Ultramicroscopy 118 (2012) 11–16

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Ultramicroscopy

0304-39

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n Corr

E-m

lexx.ma

(M. Epp

journal homepage: www.elsevier.com/locate/ultramic

Improving the reliability of the background extrapolation in transmissionelectron microscopy elemental maps by using three pre-edge windows

Tobias Heil, Benedikt Gralla, Michael Epping, Helmut Kohl n

Physikalisches Institut and Interdisziplinares Centrum fur Elektronenmikroskopie und Mikroanalyse (ICEM), Universitat Munster, Wilhelm-Klemm-Str. 10, 48149 Munster, Germany

a r t i c l e i n f o

Article history:

Received 16 November 2011

Received in revised form

23 April 2012

Accepted 28 April 2012Available online 7 May 2012

Keywords:

EFTEM

Elemental mapping

Four-window method

Signal-to-noise ratio

Chi-squared test

91/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.ultramic.2012.04.009

esponding author. Tel.: þ49 251 8333640; fa

ail addresses: tobiasheil@uni-muenster.de (T.

trix@uni-muenster.de (B. Gralla), michael.epp

ing), kohl@uni-muenster.de (H. Kohl).

a b s t r a c t

Over the last decades, elemental maps have become a powerful tool for the analysis of the spatial

distribution of the elements within specimen. In energy-filtered transmission electron microscopy

(EFTEM) one commonly uses two pre-edge and one post-edge image for the calculation of elemental

maps. However, this so called three-window method can introduce serious errors into the extrapolated

background for the post-edge window. Since this method uses only two pre-edge windows as data

points to calculate a background model that depends on two fit parameters, the quality of the

extrapolation can be estimated only statistically assuming that the background model is correct. In this

paper, we will discuss a possibility to improve the accuracy and reliability of the background

extrapolation by using a third pre-edge window. Since with three data points the extrapolation

becomes over-determined, this change permits us to estimate not only the statistical uncertainly of the

fit, but also the systematic error by using the experimental data. Furthermore we will discuss in this

paper the acquisition parameters that should be used for the energy windows to reach an optimal

signal-to-noise ratio (SNR) in the elemental maps.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In a transmission electron microscope, electrons passingthrough the specimen can scatter inelastically at the atoms ofthe specimen. Depending on the chemical properties of the atom,the electron looses a characteristic amount of energy because ofthis process. Higher energy losses are also possible with a transferof kinetic energy to the excited atomic electron. In electronenergy-loss spectroscopy (EELS) [1] the chemical composition ofthe illuminated area of the specimen is determined using theseionization edges. The acquisition of EFTEM images on these edges(post-edge images) enables us to map the chemical compositionwith a spatial resolution on the nanometer scale. However, thepost-edge image alone is not sufficient for creating such anelemental map, because other scattering processes create a back-ground signal in addition to the element specific signal. It is,however, possible to estimate the intensity of this background byacquiring images with energy losses lower than that of the post-edge image. With these pre-edge images and a model for thebackground, the background signal within the post-edge windowcan be extrapolated.

ll rights reserved.

x: þ49 251 8333602.

Heil),

ing@uni-muenster.de

Over the last decades the three-window method, commonlyused in conjunction with an inverse power-law model dependingon two fit parameters as background model in the energy-lossregime above 100 eV, has become a standard procedure for theacquisition of elemental maps. Nowadays other chemical-sensi-tive techniques are available that surpass the EFTEM elementalmapping in certain aspects, but they have their own limitation aswell. For example, with STEM EELS [2,3] a higher spatial resolu-tion is reachable, but these images usually suffer from a muchlower number of pixels obtained within a reasonable imageacquisition time. EFTEM spectrum imaging [4,5] has the benefitthat more data points of the spectrum are available for thesubsequent analysis, but since more images have to be acquired,either the exposure time increases or the SNR of the elementalmap decreases.

