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Least-squares analysis of photoemission data
Journal of Electron Spectroscopy and Related Phenomena, 37 (1985) 57-67 Elsevier Science Publishers B.V., Amsterdam - Printed in The ~et~er~nds
LEAST-SQUARES ANALYSIS OF PHOTOEMISSION DATA
AT & T Bell Luboratories,
Murray Hill, NJ 07974
(U.S.A.)
{Received 20 March 1985)
ABSTRACT Least-squares fitting with a mathematical model function provides the most effective technique for extracting numerical information from photoemission data. With a model function that properly contains the physics of the phot~~i~ion process, averlapping lines can be separated, and such parameters as the binding energy, lifetime width, phonon broadening, and, in metals, the singularity index are readily determined. The validity of a model function can be judged by inspection of the spectrum of the residuals.
‘The peak positions and linewidths in a photoemission spectrum can be determined reasonably well by inspection when all lines are well separated. However, the spectrum contains a great deal of additional information that can be extracted with more sophisticated analysis. The linewidth contains cont~but~~ns from a ember of distrait physical processes, e.g., the l~etim~ of the core hole, the vibrational excitation of the lattice, and the inherent resolution of the photoelectron spectrometer, These contributions cannot be separated without additional information, some of which resides in the details of the lineshape. Careful analysis of the shape of the lines may also allow us to determine whether any asymmetry is due to unresolved ~ornpo~e~~ or to final state effects. The in~~~at~on content of the data is not exhausted by using the location of a few points (e.g,, the peak and half-height positions); our analysis should use all the data points, This can be accomplished by fitting a theoretical function to the data using the method of least-squares. The validity of this approach depends critically on the nature of this theoretical function. The results are useful only if the function is based on the pbysic~ processes of photoemission. We first survey the physical processes [l, 21 that contribute to the photoemission spectrum we record in the laboratory. A core level photoemission 0368-2048/85/$03.30
0 1985 Elsevier Science Publishers B.V.
snss
4 I Sn4d
A
sn A
# 32
30
28
26
24
22
BINDING ENERGY @VI
Fig. 1. Comparison of the lineshapes of the Sn 4d core level in the semiconductor and in metallic Sn.
SnSz
inherently has a Lorentzian shape whose width is the inverse of the core hole lifetime. Another contribution to the lineshape comes from phonons, the vibrational response of the host lattice, which produce further broadening of essentially Gaussian character. In metals, the final state screening response of the conduction electrons profoundly alters the shape pair of the photoemission line. Screening occurs via electron-hole excitations at the Fermi level. The spectrum of these excitations diverges at zero energy, contributing a one-sided power law singularity to the shape of the main peak itself [3-51. In insulators, final state relaxation tends to produce additional line structure, generally referred to as shake-up satellites. These represent final states in which outer electrons have been excited by the creation of the core hole. Some of the lineshape effects described above can be identified in Fig. 1, which shows the Sn 4d lines in X-ray photoelectron spectroscopy (XPS) data for the large gap semiconductors SnS2, fitted with phonon-broadened Lorentzians, and for the metal Sn, fitted with a function incorporating the effects of the metallic screening response [6]. The semiconductor lattice has a stronger vibrational response, making the Sn 4cl peaks broader in that spectrum, while the spectrum for the metal shows the characteristic asymmetry due to final-state screening. Above we have described the physical processes that determine the shape of the peaks in the core level photoemission spectrum. Another essential part of the mathematical function to be fitted to the data is a suitable background function. The background in a photoemission spectrum is not well peak
59
d 3P \
0 0
I-
0t
--
1
900
1
GOD
I
700
/
600
I
500
1
400
300
/
I
200
100
BINDING ENERGY (eV)
Fig. 2. Wide XPS scan for an evaporated Pd film. Note the sudden increase in background at each core electron photoemission line.
approximated by a straight line; rather, it has a complex shape determined by the interaction of the photoelectron with the solid from which it emerges [7]. This background is seen most clearly in a wide scan, as in Fig. 2. The background is due to photoelectrons created deep within the bulk of the solid, i.e., more than one mean free path from the surface. As these electrons diffuse toward the surface they excite plasmons and interband transitions. They emerge with reduced kinetic energy and appear in the spectrum at larger binding energy; the shape of the background therefore reflects the nature of the available excitations in the solid. In some materials discrete multiple plasmon losses are resolved; in others the energy-loss tail is largely featureless, as in Fig. 2. Another contribution to the background comes from the secondary electrons, which, because of their low kinetic energy, become important only at lower photon energy, or when data are taken close to the work function cut-off. We have considered thus far the photoemission spectrum produced by a perfectly monochromatic source and an ideal spectrometer. In reality this spectrum is modified by the instrumental response function, which includes not only aberrations in the detection system but also the finite width of the light source. Note that when no X-ray monochromator is used, this response function includes the entire spectrum of the X-ray source.
