Microsc. Microanal. 20, 798–806, 2014 doi:10.1017/S1431927614000567
© MICROSCOPY SOCIETY OF AMERICA 2014
Highly Automated Electron Energy-Loss Spectroscopy Elemental Quantification Raman D. Narayan,1,* J. K. Weiss,1 and Peter Rez2 1 2
AppFive LLC, 1095 West Rio Salado Parkway, Suite 110, Tempe, AZ 85281, USA Department of Physics, Arizona State University, PO Box 871504, Tempe, AZ 85287, USA
Abstract: A model-based fitting algorithm for electron energy-loss spectroscopy spectra is introduced, along with an intuitive user-interface. As with Verbeeck & Van Aert, the measured spectrum, rather than the single scattering distribution, is fit over a wide range. An approximation is developed that allows for accurate modeling while maintaining linearity in the parameters that represent elemental composition. Also, a method is given for generating a model for the low-loss background that incorporates plural scattering. Operation of the userinterface is described to demonstrate the ease of use that allows even nonexpert users to quickly obtain elemental analysis results. Key words: EELS, quantification, model, fitting, plural scattering, multiple scattering
I NTRODUCTION Electron energy-loss spectroscopy (EELS) is a very powerful tool for elemental analysis in materials and biological sciences, but the complexity of its application can be an obstacle to its widespread use. We present a quantitative procedure for EELS elemental analysis that requires minimal user input. In energy-loss spectra the inner shell edges that are used for elemental quantification are superimposed on a slowly varying background. If energy-loss spectroscopy is to be successfully used to accurately determine chemical composition, accurate and reliable background models are essential. Traditionally, a background of the form AE–r is fitted in the region before each core-loss edge and extrapolated over the energy range needed for integration over the core-loss signal (Egerton, 1996). Although the asymptotic form of cross-section models gives some justification for this procedure, the effects of any systematic error are greatly magnified by the extrapolation process. This is especially true for thicker specimens, where the effect of multiple scattering presents a serious problem in any attempt to model the background. In addition to the failure of the commonly used powerlaw background model over wide energy ranges, there are other drawbacks to traditional EELS quantification. If the core-loss spectrum is deconvolved, its interpretation may be complicated by artifacts. Results are also dependent on the choice of fitting or integration windows. Since windows for background determination and edge integration are needed for each core-loss edge, EELS quantification can be very difficult when the onsets of multiple edges occur close together in the spectrum (Verbeeck & Van Aert, 2004). revised September 13, 2013; accepted February 23, 2014 *Corresponding author. raman@appfive.com
Another approach has been to model the entire coreloss region, including the background simultaneously, as in Manoubi et al. (1990) and in the EELSModel (Verbeeck & Van Aert, 2004) and Hyperspy (The Hyperspy Developement Team, 2013) programs. Here a linear combination of the background and core-loss edges is fitted to the spectrum. Verbeeck & Van Aert (2004) showed that the accuracy of their quantification procedure approaches theoretical limits and successfully modeled spectra where edges are close together. Theoretical core-loss edges are convolved with a measured low-loss spectrum to account for plural scattering, following the Fourier–ratio method. In model-based EELS quantification, an accurate background model is even more important than in the conventional approach, since it must be applied across the entire core-loss region simultaneously. This is especially true when one must fit two edges that are far apart in the spectrum. Manoubi et al. (1990) convolved a power-law background with the low-loss, but noted complications owing to artifacts from the artificial onset of the background model, particularly for core-loss edges at lower energy-loss. Verbeeck & Van Aert (2004) mentioned this issue with Manoubi’s method, and respond by not convolving the background model, but instead added parameters so the power-law can vary with energy-loss. We propose the following enhancements to model-based EELS quantification. First, a more accurate calculation of multiple scattering in the elemental edges is provided (see Modeling the Core-Loss Edges section), an improvement to the Fourier–ratio method. Second, given that the modeling of the background is critical to accurate quantification and multiple scattering makes a significant contribution to the pre-edge background, an alternate procedure is given for obtaining a background model that takes into account multiple scattering (see Modeling the Background section). Lastly, an intuitive program for EELS quantification is introduced
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(see Workflow for Elemental Analysis section), where we aim to optimize this approach such that a minimal amount of input is required from the user. As with Verbeeck & Van Aert (2004), we fit the observed spectrum rather than the calculated single scattering distribution (SSD). The low-loss region is extrapolated using either a power-law or one of several more detailed functions. For each element to be quantified, a model EELS profile is constructed that includes all edges that will be present in the core-loss region. This allows the user to proceed without having to identify all the individual core-loss edges to fit. The approach can be helpful for more challenging spectra where edges overlap, or where there might be some ambiguity as to which elements are present. Core-loss edges are calculated from the Hartree–Slater model (Leapman et al., 1980) that can be used over a wider range of elements and edges than hydrogenic cross-sections (Egerton, 1979). In its simplest implementation, the procedure takes as input (in addition to instrumental parameters) a complete EELS spectrum that includes the low-loss region and zeroloss peak, and a list of elements to model. A routine for splicing parts of the spectrum together is also included. We hope that this work will make EELS quantification more practical for a broad community of potential users and will improve the consistency of their results. In the following sections, the essential and optional inputs and some of the theory for the fitting procedure are discussed. In particular, an approximation is given that allows one to accurately model plural scattering in the elemental edges (see Modeling the Core-Loss Edges section), while using a single linear parameter for the abundance of each element, and a model for the low-loss background is introduced that incorporates plural scattering (see Modeling the Background section). In the Results section the benefit gained from the background model from the Modeling the Background section is demonstrated and the user-interface is described. Future work is discussed in the Discussion section.
