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Improved background-fitting algorithms for ionization edges in electron energy-loss spectra
Ultramicroscopy 92 (2002) 47–56
Improved background-fitting algorithms for ionization edges in electron energy-loss spectra R.F. Egerton*, M. Malac Physics Department, Faculty of Science, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 21 June 2001; received in revised form 31 October 2001; accepted 13 November 2001
Abstract We discuss improved procedures for fitting a power-law background to an ionization edge in an electron energy-loss spectrum. They place constraints on the background, both above and below the ionization-threshold energy, and are of particular advantage in the case of weak edges arising from low elemental concentrations. The algorithms are currently implemented as short Calculator programs for Gatan EL/P software. Their advantages and limitations are discussed, in comparison with multiple-least-squares and spatial-difference techniques. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Electron energy-loss spectroscopy (EELS); Elemental analysis; Background fitting
1. Introduction Electron energy-loss spectroscopy (EELS) has proved to be a useful analytical tool in conjunction with transmission electron microscopy. A common use is for elemental analysis, as an alternative to energy-dispersive X-ray (EDX) spectroscopy. EELS offers significant advantages for the quantitative measurement of light elements, or when very high spatial resolution is required [1]. However, practical difficulties arise from the fact that the element-specific ionization edges are superimposed on a relatively large background arising from the excitation of electrons of lower binding energy. This spectral background must be subtracted at each ionization edge in order to directly measure characteristic intensities and elemental ratios. A *Corresponding author. Tel.: +1-780-492-5095; fax: +1780-492-0714. E-mail address: [email protected] (R.F. Egerton).
background-subtracted core-loss intensity profile is also required for structural analysis, based on the near-edge or extended energy-loss fine structure (ELNES or EXELFS). Fortunately, the pre-edge background often approximates to a power law AE r , where E represents energy loss and both A and r can be taken as constants. The simplest method of elemental analysis therefore consists of fitting the background to an AE r function in a pre-edge fitting region, calculating (by extrapolation) and subtracting the background contribution beneath the ionization edge, and then measuring the integrated core-loss intensity Ic within an integration region which starts at the threshold energy Ec and extends over an energy range D beyond the edge. Repeating this procedure for the ionization edges of two elements, A and B, their atomic ratio is given [2] by B N B =N A ¼ ðIcB =IcA ÞðsA c =sc Þ;
0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 1 5 5 - 3
ð1Þ
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R.F. Egerton, M. Malac / Ultramicroscopy 92 (2002) 47–56
B where sA c and sc are core-loss (ionization) crosssections which can be calculated for most edges, knowing D and the angular range b of scattering collected by the electron spectrometer. We recently used Eq. (1) to measure concentrations of boron (in a carbon matrix) down to 0.2%, with an estimated accuracy of 10% [3]. However, when this procedure is applied to low-concentration elements, the measured core-loss integral is highly sensitive to the accuracy of the background fitting. Since the core-loss intensity decays towards zero for EbEc ; the extrapolated background ought to converge to the measured spectral intensity for energy losses well above the edge threshold Ec : In practice, the calculated background sometimes diverges from the spectral data, or may converge and then exceed the measured intensity, giving (after background subtraction) a negative core-loss intensity, which is clearly unphysical. Several possible causes of such aberrant behaviour can be identified:
1. Statistical noise within the fitting region gives rise to errors in A and r which are ‘‘amplified’’ by extrapolation [1,4]. 2. Even in the case of a very thin monatomic specimen, calculations and experiments show that the background exponent r gradually increases with energy loss (for small b), an effect which can be interpreted in terms of decreasing collection efficiency of the spectrometer [5]. 3. In thicker specimens, AE r behaviour is distorted due to plural scattering contributions to the background [1]. 4. If the ionization edge of a minor (low concentration) element is preceded (e.g. within 100 eV) by that of a major element, fine-structure oscillations in the major-element background may render the AE r approximation inappropriate. 5. Spectral artifacts, due to electron scattering within the spectrometer or detector-gain variations in the case of a parallel-recording spectrometer, may cause the background to deviate from a smooth decay [1]. The remove background procedure employed in the widely used Gatan EL/P energy-loss software
assumes an AE r background and allows no choice in fitting procedure, other than the position and width of the pre-edge fitting window. Non-convergence and (sometimes) negative intensities are therefore common in the case of minor edges. For measurements of boron concentration [3], we used the EL/P remove background routine but were often obliged to deliberately miscalibrate the energy-loss scale (or subtract an adjustable constant from the spectral data) in order to achieve a convergent background. Such measures, besides being inelegant, involve human judgment about what the backgroundsubtracted core-edge intensity should look like. Clearly it is desirable to replace this subjective procedure with an automatic one which is operator independent. Several researchers have investigated alternative elemental-analysis procedures. Some of these involve replacing the power-law approximation with other functional forms, which may involve more than two fitting parameters [6,7]. Digital filters have also been used on energy-loss spectral data [8–10] as well as EDX spectra [11] and have been successful in revealing weak ionization edges, especially in data which is relatively free of noise and recording artifacts. First- or second-difference recording is advantageous in the case of parallelrecording detectors, where it can reduce the influence of detector-gain variations. This method has demonstrated the presence of elements at concentrations below 100 ppm in favourable cases, namely edges with sharp white-line peaks at the ionization threshold [12–14]. Multiple-least-squares (MLS) fitting to spectra, often recorded in first- or second-difference mode, has been applied to EELS elemental quantification [13–19]. MLS fitting is normally carried out ‘‘offline’’, using a separate computer from the one used for the data acquisition. It requires additional spectra recorded from elemental standards, preferably with each element in a similar chemical environment to ensure similar ELNES, and measured using the same EELS system to ensure the same energy resolution [20]. Spectrum deconvolution may be necessary to allow for differences in thickness between the elemental standards and specimen being analysed.
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A spatial-difference technique has also proved useful for detecting and measuring low elemental concentrations [21]. Two spectra are recorded from separate regions of a specimen and (after scaling if necessary) the data are subtracted to yield information on the difference in elemental composition between the two regions. If an element is absent from one region but present in the other, this technique can be used to measure elemental concentration. The two regions are preferably adjacent, so that the specimen thickness and overall composition are similar; in that case, data subtraction makes exact allowance for the background underlying the ionization edge, even if it does not approximate to a power law or any other smooth function.
