Practical Methods for Detecting Peaks in Auger Electron ... No.3/Vol... · Journal of Surface Analysis Vol.14, No. 3 (2008) pp. 225-242 Furukawa et al., Practical Methods for Detecting

Journal of Surface Analysis Vol.14, No. 3 (2008) pp. 225-242 Furukawa et al., Practical Methods for Detecting Peaks in Auger Electron Spectroscopy And X-Ray Photo-electron Spectroscopy

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Technical Report

Practical Methods for Detecting Peaks in Auger Electron Spectroscopy and X-Ray Photoelectron Spectroscopy

Y. Furukawa,a,* Y. Nagatsuka,b Y. Nagasawa,b S. Fukushima,c M. Yoshitake,c and A. Tanakad aDENKI KAGAKU KOGYO K. K.,3-5-1 Asahimachi, Machida-shi, Tokyo 194-8560, Japan

bJEOL Ltd., 3-1-2 Musashino, Akishima, Tokyo 196-8558, Japan cNational Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan

dULVAC-PHI, Inc., 370 Enzo, Chigasaki, Kanagawa 253-8522, Japan *[email protected]

(Received: September 25, 2007; Accepted: December 7, 2007)

Three kinds of peak detecting algorithms for AES and XPS spectrum are proposed. These peak detecting

methods are composed of three stages of algorithms: rough estimation of the background; direct calculation of the peak and background relation at the candidate peak; and application of the second derivative curve. This report provides concrete methods of finding peaks in a measured spectrum of surface analysis based on an empirical investigation of how to detect significant signals among faint ones. Algorithms and charac-teristics of the respective peak detecting methods are discussed.

1. Introduction

The presence of peaks in the spectrum obtained from the original data must first be determined as the primary source of information, when a qualitative or quantitative analysis is carried out for AES and XPS, especially when making a decision on the presence of elements.

Surface chemical analysts use many different methods to detect peaks in data, but as there may be no uniquely defined perfect method, analysts sometimes pick peaks visually. However, since scientists often rely on com-puters to perform basic analyses of data, scientists and instrument manufactures have developed their own peak detecting algorithms [1]. Their algorithms and techniques rely on straightforward methods for calculating the peak locations.

There are different ways to write algorithms to detect peaks, but their algorithms are not sufficiently clarified, nor do they always work for poorly resolved peaks. Since each method has relative merits, it is desirable for analysts to understand each algorithm and its perform-ance in order to select the best or alternate method.

It is also important to track accurately and verify what has been done to the data. Data files processed by pro-grams should leave a comprehensive audit trail to ensure

traceability. To satisfy the strictest compliance standards, the appropriate audit trail should contain a complete de-scription of the processing methods and equations ap-plied to the data as well as program version.

This report provides concrete methods of finding peaks in a measured spectrum of surface analysis based on an empirical investigation of how to detect significant signals among faint ones. The purpose of this report is to give analysts a practical guide for finding peaks in a spectrum using their algorithms.

In the activity that relates to the international stan-dardization in ISO TC201-SC3-WG4 (Surface Chemical Analysis-Data Management and Treatment- Peak Detec-tion), three peak detecting methods have been proposed and discussed.

2. General considerations regarding peak detection

Described herein are basic data processing methods and specific aspects of detecting peaks such as peak de-tection criteria, the detection of poorly resolved peaks, small and broad peaks and overlapped peaks with val-leys.

Copyright (c) 2008 by The Surface Analysis Society of Japan


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2.1. Peak detection criteria In order to define peak detection criteria, it should be

assumed that the data obtained with a pulse-counting detector will obey statistical theory based on the Poisson distribution. Then, empirical conditions for the detection criteria can be naturally defined, and concrete techniques for practical peak detection methods can be obtained.

“Peak detection criteria” is specifically referred to empirically determined criteria for detecting peaks in AES and XPS spectra, which are normally acquired without distortion due to instrument error, except for random noise.

Strictly speaking, the peak in question may have dif-ferent background intensities on its lower and higher energy sides. In such a case, the presence of the peak should be decided by setting conditions on both sides of the peak independently. However, this makes it too com-plex to develop the argument further, so sometimes the peak is assumed to have a nearly flat or linear back-ground on both sides, and it is further assumed that the peak can be decided by a single decision inequality.