M ATERIALS AND M ETHODS Required Input The following are required from the user:
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3. treatment of near-edge structure (see Workflow for Elemental Analysis section); 4. whether to fit an additive constant.
Fitting Region Beginning with the full spectrum, the first decision to be made is where in the spectrum to start fitting. Ideally, one might like to model the entire spectrum, but constructing an accurate model for the low-loss region is not practical since it would involve a detailed application of solid state theory. If not specified by the user, a cutoff energy-loss is chosen such that at least one edge for each element of interest occurs at higher energy-loss (>90 eV). It is also necessary that fitting begin in a smooth region of the spectrum—otherwise it will be difficult to apply a background model. The program also avoids, when possible, a situation where there are large energy differences between edges since it is more difficult for a background model to be applied successfully over a wide region of energy-loss.
Modeling the Core-Loss Edges In the limit of a very thin specimen, the EELS spectrum is modeled as j
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S(E) is the full spectrum, L(E) the low-loss extrapolated using the background model, and Ki(E) the core-loss edges for each element. The included core-loss edges, or angle-integrated differential cross-sections dσ/dE, are those whose binding energies EB are within the range, represented in equation (1) by Emin and Emax, chosen for modeling the core-loss region. The multiplicative parameters ai scale with the concentration of each element. Both multiple scattering and instrument resolution are neglected in equation (1). When the full EELS spectrum is recorded, instrumental resolution and multiple scattering can be included in the model. Separating the SSD into low-loss and core-loss terms: sðνÞ ¼ lðνÞ + ai ki ðνÞ;
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1. full spectrum including zero-loss peak and low-loss (core-loss spectra can be modeled, but less accurately); 2. accelerating voltage; 3. collection half angle; 4. elements: All elements known to be present in spectrum.
where s(ν), l(ν), and k(ν) are the Fourier transforms of the quantities represented by the same letters in equation (1). In Fourier space, the plural-scattered, measured spectrum, j(ν) is related to the single scattering spectrum s(ν) as
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(Egerton, 1996: 247) where I0 is the integrated intensity of the zero-loss peak or instrumental response function z(ν). Substituting into equation (3) for plural scattering we get
There are a number of optional inputs that change aspects of the fitting procedure: 1. energy-loss loss at which to start fitting (see Fitting Region section); 2. model for pre-edge background (see Modeling the Background section);
jðνÞ ¼ zðνÞ exp½sðνÞ=I0 ;
jðνÞ ¼ zðνÞ exp½flðνÞ + ai ki ðνÞg=I0 ;
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where l(ν) is our best approximation for the SSD of the lowloss spectrum.