2. Methods In each procedure described below, we place additional constraints on the extrapolated background so that it is forced to converge to the data at some energy beyond an ionization edge (E > Ec ). Our aim was to improve the reliability and sensitivity of elemental analysis, using algorithms which are simple enough to be incorporated in existing EELS data-acquisition systems. We therefore made use of Gatan EL/P software (commonly used for spectrum acquisition, calibration and processing), in particular the Calculator option provided in the Process menu of EL/P. This is effectively a programmable calculator, into which simple text programs can be loaded and run. It allows energy windows (Intervals) to be defined via the mouse cursor and variable quantities (R1–R89) to be evaluated for use within the program. 2.1. Tied background The simplest procedure is to require that the background intersect the data at some energy well beyond the edge (EbEc ). Since the spectrum contains statistical noise, making the background match the measured electron intensity within a single data channel could introduce a systematic error. Rather, we can ensure that the background
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matches the experimental data within a post-edge energy window (containing many channels), either through least-squares fitting or by making the integrals of the spectrum and background equal over this energy window. We will call this a tied background, since it is forced to match the data in both a pre- and post-edge region. Background subtraction then involves interpolation, rather than extrapolation, making the procedure more noise-immune [1,7]. A standard procedure for power-law fitting uses linear least-squares (LLS) fitting to logarithms of the intensity (J) and energy loss (E), based on standard formulas. If y ¼ logðJÞ and x ¼ logðEÞ; the fitting parameters are given [22] by r ¼ ½SxSð1=yÞ NSðx=yÞ=D;
ð2Þ
logðAÞ ¼ ½NSðx2 =yÞ SxSðx=yÞ=D;
ð3Þ
D ¼ ½Sðx2 =yÞSð1=yÞ fSðx=yÞg2 ;
ð4Þ
where N is the number of data points being fitted. For a tied background, we simply take each sum in Eqs. (2)–(4) to include channels in both the preand post-edge regions, N being the total number of channels involved. These equations assume that the statistical fluctuation s in each data value is given by Poisson statistics (s ¼ y1=2 ). When we tried using simpler LLS formulae [1] which assume a constant value of s; we obtained almost identical values of A and r: Recognizing that A and r may change slowly with energy loss, the values given by Eqs. (2)–(4) can be regarded as average or effective values for use in the ionization-edge region, between the pre- and post-edge windows. Fig. 1 shows the listing of a Gatan Calculator program (Afit) which fits data within two energy windows (designated intervals 2 and 3 in the program) to an AE r function. After loading this program, the operator defines the pre-edge (2) and post-edge (3) interval by use of the mouse cursor. Upon running the program, intensity and energyloss data (and the number of channels) are read for these two regions, and values of A and r are calculated according to Eqs. (2)–(4). The value of r is displayed (as variable D) in the calculator window, then the AE r function is loaded into memory Z and the background-stripped data into
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Fig. 2. Energy-loss spectrum of a thin-film calibration specimen, showing a fluorine K-edge at 686 eV. Top: raw data recorded using a Gatan PEELS system. Middle: power-law background provided by Cfit: Bottom: background-subtracted fluorine-K intensity distribution. The energy windows used for background fitting (Table 1) are also shown.
The ratio of two integrals I2 and I3 ; corresponding to intervals 2 and 3, is therefore I 2 =I 3 EðD2 =D3 Þ½ðE2max E2min Þ=ðE3max E3min Þr=2 : Fig. 1. Listing of the EL/P Calculator program Afit; which evaluates a tied background by logarithmic least-squares fitting.
memory Y; as depicted in Fig. 2. The core-loss intensity Ic can be measured by using the mouse cursor to define an energy window D of suitable width in memory Y: For reasons that will become apparent, we also implemented the power-law interpolation using a formula based on spectral areas [23]. This formula is exact for r ¼ 2 and can yield extrapolatedbackground integrals with o1% error for ro5 [1]. It is based on an observation that, for a power-law function J ¼ AE r ; the integral I over an energy range of width D is approximately equal to the product of as in D and the geometric mean of J evaluated at the energy limits E max and E min : I ED½JðE max ÞJðE min Þ1=2 ¼ ðE max E min ÞA½ðE max Þr ðE min Þr 1=2 :
ð5Þ
ð6Þ
Based on Eq. (6), the power-law exponent can be evaluated from defined energies and measured integrals: rE2 ln½ðI2 D3 Þ=ðI3 D2 Þ=ln½ðE3max E3min Þ=ðE2max E2min Þ: ð7Þ Eq. (7) represents a generalization of a ‘‘two-area’’ formula [23,1] which was designed to apply to two contiguous regions (E2max ¼ E3min ). The power-law parameter A can be evaluated by equating the measured integral I2 to its analytical counterpart: I2 E½A=ð1 rÞ½ðE2max Þ1r ðE2min Þ1r :
ð8Þ
We wrote an EL/P Calculator program (Bfit1) to calculate A and r from Eqs. (7) and (8) and then display the background and the backgroundsubtracted core-loss intensity. The program first extracts the integrals and energies corresponding to a pre-edge (2) and post-edge (3) regions. The calculated background and core-loss intensity are loaded into memories Y and Z; just as with Afit:
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2.2. Convergent background
2.3. Constrained background
The programs Afit or Bfit1 are useful for quickly stripping the background, to see if a minor edge is present, and for obtaining a first estimate of the core-loss integral Ic . We have assumed that the core-loss intensity falls to zero at the post-edge region, a reasonable approximation if the postedge region is far beyond the edge. However, that situation may not be feasible, due to limited energy range of the spectral data or to the presence of additional ionization edges at larger threshold energy. If we knew how fast the core-loss intensity fell with energy loss, we could calculate its contribution within the post-edge window and correct the values of A and r accordingly. As a simple approximation, we can take the core-loss intensity to be a power-law function BE s : This is reasonable for K-edges [5] and even for the ‘tails’ of other edges, taking energies well beyond the edge. Although s depends on energy loss and collection angle b; its value does not differ greatly from r unless b is unusually small [24]. Our modified procedure for background fitting is therefore as follows. We define a core-loss region (4), which typically starts just beyond the first maximum in core-loss intensity (just above threshold for K or white-line edges, at higher energy for other edges), and use values of A and r given by Bfit1 to calculate the background integral Ib ð4Þ within region (4), employing Eq. (8) with subscript 4. We obtain the core-loss integral Ic ð4Þ ¼ I4 Ib ð4Þ , where I4 is the total integral in region (4) determined from the spectral data. Taking the core-loss intensity as BE s with s ¼ r; the core-loss contribution Ic ð3Þ within the postedge interval (3) is estimated, employing Eq. (8) with subscript 3. We then obtain revised estimates of A and r by taking I3 in Eq. (7) to be the measured integral (within region 3) minus the core-loss component Ic ð3Þ: These new values of A and r lead to revised estimates of Ib ð4Þ and Ic ð4Þ: This procedure can be repeated until the values of A; r and Ic ð4Þ converge; in the discussion below, the corresponding Calculator programs will be referred to as Bfitn; where n indicates the number of iterations used.
The assumption of a BE s core-loss intensity distribution (with s ¼ r) can be avoided by incorporating information about the actual shape of the ionization edge. In MLS fitting, this information is the core-loss intensity distribution Jc ðEÞ; but since we are dealing here with energyloss integrals, we can more easily use the ratio C of core-loss integrals within two defined energy ranges (designated as 5 and 6, as in Fig. 2): C ¼ Ic ð6Þ=Ic ð5Þ:
ð9Þ
If the specimen is very thin (say, toL=4 where L is the ‘‘plasmon’’ mean-free path for low-loss inelastic scattering) plural scattering does not greatly distort the shape of the core-loss profile. If so, this ratio should equal the ratio of core-loss scattering cross-sections (for the appropriate energy intervals) and a value of C can be derived from CEsc ð6Þ=sc ð5Þ:
ð10Þ
These cross-sections are readily calculated using hydrogenic or Hartree–Slater computer programs [1,5,25] or can be obtained from parameterized tabulations [26,27] or experimental measurements [28,29]. If the specimen is somewhat thicker (0:25o t=Lo1), the effect of plural (low-loss+core-loss) scattering becomes appreciable. But provided both core-loss energy windows commence at the edge threshold Ec ; the effect of this plural scattering can be included approximately by multiplying the cross-section ratio by a ratio of two intensities Il integrated within corresponding windows in the low-loss region of the spectrum: C E½Il ðD5 Þ=Il ðD6 Þ½sc ð6Þ=sc ð5Þ ¼ R½sc ð6Þ=sc ð5Þ:
ð11Þ
These windows include the zero-loss peak and extend up to an energy loss equal to the energy range of intervals 5 and 6 (E ¼ D5 or D6 ). For t=Lo0:8; Eq. (11) should be valid to within 20% provided both energy windows extend at least 40 eV from the edge threshold [2]. If the goal is to determine elemental composition (rather than a precise background subtraction) an accurate value of R is probably not imperative, since errors in R
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will tend to cancel when taking the core-loss ratio in Eq. (1). For even thicker specimens, the effect of plural scattering on the core-loss intensity could be removed by Fourier-ratio deconvolution [1]. However, such specimens are normally too thick for elemental analysis, unless the latter is based on high-energy (Ec > 1000 eV) edges [30]. Constraining the background fit to satisfy Eq. (9) involves an iterative procedure. After setting up suitable core-loss intervals (5 and 6), and starting from the value of r provided by Bfit1; values of Ib ð6Þ; Ib ð5Þ; Ic ð6Þ and Ic ð5Þ are calculated and the ratio Ic ð6Þ=Ic ð5Þ is compared with its expected value C; given by Eq. (10) or (11). If the measured ratio is too small, the value of r is incrementally increased until Ic ð6Þ=Ic ð5Þ matches the expected value (to within 5%). If r were initially too high, r would be incrementally decreased. We have written a Calculator program Cfit which performs these operations.