60
The problem of data analysis is now reduced to that of constructing a mathematical model function that incorporates a suitable description of the line spectrum and the background. A properly constructed model function reflects the physical contributions to photoemission, so that the parameters of the model relate directly to quantities of interest. We emphasize the importance of a physically motivated model function; there is little profit in fitting an arbitrary construct to data.
MODEL
FUNCTIONS
It is appropriate to begin the discussion of model functions with the observation that the correct way of combining the effects of two broadening processes is through convolution of the lineshapes produced by each process alone. Thus a spectrum containing the influence both of a finite core hole lifetime and of phonon broadening may be calculated as f(w)
= jg(w’)h(w _m
- o’)dw’,
(1)
where g(w) is a Gaussian and h(w) is a Lorentzian funtion. The result of this convolution, known as the Voigt function [S] , is applicable to the photoemission spectra of insulators and semiconductors, which have symmetrical peaks (see Fig. la). The Voigt function is sometimes approximated by a sum of Gaussian and Lorentzian (G-L) lines with the same full-widths at halfmaximum (FWHM) [g-11]. This approach yields the binding energy, the net linewidth, and the mixing ratio of Gaussian and Lorentzian components. Using this information, it is possible to obtain the widths of the Gaussian and Lorentzian of the corresponding Voigt function [11]. The most important drawback of this representation lies in its oscillatory deviations from the Voigt function (by as much as one percent of the peak height); these oscillations make it difficult to judge the quality of the fit. The spectrum of the metallic screening process, w”-r, is difficult to handle numerically, because of the divergence at w = 0. Fortunately the convolution of this one-sided power-law function wish a Lorentzian is available in closed form due to the work of Doniach and Sunjic [4]. f(E) =
cos [no/2 + (1 -a) @2
+ y
arctan(e/T)]
)‘l -a)/2
(2)
where E is energy measured from the position of the unperturbed Lorentzian, y the half-width at half maximum of the Lorentzian, and cx the singularity index, which, typically, assumes values in the range from 0 to 0.5. The convolution of the D-S lineshape with a Gaussian provides a function suitable to represent all peaks in a core electron photoemission spectrum,
61
including the satellites and plasmons. When there is no metallic screening, CY= 0, and the D-S lineshape reduces to a Lorentzian. Under no circumstances should a background of any type be subtracted from data prior to fitting. This is especially important when there are asymmetric lines, with long intrinsic tails of the type encountered in met,als, but even the tails of the Lorentzian profile can be compromised by injudicious background subtraction. The background should always be determined as the lines are being fitted. We first consider the representation of the step-like background illustrated in Fig. 2. Backgrounds of this kind are often approximated by assuming that every delta function line element has a step function tail. The background, then, is at every energy proportional to the integral of the line spectrum at lower binding energy. This formulation has the merit of simplicity, but seldom if ever corresponds exactly to physical reality [7]. It may, nevertheless, provide an adequate approximation, when the background intensity is small compared to that of the peaks. If there are no extrinsic excitations at small energies, it may be appropriate to shift the integrated spectrum to higher binding energy. When discrete excitations, e.g., plasmons, are resolved in the background near the main line, it is better to include an explicit representation of them in the model function. X-ray tube satellites, such as the Ka,,, and K/3induced photoemission, can be treated in the same way. The other commonly-encountered background term is that due to secondary electrons. This is usually well represented by a quadratic or cubic function whose coefficients are also left to be determined in the fitting. A linear background is not, in general, an adequate representation. The final step in the calculation of the spectrum is the convolution of the model function with the instrumental response function, which must be determined in independent measurements. In practice the response function is best determined from the shape of the Fermi cut-off in a metal like Ag, which has a featureless occupied conduction band with essentially constant density of states over a region of almost 4eV. This valence band spectrum can be used in two ways to determine the resolution function. First, if the thermal width (0.025 eV at room temperature) of the Fermi edge is small compared to the resolution, one can consider the conduction band cut-off to be a step function. The response function is then obtained simply as the derivative of the measured spectrum. Alternatively, one could fit the spectrum with a convolution of a Fermi function with a parametrized representation of the response function; this method is preferred when the resolution is comparable to the thermal width or when, as is usually the case, poor statistics make it difficult to determine the spectrum’s derivative. As a variation of the latter method one could fit the spectrum of a core level whose Lorentzian width and asymmetry are well known, treating the parameters of a mathematical resolution function as variables. The Ag3d5,* line with I’ = 0.27 eV and (Y= 0.06 is suitable for this approach. These determinations
62
of the response function are adequate when the photon source is a synchrotron or a monochromatized X-ray anode. For an instrument using unmonochromatized X-rays the response function includes the entire spectrum of the X-ray source. The spectra from X-ray tubes with Mg or Al anodes contain non-diagram lines of significant strength, the Kor,,, doublets at 9 and 11 eV, respectively, above the main Kar,,, lines. The Ka,,, emission can usually be ignored. The K/3 emission lines, which are also quite weak, lie 50 and 72 eV above the main emission line for Mg and Al anodes, respectively. For each XPS peak these extra X-ray lines produce satellites at the correspondingly smaller binding energies. The satellites of strong photoemission lines can interfere with the study of weaker lines. This problem is particularly severe when satellites of strong lines fall into the valence band region.