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Separating the two terms in the exponent in equation (4), we have jðνÞ ¼ jL ðνÞ exp½ai ki ðνÞ=I0 ;
(5)
where the plural-scattered, extrapolated low-loss spectrum jL(ν) is defined as jL ðνÞ zðνÞ exp½lðνÞ=I0 :
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Equation (5) provides a model that can be applied to an EELS spectrum if both the low-loss and core-loss regions are acquired. However, since there is much less intensity in the core-loss edges than in the low-loss it is possible to expand the exponential in equation (5) as a power series: " # ai ki ðνÞ 1 ai ki ðνÞ 2 1 ai ki ðνÞ 3 jðνÞ ¼ jL ðνÞ 1+ + + ¼ : I0 2! I0 3! I0 (7) Keeping the first two terms in the expansion, the parameters ai become linear parameters which is beneficial since linear fitting is faster and less error prone (Lawton & Sylvestre, 1971): jðνÞ jL ðνÞ+ai jL ðνÞki ðνÞ=I0 :
(8)
Equation (8) corresponds to the familiar Fourier–ratio method (Egerton & Whelan, 1974). The inverse Fourier transform of equation (8) provides an approximate model that is linear in parameters ai that multiply the convolved core-loss edges. It is possible to maintain the quadratic term in equation (7) without sacrificing linearity in parameters ai. We introduce the following approximation, which makes use of the Fourier transform of the measured spectrum jexper(ν): jðνÞ jL ðνÞ + ai ki ðνÞ
jL ðνÞ + jexper ðνÞ : 2I0
(9)
The core-loss edges are essentially being convolved with the average of the extrapolated low-loss and measured spectrum. Substituting equation (7) for jexper(ν) in equation (9), we get ai ki ðνÞ jðνÞ jL ðνÞ + 2I0 " ( )# ai ki ðνÞ 1 ai ki ðνÞ 2 jL ðνÞ + jL ðνÞ 1 + + + ¼ ; ð10Þ I0 2! I0 "
# ai ki ðνÞ 1 ai ki ðνÞ 2 1 ai ki ðνÞ 3 ¼ jL ðνÞ 1 + + + + ¼ : I0 2 I0 4 I0 (11) The difference between our estimate given by eqaution (9) and the spectrum including all orders of multiple scattering of the core-loss edges in equation (5) is quantified by comparing the estimated expansion [equation (11)] with the
correct expansion in equation (7). This deviation occurs at the cubic term, and is ΔjðνÞ 1 ai ki ðνÞ 3 : (12) jL ðνÞ 12 I0 The quality of this approximation can be evaluated for a given specimen thickness t by replacing in equation (12) the fitted edges aiki(ν) with their integrated intensity IK, and replacing j(ν) and jL(ν) with their integrated intensities I and IL. IK represents the intensity of the core-loss edges over the entire core-loss region, rather than over a small energy-loss window, since we are fitting the whole core-loss spectrum. Equation (12) becomes ΔI 1 IK 3 : (13) IL 12 I0 From Poisson statistics: IK 1 - exp½ - t=λK ; ¼ exp½ - t=λtotal I0
(14)
where λK and λtotal are the mean free paths of the core-loss edges and the total inelastic mean free path, respectively. Since t=λK 1 for many transmission elecron microscopic specimens, exp[ − t/λK] can be approximated using a power series, giving IK =I0 t=λK exp½t=λtotal : (15) Considering, for example, the total Si L-edge at an accelerating voltage of 200 keV, assuming all electrons are collected, and t = λtotal, equation (14) gives IK =I0 0:21 using a total inelastic mean free path from Chou & Libera (2003) and total cross-sections from the Hartree–Slater model (Leapman et al., 1980). Applying this value to equation (13) we get ΔI=IL 8 ´ 10 - 4 . For comparison, the integrated first- and second-order terms in equation (7) have values of 0.21 and 0.022 if ai ki ðνÞ ! IK . The approximation given by equation (9) captures the second term in equation (7) whereas the Fourier–ratio method [equation (8)] does not. This secondorder term corresponds to electrons that undergo two core-loss events, therefore equation (9) will yield a greater increase in accuracy when there are core-loss edges with greater cross-sections, such as Si-L23. It is then more important when fitting at lower energy-losses, since edges at lower binding energies generally have greater cross-sections. The inverse Fourier transform of equation (9) JL ðEÞ + Jexper ðEÞ JðEÞ JL ðEÞ + ai Ki ðEÞ ´ ; (16) 2I0 is the model that will be fitted to the measured spectrum. This is linear in parameters ai, though there are nonlinear parameters implicitly contained in the extrapolated low-loss JL(E). The nonlinear parameters in JL(E) are fitted using a Levenberg–Marquardt algorithm (Press et al., 2007), and ai are included using linear fitting.