3. Results We have tested the fitting algorithms using several thin-film specimens. Energy-loss spectra were recorded using a Gatan 666 parallel-recording spectrometer (collection semi-angle of b ¼ 11 mrad) attached to a JEOL-2010 TEM operated at 200 kV, except for Fig. 5 which was obtained from a Gatan GIF system attached to a JEOL-3000F. We used pre-edge windows between 10 and 40 eV in width, depending on the occurrence of fine structure in the pre-edge background. Fig. 2 shows a K-ionization of fluorine recorded from a light-element calibration specimen containing 28 at% F [31]. The results of running different background-fitting procedures are given in Table 1. Because of the high fluorine concentration, EL/P power-law extrapolation provided a reasonable-looking background. Least-squares interpolation (Afit) gave a significantly lower value of core-loss intensity Ic (integrated over 100 eV range); the core-loss intensity is not close to zero in region 3 so the background is over-subtracted. Bfit1 gave very similar results to Afit; confirming
Table 1 Comparison of background-fitting programs at the 686 eV fluorine K-edge in a spectrum (Fig. 2) recorded from thin-film calibration specimen containing 28% fluorine [31] Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit1 Bfit2 Bfit3 Bfit4 Cfit
3.88 3.41 3.41 3.92 3.93 3.93 3.91
18.6 0.92 0.94 2.82 2.97 3.01 13.7
586 496 492 615 632 636 586
EL/P is the Gatan power-law extrapolation, based on a 626– 666 eV pre-edge interval. Afit is a power-law interpolation, using a 1100–1140 eV post-edge interval. Bfit1–Bfit4 are based on Eqs. (5)–(8) with 1–4 iterations and a 770–810 eV core-loss fitting window. Cfit is based on Eq. (9) with C ¼ 1:62 for 686– 786 and 686–886 eV windows. The core-loss interval for measurement of Ic is 686–786 eV in each case.
the validity of the area formulae, Eqs. (5)–(7), for a major edge. With 3 or 4 iterations, Bfit provided values of r and A close to their asymptotic values and a value of Ic about 8% higher than the EL/P value. Running Cfit; with a value of C obtained from SIGMAK3 hydrogenic cross-sections [1], resulted in a core-loss intensity identical to that obtained from EL/P, suggesting that simple extrapolation can often be quite adequate for major edges. The situation for a minor edge is shown in Fig. 3 and Table 2. The specimen was a 10 nm carbon film onto which a small amount of boron was deposited by electron-beam evaporation. To measure boron thickness down to 0.01 nm, we made use of a quartz-crystal microbalance (directly exposed to the boron flux) and an atomic-beam attenuator placed in front of the carbon substrate [3]. This attenuator consisted of a rapidly rotating shutter containing a 51 cutout so that the boron to the substrate was reduced by a factor of 100. The specimen analysed in Table 2 contained 0.18 at% boron. The EL/P background intersected our spectral data, leading to a negative value of Ic : Although values of r and A converged quite well within 3 iterations, a fourth iteration was required to get Ic within 1% of its asymptotic value. Such a
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Fig. 3. Energy-loss spectrum of a 20 nm carbon film containing 0.18% boron. Top: raw data. Slightly below: power-law background provided by Cfit: Below: background-subtracted boron-K intensity distribution, magnified by a factor of 32. The energy windows used for background fitting are shown.
Table 2 Comparison of background-fitting programs at the 188 eV boron K-edge in a spectrum (Fig. 3) recorded from a carbon film containing 0.18% boron [3] Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit1 Bfit2 Bfit3 Bfit4 Cfit
3.69 3.87 3.88 3.92 3.93 3.93 3.91
0.89 2.27 2.32 2.82 2.97 3.01 13.7
792 692 552 848 926 946 905
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within the accuracy of measurement for such low elemental concentrations. Fig. 4 shows a core-loss spectrum recorded from a carbon film onto which was evaporated a small amount (1%) of calcium fluoride. Since the relatively weak calcium L-edge occurrs only 60 eV above the carbon-K threshold, power-law background fitting is made problematic by finestructure oscillations in the carbon-K intensity. But by choosing a 10 eV window just before the Ca-L threshold and using both Bfit4 and Cfit; we obtained values of Ic within 5% of each other (and about 35% less than that from the EL/P background extrapolation) (Table 3). The last example is a spectrum (Fig. 5) recorded using a Gatan GIF system from a thin film of cobalt fluoride evaporated onto a carbon support. Because the specimen was analysed at low temperature (100 K) with a defocused electron beam, loss of fluorine should be negligible during the spectroscopy and we can expect the composition to be close to CoF2. EELS analysis using Bfit4 with SIGMAK3 and SIGMAL3 cross-sections (for the F-K and Co-L edges, respectively) gave fluorine/cobalt atomic ratios of 2.22, 1.99 and 1.99 for 50, 60 and 80 eV integration windows, respectively.
EL/P is the Gatan power-law extrapolation, Afit is power-law interpolation and Bfit12Bfit4 are based on Eqs. (5)–(8) with 1– 4 iterations. The pre-edge interval was 176–188 eV, post-edge interval 250–262 eV and the core-loss interval (for measurement of Ic ) 188–238 eV. Cfit is based on Eq. (9) with C ¼ 1:60 for 190–220 and 190–250 eV windows.
result is to be expected: Ic is being obtained from the difference of nearly equal I4 and Ib ð4Þ and so is sensitive to small differences in the background parameters. This same sensitivity results in a 16% difference in Ic between least-squares and areaformula background fitting (Afit and Bfit1). Using a value of C obtained from SIGMAK3, Cfit gave a value of Ic about 9% lower that Bfit4; probably
Fig. 4. Energy-loss spectrum of a carbon specimen containing about 1% calcium. Top: raw data. Slightly below: power-law background provided by Cfit: Below: background-subtracted Ca-L intensity distribution. Energy intervals used for background fitting are shown.
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Table 3 Comparison of background-fitting programs for the calcium L23-edge in a spectrum (Fig. 4) recorded from an amorphous specimen containing 1.0% calcium Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit4 Cfit
4.19 3.90 4.04 4.01
13.4 2.46 5.42 4.67
230 142 174 166
EL/P is the Gatan power-law extrapolation, with 335–345 eV pre-edge window. Afit is a power-law interpolation, using 425– 440 eV post-edge interval. Bfit4 is based on Eqs. (5)–(8) with 4 iterations. Cfit is based on Eq. (9) with C ¼ 1:51 for core-loss windows as shown in Fig. 4. A 346–396 eV interval was used to measure Ic :
Fig. 5. Energy-loss spectrum of a specimen containing cobalt fluoride, showing Bfit4 background fit to the cobalt edge, based on 754–774 eV pre-edge window, 925–950 post-edge window and 825–850 eV core-loss fitting window.
