (E, ) m ] ]]]]]] s 5 n( p 2 2) i 51 , 51 2
(24)
The eigenvalues and eigenvectors of the M matrix [Eq. (18)] of the simulated spectra were evaluated
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and only two non-zero eigenvalues were found, as expected, for this error-free two component system. The two eigenvectors V1 (E) and V2 (E) corresponding to the two nonzero eigenvalues are plotted in Fig. 3. Values of j 1i and j 2i for each spectrum (i51 to n) were determined using Eq. (23) and are shown in a j 1 versus j 2 plot as points on the BC line segment, Fig. 4. These results were then used to reconstruct each simulated spectra according to Eq. (21) and then compared using Eq. (24). The model spectra were found to be in perfect agreement with the original spectra (s 55.45310 217 ). The next step retrieves the original functions, f1 (E) and f2 (E), and the value of k used in the simulations. The spectra of the pure components are also linear combinations of the eigenvectors V1 (E) and V2 (E). f1 (E) 5 h11V1 (E) 1 h21V2 (E)
(25)
f2 (E) 5 h12V1 (E) 1 h22V2 (E) Each of the observed vectors, Yi (E), is completely
Fig. 4. Each score on segment BC successfully reconstructs the corresponding original simulated spectrum. The scores at A and D represent the limits for the retrieval of the pure component functions. For the analysis of the simulated spectra described in Section 4, the scores h12 , h22 and h11 , h21 successfully retrieve the pure component functions used in the simulations. The solid line is defined by Eq. (30).
described by the scores ( j 1i , j 2i ) (i 5 1 to n) of the eigenvectors and is represented as a point in the ( j 1 j 2 )-plane. The scores ( j 1i , j 2i ) on the line segment BC shown in Fig. 4 were obtained from the analysis of the simulated spectra shown in Fig. 1b. Since the vectors f1 (E) and f2 (E) are also linear combinations of the eigenvectors, V1 (E) and V2 (E), then the scores (h11 , h21 ) and (h12 , h22 ) that describe the component spectra are also found in the ( j 1 , j 2 )-plane. The next step is to restrict the area in the ( j 1 , j 2 )-plane where (h11 , h22 ) can be found. Since the unknown vectors f1 (E) and f2 (E) have nonnegative elements, the scores (h11 , h21 ) and (h12 , h22 ) must be in the region of the ( j 1 , j 2 )-plane which satisfies:
j 1 n1 , 1 j 2 n2 , $ 0 Fig. 3. The eigenvectors V1 (E) and V2 (E) obtained from the covariance matrix of the simulated spectra shown in Fig. 1b. These eigenvectors along with the scores derived from the analysis contain all of the information to reconstruct the original spectra and to reproduce each of the component spectra.
(26)
for all , where nj , is the , th element of the jth eigenvector Vj (E). To find the set of points in the ( j 1 , j 2 )-plane that satisfies this condition, it is important to note that while all of the elements of the eigenvec-
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
tor V1 (E) are nonnegative, some of the elements of the V2 (E) eigenvector are negative because of the orthogonality imposed on eigenvectors, (Fig. 3). Let
n1 2 , 5V2 , n1 1 , 5 v1 ,
(V2 , $ 0)
(27)
(V2 , $ 0)
2 where n 1 1 , and n 2 , are the elements, ,, of the vectors V1 (E) and V2 (E) for which V2 , $ 0. Let
n n
2 2, 2 1,
5V2 , 5 v1 ,
(V2 , $ 0) (V2 , $ 0)
(28)
2 where n 2 2 , and n 2 , are the elements, ,, of V1 (E) and V2 (E) for which V, , 0. The set of points in the ( j 1 , j 2 )-plane that satisfy Eq. (26) then is given by:
j 2 $ zj 1 j 2 # tj 1 where j1 $ 0
U U
n1 1, z 5 2 min ] n1 2, n2 1, t 5 2 min ] n2 2,
U U
Yi (E) m 5 (h11 1 h12 k/cos ui )V1 (E) 1 (h21 1 h22 k/cos u i )V2 (E) ]]]]]]]]]]]] 1 1 k/cos ui
(29)
(33)
When Eq. (33) is compared with Eq. (21), it is evident that
h11 1 h12 k / cos ui j 1i 5 ]]]]] 1 1 k / cos ui h21 1 h22 k / cos ui j 2i 5 ]]]]] 1 1 k / cos ui
(34)
If measurements are made at several angles, then (h11 , h21 ), (h12 , h22 ), and k can be found from this set of linear equations written in the form shown in Eq. (35).
h11 cos ui 1 h12 k 2 kj 1i 5 j 1i cos ui h21 cos ui 1 h22 k 2 kj 2i 5 j 2i cos ui
and
203
(35)
Values of h11 , h21 , h12 , h22 , and k are found from this set of equations as follows:
The relationships represented by Eq. (29) are plotted in Fig. 4 as solid lines passing through the origin and the points A and D, respectively. The scores (h11 , h12 ) and (h12 , h22 ) that are needed to determine the vectors f1 (E) and f2 (E) of the pure components lie in the area enclosed by these boundary lines. The region containing the desired scores, hij can be further restricted by making use of the properties of the vectors Yi (E) and fi (E). Note that the scores ( j 1i , j 2i ) in Fig. 4 fall on a straight line defined by: c1 j1 1 c2 j2 5 1
(30)
p , 51
where c i 5 o ni , , which follows from the restrictions that all Yi (E) are normalized to unity so that all linear combinations j 1V1 (E) 1 j 2V2 (E) that satisfy Eq. (2) also satisfy Eq. (30). Since it is assumed that the vectors fi (E) are also normalized to unity, the desired scores (h11 , h21 ) and (h12 , h22 ) must fall on the line as well. More specifically, the score (h11 , h21 ) will lie on the line segment CD, and the score (h11 , h22 ) on the segment AB. The desired scores (h11 , h21 ) and (h12 , h22 ) can be found by using the angular dependence of a (ui ) and b (ui ) given by the model, Eq. (15). Substitution of Eq. (25) into Eq. (15) gives:
Fig. 5. Open circles represent the original functions used in the simulations, and the solid lines show the functions retrieved from the self-modeling. The bulk function f1 (E) was retrieved using the score h11 , h21 and surface function f2 (E) was retrieved using the score h12 , h22 (see Fig. 4).