However, since the three-window method uses only twopre-edge windows, only two values for the extrapolation areavailable. While this is sufficient for an extrapolation that usestwo fit parameters, as the power-law model does, there is nopossibility to check for the quality of the extrapolated backgroundby comparing the experimental data with the calculated fit.Therefore, this approach contains a fundamental problem in thebackground extrapolation, since it can introduce a systematicerror into the results. An estimate for the statistical error can bedone [6], but a deviation of the background from a perfect modelbackground or strong noise due to a poor signal-to-noise ratio in

T. Heil et al. / Ultramicroscopy 118 (2012) 11–1612

the EFTEM images will remain unnoticeable. To account for thesefactors we use a third pre-edge window for the backgroundestimation. Furthermore this modification offers the possibilityto use the w2 test to determine the goodness of the fit bycomparing the fit with the experimental data [7]. With thistest, discrepancies between the experimental data and the back-ground model can be detected that can be caused by otherionization edges for example. We will discuss this test later inthis publication.

Acquiring even more images would naturally further improvethe reliability of the extrapolation, but at the same time thiswould also mean a further increase in necessary acquisition timeor a decrease in the signal-to-noise ratio in the elemental mapdue to the shorter exposure time for the single images [8].Therefore, we settle for three pre-edge windows, since three isthe minimum number of images needed for the control methodswe want to present here. Due to the additional pre-edge windowcompared to the three-window method, our modified elementalmapping method will be referred to as the four-window method.

2. Experimental

We used a Libra 200FE microscope with a corrected in-columnOMEGA type imaging filter for the image acquisition. It has aspherical and chromatic aberration constant of 1.2 mm and anenergy resolution of 0.7 eV. We used a Gatan Ultrascan 4000,model 895 CCD-camera. All images were acquired with a binningfactor of 2, resulting in a size of 2048�2048 pixels. However,only the center 1024�1024 pixels of the images where used inorder to eliminate distorting effects caused by the border pixels.For the acquisition of the images we used a dark reference imagethat was calculated out of three images to increase the quality ofthe images [9].

Fig. 1. Illustration of the four-window-method for the background correction in eleme

3. Comparing the four-window method to the three-windowmethod

Lozano-Perez et al. have already demonstrated that the statis-tically calculated systematic error sExt can be reduced by usingthe four-window method (Fig. 1) compared to the three-windowmethod [10]. The error sExt is related to the parameter h by

h¼ 1þs2

Ext

IBS

, ð1Þ

where IBSis the intensity of the calculated background in the post-

edge (signal) window [6]. Therefore, the parameter h, thatdescribes the quality of the extrapolated background fit, can bereduced as well by using a third pre-edge window (Fig. 2).

In their investigation, Lozano-Perez et al. increased the pre-edge acquisition time by adding the third pre-edge windowwithout decreasing the acquisition time of the other two pre-edge windows. For an acquisition time t of one window, theacquisition time for the three pre-edge windows equals 3t incomparison to the 2t of the two pre-edge windows. This per secould already explain the smaller error sExt. However, they alsocompared the resulting error using three pre-edge windows tothe error of two pre-edge windows with doubled acquisition time,leading to a complete acquisition time of 4t. Despite the addi-tional acquisition time, the resulting sExt is almost identical.Lozano-Perez et al. concluded therefore that using three pre-edgewindows with a complete acquisition time of 3t is preferable tothe double-acquired two pre-edge windows with a completeacquisition time of 4t. Furthermore it can be deduced that thesame resulting sExt using regular two pre-edge windows with acomplete acquisition time of 2t can be reached using three pre-edge windows with a complete acquisition time of only 3/2 t. Or,if the same complete acquisition time is used for both, sExt

becomes smaller with three pre-edge windows.

ntal maps, using three pre-edge images (E1, E2, E3) and one post-edge image (ES).

Fig. 2. Map of the calculated h for an iron elemental map of an iron-chromium-layer system using two (a) and three (b) pre-edge images. The regions with a higher h

consist of chromium, the other regions consist of iron. In this example, the parameter h is reduced by 15% on average by using a third pre-edge window. The individual

images have been acquired with a Libra 200FE.