3. Peak detection methods

The proposed peak detection methods are composed of three kinds of algorithms: for making a rough estimation of the background, for using the second derivative curve, and for directly calculating the peak and background relation at the candidate peak. The algorithms of these three methods are described in the following.

3.1. Peak detection using rough estimation of spec-trum background

This method firstly assumes that the background curve of a spectrum is generally gentle and the total spectrum region containing peaks is much narrower than the re-gion without peaks, and then makes a rough estimation of the background intensity for each point of the spec-trum.

As the background intensity changes rather gently compared with the intensity near the peak, it can be ap-proximately expanded by using the 2m+1 points of data which cover the region with several times of the typical full width at half maximum of the peak,* w. The number

of averaging points, 2m+1, is selected so as to be equal to several times of the number of points in the typical full width at half maximum of the average peak of the spec-trum.

The background bi (i=1,T) (where, T is the total sam-pling points) then can be approximately written by using the given spectrum data yi as follows:

��

��m

mjjiji yhb (1)

where hj is the coefficient of the simple moving average, and is expressed as hj=1/(2m+1).

The variance of bi is expressed by using such expan-sion as follows [2]:

� � � ��

�� m

mj

m

mj

m

jljijiljjijbi CovhhVarh ,

22� (2)

where (Var)i+j is the variance of yi+j and (Cov)[i+j],[j+l] the covariance of yi+j and yj+l. If the random nature of the spectrum data yi which obeys statistical theory based on the Poisson distribution is assumed,

� � jiji yVar �� (3)

Further, each yi is measured independently, and may have no correlation with each other, then:

� �� 0, �� ljjiCov (4)

Then, the variance of bi can be approximately expressed as

��

��

��m

mjjiji

m

mjjbi y

myh 2

22

)12(1� (5)

If the variance of ni=yi-bi is defined as �ni2, it can be es-

timated as follows:

��

��

m

mjjiibiyini y

my 2

222

)12(1�� (6)

Therefore, the final inequality to judge a peak is given by using the critical value k�2~3,

*The typical full width at half maximum of the average peak: Generally, spectra acquired with an instrument such as AES or XPS include peaks with certain range of FWHM. “The FWHM of the average peak” represents the approximate value for the FWHMs of such peaks. For example, it is 20 eV for AES and 10 eV for XPS, which may be allowed to adopt as default values.


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��

��

m

mjjiinii y

mykkn 2)12(

1� (7)

or, in a more familiar expression,

niii kby �� (8)

where bi+k�ni defined as the noise threshold curve. The local maximum yi satisfying the inequality Eq. (8) is re-garded as a peak (see Fig. 1.).

Furthermore, in order to apply this procedure more ef-fectively for the practical situations, some exceptional cases should be taken into consideration. In one of such situations, there is the case that some especially broad peaks in the spectrum do not satisfy inequality Eq. (8).

In order to cope with these situations, a peak, which satisfies the following inequality, is included if its Si

* is greater than a certain value Si0

**.

0ii SS � (9)

The approximate value of Si0 is roughly estimated by assuming it as an ideal triangle peak with the typical full

width w at half maximum of the peak and height k�ni (noise threshold curve),� then its peak area S would be, 1/2 2 w k�ni=wk�ni.

wkS nii ��0 (10)

In fact, if the peak area Si is greater than the value above, the peak is found to be real one in most of the cases [3].

For poorly resolved peaks with deep valleys, the fol-lowing data processing is effective. If the spectrum has plural peaks above noise threshold curve k�ni and valleys that do not cross the noise threshold curve, and if valley depth D*** of the minor local maximum in the spectra exceeds the noise fluctuation k�ni, the peaks are regarded as real.

nikD �� (11)

By adding such exceptional cases, this peak-detection method becomes effective in the actual spectrum proc-essing.

bi+k�ni : noise threshold curve

yi

Counts

Energy

Si

bi D

valley depth

bi+k�ni : noise threshold curve

yi

Counts

Energy

Si

bi D

valley depth

Fig. 1. Schematic diagram of yi, bi and bi+k�ni. *Si: The software evaluates the (i-th) peak area by summing the difference from the measured spectrum value to the background value at every successive position while the difference is positive. **Si0: This is obtained as the area of the triangle (typical noise peak) by multiplication of typical FWHM and k�ni. ***Valley depth, D: The software evaluates the valley depth by searching a local maximum that exceeds the deviation of the noise height and, just after finding the local maximum, a local minimum that exceeds the deviation of the noise height, and calculating the difference between the local maximum and the local minimum.