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
204
h11 h12 k k 5 (K9K)21 K9y h22 k h21
The values obtained for the scores (h11 , h21 ) and (h12 , h22 ) are shown in Fig. 4. The value of k was found to be 0.150 which is in a perfect agreement with the value of 0.150 used in simulating the spectra. Equations given by Eq. (25) were used to retrieve f1 (E) and f2 (E), and the results are compared in Fig. 5 with the original functions used in the simulations. As noted earlier, the error between the simulated and reconstructed spectra according to Eq. (24) was found to be small in this error-free case. An evaluation of possible sources of noise and experimental errors is made in the next section, before the self-modeling technique is used to analyze the experimental data shown in Fig. 1a.
**
cos ui ? cos up where K 5 0 0 0
*
j 1i cos ui ? j 1p cos up and y 5 j 2i cos ui ? j 2p cos up
1 1 1 0 0 0
* *
2 j 1i ? 2 j 1p 2 j 2i ? 2 j 2p
0 0 0 1 1 1
0 0 0 cos ui ? cos up
* (36)
5. Error analysis The most significant sources of error in the angle
Fig. 6. Eigenvectors V1 (E) and V2 (E) from the covariance matrix of twenty sets of simulated spectra which include random estimated errors. Each of the twenty sets consists of six simulated spectra; one set for each angle. The open circles represent the eigenvectors for the error free simulation. The eigenvector V1 (E) for the major or bulk component is insensitive to the experimental errors considered here. The eigenvector for the minor of surface component V2 (E), however, is more sensitive to experimental error as is evident by the addition of noise and in some cases a reversal in sign.
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resolved photoelectron spectra under discussion are (i) the precision in setting the detector angle, (ii) the signal-to-noise ratio of the photoelectron intensity and (iii) the changes in the energy calibration with changes in angle that may arise from instrumental causes or variations in surface charging of insulating or semiconducting specimens. Each of these sources of error were evaluated separately and in combination by incorporating estimated values into the simulations. The detector angle is reproducible to within 628 1 . Since photoelectron intensities of 50 000 to 100 000 counts per channel are routine, a S /N ratio of 0.002 was used in the simulations. An internal reference (such as ‘residual carbon’) can be used to calibrate the energy scale of each spectrum to within 60.05 eV. The preset angles used in the spectrum simulations were varied over the 628 limits
Fig. 7. Scores obtained from the analysis of twenty sets of simulated spectra containing errors in the bulk function f1 (E) and surface function f1 (E) used to simulate the spectra. The solid circles are the scores j 1i , j 2i , used in reconstructing the original simulated spectra. The scores h11 , h21 used to retrieve the bulk function and the scores h12 , h22 used to retrieve the surface function for each set of simulated spectra are shown as open circles and open squares, respectively. The solid line is defined by Eq. (30) for the error free case. The difference in scatter between the scores corresponding to the bulk and surface component functions is reflected in the retrieved functions shown in Fig. 8.
205
while the preset angles of Eq. (36) used in the analysis were held constant. The propagation of each source of error was evaluated separately by introducing random error in angle within 628 of the preset angle, a noise function with S /N50.002, and a random shift in energy scale with limits equivalent to 60.05 eV. Of the sources of error considered, the random shifts in the energy scale caused the most significant variation in the retrieved values of k and on the surface component spectrum, f2 (E). None of the estimated errors had much of an effect on the retrieved bulk component spectrum, f2 (E). All three sources of error were incorporated into twenty simulations and the results of the analysis of these sets of spectra are described in detail to provide a basis for the assessment of error in the selfmodeling of experimental data. The eigenvectors V1 (E) and V2 (E) corresponding to the two largest eigenvalues of the covariance matrix, M, are plotted in Fig. 6. The plot of V1 (E) shows the results for each of the twenty simulations along with the values obtained without error. In contrast to eigenvector V1 (E), a significant spread of values was found for V2 (E). Furthermore, approximately half of the simulations resulted in V2 (E) values with a sign opposite to that found for the error free case. This sign difference did not affect the self-modeling results, since the scores, j ij , also changed sign, Fig. 7. The scores used in reconstructing each simulated spectrum according to Eq. (21), were found clustered about the error free scores, Fig. 7. Some scatter was found in the h11 , h21 values used to retrieve the bulk function, f1 (E), but considerably more scatter was found in h12 , h22 values used in the retrieval of the surface function, f2 (E), Fig. 8. Based on the analysis of simulations using separate errors, as mentioned above, the most significant deviations of the h12 , h22 scores from the error free values is attributed to shifts in the energy calibration close to the 60.05 eV limit at one or more of the detector angles. This result teaches that the successful application of selfmodeling resolving surface / bulk spectra requires that the energy calibration be kept constant to 60.05 eV or better as the detector angle is changed. Values of k ranged from 0.04 to 0.20 and averaged 0.14 60.05. The average value of s for the fits to the simulated spectra was 3.6260.11310 24 , which will serve as a
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Fig. 8. Bulk, f1 (E), and surface, f2 (E), functions retrieved from twenty simulated spectra containing errors in the experimental parameters. The open circles represent the error free surface and bulk functions. For the bulk component, the self-modeling is relatively insensitive to the estimated errors that are applicable to the acquisition of XPS spectra with the SCIENTA ESCA instrument. In all but three cases, the self-modeling was found to be relatively robust to these errors. The three shifted surface functions which also exhibit exceptional line shapes are attributed to shifts in the energy scale at one or more of the detector angles by amounts close to the 60.05 eV limit set in the simulations.
criterion for assessing the validity of the self-modeling analysis of the experimental data as described below.
6. Self-modeling of experimental spectra (surface silanols on SiO 2 ) The C(1s) peak from residual organic carbon (‘adventitious carbon’) was taken at each detector angle to serve as an internal energy standard. A shift in the energy scale was not required for any of the angles in this case, since the variation in the C(1s) peak energy was less 60.05 eV for the range of angles used in the acquisition of the spectra, Fig. 9. The eigenvectors, V1 (E) and V2 (E) and the scores j ij , found in the analysis of the O(ls) spectra of hydrated
SiO 2 are shown in Figs. 10 and 11, respectively. Note that both the eigenvectors and scores exhibit features similar to those obtained in the analysis of the simulated data. The error between the original and reconstructed spectra was found to be 7.23 10 25 , which is less than that estimated from the sources of experimental error described above. This result suggests that it is reasonable to proceed with the self-modeling of the experimental spectra. The scores of hij used to find f1 (E) and f2 (E) are also shown in Fig. 11. These scores were found to be close to the line defined by Eq. (30) ensuring that the f1 (E) and f2 (E) functions have normalized values close to unity as required by Eq. (1). These resolved bulk and surface component spectra are shown in Fig. 12. A quantitative description of the line shapes of the component spectra was obtained by fitting each to the following Voigt function.