LEAST-SQUARES
ADJUSTMENT
In general, data-fitting consists of determining the optimal values of a set of parameters, pi, that appear in a suitable model function. In the method of least-squares, the optimal values of pi are those that minimize R, the sum of the squares of the differences between the data points and the mathematical function. Computer programs that carry out least-squares optimization are readily available [ 121. The method is conceptually simple. It may be helpful to think of R as a surface in a multidimensional space of the adjustable parameters, pi. The point of best fit is a minimum in R, where the partial derivatives of R with respect to the parameters all vanish. The program begins with a set of initial values of pi (usually a “good guess”), and numerically determines the partial derivatives of R. From this information is calculated the correction vector 6i, where the new set of parameter values pi + 61 is estimated to yield the greatest decrease in R. This process is then repeated until a minimum value of R is reached. As there may well be a number of local minima in parameter space, a meaningful result is most likely if one begins with a set of reasonable parameters containing as much of the physical information as is known. An estimate of the reliability of the final values of the parameters is obtained from the shape of R at the minimum. If R rises steeply in all directions away from the minimum, then all parameters are well determined. If, on the other hand, it rises slowly in some direction, i.e., if it is more like a valley than a circular depression, then some parameter may be inherently poorly determined. The reliability of the parameters is therefore gauged by the curvature of R, i.e., by the second partial derivatives of R. Small derivatives mean large uncertainty. A valuable output of fitting programs is a matrix of parameter correlation coefficients. It shows the extent to which one can balance out a change in one parameter by changing another one, without altering the overall quality of the fit. A correlation coefficient with
63
a magnitude greater than 0.95 is a warning that the two parameters involved may not be well determined. The magnitude of R is an immediate measure of the success of the least squares optimization. If the data are obtained by counting photoelectrons, then the statistical uncertainty of each data point is given by the square root of the number of electrons counted. Therefore, R, the sum of the squares of the errors, should be equal to the sum of all the counts in the fitted spectrum. With an adequate model function it is not difficult to approach this condition. The most valuable information about the quality of the fit is obtained from a plot of the residuals themselves. When an ideal fit has been achieved the residuals contain only the random statistical fluctuations of the data. Any systematic oscillation of the residuals plotted as a function of binding energy implies that the data cannot be adequately described by the model function. This is an invitation to generate a better model function.
CASE STUDIES
It is important to remember that the results of fitting are significant only if the model function is based on the underlying physical reality. The following examples illustrate this concern. In Fig. 3 we show the results obtained by fitting two different model functions to the 3d5,* line of metallic tin. The sample was prepared by vacuum evaporation and was free of surface contamination. The data were taken with a HP 5950A spectrometer at a resolution of 0.55 eV FWHM. In Fig. 3a the model function is a D-S line, appropriate for a metal. The residuals show that it is able to reproduce the data in adequate detail. In Fig. 3b the model contains two G-L lines, one to fit the peak, the other to fit the tail, plus an integral background. R is almost twice as large as in Fig. 3a and the residuals show that the model function is not a good representation of the data. The many-body tail has been made up in part by the second component, with the rest put into the integral background. The net result is a fit not worse than many that are published, but one that displays its problems so clearly that one has no difficultly recognizing that it is of no value. A more subtle example is provided by data in Fig. 4, on polycrystalline gold taken with monochromatized Al Ka radiation at an instrumental resolution of 0.25 eV FWHM [13]. The data contain, on the low binding energy side of the line, a weak, unresolved component due to the surface atoms. The shift of this component relative to the bulk signal is not readily estimated without numerical analysis. In Fig. 4a we show the results of fitting the data with two D-S lines with the same shape [13]. The residuals show that this model is able to account fully for the observations. The
64
02
02
0
0
484
486
488
482
BINDING
BINDING ENERGY (e”,
482
484 ENERGY
IeVl
Fig. 3. Sn 3d5,? spectrum from a tin layer vacuum-evaporated onto a carbon substrate, taken with monochromatized Al KCY radiation. In (a) the data are fitted with a D-S function and in (b) with two G-L lines. The model used in (a) has 7 adjustable parameters, that in (b) has 9. The residuals show clearly that the model function used in (b) is unsuitable.
I
I
I
II
I
1
I
I
65
64
63
,
I
I
62 BINDING
ENERGY
I
’ t
65
64
.
,-
J
63
62
(6V)
Fig. 4. Au 4f spectrum from a vacuum-evaporated layer, taken with monochromatized Al KCY radiation at 0.25 eV FWHM resolution. Note that the surface signal at the low binding energy side of the line cannot be located accurately by inspection. In (a), a fit with two D-S lines of identical shape yields a surface atom core level shift of -0.395 eV and an (Y of 0.05. In (b), a fit with OLconstrained to zero is significantly worse, and yields false values for the lifetime width and the surface shift.