4. Discussion To illustrate the utility of having more than a single fitting program, we now discuss some differences between the algorithms just described and also compare them with other procedures used in core-loss EELS analysis. Traditional background fitting requires only a single fitting window, whose upper limit E max is usually set a few eV below the ionization edge
being analysed (to avoid the effect of tails of the detector response function) and whose starting energy E min can be varied by the operator. The background underneath the ionization edge is estimated by extrapolation from lower energy loss, and is subject to both systematic and statistical errors [4]. In the case of a major ionization edge, arising from an element of high concentration, these errors need not be severe and satisfactory elemental analysis is often achieved. But in the case of a weak edge representing a low-concentration element, the background-subtracted intensity distribution may not match the expected shape of the corresponding ionization edge and may even become negative. Although the fitting parameters (A and r) can be re-adjusted to achieve a reasonable-looking fit, this procedure is inconvenient within the current Gatan EL/P software and involves a subjective judgment. The programs Afit and Bfit1 require two fitting windows: a pre-edge interval set just below the ionization threshold Ec and a post-edge interval considerably above Ec : The background underneath the edge is estimated by interpolation, and is less error-prone [1,7]. Subtraction of this background reveals the existence and approximate shape of a weak ionization edge, its noise content and the possible presence of spectral artifacts. The integrated core-loss intensity can be used to estimate elemental concentration, particularly if the post-edge region is set far beyond the edge threshold. However, these two programs may give a misleading result for a major (prominent) ionization edge, for which the extrapolated-background procedure is usually adequate. A more realistic core-loss energy distribution is obtained by running Bfit4; which requires choosing a third fitting window, above the ionization threshold. For a weak ionization edge, this is most easily done after running Afit and displaying the approximate core-loss distribution in memory Y: The starting energy E min should exceed the maximum in the intensity distribution: just above the threshold for an edge which rises abruptly but considerably beyond the edge in the case of a delayed edge (rounded intensity profile). By making a correction for core-loss intensity in
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the post-edge interval, Bfit4 should yield more accurate core-loss integrals and elemental ratios. The energy window used for elemental quantification, via Eq. (1), need not be the same as the coreloss fitting window. The program Cfit is designed to ensure a coreloss intensity distribution of the correct overall shape but involves four fitting regions: two within the core-loss region, in addition to the pre-edge and post-edge windows. It also requires a value for the expected ratio C of core-loss intensity within the two core-loss regions. For very thin specimens (t=Lo0:25 where L is the plasmon mean free path), this ratio can be taken as the ratio of two calculated ionization cross-sections. For thicker specimens, the ratio can be corrected to allow for the effect of plural scattering. The integration window used for elemental quantification can be one of the core-loss fitting windows, or else chosen independently. Cfit can be directly compared with the MLS procedure which has been successfully used to isolate core-edge intensities present in energy-loss spectra [13–20]. In both cases, information about the core-edge shape is incorporated into the fitting procedure: as a single integrated-intensity ratio in Cfit or as an array of core-loss intensities in the MLS procedure. In principle, the larger amount of information present in a core-loss profile should be an advantage. However, problems can arise in the near-edge region, where the pronounced intensity modulations (ELNES) can differ for elements present in different chemical environments. For this reason, the ELNES region (where spectral intensity is highest) is often excluded from the MLS fitting, with some sacrifice in terms of signal/ noise ratio. In contrast, the core-loss fitting regions used in Cfit can include the ELNES region, based on the observation that the chemical and crystallographic environment of an element changes the integrated core-loss intensity very little, typically by o5% for D > 30 eV [32]. The shape of the core-loss intensity is also affected by specimen thickness [33]. In the MLS procedure, different thicknesses (between the analysed and standard specimens) are dealt with using spectral convolution or deconvolution [17]. In the Cfit algorithm, a simple correction
55
factor, as in Eq. (10) or (11), should give satisfactory elemental ratios for specimens of typical thickness. Limitations of the power-law background fitting appear in the case of closely spaced ionization edges, where modulations in the background intensity (due to a lower energy edge) will distort the core-loss profile and may introduce an error into the integrated core-loss intensity, particularly for weak ionization edges. Here, spatial-difference methods [21] may be preferable, provided there are specimen regions with and without the element of interest. MLS fitting can also deal with closely spaced edges, provided two spectral standards are used for each ionization edge: one for the background and the other for the analysed element [17]. An important feature of the algorithms described in this paper is that they can be executed on-line by the computer that handles spectrum acquisition. The advantage of on-line processing is that it enables artifacts in the data to be detected and eliminated while a specimen in still in the TEM, and before spending time on subsequent spectral processing. For example, photodiode arrays used for parallel recording suffer from interchannel gain variations which are only partially compensated by gain normalization. Gain variations may not be visible in the raw data but can appear prominently after background subtraction, in the form of a distortion of the shape of an ionization edge. If such artifacts appear in background-subtracted data, the spectrum needs to be re-recorded with the ionization edge shifted to a different part of the array. Another artifact arises from apertures present inside the spectrometer, designed to reduce electron backscattering at the diode-array detector by absorbing the intense zero-loss peak. Under some conditions, this vignetting effect creates a partial shadow on the detector, reducing electron intensity at the low-energy end of a core-loss spectrum and distorting the power-law background. Such an effect shows up as a poor fit to the AE r background; if detected on-line, it can be eliminated by shifting the ionization edge of interest towards the middle of the array.
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5. Conclusions We have written and tested simple algorithms for subtracting the background to ionization edges in electron energy-loss spectra. They are useful for quantitative analysis of elements present in low concentration (a few percent or lower) or in connection with core-loss fine-structure analysis. The programs are readily incorporated into existing software found on most EELS systems. Being based on analytical formulae, the algorithms are executed very rapidly. They could therefore be used in connection with STEM elemental mapping, where core-loss background subtraction has to be performed at each pixel. Also, since they make use of integrated intensity within fixed energy windows (rather than individual spectral channels), the algorithms should be applicable to fixed-beam energy-filtered (EFTEM) imaging. The procedures described in this publication have been made available to the Gatan company, for inclusion in future software. However, users of current Gatan systems can download the Gatan Calculator programs Afit; Bfit4 and Cfit from http://laser.phys.ualberta.ca/Begerton/index.html as Macintosh-binary files.
References [1] R.F. Egerton, Electron Energy-loss Spectroscopy in the Electron Microscope, 2nd Edition, Plenum Publishing Co., New York, 1996, pp. 272–284. [2] R.F. Egerton, Ultramicroscopy 3 (1978) 243. [3] Y. Zhu, R.F. Egerton, M. Malac, Ultramicroscopy 87 (2001) 135. [4] R.F. Egerton, Ultramicroscopy 9 (1982) 387. [5] R.F. Egerton, Ultramicroscopy 4 (1979) 169. [6] F.P. Ottensmeyer, Ann. NY Acad. Sci. 483 (1986) 339. [7] C. Quintana, J.P. Lechaire, N. Bonnet, C. Risco, J.L. Carrascosa, Microsc. Res. Tech. 53 (2001) 147.