T. Heil et al. / Ultramicroscopy 118 (2012) 11–16 13

A drawback of the four-window method is that it adds anotherimage to the necessary drift correction of the EFTEM imageswith respect to each other. But since this drift correction can beperformed automatically during the post-processing of the images[11] or even during their acquisition [9], this can be considered aminor problem.

As for the three-window method it is of high importance forthe four-window method to choose the instrumental parametersthat will provide an optimal SNR. Those parameters include thehigh voltage, the illumination cone angle, the objective aperture,the defocus, the magnification and the width, position andnumber of the energy windows. While there are many publica-tions about the optimal parameters for the three-window methodusing two pre-edge windows [6,8,11,12], there are only a few thatalso investigate the four-window method [8,13,14]. Most of theoptimal parameters for the three-window method can be used inthe same way for the four-window one, but the width andposition of the energy windows require a separate investigation,since there is one more pre-edge window to be placed. The bestposition for this window should be the one that leads to thelargest SNR. Therefore, we shall first discuss the determination ofthe SNR and then deduce from those equations how the positionof the third window influences this value.

4. Background and SNR calculation using more than two pre-edge windows

There are different possibilities for the calculation of the post-edge image background [15,16]. Since the process of the backgroundextrapolation for three (or more) pre-edge windows differs from theone using only two pre-edge windows, we shall now shortlydescribe the weighted-least-square-fit method. A more detaileddescription can be found in other publications [8,17].

We assume an inverse power-law model for the background [1]

IBðEÞ ¼ AE�r , ð2Þ

where IB is the intensity of the background, A and r are fitparameters and E is the energy loss of the energy window (Fig. 1).This equation can be transformed into a logarithmic form:

ln IBiðEiÞ ¼ ln A�rln Ei, i¼ 1,. . .,n, ð3Þ

where n equals the number of pre-edge windows. This can berewritten as

yi ¼ a�rxi,

with xi ¼ ln Ei, yi ¼ ln IBi, and a¼ ln A:

The weighted square deviation between the calculated back-ground and the experimental data can be calculated with

e2 ¼Xn

i ¼ 1

oi½yi�aþrxi�2: ð4Þ

Assuming a Poisson distribution, the weighting coefficient oi

has the form:

oi ¼1

varðyiÞffi IBi

: ð5Þ

Now a and r can be calculated by minimizing the weightedsquare deviation. With the system of equations:

@e2

@a¼�2

Xn

i ¼ 1

oiðyi�aþrxiÞ ¼ 0 ð6Þ

@e2

@r¼ 2

Xn

i ¼ 1

oiðyi�aþrxiÞxi ¼ 0 ð7Þ

Fig. 3. Illustration of the effect that a third pre-edge window has on the

extrapolation error. The A �E�r background, the statistical noise of the measure-

ment and the maximal extrapolation difference using two windows is displayed in

gray, the modified noise value and the extrapolation difference using three

windows is displayed in black. In (a) the additional window (E3) has the same

energy loss as the window that is placed far away from the ionization edge (E1), in

(b) the additional window has the same energy loss as the near-edge window (E2).

The difference between the extrapolated values in the post-edge area is signifi-

cantly larger in the second case.

T. Heil et al. / Ultramicroscopy 118 (2012) 11–1614

we obtain

a¼

Pioiyi

Pioixi

2�P

ioixi

PioiyixiP

ioixi2P

ioi�P

ioixi

� �2ð8Þ

r¼

Pioiyi

Pioixi�

Pioi

PioiyixiP

ioixi2P

ioi�P

ioixi

� �2: ð9Þ

For the numerical calculation of the fit parameters, severaliteration steps are necessary. In the first step the experimentalvalue of IBi

is used for oi. In all following steps, the calculatedvalues of a and r from the prior iteration are used to calculate oi.The iteration is stopped if the square deviation e2 reaches itsminimum.