0 012
B
0.04
006
0 004 -0.04
000 31
30 BINDING
23 ENERGY (6”)
004
006 31
JO
29 BINDING ENERGY (W
26
Fig. 5. [5] Ge 3d spectrum of a Ge(ll1) surface taken with 46 eV synchrotron radiation. In (a), the data are fitted with three spin-orbit doublets of identical shape, made up of Voigt functions. The background has a general quadratic form which was adjusted in the process of fitting. In (b), the same data fitted are with two spin-orbit doublets.
singularity index has a value of 0.05, appropriate for a metal. In Fig. 4b we show the results of fitting the same data with the singularity index set to zero. An integral background is now required to make up the many-body tail. The fit is not nearly so good, and R is almost twice as large as in Fig. 4a. Moreover, the values of all the parameters have been compromised by this approach. The lifetime width is 0.42eV instead of 0.32 eV, resulting in a significant misfit at the low energy knee of the line pair, and the surface shift is -0.37 eV instead of - 0.395 eV. These changes are larger than the mathematical errors estimated by the fitting programs, demonstrating that parameter values have significance only within the confines of their model. If the model is not appropriate for the data, then the output has no physical significance. For data taken near threshold the background of degraded electrons becomes dominant. In Fig. 5a we examine the photoemission spectrum obtained from a clean Ge(ll1) surface, using synchrotron radiation with 46 eV photon energy [14]. The background is represented by a general quadratic, and the lines that make up the three spin-orbit pairs are Voigt functions. The two weaker components are due to surface reconstruction. We note that the quadratic is able to give an adequate representation of the
27
66
background over the entire range. The structureless spectrum of residuals confirms that the model is adequate. An attempt to fit the same data with only two spin-orbit doublets yields the result shown in Fig. 5b. The residuals show large systematic fluctuations, and unrealistically large Gaussian linewidths are required.
DECONVOLUTION
Another method that is often used to deal with resolution-limited data is deconvolution [ 15-171. Here the term deconvolution denotes the inverse of convolution, i.e., the removal of a broadening effect. It requires accurate knowledge of the functional form of the broadening, but requires no assumptions about the underlying lineshapes that are to be revealed. It is able to provide significant sharpening of the data, but does not lead directly to numerical values. Deconvolution tends to distort lineshapes and may introduce spurious structure into the output. In photoelectron spectroscopy deconvolution is of help mainly in valence band spectra, when a mathematical description of the underlying band structure is not available. CONCLUSION
Core level photoemission spectra can be accurately described by a convolution of functions: the spectrometer’s response function; Gaussian functions for phonon and inhomogeneous broadening; Lorentzian functions for lifetime broadening; and, in metals, a power-law singularity. The background in XPS can usually be described by the integral of the spectrum at lower binding energy, or by a set of discrete plasmons. In UV photoemission a quadratic provides an adequate representation of the secondary electron spectrum. Least-squares minimization can be used to determine the values of the parameters of a chosen model, and the spectrum of residuals can be examined for systematic oscillations, which indicate when the model is inadequate. For a well-chosen model, the parameters determined in this way provide precise and direct physical information about the sample studied. The use of the computer in fitting data, then, greatly increases the power of photoemission as an analytical tool.
REFERENCES 1
2
C.S. Fadley, in C.R. Brundle and A.D. Baker (Eds.), Electron Spectroscopy: Theory, Techniques and Applications, Vol. 2, Academic Press, London, 1978, Chap. 1, pp. l-156. R.L. Martin and D.A. Shirley, in C.R. Brundle and A.D. Baker (Eds.), Electron
67
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Spectroscopy: .Theory, Techniques and Applications, Vol. 1, Academic Press, London, 1977, Chap. 2, pp. 76-117. G.D. Mahan, Phys. Rev., 163 (1967) 612. S. Doniach and M. Sunjic, J. Phys. C, 3 (1970) 285. G.K. Wertheim and P.H. Citrin, in M. Cardona and L. Ley (Eds.), Photoemission in Solids I, Vol. 26, Springer-Verlag, Berlin, 1978, Chap. 5, pp. 197-234. G.K. Wertheim and S. Hiifner, Phys. Rev. Lett., 35 (1978) 53. S. Tougard and B. Jorgensen, Surf. Sci., 143 (1984) 482. W. Voigt, Munch. Ber., (1912) 603. E.E. Whiting, J. Quant. Spectrosc. Radiat. Transfer, 8 (1968) 1379. J.F. Kielkopf, J. Opt. Sot. Am., 63 (1973) 987. G.K. Wertheim, M.A. Butler, K.W. West and D.N.E. Buchanan, Rev. Sci. Instrum., 45 (1974) 1369. D.W. Marquardt, J. Sot. Indust. Appl. Math., 11 (1963) 431. P.H. Citrin, G.K. Wertheim and Y. Baer, Phys. Rev. B, 27 (1983) 3160. S.B. DiCenzo, P.A. Bennett, D. Tribula, P. Thiry, G.K. Wertheim and J.E. Rowe, Phys. Rev. B, 31 (1985) 2330. G.K. Wertheim, J. Electron Spectrosc. Relat. Phenom., 6 (1975) 239. J.D. Allen and F.A. Grimm, Chem. Phys. Lett., 66 (1979) 72. A.F. Carley and R.W. Joyner, J. Electron Spectrosc. Relat. Phenom., 16 (1979) 1.