Simulated Spectra The effectiveness of our procedure for including multiple scattering in the model for the low-loss background,
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Figure 1. Simulated boron nitride spectrum of thickness of two inelastic mean free paths. It is simulated for accelerating voltage 200 keV and collection half angle 200 mrad. An artificial gain change is applied at 180 eV to make the core-loss edges visible.
described below in the Modeling the Background section, is evaluated using simulated spectra. Simulated boron nitride (BN) spectra are generated using a Monte Carlo procedure where each electron is tracked individually. Each of 108 incident electrons begins with an energy of 200 keV, and its energy is slightly altered according to a normal distribution to simulate the spread in energies that would be seen in the zero-loss peak. The number of inelastic collisions the electron will suffer is drawn randomly from a Poisson distribution, and this depends on the specimen thickness relative to the inelastic mean free path. The simulated low-loss spectrum is based on the plasmon-pole approximation (Ritchie & Howie, 1977; Raether, 1980), and the Boron and Nitrogen K-edges are the same Hartree–Slater differential crosssections used for fitting (Leapman et al., 1980). The simulated spectra do not include all the features seen in real EELS spectra, such as those caused by band structure, but this is not important for the present purpose. The effects of multiple scattering are readily apparent in the BN spectrum shown in Figure 1, where a specimen thickness of two inelastic mean free paths is simulated. The plasmon peak at 25 eV repeats at 50 and 75 eV for two and three plasmon excitations. Multiple scattering also has the effect of softening the onset of the B-K and N-K core-loss edge, which would have sawtooth-like profiles for a thin specimen.
Modeling the Background Modeling the low-loss, or pre-edge, background is arguably one of the most difficult aspects of model-based and conventional EELS elemental analysis. It is essential for the background to be fitted accurately, since the calculated composition is proportional to the intensity of the core edges, and this intensity is only known when the background intensity is accounted for. It should be noted that in model-based quantification, the models for the outer core-loss edges constitute part of the background for edges with higher binding energies. For example, in a specimen containing boron, carbon, and
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nitrogen, the boron and carbon K-edges contribute to the background for the nitrogen K-edge. This is not the case in the conventional approach, where a background model is fitted independently for each core-loss edge in an energy window preceding the onset of each edge. This can lead to a quite nonphysical model where there is no correlation between the independent background models. To model a measured spectrum, as opposed to its SSD, one might extrapolate the measured low-loss spectrum using an analytic model function such as a power-law (Egerton, 1996), which is known to approximately match the energy-loss dependence in the tail of ionization differential cross-sections. This can be unrealistic for thicker specimens, as both experimental observations and computational modeling show that multiple scattering of the low-loss excitations leads to significant changes in the background, in particular a deviation from the well-known inverse power-law. We propose treating plural scattering in the background model by extrapolating the SSD rather than the raw spectrum, and then reversing the deconvolution used to obtain the SSD. The rationale for this approach is that an analytic background model can be more successfully applied to the SSD, where there is ideally no plural scattering, than to the raw spectrum. As with Manoubi et al. (1990) and Verbeeck & Van Aert (2004), the parameters for the background model are fitted to the entire core-loss spectrum simultaneously with those for the elemental composition [ai in equation (16)]. Before introducing the procedure, it is helpful to recall that the plural-scattered, measured spectrum j(ν) is given by equation (3) in Fourier space. An approximate model for plural scattering is calculated by the following steps: 1. First, the SSD is obtained using Fourier–log deconvolution. The SSD is reconvolved with the zero-loss peak to suppress noise, as described by Egerton (1996: 248). In terms of Fourier transforms of the zero-loss peak z(ν) and measured spectrum j(ν), the smoothed SSD, s0 ðνÞ, is s0 ðνÞ ¼ zðνÞ ln½jðνÞ=zðνÞ:
(17)
2. Starting at the energy-loss where one chooses to begin fitting, the smoothed SSD is extrapolated using a powerlaw or more sophisticated model. The Fourier transform of the extrapolated SSD will be represented by l0 ðνÞ, since it represents the low-loss spectrum. l0 ðνÞ is our model for a single-scattered low-loss spectrum, or alternatively, an SSD without core-loss edges. 3. Reversing the deconvolution, an approximation for the Fourier transform of the extrapolated, plural-scattered low-loss spectrum j0L ðνÞ is obtained by replacing s(ν) in equation (3) with the extrapolated SSD l0 ðνÞ, giving j0L ðνÞ ¼ I0 exp½l0 ðνÞ=I0 :
(18)
This constitutes an approximation since the noisereducing function z(ν) is applied to the SSD [equation (17)] before reversing the deconvolution. In Figure 2 the relevance of the above procedure is illustrated, by comparing the above background model to
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Figure 2. Plural scattering background model versus an unconvolved power-law. The top panel shows a simulated boron nitride spectrum (see Simulated Spectra section) and the same spectrum with core-loss edges excluded (actual low-loss). The accelerating voltage is 200 keV, collection half angle 200 mrad, and thickness two inelastic mean free paths. The plural and raw background curves are the background models optimized along with core-loss edges to fit the spectrum. “Plural background” is due to the procedure descibed in Modeling the Background section, convolving a power-law, and “raw background” results from simply applying a power-law to the entire core-loss region. Residuals for both background models with respect to the actual low-loss spectrum are shown in the lower panel.