To calculate the background extrapolation error sExt one canuse the following equation with the previously determinedparameters a and r [6]:

s2Ext ¼

@IBS

@a

� �2

varðaÞþ@IBS

@r

� �2

varðrÞþ2@IBS

@a

� �@IBS

@r

� �covarða,rÞ:

ð10Þ

With [17]

@IBS

@a

� �¼ expða�rxSÞ,

@IBS

@r

� �¼�xBS

expða�rxSÞ,

varðaÞ ¼mo2P

ioiðxi�mo1Þ,

varðrÞ ¼1P

ioiðxi�mo1Þ,

covarða,rÞ ¼mo1P

ioiðxi�mo1Þand

mok ¼

Pix

ki oiP

ioi

the equation can be written as

s2Ext ¼ expð2a�2rxSÞ

x2S

Pioi�2xS

Pioixiþ

Pioix

2iP

ioi

Pioiðxi�mw1Þ

2

¼ 2 expð2a�2rxSÞ

Pioiðxi�xSÞ

2

Pj

Pkojokðxj�xkÞ

2: ð11Þ

Finally, the SNR can be calculated using the equation

SNR¼ICS

NS¼

ICSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiICSþ IBSþs2

Ext

q , ð12Þ

where NS is the noise of the post-edge window, ICSis the

background-corrected intensity and IBSis the intensity of the

background [1].

5. Optimal positioning of the energy windows

With three pre-edge windows the task of positioning theenergy windows is more complicated than for two pre-edgewindows. The positions of two windows are directly related tothe maximal usable pre-edge area. The first one should be placedas far away as possible and the second one as close as possible tothe ionization edge in order to obtain the best SNR [6]. The thirdwindow on the other hand can be freely placed between thesetwo windows. However, its position has a measurable influenceon the resulting SNR and placing it directly in the middle betweenthe two other windows is rarely the best choice.

A general tendency for the optimal placement of the thirdwindow can be deduced by considering the effect of thiswindow on the extrapolation when its energy loss equals one of

the two other pre-edge windows and can therefore be usedas a respective second measurement. To describe this procedure,the weighting coefficient oi of the corresponding energywindows is doubled. A rearranged version of Eq. (11) for thespecial case of two pre-edge images (i¼2) shows which weight-ing coefficient has a larger influence on the background extra-polation error sExt:

s2Ext ¼ 2 expð2a�2rxSÞ

1o2

xsx1�1

� �2þ 1

o1

xs�x2x1

� �2

x2x1�1

� �2, ð13Þ

where x1 is the logarithmic energy loss of the window that isplaced far away from the ionization edge and x2 is the logarithmicenergy loss of the window that is placed close to the ionizationedge. Since x2 is per definition larger than x1, it can be concludedthat the expression which is divided by o2 is always larger thanthe one that is divided by o1. Therefore, it should always be morebeneficial to increase the weighting coefficient o2 of the windowthat is placed near to the ionization edge.

This effect can also be illustrated graphically. Using twoimages instead of one reduces the statistical noise intensity ofthe measured value by the factor of

ffiffiffi2p

, thus modifying theresulting maximal extrapolation error for the background accord-ingly. If the noise is reduced for the pre-edge window that isplaced far away from the ionization edge, the effect on theextrapolation in the post-edge area is negligible (Fig. 3a). If thisis done for the other pre-edge window, the maximal extrapolationerror in that area is reduced significantly (Fig. 3b).

T. Heil et al. / Ultramicroscopy 118 (2012) 11–16 15

6. The v2 test

The calculation of the background extrapolation error sExt ofthe weighted least square considers only the background modeland assumes that the noise follows a Poisson distribution.The measured values influence this parameter only through theweighting of the different energy-loss positions. A comparisonbetween the fit values and the experimental data is not con-ducted. In order to check for systematic errors, a w2 test can beused [7]. In this test, the variance of the fit is compared to themeasured value. w2 increases for larger deviations of these values.This test has already been introduced to electron microscopy forchecking fits in EELS [18], but since it cannot be used with onlytwo pre-edge data points, it is fairly unknown in EFTEM.The equation for the (normalized) w2 is

w2 ¼1

nXn

i ¼ 1

ðIExpi�IBiÞ2

varðIExpiÞ

, ð14Þ

where n is the number of pre-edge windows n minus 2, IExpiis the

experimental value and IBiis the calculated value. The variance

varðIExpiÞ differs for each CCD camera type and binning value, but can

be calculated with little effort since varðIExpiÞ ¼ s2, where s is the

standard deviation of the experimental value. If a specimen with athickness larger than the mean free path is investigated, the variancecan be further influenced by the objective aperture [19].