LEAST-SQUARES ANALYSIS OF PHOTOEMISSION DATA
AT & T Bell Luboratories,
Murray Hill, NJ 07974
(U.S.A.)
{Received 20 March 1985)
ABSTRACT Least-squares fitting with a mathematical model function provides the most effective technique for extracting numerical information from photoemission data. With a model function that properly contains the physics of the phot~~i~ion process, averlapping lines can be separated, and such parameters as the binding energy, lifetime width, phonon broadening, and, in metals, the singularity index are readily determined. The validity of a model function can be judged by inspection of the spectrum of the residuals.
‘The peak positions and linewidths in a photoemission spectrum can be determined reasonably well by inspection when all lines are well separated. However, the spectrum contains a great deal of additional information that can be extracted with more sophisticated analysis. The linewidth contains cont~but~~ns from a ember of distrait physical processes, e.g., the l~etim~ of the core hole, the vibrational excitation of the lattice, and the inherent resolution of the photoelectron spectrometer, These contributions cannot be separated without additional information, some of which resides in the details of the lineshape. Careful analysis of the shape of the lines may also allow us to determine whether any asymmetry is due to unresolved ~ornpo~e~~ or to final state effects. The in~~~at~on content of the data is not exhausted by using the location of a few points (e.g,, the peak and half-height positions); our analysis should use all the data points, This can be accomplished by fitting a theoretical function to the data using the method of least-squares. The validity of this approach depends critically on the nature of this theoretical function. The results are useful only if the function is based on the pbysic~ processes of photoemission. We first survey the physical processes [l, 21 that contribute to the photoemission spectrum we record in the laboratory. A core level photoemission 0368-2048/85/$03.30
0 1985 Elsevier Science Publishers B.V.
snss
4 I Sn4d
A
sn A
# 32
30
28
26
24
22
BINDING ENERGY @VI
Fig. 1. Comparison of the lineshapes of the Sn 4d core level in the semiconductor and in metallic Sn.
SnSz
inherently has a Lorentzian shape whose width is the inverse of the core hole lifetime. Another contribution to the lineshape comes from phonons, the vibrational response of the host lattice, which produce further broadening of essentially Gaussian character. In metals, the final state screening response of the conduction electrons profoundly alters the shape pair of the photoemission line. Screening occurs via electron-hole excitations at the Fermi level. The spectrum of these excitations diverges at zero energy, contributing a one-sided power law singularity to the shape of the main peak itself [3-51. In insulators, final state relaxation tends to produce additional line structure, generally referred to as shake-up satellites. These represent final states in which outer electrons have been excited by the creation of the core hole. Some of the lineshape effects described above can be identified in Fig. 1, which shows the Sn 4d lines in X-ray photoelectron spectroscopy (XPS) data for the large gap semiconductors SnS2, fitted with phonon-broadened Lorentzians, and for the metal Sn, fitted with a function incorporating the effects of the metallic screening response [6]. The semiconductor lattice has a stronger vibrational response, making the Sn 4cl peaks broader in that spectrum, while the spectrum for the metal shows the characteristic asymmetry due to final-state screening. Above we have described the physical processes that determine the shape of the peaks in the core level photoemission spectrum. Another essential part of the mathematical function to be fitted to the data is a suitable background function. The background in a photoemission spectrum is not well peak
59
d 3P \
0 0
I-
0t
--
1
900
1
GOD
I
700
/
600
I
500
1
400
300
/
I
200
100
BINDING ENERGY (eV)
Fig. 2. Wide XPS scan for an evaporated Pd film. Note the sudden increase in background at each core electron photoemission line.
approximated by a straight line; rather, it has a complex shape determined by the interaction of the photoelectron with the solid from which it emerges [7]. This background is seen most clearly in a wide scan, as in Fig. 2. The background is due to photoelectrons created deep within the bulk of the solid, i.e., more than one mean free path from the surface. As these electrons diffuse toward the surface they excite plasmons and interband transitions. They emerge with reduced kinetic energy and appear in the spectrum at larger binding energy; the shape of the background therefore reflects the nature of the available excitations in the solid. In some materials discrete multiple plasmon losses are resolved; in others the energy-loss tail is largely featureless, as in Fig. 2. Another contribution to the background comes from the secondary electrons, which, because of their low kinetic energy, become important only at lower photon energy, or when data are taken close to the work function cut-off. We have considered thus far the photoemission spectrum produced by a perfectly monochromatic source and an ideal spectrometer. In reality this spectrum is modified by the instrumental response function, which includes not only aberrations in the detection system but also the finite width of the light source. Note that when no X-ray monochromator is used, this response function includes the entire spectrum of the X-ray source.