that obtained by simply applying a power-law to the entire core-loss region. Here the above procedure involves applying a power-law to the SSD rather than the measured spectrum. Both background models are fitted, along with core-loss edges, to a simulated BN spectrum (see Simulated Spectra section) shown in the top panel, which has a thickness of two inelastic mean free paths. Also shown in the top panel of Figure 2 is the same simulated spectrum excluding core edges (actual low-loss), which includes electrons that have undergone multiple low-loss events. This is obtained by excluding simulated electrons that suffer a core-loss event. For a measured spectrum, this low-loss component would not be available after the onset of the B-K-edge at 190 eV. The bottom panel shows the difference from the spectrum excluding core edges for our background model (plural background) and the unconvolved power-law (raw background). The success of our procedure is demonstrated by the fact that our background model tracks the actual low-loss much more closely.
Figure 3. Simulated boron nitride spectra for thickness of 1.55 inelastic mean free paths and collection half angles 200 (top) and 5 mrad (bottom). Fitted curves are shown for the case where the background model incorporates plural scattering.
Although the SSD is used to obtain the model for the low-loss background, the theoretical cross-sections are still being fitted to the measured spectrum rather than its SSD.
RESULTS Background Model In this section the fitting procedure described above is applied to simulated EELS spectra, and the benefit of the method from the Modeling the Background section for considering plural scattering in the background model is demonstrated. Spectra are simulated (see Simulated Spectra section) for values of specimen thickness up to three mean free paths, and for collection half angles of 200 and 5 mrad. A simulated spectrum is shown for both collection angles in Figure 3, along with the fitted model. The fitted boron to nitrogen (B/N hereafter) ratios are plotted in Figure 4. Also plotted are B/N ratios obtained by extrapolating the measured low-loss with a power-law (see Fig. 2) rather than including multiple scattering as described in the Modeling the Background section. In both cases, the power-law is fitted over the entire core-loss region. The plotted data points represent the average B/N ratio for ten simulated spectra corresponding to the same thickness. Error bars are equal to the standard deviation in the mean B/N ratio at each thickness: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u X σx 1 σ x ¼ pffiffiffiffi ¼ t (19) ðxi - xÞ2 ; NðN - 1Þ i¼1 N (Lyons, 1991) where x is the B/N ratio and N = 10.
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iii. Do nothing: include the channels containing near-edge structure as with the rest of the fitted core-loss region. This most simplistic option can introduce significant bias, since the Hartree–Slater cross-sections do not include near-edge structure (Manoubi et al., 1990).
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Figure 4. Variation of boron to nitrogen (B/N) ratio with specimen thickness for simulated collection half angles 200 (top) and 5 mrad (bottom) mrad. Points connected by a solid line represent the background model that incorporates plural scattering as described in the Modeling the Background section, whereas for the dotted line the background model is extrapolated from the raw (simulated) spectrum.
For both collection angles, the B/N ratio varies less with thickness when plural scattering is incorporated in the background model using the SSD. This is evidence of the importance of using a background model that accounts for the effects of plural scattering.
Workflow for Elemental Analysis In this section a sample workflow is given to illustrate the ease of use of this EELS quantification procedure. 1. After importing a CsI spectrum (Ahn et al., 1983) from a two-column text file, one arrives at the “setup” screen, shown in Figure 5. 2. Set the acceleration voltage and collection half angle (a and b from Fig. 5). 3. Select elements cesium and iodine from the periodic table (Fig. 5, c). Interactive edge onset markers appear (l). These can be adjusted to change the binding energies for the fitted theoretical edges. 4. (Optional) Choose how to treat the near-edge structure. There are three choices: i. Apply a nonparametric fit to the near-edge structure, using the measured spectrum to approximate plural scattering. This is similar to the treatment by Verbeeck et al. (2006). ii. Masking: exclude the channels in the near-edge region from the fit.