[8] N.J. Zaluzec, Ultramicroscopy 18 (1985) 185. [9] N. Bonnet, J. Michel, D. Wagner, G. Balossier, Ultramicroscopy 41 (1992) 105. [10] J. Michel, N. Bonnet, D. Wagner, G. Balossier, P. Bonhomme, Ultramicroscopy 48 (1993) 121. [11] N.J. Zaluzec, Ultramicroscopy 28 (1989) 226. [12] R.D. Leapman, D.E. Newbury, Anal. Chem. 13 (1993) 2409. [13] H. Shuman, A.P. Somlyo, Ultramicroscopy 21 (1987) 23. [14] Y.-Y. Wang, R. Ho, Z. Shao, A.P. Somlyo, Ultramicroscopy 41 (1992) 11. [15] J.D. Steele, J.M. Titchmarsh, J.N. Chapman, J.H. Paterson, Ultramicroscopy 17 (1985) 273. [16] H. Shuman, P. Kruit, Rev. Sci. Instrum. 56 (1985) 231. [17] R.D. Leapman, C.R. Swyt, Ultramicroscopy 26 (1988) 393. [18] R.D. Leapman, S.B. Andrews, J. Microsc. 165 (1992) 225. [19] R.D. Leapman, J.A. Hunt, R.A. Buchanan, S.B. Andrews, Ultramicroscopy 49 (1993) 225. [20] A. Tence, M. Quartuccio, C. Colliex, Ultramicroscopy 58 (1995) 42. [21] H. Mullejans, J. Bruley, Ultramicroscopy 53 (1994) 351. [22] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969, p. 114. [23] R.F. Egerton, in: O. Johari (Ed.), Scanning Electron Microscopy/1980/I, SEM Inc., Chicago, 1980, p. 41. [24] D.M. Maher, D.C. Joy, R.F. Egerton, P. Mochel, J. Appl. Phys. 50 (1979) 5105. [25] R.D. Leapman, P. Rez, D.F. Mayers, J. Chem. Phys. 72 (1980) 1232. [26] R.F. Egerton, Ultramicroscopy 50 (1993) 13. [27] P. Rez, Ultramicroscopy 9 (1982) 283. [28] F. Hofer, in: L Reimer (Ed.), Energy-Filtering Transmission Electron Microscopy, Springer Series in Optical Sciences, Vol. 71, Springer, Berlin, 1995 (Chapter 4). [29] C.C. Ahn, O.L. Krivanek, EELS Atlas, Gatan Inc., 780 Commonwealth Drive, Warrendale, PA 15086, USA. [30] R.F. Egerton, Y.-Y. Yang, F.Y.Y. Chen, Ultramicroscopy 38 (1991) 349. [31] M. Malac, R.F. Egerton, Microsc. Microanal. 5 (1999) 29. [32] X. Weng, P. Rez, Ultramicroscopy 25 (1988) 345. [33] N.J. Zaluzec, in: G.W. Bailey (Ed.), 41st Annual Proceedings of the Electron Microscopic Society of America, San Francisco Press, San Francisco, 1983, p. 388.
Improved background-fitting algorithms for ionization edges in electron energy-loss spectra R.F. Egerton*, M. Malac Physics Department, Faculty of Science, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 21 June 2001; received in revised form 31 October 2001; accepted 13 November 2001
Abstract We discuss improved procedures for fitting a power-law background to an ionization edge in an electron energy-loss spectrum. They place constraints on the background, both above and below the ionization-threshold energy, and are of particular advantage in the case of weak edges arising from low elemental concentrations. The algorithms are currently implemented as short Calculator programs for Gatan EL/P software. Their advantages and limitations are discussed, in comparison with multiple-least-squares and spatial-difference techniques. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Electron energy-loss spectroscopy (EELS); Elemental analysis; Background fitting
1. Introduction Electron energy-loss spectroscopy (EELS) has proved to be a useful analytical tool in conjunction with transmission electron microscopy. A common use is for elemental analysis, as an alternative to energy-dispersive X-ray (EDX) spectroscopy. EELS offers significant advantages for the quantitative measurement of light elements, or when very high spatial resolution is required [1]. However, practical difficulties arise from the fact that the element-specific ionization edges are superimposed on a relatively large background arising from the excitation of electrons of lower binding energy. This spectral background must be subtracted at each ionization edge in order to directly measure characteristic intensities and elemental ratios. A *Corresponding author. Tel.: +1-780-492-5095; fax: +1780-492-0714. E-mail address: [email protected] (R.F. Egerton).
background-subtracted core-loss intensity profile is also required for structural analysis, based on the near-edge or extended energy-loss fine structure (ELNES or EXELFS). Fortunately, the pre-edge background often approximates to a power law AE r , where E represents energy loss and both A and r can be taken as constants. The simplest method of elemental analysis therefore consists of fitting the background to an AE r function in a pre-edge fitting region, calculating (by extrapolation) and subtracting the background contribution beneath the ionization edge, and then measuring the integrated core-loss intensity Ic within an integration region which starts at the threshold energy Ec and extends over an energy range D beyond the edge. Repeating this procedure for the ionization edges of two elements, A and B, their atomic ratio is given [2] by B N B =N A ¼ ðIcB =IcA ÞðsA c =sc Þ;
0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 1 5 5 - 3
ð1Þ
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R.F. Egerton, M. Malac / Ultramicroscopy 92 (2002) 47–56
B where sA c and sc are core-loss (ionization) crosssections which can be calculated for most edges, knowing D and the angular range b of scattering collected by the electron spectrometer. We recently used Eq. (1) to measure concentrations of boron (in a carbon matrix) down to 0.2%, with an estimated accuracy of 10% [3]. However, when this procedure is applied to low-concentration elements, the measured core-loss integral is highly sensitive to the accuracy of the background fitting. Since the core-loss intensity decays towards zero for EbEc ; the extrapolated background ought to converge to the measured spectral intensity for energy losses well above the edge threshold Ec : In practice, the calculated background sometimes diverges from the spectral data, or may converge and then exceed the measured intensity, giving (after background subtraction) a negative core-loss intensity, which is clearly unphysical. Several possible causes of such aberrant behaviour can be identified:
1. Statistical noise within the fitting region gives rise to errors in A and r which are ‘‘amplified’’ by extrapolation [1,4]. 2. Even in the case of a very thin monatomic specimen, calculations and experiments show that the background exponent r gradually increases with energy loss (for small b), an effect which can be interpreted in terms of decreasing collection efficiency of the spectrometer [5]. 3. In thicker specimens, AE r behaviour is distorted due to plural scattering contributions to the background [1]. 4. If the ionization edge of a minor (low concentration) element is preceded (e.g. within 100 eV) by that of a major element, fine-structure oscillations in the major-element background may render the AE r approximation inappropriate. 5. Spectral artifacts, due to electron scattering within the spectrometer or detector-gain variations in the case of a parallel-recording spectrometer, may cause the background to deviate from a smooth decay [1]. The remove background procedure employed in the widely used Gatan EL/P energy-loss software
assumes an AE r background and allows no choice in fitting procedure, other than the position and width of the pre-edge fitting window. Non-convergence and (sometimes) negative intensities are therefore common in the case of minor edges. For measurements of boron concentration [3], we used the EL/P remove background routine but were often obliged to deliberately miscalibrate the energy-loss scale (or subtract an adjustable constant from the spectral data) in order to achieve a convergent background. Such measures, besides being inelegant, involve human judgment about what the backgroundsubtracted core-edge intensity should look like. Clearly it is desirable to replace this subjective procedure with an automatic one which is operator independent. Several researchers have investigated alternative elemental-analysis procedures. Some of these involve replacing the power-law approximation with other functional forms, which may involve more than two fitting parameters [6,7]. Digital filters have also been used on energy-loss spectral data [8–10] as well as EDX spectra [11] and have been successful in revealing weak ionization edges, especially in data which is relatively free of noise and recording artifacts. First- or second-difference recording is advantageous in the case of parallelrecording detectors, where it can reduce the influence of detector-gain variations. This method has demonstrated the presence of elements at concentrations below 100 ppm in favourable cases, namely edges with sharp white-line peaks at the ionization threshold [12–14]. Multiple-least-squares (MLS) fitting to spectra, often recorded in first- or second-difference mode, has been applied to EELS elemental quantification [13–19]. MLS fitting is normally carried out ‘‘offline’’, using a separate computer from the one used for the data acquisition. It requires additional spectra recorded from elemental standards, preferably with each element in a similar chemical environment to ensure similar ELNES, and measured using the same EELS system to ensure the same energy resolution [20]. Spectrum deconvolution may be necessary to allow for differences in thickness between the elemental standards and specimen being analysed.