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3.2. Peak detection using threshold curve of second derivative

This method has the same effect as subtracting the background from the spectrum, by making use of the second derivatives. And because it has no arbitrariness in the background subtracting procedure, it may be a rela-tively convenient algorithm when we use it with the aid of computers.

By making use of the moving polynomial approxima-tion procedure (Savitzky-Golay method), it is possible to calculate the second derivative spectrum di from the original spectrum yi as follows:

��

��n

njjiji ygd (12)

where, gj (j = -n, n) is the Savitzky-Golay coefficients for the second derivative, and if 2n+1 points of data cover roughly the half width of the typical peak, the obtained second derivative will faithfully represent the true second derivative curve. If it is expressed as mentioned above, the variance �i

2 of di can be obtained as follows:

� � � ��

�� n

nj

n

nj

n

jljijiljjiji CovggVarg ,

22� (13)

where (Var)i+j is the variance of yi+j and (Cov)[i+j],[j+l] the covariance of yi+j and yj+l. Again, if we assume the ran-dom nature of the spectrum data yi which obeys statisti-cal theory based on the Poisson distribution, then:

� � jiji yVar �� (14)

Further, each yi is measured independently, and may have no correlation with each other, then:

� �� 0, �� ljjiCov (15)

Thus, the variance �i2 of di can be approximately calcu-

lated as follows:

ji

n

njji yg �

�� 22� (16)

As the peak in the spectrum corresponds to the local minimum of the spectrum, we judge the peak is real one if the local negative minimum dmin is less than (in abso-lute values, greater than) its noise fluctuation.

Therefore, for the peak judge inequality, if the follow-ing inequality Eq. (17) is satisfied, we admit the peak is detected at the position that gives dmin in the second de-rivative if the following inequality is satisfied [3,4],

ikd ��min (17)

where k�i defined as the noise threshold curve.

Counts

Energy

Energy

measured spectrum

Second derivative spectrum

noise threshold curve

Counts

Energy

Energy

measured spectrum

Second derivative spectrum

noise threshold curve

Fig. 2. Schematic diagram of a peak position, second derivative curve and threshold noise curve.


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3.3. Peak detection by directly calculating peak and background relations at the candidate peaks

In this method, the candidate peaks are detected by the second derivative treatment and judged by comparison with criteria.

In this case, the candidate peaks are detected following the steps described below: (1) Calculate the second derivative curve (e.g. by using

Savitzky-Golay method [4]) and calculates the stan-dard deviation (�i) for the second derivative curve.

(2) Pick up the candidate peak positions satisfying the local minimum (nagative value) of the second deriva-tive curve is negative.

(3) Let p be a candidate peak position, and w be the typical full width at half maximum of the peak* in the original spectrum, where w is usually given in the peak detection condition (see Fig. 3.). If there exists a positive local maximum at x=p1 in the second deriva-tive spectral range, p-3w�x<p in the nearest candidate peak (if does not exist, the position, p1=p-3w is re-garded as the position), the position, p1 is regarded as a midway one to the left-side background, and fur-thermore if there exists a local minimum in the smoothed spectrum or a zero cross position in the second derivative spectrum at x=q1 in the spectral range, p1-2w�x<p1 in the nearest candidate peak (if

does not exist, the position, q1=p1-2w is regarded as the position), the distance, l1=p-q1 corresponds to the left-side background position p-l1 with intensity B1.

(4) Likewise, If there exists a positive local maximum at x in the second derivative spectral range, p<x�p+3w in the nearest candidate peak (if does not exist, the position, p2=p+3w is regarded as the position), the position, x=p2 is regarded as a midway one to the right-side background, and furthermore if there exists a local minimum peak position in the smoothed spec-trum or a zero cross position in the second derivative spectrum in the spectral range, p2�x<p2+2w in the nearest candidate peak (if does not exist, the position, q2=p2+2w is regarded as the position), the distance, l2=q2-p corresponds to the right-side background po-sition p+l2 with intensity B2.