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
Fig. 9. C(1s) spectra taken at the detector angles reported in Fig. 1. The spectra have been normalized to the same peak heights for purposes of comparison.
207
Fig. 11. Scores obtained from the analysis of the experimental spectra shown in Fig. 1a of O(1s) from hydrated SiO 2 The h11 , h21 and h12 , h22 scores are the results used for the self-modeling of the bulk and surface component spectra, respectively. Note that the math solution limits A and D that were shown in Figs. 4 and 7 are not included here.
]] 2 w 4, n(2) œ y 5 A m ] ]]]]] 1 (1 2 m) ]]] ]w Œp p 4(x 2 x c )2 1 w 2
FSD S
4, n(2)(x 2 x c )2 3 exp 2 ]]]]] w2
DG
(37)
where A is the peak area, m is the mixing factor, w is the peak width and x c the peak position. Since the incremental values of x are equivalent to 0.05-eV increments in binding energy, the spectral peak area is A / 0.05. The fitted functions of the self-modeled spectra are shown Fig. 12 as solid lines. The following table shows the peak parameters for each component spectrum.
Fig. 10. Eigenvectors V1 (E) and V2 (E) obtained from the covariance matrix of the experimental spectra of O(1s) from hydrated SiO 2 shown in Fig. 1a.
x2
Component A / 0.05
x c (eV)
w (eV)
m
Bulk
533.178 60.002 533.479 60.007
1.152 60.002 1.580 60.006
0.142 1.45610 60.006 0.373 7.71610 28 60.016
Surface
0.9974 60.0002 1.049 60.005
28
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7. Conclusions
Fig. 12. Open circles show the bulk f1 (E), and surface, f2 (E), component spectra, obtained from the self-modeling of the experimental shown in Fig. 1a of the O(l s) line from hydrated SiO 2 . The solid lines represent Voigt function (see Section 6) fits to the spectra.
The surface component spectrum is broader than that found for the bulk component suggesting possible inhomogeniety in the chemical environment of the surface oxygen species. The 0.30-eV peak separation is small compared to the peak widths making the self-modeling preferable to regression methods of curve resolution. The fits of experimental spectra with the surface and bulk component spectra are shown in Fig. 13. The error between the original and calculated spectra, as defined by Eq. (24) was found to be s 57.4310 25 . A value of k50.299 was found and is significant as a calibration factor in the quantification of surface silanols by XPS (i.e., this value of k represents 64.6 OH groups per nm 2 ). The analysis of several experiments, however, is needed to establish the precision of the quantitative analysis of silanols by the method developed in this paper.
The separation of XPS spectra into surface and bulk contributions was successfully demonstrated using a method that does not require any knowledge of line shapes. While the method does depend upon the assumption of a model for the angular dependence of the bulk and surface photoelectron signals, it provides a reasonable alternative to regression analysis that requires tenuous assumptions about line shapes and fitting constraints. While efforts should be made to keep all of the major sources of experimental error to a minimum, it is especially important to minimize shifts in the energy scale as the detector angle is changed. Alternatively, the energy should be calibrated at each angle and the spectra energy shifted to a common reference value before initiating self-modeling. While the separation of the O(1s) signal of surface silanols from the bulk oxide of a hydrated silicon oxide gave reasonable results, this success is not offered as a proof of the uniqueness of the outcome. The authors, however, believe that the self-modeling method of curve resolution deserves attention in the analysis of systems where energy resolution of bulk and surface signals is an issue. For example, quantization of surface hydroxyl groups on other oxides are of fundamental and practical interest in adhesion, catalysis and dispersion. Although these systems include high surface area oxides for which the self-modeling is not applicable, analysis of polished oxide surfaces can provide reference parameters and constraints for resolution of oxide and hydroxyl components by regression analysis of spectra from high surface area compounds.
Acknowledgements The authors would like to acknowledge allocation of time and services in the SCIENTA ESCA laboratory. Technical assistance from A.C. Miller during the experimental part of this work and a review of the mathematical procedures by G.D. Harlow are greatly appreciated. This work was supported in part by the Department of Energy under Grant No. DEFG02-86ER13580 and by Lucent Technologies, Inc.
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209
Fig. 13. Open circles represent the original experimental spectra of Fig. 1a O(1s) from hydrated SiO 2 , solid line the fitted spectra, dashed line the resolved bulk component and the dotted line the surface component, respectively.
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References [1] D.L. Massart, B.G.M. Vandeginste, S.N. Deming, Y. Michotte, L. Kaufman, Chemometrics: A Textbook, in: B.G.M. Vandeginste, L. Kaufmann (Eds.), Data Handling in Science and Technology, Vol. 2, Elsevier, Amsterdam, 1988. [2] M.A. Sharaf, D.L. Illman, B.R. Kowalski, in: P.J. Elving, J.D. Vinefordner (Eds.), Chemometrics in Chemical Analysis, Vol. 82, John Wiley & Sons, New York, 1986. [3] E.R. Malinowski, D.G. Howery, Factor Analysis in Chemistry, Robert E. Krieger Publishing Co., Inc, Malabar, Florida, 1980. [4] D.M. Haaland, E.V. Thomas, Anal. Chem. 60 (1988) 1193. [5] D.M. Haaland, E.V. Thomas, Anal. Chem. 60 (1988) 1202. [6] P. Geladia, B.R. Kowalski, Anal. Chim. Acta 185 (1986) 1. [7] W.H. Lawton, E.A. Sylvestre, Technometrics 13 (1971) 617. [8] E.A. Sylvestre, W.H. Lawton, M.S. Maggio, Technometrics 16 (1974) 353. [9] R.A. Gilbert, J.A. Llewellyn, W.E. Swartz Jr., J.W. Palmer, Appl. Spectros. 36 (1982) 428.
[10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21] [22]
J.M. Halket, J. Chromatogr. 186 (1979) 443. M.A. Sharaf, B.R. Kowalski, Anal. Chem. 53 (1981) 518. M.A. Sharaf, B.R. Kowalski, Anal. Chem. 54 (1982) 1291. F.J. Knoor, J.H. Futrell, Anal. Chem. 51 (1979) 1236. N. Ohta, Anal. Chem. 45 (1973) 553. C.S. Fadley, R.J. Baird, W. Sickhaus, T. Novakov, S.A.L. Bergstrom, J. Elect. Spec. Relat. Phenom. 4 (1974) 93. C.S. Fadley, Angle-resolved X-ray Photoelectron Spectroscopy, in: Progress in Surface Science, Vol. 16, Pergamon Press, Elmsford, NY, 1984. S. Hofmann, Surf. and Interface Anal. 2 (1980) 148. S. Hofmann, J.M. Sanz, Surf. and Interface Anal. 6 (1984) 75. C.R. Brundle, A.D. Baker, Electron Spectroscopy, Theory, Techniques, and Applications, Pergamon, Oxford, 1978. W. Kern, J. Electrochem. Soc. 137 (1990) 1887. R.K. Iler, The Colloid Chemistry of Silica and Silicates, Cornell University Press, Ithaca, NY, 1955. MATLAB: The Language of Technical Computing, The Math Works, Inc., Natick, Massachusetts.