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A spatial-difference technique has also proved useful for detecting and measuring low elemental concentrations [21]. Two spectra are recorded from separate regions of a specimen and (after scaling if necessary) the data are subtracted to yield information on the difference in elemental composition between the two regions. If an element is absent from one region but present in the other, this technique can be used to measure elemental concentration. The two regions are preferably adjacent, so that the specimen thickness and overall composition are similar; in that case, data subtraction makes exact allowance for the background underlying the ionization edge, even if it does not approximate to a power law or any other smooth function.
2. Methods In each procedure described below, we place additional constraints on the extrapolated background so that it is forced to converge to the data at some energy beyond an ionization edge (E > Ec ). Our aim was to improve the reliability and sensitivity of elemental analysis, using algorithms which are simple enough to be incorporated in existing EELS data-acquisition systems. We therefore made use of Gatan EL/P software (commonly used for spectrum acquisition, calibration and processing), in particular the Calculator option provided in the Process menu of EL/P. This is effectively a programmable calculator, into which simple text programs can be loaded and run. It allows energy windows (Intervals) to be defined via the mouse cursor and variable quantities (R1–R89) to be evaluated for use within the program. 2.1. Tied background The simplest procedure is to require that the background intersect the data at some energy well beyond the edge (EbEc ). Since the spectrum contains statistical noise, making the background match the measured electron intensity within a single data channel could introduce a systematic error. Rather, we can ensure that the background
49
matches the experimental data within a post-edge energy window (containing many channels), either through least-squares fitting or by making the integrals of the spectrum and background equal over this energy window. We will call this a tied background, since it is forced to match the data in both a pre- and post-edge region. Background subtraction then involves interpolation, rather than extrapolation, making the procedure more noise-immune [1,7]. A standard procedure for power-law fitting uses linear least-squares (LLS) fitting to logarithms of the intensity (J) and energy loss (E), based on standard formulas. If y ¼ logðJÞ and x ¼ logðEÞ; the fitting parameters are given [22] by r ¼ ½SxSð1=yÞ NSðx=yÞ=D;
ð2Þ
logðAÞ ¼ ½NSðx2 =yÞ SxSðx=yÞ=D;
ð3Þ
D ¼ ½Sðx2 =yÞSð1=yÞ fSðx=yÞg2 ;
ð4Þ
where N is the number of data points being fitted. For a tied background, we simply take each sum in Eqs. (2)–(4) to include channels in both the preand post-edge regions, N being the total number of channels involved. These equations assume that the statistical fluctuation s in each data value is given by Poisson statistics (s ¼ y1=2 ). When we tried using simpler LLS formulae [1] which assume a constant value of s; we obtained almost identical values of A and r: Recognizing that A and r may change slowly with energy loss, the values given by Eqs. (2)–(4) can be regarded as average or effective values for use in the ionization-edge region, between the pre- and post-edge windows. Fig. 1 shows the listing of a Gatan Calculator program (Afit) which fits data within two energy windows (designated intervals 2 and 3 in the program) to an AE r function. After loading this program, the operator defines the pre-edge (2) and post-edge (3) interval by use of the mouse cursor. Upon running the program, intensity and energyloss data (and the number of channels) are read for these two regions, and values of A and r are calculated according to Eqs. (2)–(4). The value of r is displayed (as variable D) in the calculator window, then the AE r function is loaded into memory Z and the background-stripped data into
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50
Fig. 2. Energy-loss spectrum of a thin-film calibration specimen, showing a fluorine K-edge at 686 eV. Top: raw data recorded using a Gatan PEELS system. Middle: power-law background provided by Cfit: Bottom: background-subtracted fluorine-K intensity distribution. The energy windows used for background fitting (Table 1) are also shown.
The ratio of two integrals I2 and I3 ; corresponding to intervals 2 and 3, is therefore I 2 =I 3 EðD2 =D3 Þ½ðE2max E2min Þ=ðE3max E3min Þr=2 : Fig. 1. Listing of the EL/P Calculator program Afit; which evaluates a tied background by logarithmic least-squares fitting.
memory Y; as depicted in Fig. 2. The core-loss intensity Ic can be measured by using the mouse cursor to define an energy window D of suitable width in memory Y: For reasons that will become apparent, we also implemented the power-law interpolation using a formula based on spectral areas [23]. This formula is exact for r ¼ 2 and can yield extrapolatedbackground integrals with o1% error for ro5 [1]. It is based on an observation that, for a power-law function J ¼ AE r ; the integral I over an energy range of width D is approximately equal to the product of as in D and the geometric mean of J evaluated at the energy limits E max and E min : I ED½JðE max ÞJðE min Þ1=2 ¼ ðE max E min ÞA½ðE max Þr ðE min Þr 1=2 :
ð5Þ
ð6Þ
Based on Eq. (6), the power-law exponent can be evaluated from defined energies and measured integrals: rE2 ln½ðI2 D3 Þ=ðI3 D2 Þ=ln½ðE3max E3min Þ=ðE2max E2min Þ: ð7Þ Eq. (7) represents a generalization of a ‘‘two-area’’ formula [23,1] which was designed to apply to two contiguous regions (E2max ¼ E3min ). The power-law parameter A can be evaluated by equating the measured integral I2 to its analytical counterpart: I2 E½A=ð1 rÞ½ðE2max Þ1r ðE2min Þ1r :
ð8Þ
We wrote an EL/P Calculator program (Bfit1) to calculate A and r from Eqs. (7) and (8) and then display the background and the backgroundsubtracted core-loss intensity. The program first extracts the integrals and energies corresponding to a pre-edge (2) and post-edge (3) regions. The calculated background and core-loss intensity are loaded into memories Y and Z; just as with Afit:
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51
2.2. Convergent background
2.3. Constrained background
The programs Afit or Bfit1 are useful for quickly stripping the background, to see if a minor edge is present, and for obtaining a first estimate of the core-loss integral Ic . We have assumed that the core-loss intensity falls to zero at the post-edge region, a reasonable approximation if the postedge region is far beyond the edge. However, that situation may not be feasible, due to limited energy range of the spectral data or to the presence of additional ionization edges at larger threshold energy. If we knew how fast the core-loss intensity fell with energy loss, we could calculate its contribution within the post-edge window and correct the values of A and r accordingly. As a simple approximation, we can take the core-loss intensity to be a power-law function BE s : This is reasonable for K-edges [5] and even for the ‘tails’ of other edges, taking energies well beyond the edge. Although s depends on energy loss and collection angle b; its value does not differ greatly from r unless b is unusually small [24]. Our modified procedure for background fitting is therefore as follows. We define a core-loss region (4), which typically starts just beyond the first maximum in core-loss intensity (just above threshold for K or white-line edges, at higher energy for other edges), and use values of A and r given by Bfit1 to calculate the background integral Ib ð4Þ within region (4), employing Eq. (8) with subscript 4. We obtain the core-loss integral Ic ð4Þ ¼ I4 Ib ð4Þ , where I4 is the total integral in region (4) determined from the spectral data. Taking the core-loss intensity as BE s with s ¼ r; the core-loss contribution Ic ð3Þ within the postedge interval (3) is estimated, employing Eq. (8) with subscript 3. We then obtain revised estimates of A and r by taking I3 in Eq. (7) to be the measured integral (within region 3) minus the core-loss component Ic ð3Þ: These new values of A and r lead to revised estimates of Ib ð4Þ and Ic ð4Þ: This procedure can be repeated until the values of A; r and Ic ð4Þ converge; in the discussion below, the corresponding Calculator programs will be referred to as Bfitn; where n indicates the number of iterations used.