(5) If the background curve near the peak can be ap-proximated by a straight line, the background inten-sity B at the peak position is calculated as

� � � �211221 lllBlBB �� (18)

The background intensity, B, at the candidate peak position is estimated by adding the fractional back-ground intensities from both sides of the peak, B1 and B2, which are inversely proportional to the distances

Counts measured spectrum

2nd derivative curve

B1B2

q1 p1 p p2 q2

W*

l1 l2

P Counts measured spectrum

2nd derivative curve

B1B2

q1 p1 p p2 q2

W*W*

l1l1 l2l2

P

Fig. 3. Schematic diagram of a peak position and its background at both sides of the peak. *The typical full width at half maximum of the average peak : refer to 3.1.


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from the peak to both sides of background positions l1 and l2.

Let P denote the peak intensity with the back-ground and N the net peak intensity, then, N=P-B, and the variance �N

2 of N is calculated as follows [5]:

222BpN �� , (19)

where

22

2121

2 ,, BBP BBp �� , (20)

and

� �� 22

2211

21

2212

2BBB llllll �� , (21)

then, �N2 is calculated as follows:

� � � �2212

12221

2 lllBlBPN �� (22)

Therefore, the peak decision condition is given as fol-lows:

NkN �� (23)

If the second derivative curve does not cross the hori-zontal axis within a distance of 3 times the peak width (full width at half maximum of a typical peak) from the candidate peak position on both sides of the peak, we accept the position with a distance of 3 times the peak width from the candidate peak position as a background position for the peak.

4. Some practical examples 4.1. General

Some practical examples applied to the typical spectra of AES and XPS using codes written for computing are shown for the three peak detection methods. Fig. 4. Fig-ures 5 and 6 show AES spectra, Figs. 7-9 show XPS spectra. Each processing condition is shown below, where T, Bsmp, Smp, Step and Typw stand for the total sampling points, smoothing points of background, smoothing points of spectrum, sampling step width and typical peak width of the spectrum, respectively, and k is the critical value shown in each peak detection formula.

4.2. Processing conditions 4.2.1. AES spectrum

T=2001, Step=1 eV, Typw=20 eV Figure 4: Bsmp=31, k=2.5 Figure 5: Smp=13, k =1.4 Figure 6: Smp=13, k = 2.5

4.2.2. XPS spectrum T=1001, Step=1 eV, Typw=10 eV Figure7: Bsmp=31, k=3 Figure 8: Smp=7, k=2.5 Figure 9: Smp=11, k=3

5. Summary 5.1. Relative merits of each method

Proposed peak detection methods for a spectrum are not fully perfect but if the parameters such as Smp, Bsmp, Step and Typw for the detecting algorithms are given properly in automatic computing, finely good results are obtained for practical uses as shown in Figs.4-9. Sum-marized relative merits of each method are shown in Ta-ble 1.

6. Acknowledgement

The author would like to thank Dr. Yoshitaka Na-gatsuka for developing these peak detection algorithms. Local members of TC201SC3WG4 in Japan are also acknowledged for their developing codes for computing and useful advice and fruitful discussion.

7. References [1] Y. Nagatsuka, K. Yoshida, Y. Nagasawa, and

J.Suzumi, J. Surf. Anal. 3, 602 (1997). [2] P. Gans, Data Fitting in the Chemical Sciences, pp.

17-18, Wiley (1992). [3] Y. Nagatsuka, K. Yoshida, Y. Nagasawa, and Y. Ono,

J. Surf. Anal. 5, 60 (1999). [4] A. Savitzsky and M. J. E. Golay, Annal. Chem. 36,

1627 (1964). [5] H. A. Liebhafsky, M. G. Pfeiffer, and P. D. Zemany,

Annal. Chem. 27, 1257 (1955).


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Table 1 Relative merits of each method.