A self-modeling approach to the resolution of XPS spectra into surface and bulk components a, b a Gary W. Simmons *, David L. Angst , Kamil Klier a
Department of Chemistry and Zettlemoyer Center of Surface Studies, Lehigh University, Bethlehem, PA 18015, USA b Lucent Technologies Research Laboratory, Breinigsville, PA 18031, USA Received 10 March 1999; accepted 26 June 1999
Abstract A method of resolving XPS spectra into surface and bulk component spectra for cases of moderate energy resolution is designed based on an analysis of a family of spectra acquired at different polar angles from flat specimens. Assumptions about line shapes are not required, but an analytical model of the angular dependence of the bulk and surface XPS signals is needed when the component spectra overlap in the range of binding energies of interest. Recommendations are made of the error limits of experimental variables for the successful application of the self-modeling method. The method was used successfully in the separation and quantitative analysis of the O(1s) XPS spectra of surface silanols from bulk silicon dioxide of a fully hydrated silicon dioxide surface. The surface (silanol) and bulk (oxide) components were found to be separated by 0.30 eV, and the surface component was found to be broader (1.58 eV) than the bulk component (1.15 eV). 1999 Elsevier Science B.V. All rights reserved.
1. Introduction X-ray photoelectron spectroscopy (XPS) has been employed widely for studies of the chemical and physical properties of surfaces. Chemical information is limited, however, for cases when photoemission spectra of different chemical species are not clearly energy resolved. A typical example of this limitation is shown in Fig. 1a of the O(1s) spectrum from a hydrated silicon oxide surface. These spectra, normalized to unit areas, were taken at different photoelectron detector angles. The relative increase in intensity on the high binding energy side of the O(1s) line with increasing angle can be attributed to oxygen such as surface silanol groups, ‘strained’ *Corresponding author.
oxygen in the oxide surface and oxygen in residual organic compound(s). Since steps were taken to fully hydrate and clean the surface of this specimen, it is assumed that the spectra shown in Fig. 1a represents mostly bulk silicon oxide and surface silanols. Quantitation of the silanol coverage requires curveresolving methods to determine the relative areas of the silanol and oxide O(1s) photoemission lines. Because of the relatively small chemical shift between silanol and oxide species, regression methods for resolving these two components require assumptions about line shapes and fitting constraints. Analysis of the spectra shown in Fig. 1a using such methods have not given unequivocal results. More satisfactory curve resolution, however, can be obtained using a self-modeling method that is based on principal component analysis. Although restricted to
0368-2048 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 99 )00066-3
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Fig. 1. (a) Oxygen (1s) XPS spectra of hydrated silicon oxide surface as a function of detector angle: 08, 458, 658, 758, 808, and 858. (b) Simulated spectra comprised of the sum of two arbitrary functions representing bulk and surface components.
two component systems, surface and bulk components in this case, self-modeling can be used to obtain the spectrum of each component without assumptions about their line shapes. The developments of principal component analysis is well documented [1–14]. The method described in this paper follows closely that given by Sylvestre et al. [7,8] and presents an analysis of angle-resolved spectra by this method for the first time. It is assumed that the energy (E) spectrum of a two component system has the form: Y(E) 5 a f1 (E) 1 b f2 (E)
(1)
where the spectrum has been normalized to unit area, f1 (E) and f2 (E) are unknown, nonnegative, linearly independent functions also normalized to unit area, and a and b are nonnegative variables independent of E. The normalization is taken over the binding energy range of L1 to L2 . L2
E Y(E) dE 5 1;
L1
L2
and
E f (E) dE 5 1 i
L1
(2)
This normalization requires that a 1 b 5 1 and makes Y(E) a convex combination of f1 (E) and f2 (E). In the discussion that follows, f1 (E) is the photoelectron spectrum from the bulk and f2 (E) the spectrum from the surface. It is important to realize that fi (E) may represent one or more chemical species. When more than one chemical species is present, other resolution methods may be employed after the surface and bulk spectra have been separated. The variables a and b in Eq. (1) represent the composition of a two component system, and the resolution of the spectral line for each of the pure components is subject to the restriction that there is no region in the binding energy range L1 to L2 for which f1 (E) and f2 (E) are simultaneously identically zero: f1 (Ej ) 5 0 and f1 (Ek ) . 0 and
f2 (Ej ) . 0 f2 (Ek ) 5 0.
(3)
This restriction is not met for the spectra shown in Fig. 1a, where the surface and bulk spectra overlap over most of the O(1s) photoelectron range of
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
199
binding energies. The restriction given by Eq. (3), however, can be lifted when a and b are known functions [8]. Since the relative intensities of the surface and bulk signals change with detector angle, the variables a and b are functions of this angle. The form of the a(u) and b(u) functions used in lifting the restrictions [Eq. (3)] are, however, dependent on the model chosen to describe the angular dependence. Although this model dependence may be considered a limiting feature, it does offer a more rational approach to spectrum resolution than guessing at line shapes for the O(1s) photoemission from the surface and bulk components; an even more unpleasant task if the surface and bulk components each have several overlapping photoemission lines. The objective of this paper is to demonstrate a method for separating the surface and bulk components of angle resolved photoelectron spectra without any assumption about the line shapes of either component. Firstly, various surface layer models are considered to arrive at possible angular dependent functions, a(u) and b(u). Secondly, a model considered to be applicable to a monolayer of silanols on silicon dioxide is used to simulate spectra for evaluating the self-modeling resolution technique. Thirdly, an assessment of the effects of experimental errors is made, and finally, the technique is used to separate the surface and bulk components of the experimental spectra shown in Fig. 1a.
Fig. 2. Geometry used to model the angular dependence of the surface and bulk photoelectron intensities.