The assumption of a BE s core-loss intensity distribution (with s ¼ r) can be avoided by incorporating information about the actual shape of the ionization edge. In MLS fitting, this information is the core-loss intensity distribution Jc ðEÞ; but since we are dealing here with energyloss integrals, we can more easily use the ratio C of core-loss integrals within two defined energy ranges (designated as 5 and 6, as in Fig. 2): C ¼ Ic ð6Þ=Ic ð5Þ:
ð9Þ
If the specimen is very thin (say, toL=4 where L is the ‘‘plasmon’’ mean-free path for low-loss inelastic scattering) plural scattering does not greatly distort the shape of the core-loss profile. If so, this ratio should equal the ratio of core-loss scattering cross-sections (for the appropriate energy intervals) and a value of C can be derived from CEsc ð6Þ=sc ð5Þ:
ð10Þ
These cross-sections are readily calculated using hydrogenic or Hartree–Slater computer programs [1,5,25] or can be obtained from parameterized tabulations [26,27] or experimental measurements [28,29]. If the specimen is somewhat thicker (0:25o t=Lo1), the effect of plural (low-loss+core-loss) scattering becomes appreciable. But provided both core-loss energy windows commence at the edge threshold Ec ; the effect of this plural scattering can be included approximately by multiplying the cross-section ratio by a ratio of two intensities Il integrated within corresponding windows in the low-loss region of the spectrum: C E½Il ðD5 Þ=Il ðD6 Þ½sc ð6Þ=sc ð5Þ ¼ R½sc ð6Þ=sc ð5Þ:
ð11Þ
These windows include the zero-loss peak and extend up to an energy loss equal to the energy range of intervals 5 and 6 (E ¼ D5 or D6 ). For t=Lo0:8; Eq. (11) should be valid to within 20% provided both energy windows extend at least 40 eV from the edge threshold [2]. If the goal is to determine elemental composition (rather than a precise background subtraction) an accurate value of R is probably not imperative, since errors in R
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will tend to cancel when taking the core-loss ratio in Eq. (1). For even thicker specimens, the effect of plural scattering on the core-loss intensity could be removed by Fourier-ratio deconvolution [1]. However, such specimens are normally too thick for elemental analysis, unless the latter is based on high-energy (Ec > 1000 eV) edges [30]. Constraining the background fit to satisfy Eq. (9) involves an iterative procedure. After setting up suitable core-loss intervals (5 and 6), and starting from the value of r provided by Bfit1; values of Ib ð6Þ; Ib ð5Þ; Ic ð6Þ and Ic ð5Þ are calculated and the ratio Ic ð6Þ=Ic ð5Þ is compared with its expected value C; given by Eq. (10) or (11). If the measured ratio is too small, the value of r is incrementally increased until Ic ð6Þ=Ic ð5Þ matches the expected value (to within 5%). If r were initially too high, r would be incrementally decreased. We have written a Calculator program Cfit which performs these operations.
3. Results We have tested the fitting algorithms using several thin-film specimens. Energy-loss spectra were recorded using a Gatan 666 parallel-recording spectrometer (collection semi-angle of b ¼ 11 mrad) attached to a JEOL-2010 TEM operated at 200 kV, except for Fig. 5 which was obtained from a Gatan GIF system attached to a JEOL-3000F. We used pre-edge windows between 10 and 40 eV in width, depending on the occurrence of fine structure in the pre-edge background. Fig. 2 shows a K-ionization of fluorine recorded from a light-element calibration specimen containing 28 at% F [31]. The results of running different background-fitting procedures are given in Table 1. Because of the high fluorine concentration, EL/P power-law extrapolation provided a reasonable-looking background. Least-squares interpolation (Afit) gave a significantly lower value of core-loss intensity Ic (integrated over 100 eV range); the core-loss intensity is not close to zero in region 3 so the background is over-subtracted. Bfit1 gave very similar results to Afit; confirming
Table 1 Comparison of background-fitting programs at the 686 eV fluorine K-edge in a spectrum (Fig. 2) recorded from thin-film calibration specimen containing 28% fluorine [31] Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit1 Bfit2 Bfit3 Bfit4 Cfit
3.88 3.41 3.41 3.92 3.93 3.93 3.91
18.6 0.92 0.94 2.82 2.97 3.01 13.7
586 496 492 615 632 636 586
EL/P is the Gatan power-law extrapolation, based on a 626– 666 eV pre-edge interval. Afit is a power-law interpolation, using a 1100–1140 eV post-edge interval. Bfit1–Bfit4 are based on Eqs. (5)–(8) with 1–4 iterations and a 770–810 eV core-loss fitting window. Cfit is based on Eq. (9) with C ¼ 1:62 for 686– 786 and 686–886 eV windows. The core-loss interval for measurement of Ic is 686–786 eV in each case.
the validity of the area formulae, Eqs. (5)–(7), for a major edge. With 3 or 4 iterations, Bfit provided values of r and A close to their asymptotic values and a value of Ic about 8% higher than the EL/P value. Running Cfit; with a value of C obtained from SIGMAK3 hydrogenic cross-sections [1], resulted in a core-loss intensity identical to that obtained from EL/P, suggesting that simple extrapolation can often be quite adequate for major edges. The situation for a minor edge is shown in Fig. 3 and Table 2. The specimen was a 10 nm carbon film onto which a small amount of boron was deposited by electron-beam evaporation. To measure boron thickness down to 0.01 nm, we made use of a quartz-crystal microbalance (directly exposed to the boron flux) and an atomic-beam attenuator placed in front of the carbon substrate [3]. This attenuator consisted of a rapidly rotating shutter containing a 51 cutout so that the boron to the substrate was reduced by a factor of 100. The specimen analysed in Table 2 contained 0.18 at% boron. The EL/P background intersected our spectral data, leading to a negative value of Ic : Although values of r and A converged quite well within 3 iterations, a fourth iteration was required to get Ic within 1% of its asymptotic value. Such a
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Fig. 3. Energy-loss spectrum of a 20 nm carbon film containing 0.18% boron. Top: raw data. Slightly below: power-law background provided by Cfit: Below: background-subtracted boron-K intensity distribution, magnified by a factor of 32. The energy windows used for background fitting are shown.
Table 2 Comparison of background-fitting programs at the 188 eV boron K-edge in a spectrum (Fig. 3) recorded from a carbon film containing 0.18% boron [3] Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit1 Bfit2 Bfit3 Bfit4 Cfit
3.69 3.87 3.88 3.92 3.93 3.93 3.91
0.89 2.27 2.32 2.82 2.97 3.01 13.7
792 692 552 848 926 946 905
53
within the accuracy of measurement for such low elemental concentrations. Fig. 4 shows a core-loss spectrum recorded from a carbon film onto which was evaporated a small amount (1%) of calcium fluoride. Since the relatively weak calcium L-edge occurrs only 60 eV above the carbon-K threshold, power-law background fitting is made problematic by finestructure oscillations in the carbon-K intensity. But by choosing a 10 eV window just before the Ca-L threshold and using both Bfit4 and Cfit; we obtained values of Ic within 5% of each other (and about 35% less than that from the EL/P background extrapolation) (Table 3). The last example is a spectrum (Fig. 5) recorded using a Gatan GIF system from a thin film of cobalt fluoride evaporated onto a carbon support. Because the specimen was analysed at low temperature (100 K) with a defocused electron beam, loss of fluorine should be negligible during the spectroscopy and we can expect the composition to be close to CoF2. EELS analysis using Bfit4 with SIGMAK3 and SIGMAL3 cross-sections (for the F-K and Co-L edges, respectively) gave fluorine/cobalt atomic ratios of 2.22, 1.99 and 1.99 for 50, 60 and 80 eV integration windows, respectively.