60
The problem of data analysis is now reduced to that of constructing a mathematical model function that incorporates a suitable description of the line spectrum and the background. A properly constructed model function reflects the physical contributions to photoemission, so that the parameters of the model relate directly to quantities of interest. We emphasize the importance of a physically motivated model function; there is little profit in fitting an arbitrary construct to data.
MODEL
FUNCTIONS
It is appropriate to begin the discussion of model functions with the observation that the correct way of combining the effects of two broadening processes is through convolution of the lineshapes produced by each process alone. Thus a spectrum containing the influence both of a finite core hole lifetime and of phonon broadening may be calculated as f(w)
= jg(w’)h(w _m
- o’)dw’,
(1)
where g(w) is a Gaussian and h(w) is a Lorentzian funtion. The result of this convolution, known as the Voigt function [S] , is applicable to the photoemission spectra of insulators and semiconductors, which have symmetrical peaks (see Fig. la). The Voigt function is sometimes approximated by a sum of Gaussian and Lorentzian (G-L) lines with the same full-widths at halfmaximum (FWHM) [g-11]. This approach yields the binding energy, the net linewidth, and the mixing ratio of Gaussian and Lorentzian components. Using this information, it is possible to obtain the widths of the Gaussian and Lorentzian of the corresponding Voigt function [11]. The most important drawback of this representation lies in its oscillatory deviations from the Voigt function (by as much as one percent of the peak height); these oscillations make it difficult to judge the quality of the fit. The spectrum of the metallic screening process, w”-r, is difficult to handle numerically, because of the divergence at w = 0. Fortunately the convolution of this one-sided power-law function wish a Lorentzian is available in closed form due to the work of Doniach and Sunjic [4]. f(E) =
cos [no/2 + (1 -a) @2
+ y
arctan(e/T)]
)‘l -a)/2
(2)
where E is energy measured from the position of the unperturbed Lorentzian, y the half-width at half maximum of the Lorentzian, and cx the singularity index, which, typically, assumes values in the range from 0 to 0.5. The convolution of the D-S lineshape with a Gaussian provides a function suitable to represent all peaks in a core electron photoemission spectrum,
61
including the satellites and plasmons. When there is no metallic screening, CY= 0, and the D-S lineshape reduces to a Lorentzian. Under no circumstances should a background of any type be subtracted from data prior to fitting. This is especially important when there are asymmetric lines, with long intrinsic tails of the type encountered in met,als, but even the tails of the Lorentzian profile can be compromised by injudicious background subtraction. The background should always be determined as the lines are being fitted. We first consider the representation of the step-like background illustrated in Fig. 2. Backgrounds of this kind are often approximated by assuming that every delta function line element has a step function tail. The background, then, is at every energy proportional to the integral of the line spectrum at lower binding energy. This formulation has the merit of simplicity, but seldom if ever corresponds exactly to physical reality [7]. It may, nevertheless, provide an adequate approximation, when the background intensity is small compared to that of the peaks. If there are no extrinsic excitations at small energies, it may be appropriate to shift the integrated spectrum to higher binding energy. When discrete excitations, e.g., plasmons, are resolved in the background near the main line, it is better to include an explicit representation of them in the model function. X-ray tube satellites, such as the Ka,,, and K/3induced photoemission, can be treated in the same way. The other commonly-encountered background term is that due to secondary electrons. This is usually well represented by a quadratic or cubic function whose coefficients are also left to be determined in the fitting. A linear background is not, in general, an adequate representation. The final step in the calculation of the spectrum is the convolution of the model function with the instrumental response function, which must be determined in independent measurements. In practice the response function is best determined from the shape of the Fermi cut-off in a metal like Ag, which has a featureless occupied conduction band with essentially constant density of states over a region of almost 4eV. This valence band spectrum can be used in two ways to determine the resolution function. First, if the thermal width (0.025 eV at room temperature) of the Fermi edge is small compared to the resolution, one can consider the conduction band cut-off to be a step function. The response function is then obtained simply as the derivative of the measured spectrum. Alternatively, one could fit the spectrum with a convolution of a Fermi function with a parametrized representation of the response function; this method is preferred when the resolution is comparable to the thermal width or when, as is usually the case, poor statistics make it difficult to determine the spectrum’s derivative. As a variation of the latter method one could fit the spectrum of a core level whose Lorentzian width and asymmetry are well known, treating the parameters of a mathematical resolution function as variables. The Ag3d5,* line with I’ = 0.27 eV and (Y= 0.06 is suitable for this approach. These determinations
62
of the response function are adequate when the photon source is a synchrotron or a monochromatized X-ray anode. For an instrument using unmonochromatized X-rays the response function includes the entire spectrum of the X-ray source. The spectra from X-ray tubes with Mg or Al anodes contain non-diagram lines of significant strength, the Kor,,, doublets at 9 and 11 eV, respectively, above the main Kar,,, lines. The Ka,,, emission can usually be ignored. The K/3 emission lines, which are also quite weak, lie 50 and 72 eV above the main emission line for Mg and Al anodes, respectively. For each XPS peak these extra X-ray lines produce satellites at the correspondingly smaller binding energies. The satellites of strong photoemission lines can interfere with the study of weaker lines. This problem is particularly severe when satellites of strong lines fall into the valence band region.