2. Surface layer models
1. Photoemission from an infinitely thick substrate (x 0 → `) through a layer of finite thickness, x 1 :
The angular dependence of photoemission from surface layers has been detailed in the literature [15–19]. The essential features of these treatments are given to serve as reference for the discussion of the surface / bulk resolution technique that follows. The specimen surface is assumed to be flat and photoelectron spectra are acquired over a large range of photoelectron detector angles, u, as defined in Fig. 2. The photoemission intensity Iw in the direction u from a point located at x is Iw 5 I0 e
2w / l e 2w 1 / l s
e
(4)
where le is the mean escape depth in the material of the emitting layer and ls is the mean escape depth in
the screening layer. In the simplest approximation, le 5 ls 5 l, and backscattering and refraction are neglected. Integration of Eq. (4) for a given u over all w, w5x / cos u, between w5 0 and w0 5x 0 / cos u gives: I(u ) 5 I0 e 2x 1 / l cos u l[1 2 e 2x 0 / l cos u ]
(5)
which represents a master equation for two layers of arbitrary thicknesses x 1 and x 0 . It has all the features to describe the angular dependence of photoemission from uniformly thick layers.
I(u ) 5 I0 le 2 x 1 /lcos u
(6)
2. Photoemission from a surface monolayer of thickness t 0 (x 0 5 t 0 , x 1 5 0): I(u ) 5 I0 t 0 / cos u
(7)
3. Photoemission from an infinitely thick substrate with no overlayer (x 0 →`, x 1 50): I(u ) 5 I0 l
(8)
4. Photoemission from a thin uniform elemental layer (x 0 → t 0 ) screened by an outer layer of thickness x 1 :
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I0 t 0 2x / I(u ) 5 ]] e 1 l cos u cos u
(9)
The photoelectron emission is also a function of photoelectron diffraction effects and instrumental factors. Diffraction effects are not likely to be significant for a monolayer coverage of silanols on a substrate of amorphous silicon oxide. An angular instrument response function, F(u ), is incorporated into the model equations to account for changes in photoelectron signal specific to the geometry defined by the x-ray source, specimen and detector. The experimental photoemission is then given by I(u ) experimental 5 F(u )I(u )
(10)
where I(u ) is the photoemission from one of the specific models described above.
3. The surface silanol model The spectra shown in Fig. 1a were taken from a specimen prepared in the following manner. A silicon wafer of highly polished integrated circuit substrate material, was cleaned by a procedure used in integrated circuit manufacturing [20]. The substrate was etched for 5 min in 10% HF / water to remove oxide. The substrate was then immersed for 15 min each in two separate baths of ammonium hydroxide, hydrogen peroxide, and water (5:1:1 ratio, 758C), followed by immersion in a bath of hydrochloric acid, hydrogen peroxide and water (6:1:1 ratio, 758C) for an additional 15 min. The cleaned substrates were oxidized at 11508C in 100% oxygen using a Tamarack 100-M rapid thermal annealing unit. The thickness of the oxide formed during a 5-min oxidation was on the order of 130– ˚ The substrate was then immersed in high 150 A. purity deionized water and held at 908C for 20 days to completely hydrate the surface oxide. Residual organic carbon was reduced by a five min exposure of the substrate to UV-ozone just prior to insertion into the spectrometer. X-ray photoelectron spectra 1 1
XPS spectra were taken with a SCIENTA ESCA 300 spectrometer using monochromatic Ka X-rays from a rotating anode source and a computer controlled goniometer.
were taken of the O(1s) binding energy region for detector angles of 08, 458 658, 758 808 and 858. A zero base-line was established for each spectrum by subtracting a straight line defined by the average of the first and last ten data points. Each spectrum was then normalized to unit area, Fig. 1a. The silicon dioxide is assumed to be covered with a saturated layer of silanols (|4.6 OH groups per nm 2 ) [21], the angular dependent O(1s) photoemission from the silanol layer is assumed to be given by Eq. (7) (x 1 5 t 0 ) and from the oxide substrate by Eq. (6). Eq. (1) is now written for silicon oxide covered with a uniform monolayer of silanols as: Yi (E) 5 F(ui )I0sn O le 2t 0 / l cos ui f1 (E) 1 n OH t 0 f2 (E) / cos uid
(11)
where n O is the volume concentration of oxygen in the oxide and n OH is the volume concentration of silanols in the surface layer. Since the silanol layer has a thickness on the order of the diameter of an -OH group, then to a first approximation t 0 < l. Y(E) 5 F(u1 )I0 (n O l f1 (E) 1 n OH t 0 f2 (E) / cos ui )
(12)
The integral used to find the normalized form of Eq. (12) is written as follows: L2
L2
E Y (E) dE 5 F(u )I 1n l E f (E) dE i
L1
i
0
O
1
L1
L2
1 n OH t 0 / cos ui
E f (E) dE2 2
(13)
L1
Since eLL12 5 1 and eLL12 Yi (E) dE 5 1, the normalized form of Eq. (12) is given by Eq. (14). n O l f1 (E) 1 n OH t 0 f2 (E) / cos ui Y1 (E) 5 ]]]]]]]]] n O l 1 n OH t 0 / cos ui
(14)
Since the quantity n 0 l is a constant over the energy range of the O(1s) emission, the concentration of surface silanols can be referenced to the oxide substrate, k 5 n OH t 0 /n O l. The model O(1s) angular dependent photoemission for a monolayer of silanols on silicon oxide is given in the final form as follows:
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
S
1 Yi (E) 5 ]]]] 1 1 k / cos ui
DS
k f1 (E) 1 ]] f2 (E) cos ui
([V ] 21 [M][V ]) jm 5 l jdjm
D
201
(19)
where l j are eigenvalues of the set of equations (15)
Arbitrary functions for f1 (E) and f2 (E) (mixed Lorenzian / Gaussian) were used in Eq. (15) to simulate a set of spectra for a value of k50.150 and for angles of 08, 458, 658, 758, 808 and 858. These spectra, shown in Fig. 1b, exhibit features similar to the experimental data shown in Fig. 1a. The next section describes the procedure for retrieving the value of k and the functions f1 (E) and f2 (E) from the simulated spectra.
4. Self-modeling curve resolution of simulated spectra In the procedures that follow, assumptions are not made about the value of k nor the functions f1 (E) and f2 (E) that were used to produce the simulated data, and for this reason the resolution method can be thought of as self-modeling [7,8]. The mathematical steps are presented in sufficient detail such that they can be readily implemented using a high-performance, high-level platform such as MATLAB [22] 2 . In vector notation, Eq. (1) is written as follows:
[M]Vj 5 l jVj ,
(20)
Vj is the jth column of the matrix [V ] and djm is the Kronecker delta. The relative magnitude of the eigenvalue l j corresponding to an eigenvector Vj indicates the relative contribution of that eigenvector to the variance of the data. For the two component system under consideration, all of the variance is represented by the eigenvectors associated with two largest eigenvalues. Any remaining non-zero eigenvalues are attributed to system noise and experimental error, and the corresponding eigenvectors are therefore not included in the analysis. If Eq. (16) holds exactly, then all the spectra, Yi (E) can be expressed as linear combinations of the two eigenvectors, V1 (E) and V2 (E) associated with the two largest eigenvalues of M: Yi (E) 5 j 1iV1 (E) 1 j 2iV2 (E).