If (i) or (ii) is chosen, the range of channels to treat as nearedge structure is specified in controls e and f in Figure 5. 5. (Optional) Choose a power-law (default) or more detailed model for the low-loss background. 6. (Optional) Choose whether or not to incorporate plural scattering in the background model as described in the Modeling the Background section. 7. (Optional) Choose whether to include a uniform background intensity as a free parameter in the fitted model. By default, this decision is made automatically based on the negative energy-loss channels. 8. (Optional) The lowest energy-loss for modeling the spectrum is chosen automatically by default (j in Fig. 5), or the user can manually select the first channel (k). 9. Clicking on the quantify button (m in Fig. 5) runs the fitting algorithm, and displays the fitted model and elemental composition results as shown in Figure 6. Clicking on “Setup” (n) returns one to the setup screen (Fig. 5) where all of the above settings can be modified.
DISCUSSION The procedure for model-based EELS elemental analysis described here is meant to be appropriate for both nonexpert and advanced users. The utility of the method described in the Modeling the Background section is demonstrated in the Results section. Regarding the user-interface described in the Workflow for Elemental Analysis section, it is important to note that one can obtain elemental analysis results by supplying only the measured spectrum, accelerating voltage, collection angle, and elements to be fitted. In the remainder of this section, future improvements are discussed. These fall in two categories—adding new functionality to more accurately model difficult spectra, and further automating the user-interface. In the first category, the most obvious addition will be to support the use of core-loss edges measured from standards in place of the theoretical Hartree–Stater differential crosssections (Leapman et al., 1980). This will allow the fitting algorithm to more accurately handle spectra with overlapping edges due to different elements, when neither element has another (nonoverlapping) edge that can be modeled (Leapman & Swyt, 1988). An example of such a spectrum is shown in Figure 7. When the edge onsets are too close together, and one chooses to either mask or use nonparametric fitting for the near-edge region, the total model can become degenerate for concentrations of the two elements, since it is not clear without fitting the edge onsets how much of the intensity is due to either element. A standard specimen with band structure similar to that seen in the spectrum could allow
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Figure 5. Electron energy-loss spectroscopy (EELS) quantification “setup” screen: (a) accelerating voltage, (b) collection half angle, (c) element selection periodic table, (d) near-edge modeling option, (e, f) energy range over which to apply the near-edge model, (g) background model, (h) whether or not to use single scattering distribution to incorporate plural scattering in the background model (see Modeling the Background section), (i) whether to apply an additive constant level to the spectrum fit, (j) whether to automatically determine energy-loss channel at which to start fitting, (k) slider for manually choosing first channel to fit, (l) atomic edge onset interactive markers, (m) setup button, (n) quantify button.
one to adequately model the near-edge structure without adding extra parameters. To effectively use edges measured from standards, one would have to correct for differences in collection angle and specimen thickness between the standard spectrum and that which is being modeled. The fitting procedure might also be expanded to support quantitative elemental mapping and tomography, which might require further optimization for speed. Another important addition will be to include the beam convergence angle in the calculation of core-loss edges, as in Kohl (1985). With regard to automation, the general mindset behind the procedure described in the Workflow for Elemental Analysis section is to remove, or at least make optional, unnecessary manual settings that affect the outcome of elemental analysis. In this spirit, an algorithm could be used to automatically identify which elements are present in the spectrum. One might step through the spectrum, looking for increases in intensity that satisfy some calculated threshold. It could also
be helpful to automatically search for and apply saved standard edges. As mentioned in the Workflow for Elemental Analysis section, edge onset energies have been left to the user to change manually, and these could be added as free parameters, which might be particularly helpful if standard edges are used.
SUMMARY This paper adds to prior work on model-based EELS analysis an approximation to allow for more accurate modeling of plural scattering in the core-loss edges while maintaining linearity in the elemental composition parameters (see Modeling the Core-Loss Edges section), a method for producing a low-loss background model that incorporates the effects of plural scattering (see Modeling the Background section), and an intuitive user-interface that allows one to arrive at elemental analysis results with very little effort (see Workflow for
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Figure 6. Electron energy-loss spectroscopy (EELS) quantification “quantify” screen. The controls on the left side toggle on and of the measured spectrum and curves corresponding to final and initial models, the single scattering distribution, residuals (data minus fitted), and the background model. Fractional composition results are shown as a pie chart and in a table at the bottom of the screen.