LEAST-SQUARES
ADJUSTMENT
In general, data-fitting consists of determining the optimal values of a set of parameters, pi, that appear in a suitable model function. In the method of least-squares, the optimal values of pi are those that minimize R, the sum of the squares of the differences between the data points and the mathematical function. Computer programs that carry out least-squares optimization are readily available [ 121. The method is conceptually simple. It may be helpful to think of R as a surface in a multidimensional space of the adjustable parameters, pi. The point of best fit is a minimum in R, where the partial derivatives of R with respect to the parameters all vanish. The program begins with a set of initial values of pi (usually a “good guess”), and numerically determines the partial derivatives of R. From this information is calculated the correction vector 6i, where the new set of parameter values pi + 61 is estimated to yield the greatest decrease in R. This process is then repeated until a minimum value of R is reached. As there may well be a number of local minima in parameter space, a meaningful result is most likely if one begins with a set of reasonable parameters containing as much of the physical information as is known. An estimate of the reliability of the final values of the parameters is obtained from the shape of R at the minimum. If R rises steeply in all directions away from the minimum, then all parameters are well determined. If, on the other hand, it rises slowly in some direction, i.e., if it is more like a valley than a circular depression, then some parameter may be inherently poorly determined. The reliability of the parameters is therefore gauged by the curvature of R, i.e., by the second partial derivatives of R. Small derivatives mean large uncertainty. A valuable output of fitting programs is a matrix of parameter correlation coefficients. It shows the extent to which one can balance out a change in one parameter by changing another one, without altering the overall quality of the fit. A correlation coefficient with
63
a magnitude greater than 0.95 is a warning that the two parameters involved may not be well determined. The magnitude of R is an immediate measure of the success of the least squares optimization. If the data are obtained by counting photoelectrons, then the statistical uncertainty of each data point is given by the square root of the number of electrons counted. Therefore, R, the sum of the squares of the errors, should be equal to the sum of all the counts in the fitted spectrum. With an adequate model function it is not difficult to approach this condition. The most valuable information about the quality of the fit is obtained from a plot of the residuals themselves. When an ideal fit has been achieved the residuals contain only the random statistical fluctuations of the data. Any systematic oscillation of the residuals plotted as a function of binding energy implies that the data cannot be adequately described by the model function. This is an invitation to generate a better model function.
CASE STUDIES
It is important to remember that the results of fitting are significant only if the model function is based on the underlying physical reality. The following examples illustrate this concern. In Fig. 3 we show the results obtained by fitting two different model functions to the 3d5,* line of metallic tin. The sample was prepared by vacuum evaporation and was free of surface contamination. The data were taken with a HP 5950A spectrometer at a resolution of 0.55 eV FWHM. In Fig. 3a the model function is a D-S line, appropriate for a metal. The residuals show that it is able to reproduce the data in adequate detail. In Fig. 3b the model contains two G-L lines, one to fit the peak, the other to fit the tail, plus an integral background. R is almost twice as large as in Fig. 3a and the residuals show that the model function is not a good representation of the data. The many-body tail has been made up in part by the second component, with the rest put into the integral background. The net result is a fit not worse than many that are published, but one that displays its problems so clearly that one has no difficultly recognizing that it is of no value. A more subtle example is provided by data in Fig. 4, on polycrystalline gold taken with monochromatized Al Ka radiation at an instrumental resolution of 0.25 eV FWHM [13]. The data contain, on the low binding energy side of the line, a weak, unresolved component due to the surface atoms. The shift of this component relative to the bulk signal is not readily estimated without numerical analysis. In Fig. 4a we show the results of fitting the data with two D-S lines with the same shape [13]. The residuals show that this model is able to account fully for the observations. The
64
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BINDING
BINDING ENERGY (e”,
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Fig. 3. Sn 3d5,? spectrum from a tin layer vacuum-evaporated onto a carbon substrate, taken with monochromatized Al KCY radiation. In (a) the data are fitted with a D-S function and in (b) with two G-L lines. The model used in (a) has 7 adjustable parameters, that in (b) has 9. The residuals show clearly that the model function used in (b) is unsuitable.
I
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,
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62 BINDING
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62
(6V)
Fig. 4. Au 4f spectrum from a vacuum-evaporated layer, taken with monochromatized Al KCY radiation at 0.25 eV FWHM resolution. Note that the surface signal at the low binding energy side of the line cannot be located accurately by inspection. In (a), a fit with two D-S lines of identical shape yields a surface atom core level shift of -0.395 eV and an (Y of 0.05. In (b), a fit with OLconstrained to zero is significantly worse, and yields false values for the lifetime width and the surface shift.