(21)
Once the eigenvectors V1 (E) and V2 (E) have been determined, the model spectra Yi (E) are found by the best least squares fit for each spectrum given by Eq. (21), where j 1i , and j 2i (scores or weights) are the values which minimize the sum of squares: p
Yi (E) 5 ai f1 (E) 1 bi f2 (E)
(16)
The experimental data Yi (E) are expressed as an n3p matrix, where n is the number of spectra and p is the number of points in each spectra, Y1 (E1 ) Y5 ? Yn (E1 )
*
? ? ?
? Y1 (Ep ) ? ? ? Yn (Ep )
*
(17)
and the covariance matrix, M, of this data matrix is formed by multiplying the data matrix, Y, by its transpose, Y9 M 5 [Y9Y] /n
(18)
The covariance matrix is then diagonalized by finding a matrix [V ] such that 2
A file of the MATLAB commands and output for the analysis of the simulated spectra can be obtained by contacting the first author at [email protected].
O [Y (E ) 2 Y (E ) ] i
, s
i
2
(22)
, m
, 51
where Yi (E) s is the observed spectra and Yi (E) m the model spectra. Because of the orthogonality of the eigenvectors, the scores can be found from the following dot products:
j ij 5Vj ? Yi
(23)
Yi (E) s are then compared with the predicted Yi (E) m , and if they agree within experimental error, the form given by (Eq. 16) is verified. If the errors are large, or show lack of fit then self-modeling of the spectra is not applicable. The error can be estimated as follows [8]. p
OO n
2
[Yi (E, ) s 2 Yi (E, ) m ] ]]]]]] s 5 n( p 2 2) i 51 , 51 2
(24)
The eigenvalues and eigenvectors of the M matrix [Eq. (18)] of the simulated spectra were evaluated
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and only two non-zero eigenvalues were found, as expected, for this error-free two component system. The two eigenvectors V1 (E) and V2 (E) corresponding to the two nonzero eigenvalues are plotted in Fig. 3. Values of j 1i and j 2i for each spectrum (i51 to n) were determined using Eq. (23) and are shown in a j 1 versus j 2 plot as points on the BC line segment, Fig. 4. These results were then used to reconstruct each simulated spectra according to Eq. (21) and then compared using Eq. (24). The model spectra were found to be in perfect agreement with the original spectra (s 55.45310 217 ). The next step retrieves the original functions, f1 (E) and f2 (E), and the value of k used in the simulations. The spectra of the pure components are also linear combinations of the eigenvectors V1 (E) and V2 (E). f1 (E) 5 h11V1 (E) 1 h21V2 (E)
(25)
f2 (E) 5 h12V1 (E) 1 h22V2 (E) Each of the observed vectors, Yi (E), is completely
Fig. 4. Each score on segment BC successfully reconstructs the corresponding original simulated spectrum. The scores at A and D represent the limits for the retrieval of the pure component functions. For the analysis of the simulated spectra described in Section 4, the scores h12 , h22 and h11 , h21 successfully retrieve the pure component functions used in the simulations. The solid line is defined by Eq. (30).
described by the scores ( j 1i , j 2i ) (i 5 1 to n) of the eigenvectors and is represented as a point in the ( j 1 j 2 )-plane. The scores ( j 1i , j 2i ) on the line segment BC shown in Fig. 4 were obtained from the analysis of the simulated spectra shown in Fig. 1b. Since the vectors f1 (E) and f2 (E) are also linear combinations of the eigenvectors, V1 (E) and V2 (E), then the scores (h11 , h21 ) and (h12 , h22 ) that describe the component spectra are also found in the ( j 1 , j 2 )-plane. The next step is to restrict the area in the ( j 1 , j 2 )-plane where (h11 , h22 ) can be found. Since the unknown vectors f1 (E) and f2 (E) have nonnegative elements, the scores (h11 , h21 ) and (h12 , h22 ) must be in the region of the ( j 1 , j 2 )-plane which satisfies:
j 1 n1 , 1 j 2 n2 , $ 0 Fig. 3. The eigenvectors V1 (E) and V2 (E) obtained from the covariance matrix of the simulated spectra shown in Fig. 1b. These eigenvectors along with the scores derived from the analysis contain all of the information to reconstruct the original spectra and to reproduce each of the component spectra.
(26)
for all , where nj , is the , th element of the jth eigenvector Vj (E). To find the set of points in the ( j 1 , j 2 )-plane that satisfies this condition, it is important to note that while all of the elements of the eigenvec-
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
tor V1 (E) are nonnegative, some of the elements of the V2 (E) eigenvector are negative because of the orthogonality imposed on eigenvectors, (Fig. 3). Let
n1 2 , 5V2 , n1 1 , 5 v1 ,
(V2 , $ 0)
(27)
(V2 , $ 0)
2 where n 1 1 , and n 2 , are the elements, ,, of the vectors V1 (E) and V2 (E) for which V2 , $ 0. Let
n n
2 2, 2 1,
5V2 , 5 v1 ,
(V2 , $ 0) (V2 , $ 0)
(28)
2 where n 2 2 , and n 2 , are the elements, ,, of V1 (E) and V2 (E) for which V, , 0. The set of points in the ( j 1 , j 2 )-plane that satisfy Eq. (26) then is given by:
j 2 $ zj 1 j 2 # tj 1 where j1 $ 0
U U
n1 1, z 5 2 min ] n1 2, n2 1, t 5 2 min ] n2 2,
U U
Yi (E) m 5 (h11 1 h12 k/cos ui )V1 (E) 1 (h21 1 h22 k/cos u i )V2 (E) ]]]]]]]]]]]] 1 1 k/cos ui
(29)
(33)
When Eq. (33) is compared with Eq. (21), it is evident that
h11 1 h12 k / cos ui j 1i 5 ]]]]] 1 1 k / cos ui h21 1 h22 k / cos ui j 2i 5 ]]]]] 1 1 k / cos ui
(34)
If measurements are made at several angles, then (h11 , h21 ), (h12 , h22 ), and k can be found from this set of linear equations written in the form shown in Eq. (35).