In order to deduce if the resulting w2 indicates that the usedbackground model suits the measured values it is necessary toknow the w2 distribution as a function of n. This distribution canbe looked up in appropriate tables [7,20] or be calculated using

Fig. 4. Elemental maps of the chromium of an iron-chromium layer system. The backgr

linear model was used. While some differences between the maps can be spotted, at firs

the specimen. The histogram at the bottom shows the probability density of w2 compar

deviation s of the CCD camera with binning 2 is 1:936 � I0:522Expi

. The power law histogram

this background model fits the experimental data better than the linear model does. T

the probability function

pwðx,nÞ ¼ 1

2n=2Gðn=2Þxðn�2Þ=2e�x=2: ð15Þ

A perfect fit has the mean value of 1 for w2, but the individualvalues can differ from this. For n¼1 only about 20% of thecalculated w2 values should be between 0.7 and 1.6, with another20% higher than this and 60% lower. The best way to use thecalculated w2 values as indicator for the quality of the backgroundfit is to compare the histogram of the area of interest with thestatistical distribution of w2 (Fig. 4). If the measured histogramstrongly deviates to higher or lower values, the backgroundmodel might be faulty. A mean value below 1 indicates that thefit matches the experimental data ‘‘too well’’, which should onlybe possible if the data has a smaller variance than it should have.This can be the case if the data were smoothed for example.

While w2 can formally be calculated with any combination ofpre-edge windows it is only meaningful if the windows areindependent from each other. Therefore, this test cannot be usedif the third pre-edge window overlaps another window as it wasdescribed above. However, it can be used if the third window isplaced directly before the second window without overlap as canbe seen in Fig. 1. In this way, the resulting SNR will worsenslightly, but the possibility to check for irregularities in themeasured data can be worth this loss. We recommend to alwaysuse this window setup because the w2 test adds further reliabilityto the experimental results that cannot be gained by using theSNR calculation alone. Only if other spectral features like ioniza-tion edges of additional elements prevent the altered windowsetup or to increase a poor SNR, we would use the optimal energy

ound for the left map was calculated using a power law model, for the right map a

t sight both seem to be a reasonable representation of the chromium distribution in

ed to the calculated w2 of both background extrapolations. The measured standard

shows a much higher agreement with the probability density, which indicates that

herefore it can be deduced that the left map is the more accurate one.

T. Heil et al. / Ultramicroscopy 118 (2012) 11–1616

window positioning as described in the previous chapter. In thatcase one can acquire a single image with twice the exposure timeinstead of two images with the same energy loss. This way betterPoisson statistics can be reached for this image, but the differentexposure time will have to be considered in the extrapolation ofthe background.

7. Conclusion

Our investigation shows that the use of a third pre-edgewindow leads to an improvement in the quality of the resultingelemental maps due to the higher reliability of the backgroundextrapolation and the reduction of the statistical error of thebackground-extrapolation error sExt. In particular, elementalmaps of the higher core-loss region benefit from the additionalwindow because their pre-edge windows often feature only littledifferences in their signal intensities and therefore noise can havea larger influence in the background calculation. Furthermore it ispossible to carry out a w2 test using the experimental data and thefit to make sure that there is no major discrepancy between themeasured data and the background model.

Determining the best parameters for three pre-edge windowsis more complicated than for two pre-edge windows. Two pre-edge windows are typically placed as far apart as possible, but thethird pre-edge window has to be placed somewhere betweenthose two. This decision can have a measurable influence on theSNR of the resulting elemental map. Our investigations show thatas a rule of thumb choosing the highest possible energy loss forthis third window will lead to good results. For the w2 test thewindows must not overlap with each other.

Acknowledgments

We express our gratitude to the Deutsche Forschungsge-meinschaft (DFG) for funding the Libra 200 FE within the‘‘Großgerateinitiative’’ (Project Ko 885/8-1).

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