0 012
B
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006
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000 31
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23 ENERGY (6”)
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26
Fig. 5. [5] Ge 3d spectrum of a Ge(ll1) surface taken with 46 eV synchrotron radiation. In (a), the data are fitted with three spin-orbit doublets of identical shape, made up of Voigt functions. The background has a general quadratic form which was adjusted in the process of fitting. In (b), the same data fitted are with two spin-orbit doublets.
singularity index has a value of 0.05, appropriate for a metal. In Fig. 4b we show the results of fitting the same data with the singularity index set to zero. An integral background is now required to make up the many-body tail. The fit is not nearly so good, and R is almost twice as large as in Fig. 4a. Moreover, the values of all the parameters have been compromised by this approach. The lifetime width is 0.42eV instead of 0.32 eV, resulting in a significant misfit at the low energy knee of the line pair, and the surface shift is -0.37 eV instead of - 0.395 eV. These changes are larger than the mathematical errors estimated by the fitting programs, demonstrating that parameter values have significance only within the confines of their model. If the model is not appropriate for the data, then the output has no physical significance. For data taken near threshold the background of degraded electrons becomes dominant. In Fig. 5a we examine the photoemission spectrum obtained from a clean Ge(ll1) surface, using synchrotron radiation with 46 eV photon energy [14]. The background is represented by a general quadratic, and the lines that make up the three spin-orbit pairs are Voigt functions. The two weaker components are due to surface reconstruction. We note that the quadratic is able to give an adequate representation of the
27
66
background over the entire range. The structureless spectrum of residuals confirms that the model is adequate. An attempt to fit the same data with only two spin-orbit doublets yields the result shown in Fig. 5b. The residuals show large systematic fluctuations, and unrealistically large Gaussian linewidths are required.
DECONVOLUTION
Another method that is often used to deal with resolution-limited data is deconvolution [ 15-171. Here the term deconvolution denotes the inverse of convolution, i.e., the removal of a broadening effect. It requires accurate knowledge of the functional form of the broadening, but requires no assumptions about the underlying lineshapes that are to be revealed. It is able to provide significant sharpening of the data, but does not lead directly to numerical values. Deconvolution tends to distort lineshapes and may introduce spurious structure into the output. In photoelectron spectroscopy deconvolution is of help mainly in valence band spectra, when a mathematical description of the underlying band structure is not available. CONCLUSION
Core level photoemission spectra can be accurately described by a convolution of functions: the spectrometer’s response function; Gaussian functions for phonon and inhomogeneous broadening; Lorentzian functions for lifetime broadening; and, in metals, a power-law singularity. The background in XPS can usually be described by the integral of the spectrum at lower binding energy, or by a set of discrete plasmons. In UV photoemission a quadratic provides an adequate representation of the secondary electron spectrum. Least-squares minimization can be used to determine the values of the parameters of a chosen model, and the spectrum of residuals can be examined for systematic oscillations, which indicate when the model is inadequate. For a well-chosen model, the parameters determined in this way provide precise and direct physical information about the sample studied. The use of the computer in fitting data, then, greatly increases the power of photoemission as an analytical tool.
REFERENCES 1
2
C.S. Fadley, in C.R. Brundle and A.D. Baker (Eds.), Electron Spectroscopy: Theory, Techniques and Applications, Vol. 2, Academic Press, London, 1978, Chap. 1, pp. l-156. R.L. Martin and D.A. Shirley, in C.R. Brundle and A.D. Baker (Eds.), Electron
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3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Spectroscopy: .Theory, Techniques and Applications, Vol. 1, Academic Press, London, 1977, Chap. 2, pp. 76-117. G.D. Mahan, Phys. Rev., 163 (1967) 612. S. Doniach and M. Sunjic, J. Phys. C, 3 (1970) 285. G.K. Wertheim and P.H. Citrin, in M. Cardona and L. Ley (Eds.), Photoemission in Solids I, Vol. 26, Springer-Verlag, Berlin, 1978, Chap. 5, pp. 197-234. G.K. Wertheim and S. Hiifner, Phys. Rev. Lett., 35 (1978) 53. S. Tougard and B. Jorgensen, Surf. Sci., 143 (1984) 482. W. Voigt, Munch. Ber., (1912) 603. E.E. Whiting, J. Quant. Spectrosc. Radiat. Transfer, 8 (1968) 1379. J.F. Kielkopf, J. Opt. Sot. Am., 63 (1973) 987. G.K. Wertheim, M.A. Butler, K.W. West and D.N.E. Buchanan, Rev. Sci. Instrum., 45 (1974) 1369. D.W. Marquardt, J. Sot. Indust. Appl. Math., 11 (1963) 431. P.H. Citrin, G.K. Wertheim and Y. Baer, Phys. Rev. B, 27 (1983) 3160. S.B. DiCenzo, P.A. Bennett, D. Tribula, P. Thiry, G.K. Wertheim and J.E. Rowe, Phys. Rev. B, 31 (1985) 2330. G.K. Wertheim, J. Electron Spectrosc. Relat. Phenom., 6 (1975) 239. J.D. Allen and F.A. Grimm, Chem. Phys. Lett., 66 (1979) 72. A.F. Carley and R.W. Joyner, J. Electron Spectrosc. Relat. Phenom., 16 (1979) 1.