h11 cos ui 1 h12 k 2 kj 1i 5 j 1i cos ui h21 cos ui 1 h22 k 2 kj 2i 5 j 2i cos ui
and
203
(35)
Values of h11 , h21 , h12 , h22 , and k are found from this set of equations as follows:
The relationships represented by Eq. (29) are plotted in Fig. 4 as solid lines passing through the origin and the points A and D, respectively. The scores (h11 , h12 ) and (h12 , h22 ) that are needed to determine the vectors f1 (E) and f2 (E) of the pure components lie in the area enclosed by these boundary lines. The region containing the desired scores, hij can be further restricted by making use of the properties of the vectors Yi (E) and fi (E). Note that the scores ( j 1i , j 2i ) in Fig. 4 fall on a straight line defined by: c1 j1 1 c2 j2 5 1
(30)
p , 51
where c i 5 o ni , , which follows from the restrictions that all Yi (E) are normalized to unity so that all linear combinations j 1V1 (E) 1 j 2V2 (E) that satisfy Eq. (2) also satisfy Eq. (30). Since it is assumed that the vectors fi (E) are also normalized to unity, the desired scores (h11 , h21 ) and (h12 , h22 ) must fall on the line as well. More specifically, the score (h11 , h21 ) will lie on the line segment CD, and the score (h11 , h22 ) on the segment AB. The desired scores (h11 , h21 ) and (h12 , h22 ) can be found by using the angular dependence of a (ui ) and b (ui ) given by the model, Eq. (15). Substitution of Eq. (25) into Eq. (15) gives:
Fig. 5. Open circles represent the original functions used in the simulations, and the solid lines show the functions retrieved from the self-modeling. The bulk function f1 (E) was retrieved using the score h11 , h21 and surface function f2 (E) was retrieved using the score h12 , h22 (see Fig. 4).
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
204
h11 h12 k k 5 (K9K)21 K9y h22 k h21
The values obtained for the scores (h11 , h21 ) and (h12 , h22 ) are shown in Fig. 4. The value of k was found to be 0.150 which is in a perfect agreement with the value of 0.150 used in simulating the spectra. Equations given by Eq. (25) were used to retrieve f1 (E) and f2 (E), and the results are compared in Fig. 5 with the original functions used in the simulations. As noted earlier, the error between the simulated and reconstructed spectra according to Eq. (24) was found to be small in this error-free case. An evaluation of possible sources of noise and experimental errors is made in the next section, before the self-modeling technique is used to analyze the experimental data shown in Fig. 1a.
**
cos ui ? cos up where K 5 0 0 0
*
j 1i cos ui ? j 1p cos up and y 5 j 2i cos ui ? j 2p cos up
1 1 1 0 0 0
* *
2 j 1i ? 2 j 1p 2 j 2i ? 2 j 2p
0 0 0 1 1 1
0 0 0 cos ui ? cos up
* (36)
5. Error analysis The most significant sources of error in the angle
Fig. 6. Eigenvectors V1 (E) and V2 (E) from the covariance matrix of twenty sets of simulated spectra which include random estimated errors. Each of the twenty sets consists of six simulated spectra; one set for each angle. The open circles represent the eigenvectors for the error free simulation. The eigenvector V1 (E) for the major or bulk component is insensitive to the experimental errors considered here. The eigenvector for the minor of surface component V2 (E), however, is more sensitive to experimental error as is evident by the addition of noise and in some cases a reversal in sign.
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resolved photoelectron spectra under discussion are (i) the precision in setting the detector angle, (ii) the signal-to-noise ratio of the photoelectron intensity and (iii) the changes in the energy calibration with changes in angle that may arise from instrumental causes or variations in surface charging of insulating or semiconducting specimens. Each of these sources of error were evaluated separately and in combination by incorporating estimated values into the simulations. The detector angle is reproducible to within 628 1 . Since photoelectron intensities of 50 000 to 100 000 counts per channel are routine, a S /N ratio of 0.002 was used in the simulations. An internal reference (such as ‘residual carbon’) can be used to calibrate the energy scale of each spectrum to within 60.05 eV. The preset angles used in the spectrum simulations were varied over the 628 limits
Fig. 7. Scores obtained from the analysis of twenty sets of simulated spectra containing errors in the bulk function f1 (E) and surface function f1 (E) used to simulate the spectra. The solid circles are the scores j 1i , j 2i , used in reconstructing the original simulated spectra. The scores h11 , h21 used to retrieve the bulk function and the scores h12 , h22 used to retrieve the surface function for each set of simulated spectra are shown as open circles and open squares, respectively. The solid line is defined by Eq. (30) for the error free case. The difference in scatter between the scores corresponding to the bulk and surface component functions is reflected in the retrieved functions shown in Fig. 8.
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while the preset angles of Eq. (36) used in the analysis were held constant. The propagation of each source of error was evaluated separately by introducing random error in angle within 628 of the preset angle, a noise function with S /N50.002, and a random shift in energy scale with limits equivalent to 60.05 eV. Of the sources of error considered, the random shifts in the energy scale caused the most significant variation in the retrieved values of k and on the surface component spectrum, f2 (E). None of the estimated errors had much of an effect on the retrieved bulk component spectrum, f2 (E). All three sources of error were incorporated into twenty simulations and the results of the analysis of these sets of spectra are described in detail to provide a basis for the assessment of error in the selfmodeling of experimental data. The eigenvectors V1 (E) and V2 (E) corresponding to the two largest eigenvalues of the covariance matrix, M, are plotted in Fig. 6. The plot of V1 (E) shows the results for each of the twenty simulations along with the values obtained without error. In contrast to eigenvector V1 (E), a significant spread of values was found for V2 (E). Furthermore, approximately half of the simulations resulted in V2 (E) values with a sign opposite to that found for the error free case. This sign difference did not affect the self-modeling results, since the scores, j ij , also changed sign, Fig. 7. The scores used in reconstructing each simulated spectrum according to Eq. (21), were found clustered about the error free scores, Fig. 7. Some scatter was found in the h11 , h21 values used to retrieve the bulk function, f1 (E), but considerably more scatter was found in h12 , h22 values used in the retrieval of the surface function, f2 (E), Fig. 8. Based on the analysis of simulations using separate errors, as mentioned above, the most significant deviations of the h12 , h22 scores from the error free values is attributed to shifts in the energy calibration close to the 60.05 eV limit at one or more of the detector angles. This result teaches that the successful application of selfmodeling resolving surface / bulk spectra requires that the energy calibration be kept constant to 60.05 eV or better as the detector angle is changed. Values of k ranged from 0.04 to 0.20 and averaged 0.14 60.05. The average value of s for the fits to the simulated spectra was 3.6260.11310 24 , which will serve as a
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Fig. 8. Bulk, f1 (E), and surface, f2 (E), functions retrieved from twenty simulated spectra containing errors in the experimental parameters. The open circles represent the error free surface and bulk functions. For the bulk component, the self-modeling is relatively insensitive to the estimated errors that are applicable to the acquisition of XPS spectra with the SCIENTA ESCA instrument. In all but three cases, the self-modeling was found to be relatively robust to these errors. The three shifted surface functions which also exhibit exceptional line shapes are attributed to shifts in the energy scale at one or more of the detector angles by amounts close to the 60.05 eV limit set in the simulations.
criterion for assessing the validity of the self-modeling analysis of the experimental data as described below.