EL/P is the Gatan power-law extrapolation, Afit is power-law interpolation and Bfit12Bfit4 are based on Eqs. (5)–(8) with 1– 4 iterations. The pre-edge interval was 176–188 eV, post-edge interval 250–262 eV and the core-loss interval (for measurement of Ic ) 188–238 eV. Cfit is based on Eq. (9) with C ¼ 1:60 for 190–220 and 190–250 eV windows.
result is to be expected: Ic is being obtained from the difference of nearly equal I4 and Ib ð4Þ and so is sensitive to small differences in the background parameters. This same sensitivity results in a 16% difference in Ic between least-squares and areaformula background fitting (Afit and Bfit1). Using a value of C obtained from SIGMAK3, Cfit gave a value of Ic about 9% lower that Bfit4; probably
Fig. 4. Energy-loss spectrum of a carbon specimen containing about 1% calcium. Top: raw data. Slightly below: power-law background provided by Cfit: Below: background-subtracted Ca-L intensity distribution. Energy intervals used for background fitting are shown.
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Table 3 Comparison of background-fitting programs for the calcium L23-edge in a spectrum (Fig. 4) recorded from an amorphous specimen containing 1.0% calcium Program
Exponent r
A ( 1014)
Ic (counts 103)
EL/P Afit Bfit4 Cfit
4.19 3.90 4.04 4.01
13.4 2.46 5.42 4.67
230 142 174 166
EL/P is the Gatan power-law extrapolation, with 335–345 eV pre-edge window. Afit is a power-law interpolation, using 425– 440 eV post-edge interval. Bfit4 is based on Eqs. (5)–(8) with 4 iterations. Cfit is based on Eq. (9) with C ¼ 1:51 for core-loss windows as shown in Fig. 4. A 346–396 eV interval was used to measure Ic :
Fig. 5. Energy-loss spectrum of a specimen containing cobalt fluoride, showing Bfit4 background fit to the cobalt edge, based on 754–774 eV pre-edge window, 925–950 post-edge window and 825–850 eV core-loss fitting window.
4. Discussion To illustrate the utility of having more than a single fitting program, we now discuss some differences between the algorithms just described and also compare them with other procedures used in core-loss EELS analysis. Traditional background fitting requires only a single fitting window, whose upper limit E max is usually set a few eV below the ionization edge
being analysed (to avoid the effect of tails of the detector response function) and whose starting energy E min can be varied by the operator. The background underneath the ionization edge is estimated by extrapolation from lower energy loss, and is subject to both systematic and statistical errors [4]. In the case of a major ionization edge, arising from an element of high concentration, these errors need not be severe and satisfactory elemental analysis is often achieved. But in the case of a weak edge representing a low-concentration element, the background-subtracted intensity distribution may not match the expected shape of the corresponding ionization edge and may even become negative. Although the fitting parameters (A and r) can be re-adjusted to achieve a reasonable-looking fit, this procedure is inconvenient within the current Gatan EL/P software and involves a subjective judgment. The programs Afit and Bfit1 require two fitting windows: a pre-edge interval set just below the ionization threshold Ec and a post-edge interval considerably above Ec : The background underneath the edge is estimated by interpolation, and is less error-prone [1,7]. Subtraction of this background reveals the existence and approximate shape of a weak ionization edge, its noise content and the possible presence of spectral artifacts. The integrated core-loss intensity can be used to estimate elemental concentration, particularly if the post-edge region is set far beyond the edge threshold. However, these two programs may give a misleading result for a major (prominent) ionization edge, for which the extrapolated-background procedure is usually adequate. A more realistic core-loss energy distribution is obtained by running Bfit4; which requires choosing a third fitting window, above the ionization threshold. For a weak ionization edge, this is most easily done after running Afit and displaying the approximate core-loss distribution in memory Y: The starting energy E min should exceed the maximum in the intensity distribution: just above the threshold for an edge which rises abruptly but considerably beyond the edge in the case of a delayed edge (rounded intensity profile). By making a correction for core-loss intensity in
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the post-edge interval, Bfit4 should yield more accurate core-loss integrals and elemental ratios. The energy window used for elemental quantification, via Eq. (1), need not be the same as the coreloss fitting window. The program Cfit is designed to ensure a coreloss intensity distribution of the correct overall shape but involves four fitting regions: two within the core-loss region, in addition to the pre-edge and post-edge windows. It also requires a value for the expected ratio C of core-loss intensity within the two core-loss regions. For very thin specimens (t=Lo0:25 where L is the plasmon mean free path), this ratio can be taken as the ratio of two calculated ionization cross-sections. For thicker specimens, the ratio can be corrected to allow for the effect of plural scattering. The integration window used for elemental quantification can be one of the core-loss fitting windows, or else chosen independently. Cfit can be directly compared with the MLS procedure which has been successfully used to isolate core-edge intensities present in energy-loss spectra [13–20]. In both cases, information about the core-edge shape is incorporated into the fitting procedure: as a single integrated-intensity ratio in Cfit or as an array of core-loss intensities in the MLS procedure. In principle, the larger amount of information present in a core-loss profile should be an advantage. However, problems can arise in the near-edge region, where the pronounced intensity modulations (ELNES) can differ for elements present in different chemical environments. For this reason, the ELNES region (where spectral intensity is highest) is often excluded from the MLS fitting, with some sacrifice in terms of signal/ noise ratio. In contrast, the core-loss fitting regions used in Cfit can include the ELNES region, based on the observation that the chemical and crystallographic environment of an element changes the integrated core-loss intensity very little, typically by o5% for D > 30 eV [32]. The shape of the core-loss intensity is also affected by specimen thickness [33]. In the MLS procedure, different thicknesses (between the analysed and standard specimens) are dealt with using spectral convolution or deconvolution [17]. In the Cfit algorithm, a simple correction
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factor, as in Eq. (10) or (11), should give satisfactory elemental ratios for specimens of typical thickness. Limitations of the power-law background fitting appear in the case of closely spaced ionization edges, where modulations in the background intensity (due to a lower energy edge) will distort the core-loss profile and may introduce an error into the integrated core-loss intensity, particularly for weak ionization edges. Here, spatial-difference methods [21] may be preferable, provided there are specimen regions with and without the element of interest. MLS fitting can also deal with closely spaced edges, provided two spectral standards are used for each ionization edge: one for the background and the other for the analysed element [17]. An important feature of the algorithms described in this paper is that they can be executed on-line by the computer that handles spectrum acquisition. The advantage of on-line processing is that it enables artifacts in the data to be detected and eliminated while a specimen in still in the TEM, and before spending time on subsequent spectral processing. For example, photodiode arrays used for parallel recording suffer from interchannel gain variations which are only partially compensated by gain normalization. Gain variations may not be visible in the raw data but can appear prominently after background subtraction, in the form of a distortion of the shape of an ionization edge. If such artifacts appear in background-subtracted data, the spectrum needs to be re-recorded with the ionization edge shifted to a different part of the array. Another artifact arises from apertures present inside the spectrometer, designed to reduce electron backscattering at the diode-array detector by absorbing the intense zero-loss peak. Under some conditions, this vignetting effect creates a partial shadow on the detector, reducing electron intensity at the low-energy end of a core-loss spectrum and distorting the power-law background. Such an effect shows up as a poor fit to the AE r background; if detected on-line, it can be eliminated by shifting the ionization edge of interest towards the middle of the array.
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5. Conclusions We have written and tested simple algorithms for subtracting the background to ionization edges in electron energy-loss spectra. They are useful for quantitative analysis of elements present in low concentration (a few percent or lower) or in connection with core-loss fine-structure analysis. The programs are readily incorporated into existing software found on most EELS systems. Being based on analytical formulae, the algorithms are executed very rapidly. They could therefore be used in connection with STEM elemental mapping, where core-loss background subtraction has to be performed at each pixel. Also, since they make use of integrated intensity within fixed energy windows (rather than individual spectral channels), the algorithms should be applicable to fixed-beam energy-filtered (EFTEM) imaging. The procedures described in this publication have been made available to the Gatan company, for inclusion in future software. However, users of current Gatan systems can download the Gatan Calculator programs Afit; Bfit4 and Cfit from http://laser.phys.ualberta.ca/Begerton/index.html as Macintosh-binary files.
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