Kind of peak Peak detection method Results

(1) Single small peak 1. Background estimation 2. 2nd derivative 3. Peak and Background relations

Almost good Sometimes neglects broad peaks Sometimes neglects broad peaks

(2) Single large peak 1. Background estimation 2. 2nd derivative 3. Peak and Background relations

Good Sometimes finds spurious peaks at peak base Good

(3) Grouped small peaks 1. Background estimation 2. 2nd derivative 3. Peak and Background relations

Sometimes neglects small peaks Sometimes neglects broad peaks Sometimes neglects small peaks

(4) Shoulder peak 1. Background estimation 2. 2nd derivative 3. Peak and Background relations

Not appropriate Sometimes neglects shoulder peak Sometimes neglects shoulder peak

Notes Names of Peak detection method in Table1 are short for followings. 1. Background estimation: Peak Detection Using Rough Estimation of Spectrum Background in 3.1. 2. 2nd derivative: Peak Detection Using Threshold Curve of Second Derivative in 3.2. 3. Peak and Background relations: Peak Detection by Directly Calculating Peak and Background Relations at the Candidate Peaks in 3.3.


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(a)

(b)

Fig. 4. (a) Example of AES peak detection using the method shown in 3.1. (b) Noise threshold curve of the above peak detection.


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(a)

(b)

Fig. 5. (a) Example of AES peak detection using the method shown in 3.2. (b) Second derivative curve of the above peak detec-tion.


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(a)

(b)

Fig. 6. (a) Example of AES peak detection using the method shown in 3.3. (b) Second derivative curve of the above peak detec-tion.


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(a)

(b)

Fig. 7. (a) Example of XPS peak detection using the method shown in 3.1. (b) Noise threshold curve of the above peak detection.


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(a)

(b)

Fig. 8. (a) Example of XPS peak detection using the method shown in 3.2.(b) Second derivative curve of the above peak detec-tion.


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(a)

(b)

Fig. 9. (a) Example of XPS peak detection using the method shown in 3.3. (b) Second derivative curve of the above peak detection.


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It is more convenient to remind the readers that Eqs. (3) and (19) are the direct conclusions from the property of Poisson distribution. Z�@[

Next phrase, ”which obeys statistical theory based on the Poisson distribution” is inserted after the phrase, “If the random nature of the spectrum data yi”.

Z=>@ 2-2[

In 3.2., the authors define the p1 and p2 as the points at the second derivative's local maxima. Are they more suitable than the zero-cross points, the inflection points? Z�@[

The p1 and p2 as the points at the second derivative’s local maxima correspond to a flat position of background on both sides of a peak or flat point of valley between peaks. As red dotted line in Fig. 3 is distorted, it is modi-fied a little to appropriate shape. It seemed hard correc-tion for author to modify the second derivative curve perfectly this time.

Z=>@ 2-3[In 3.1., I don’t understand why they don’t use the

width w deduced in 3.2. for examining S. Obviously w=(p2-p1)/2, or (distance between inflection points) seems to be a better estimate for FWHM. Z�@[

The use of the typical full width at half maximum of the average peak as a default values is the feature of these three peak detection methods. For example, 20 eV for AES and rather big figure, 10 eV, for XPS are al-lowed to adopt as default values. As it is difficult to know any full width at half maximum or to calculate it before detecting peak, a default value is adopted in these three algorithms. Although the use the full width at half maximum deduced in 3.3. become possible only after detecting peaks or detecting candidate peaks, it is impos-sible to use the values before detecting peaks.

Z=>@ 2-4[

In addition to comment Z=>@ 2-3[, it seems better to unify these three methods to give a best one. Is there a special reason to leave them separately? At least when the authors write codes, they should do so. Z�@[

Unfortunately the combination of these three methods constitutes an infringement of the patents.

Z=>@ 2-5[

As long as one needs to supply values such as Smp, Bsmp, Step, and to check the appropriateness of the result, there is not much practical difference between the pre-sent method and such conventional visual inspections by human eyes. Do you think if it is possible, in the future, to write a complete and automatic code that works with-out human aid, which is the ultimate purpose of this pa-per? Z�@[

It is hard to find a proper resolution of this comment. Some answer to this comment is given in “1.Introduction”. As far as there may be no perfect method, analysts should find the best way by checking the performance of peak detecting method interactively with computer.


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