6. Self-modeling of experimental spectra (surface silanols on SiO 2 ) The C(1s) peak from residual organic carbon (‘adventitious carbon’) was taken at each detector angle to serve as an internal energy standard. A shift in the energy scale was not required for any of the angles in this case, since the variation in the C(1s) peak energy was less 60.05 eV for the range of angles used in the acquisition of the spectra, Fig. 9. The eigenvectors, V1 (E) and V2 (E) and the scores j ij , found in the analysis of the O(ls) spectra of hydrated
SiO 2 are shown in Figs. 10 and 11, respectively. Note that both the eigenvectors and scores exhibit features similar to those obtained in the analysis of the simulated data. The error between the original and reconstructed spectra was found to be 7.23 10 25 , which is less than that estimated from the sources of experimental error described above. This result suggests that it is reasonable to proceed with the self-modeling of the experimental spectra. The scores of hij used to find f1 (E) and f2 (E) are also shown in Fig. 11. These scores were found to be close to the line defined by Eq. (30) ensuring that the f1 (E) and f2 (E) functions have normalized values close to unity as required by Eq. (1). These resolved bulk and surface component spectra are shown in Fig. 12. A quantitative description of the line shapes of the component spectra was obtained by fitting each to the following Voigt function.
G.W. Simmons et al. / Journal of Electron Spectroscopy and Related Phenomena 105 (1999) 197 – 210
Fig. 9. C(1s) spectra taken at the detector angles reported in Fig. 1. The spectra have been normalized to the same peak heights for purposes of comparison.
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Fig. 11. Scores obtained from the analysis of the experimental spectra shown in Fig. 1a of O(1s) from hydrated SiO 2 The h11 , h21 and h12 , h22 scores are the results used for the self-modeling of the bulk and surface component spectra, respectively. Note that the math solution limits A and D that were shown in Figs. 4 and 7 are not included here.
]] 2 w 4, n(2) œ y 5 A m ] ]]]]] 1 (1 2 m) ]]] ]w Œp p 4(x 2 x c )2 1 w 2
FSD S
4, n(2)(x 2 x c )2 3 exp 2 ]]]]] w2
DG
(37)
where A is the peak area, m is the mixing factor, w is the peak width and x c the peak position. Since the incremental values of x are equivalent to 0.05-eV increments in binding energy, the spectral peak area is A / 0.05. The fitted functions of the self-modeled spectra are shown Fig. 12 as solid lines. The following table shows the peak parameters for each component spectrum.
Fig. 10. Eigenvectors V1 (E) and V2 (E) obtained from the covariance matrix of the experimental spectra of O(1s) from hydrated SiO 2 shown in Fig. 1a.
x2
Component A / 0.05
x c (eV)
w (eV)
m
Bulk
533.178 60.002 533.479 60.007
1.152 60.002 1.580 60.006
0.142 1.45610 60.006 0.373 7.71610 28 60.016
Surface
0.9974 60.0002 1.049 60.005
28
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7. Conclusions
Fig. 12. Open circles show the bulk f1 (E), and surface, f2 (E), component spectra, obtained from the self-modeling of the experimental shown in Fig. 1a of the O(l s) line from hydrated SiO 2 . The solid lines represent Voigt function (see Section 6) fits to the spectra.
The surface component spectrum is broader than that found for the bulk component suggesting possible inhomogeniety in the chemical environment of the surface oxygen species. The 0.30-eV peak separation is small compared to the peak widths making the self-modeling preferable to regression methods of curve resolution. The fits of experimental spectra with the surface and bulk component spectra are shown in Fig. 13. The error between the original and calculated spectra, as defined by Eq. (24) was found to be s 57.4310 25 . A value of k50.299 was found and is significant as a calibration factor in the quantification of surface silanols by XPS (i.e., this value of k represents 64.6 OH groups per nm 2 ). The analysis of several experiments, however, is needed to establish the precision of the quantitative analysis of silanols by the method developed in this paper.
The separation of XPS spectra into surface and bulk contributions was successfully demonstrated using a method that does not require any knowledge of line shapes. While the method does depend upon the assumption of a model for the angular dependence of the bulk and surface photoelectron signals, it provides a reasonable alternative to regression analysis that requires tenuous assumptions about line shapes and fitting constraints. While efforts should be made to keep all of the major sources of experimental error to a minimum, it is especially important to minimize shifts in the energy scale as the detector angle is changed. Alternatively, the energy should be calibrated at each angle and the spectra energy shifted to a common reference value before initiating self-modeling. While the separation of the O(1s) signal of surface silanols from the bulk oxide of a hydrated silicon oxide gave reasonable results, this success is not offered as a proof of the uniqueness of the outcome. The authors, however, believe that the self-modeling method of curve resolution deserves attention in the analysis of systems where energy resolution of bulk and surface signals is an issue. For example, quantization of surface hydroxyl groups on other oxides are of fundamental and practical interest in adhesion, catalysis and dispersion. Although these systems include high surface area oxides for which the self-modeling is not applicable, analysis of polished oxide surfaces can provide reference parameters and constraints for resolution of oxide and hydroxyl components by regression analysis of spectra from high surface area compounds.
Acknowledgements The authors would like to acknowledge allocation of time and services in the SCIENTA ESCA laboratory. Technical assistance from A.C. Miller during the experimental part of this work and a review of the mathematical procedures by G.D. Harlow are greatly appreciated. This work was supported in part by the Department of Energy under Grant No. DEFG02-86ER13580 and by Lucent Technologies, Inc.
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Fig. 13. Open circles represent the original experimental spectra of Fig. 1a O(1s) from hydrated SiO 2 , solid line the fitted spectra, dashed line the resolved bulk component and the dotted line the surface component, respectively.
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