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Elemental Analysis section). It is our hope that this work makes elemental analysis with EELS accessible to a broader community of scientists and technicians.
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ACKNOWLEDGMENTS We gratefully thank Professor Jo Verbeeck for taking his time to have a number of interesting and productive discussions with us on model-based EELS analysis.
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Figure 7. Edge overlap is illustrated in this TiO2 spectrum from the EELS Atlas (Ahn et al., 1983). Owing to near-edge structure, there is no smooth region between either the Ti-L23 and O-K, or the O-K and Ti-L1 edges where an atomic model for the differential cross-sections could be applied.
AHN, C.C., KRIVANEK, O.L., BURGER, R.P., DISKO, M.M. & SWANN, P.R. (1983). EELS Atlas: A Reference Collection of Electron Energy Loss Spectra Covering all Stable Elements. Tempe, AZ and Warrendale, PA: HREM Facility, Center for Solid State Science Gatan Inc. CHOU, T. & LIBERA, M. (2003). Mean free paths for inelastic electron scattering in silicon and poly(styrene) nanospheres. Ultramicroscopy 94(1), 31–35.
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EGERTON, R. (1979). K-shell ionization cross-sections for use in microanalysis. Ultramicroscopy 4(2), 169–179. EGERTON, R.F. (1996). Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd ed. New York: Plenum Press. EGERTON, R.F. & WHELAN, M.J. (1974). The electron energy loss spectrum and band structure of diamond. Philos Mag 30(4), 739–749. KOHL, H. (1985). A simple procedure for evaluating effective scattering cross-sections in STEM. Ultramicroscopy 16, 265–268. LAWTON, W.H. & SYLVESTRE, E.A. (1971). Elimination of linear parameters in nonlinear regression. Technometrics 13(3), 461–467. LEAPMAN, R. & SWYT, C. (1988). Separation of overlapping core edges in electron energy loss spectra by multiple-least-squares fitting. Ultramicroscopy 26(4), 393–403. LEAPMAN, R.D., REZ, P. & MAYERS, D.F. (1980). K, L, and M shell generalized oscillator strengths and ionization cross sections for fast electron collisions. J Chem Phys 72(2), 1232–1243.
LYONS, L. (1991). A Practical Guide to Data Analysis for Physical Science Students. Cambridge: Cambridge University Press. MANOUBI, T., TENCE, M., WALLS, M.G. & COLLIEX, C. (1990). Curve fitting methods for quantitative-analysis in electron-energy loss spectroscopy. Microsc Microanal Microstruct 1(1), 23–39. PRESS, W.H., TEUKOLSKY, S.A., VETTERLING, W.T. & FLANNERY, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, 3rd ed. Cambridge: Cambridge University Press. RAETHER, H. (1980). Excitation of Plasmons and Interband Transitions by Electrons vol. 88, Springer Tracts in Modern Physics. Berlin, Heidelberg, and New York: Springer-Verlag. RITCHIE, R.H. & HOWIE, A. (1977). Electron excitation and the optical potential in electron microscopy. Philos Mag 36, 463–481. The Hyperspy development team. (2013). Hyperspy: hyperspectral data analysis. www.hyperspy.org. VERBEECK, J. & VAN AERT, S. (2004). Model based quantification of EELS spectra. Ultramicroscopy 101(2–4), 207–224. VERBEECK, J., VAN AERT, S. & BERTONI, G. (2006). Model-based quantification of EELS spectra: Including the fine structure. Ultramicroscopy 106(11–12), 976–980.
© MICROSCOPY SOCIETY OF AMERICA 2014
Highly Automated Electron Energy-Loss Spectroscopy Elemental Quantification Raman D. Narayan,1,* J. K. Weiss,1 and Peter Rez2 1 2
AppFive LLC, 1095 West Rio Salado Parkway, Suite 110, Tempe, AZ 85281, USA Department of Physics, Arizona State University, PO Box 871504, Tempe, AZ 85287, USA
Abstract: A model-based fitting algorithm for electron energy-loss spectroscopy spectra is introduced, along with an intuitive user-interface. As with Verbeeck & Van Aert, the measured spectrum, rather than the single scattering distribution, is fit over a wide range. An approximation is developed that allows for accurate modeling while maintaining linearity in the parameters that represent elemental composition. Also, a method is given for generating a model for the low-loss background that incorporates plural scattering. Operation of the userinterface is described to demonstrate the ease of use that allows even nonexpert users to quickly obtain elemental analysis results. Key words: EELS, quantification, model, fitting, plural scattering, multiple scattering
I NTRODUCTION Electron energy-loss spectroscopy (EELS) is a very powerful tool for elemental analysis in materials and biological sciences, but the complexity of its application can be an obstacle to its widespread use. We present a quantitative procedure for EELS elemental analysis that requires minimal user input. In energy-loss spectra the inner shell edges that are used for elemental quantification are superimposed on a slowly varying background. If energy-loss spectroscopy is to be successfully used to accurately determine chemical composition, accurate and reliable background models are essential. Traditionally, a background of the form AE–r is fitted in the region before each core-loss edge and extrapolated over the energy range needed for integration over the core-loss signal (Egerton, 1996). Although the asymptotic form of cross-section models gives some justification for this procedure, the effects of any systematic error are greatly magnified by the extrapolation process. This is especially true for thicker specimens, where the effect of multiple scattering presents a serious problem in any attempt to model the background. In addition to the failure of the commonly used powerlaw background model over wide energy ranges, there are other drawbacks to traditional EELS quantification. If the core-loss spectrum is deconvolved, its interpretation may be complicated by artifacts. Results are also dependent on the choice of fitting or integration windows. Since windows for background determination and edge integration are needed for each core-loss edge, EELS quantification can be very difficult when the onsets of multiple edges occur close together in the spectrum (Verbeeck & Van Aert, 2004). revised September 13, 2013; accepted February 23, 2014 *Corresponding author. raman@appfive.com
Another approach has been to model the entire coreloss region, including the background simultaneously, as in Manoubi et al. (1990) and in the EELSModel (Verbeeck & Van Aert, 2004) and Hyperspy (The Hyperspy Developement Team, 2013) programs. Here a linear combination of the background and core-loss edges is fitted to the spectrum. Verbeeck & Van Aert (2004) showed that the accuracy of their quantification procedure approaches theoretical limits and successfully modeled spectra where edges are close together. Theoretical core-loss edges are convolved with a measured low-loss spectrum to account for plural scattering, following the Fourier–ratio method. In model-based EELS quantification, an accurate background model is even more important than in the conventional approach, since it must be applied across the entire core-loss region simultaneously. This is especially true when one must fit two edges that are far apart in the spectrum. Manoubi et al. (1990) convolved a power-law background with the low-loss, but noted complications owing to artifacts from the artificial onset of the background model, particularly for core-loss edges at lower energy-loss. Verbeeck & Van Aert (2004) mentioned this issue with Manoubi’s method, and respond by not convolving the background model, but instead added parameters so the power-law can vary with energy-loss. We propose the following enhancements to model-based EELS quantification. First, a more accurate calculation of multiple scattering in the elemental edges is provided (see Modeling the Core-Loss Edges section), an improvement to the Fourier–ratio method. Second, given that the modeling of the background is critical to accurate quantification and multiple scattering makes a significant contribution to the pre-edge background, an alternate procedure is given for obtaining a background model that takes into account multiple scattering (see Modeling the Background section). Lastly, an intuitive program for EELS quantification is introduced
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(see Workflow for Elemental Analysis section), where we aim to optimize this approach such that a minimal amount of input is required from the user. As with Verbeeck & Van Aert (2004), we fit the observed spectrum rather than the calculated single scattering distribution (SSD). The low-loss region is extrapolated using either a power-law or one of several more detailed functions. For each element to be quantified, a model EELS profile is constructed that includes all edges that will be present in the core-loss region. This allows the user to proceed without having to identify all the individual core-loss edges to fit. The approach can be helpful for more challenging spectra where edges overlap, or where there might be some ambiguity as to which elements are present. Core-loss edges are calculated from the Hartree–Slater model (Leapman et al., 1980) that can be used over a wider range of elements and edges than hydrogenic cross-sections (Egerton, 1979). In its simplest implementation, the procedure takes as input (in addition to instrumental parameters) a complete EELS spectrum that includes the low-loss region and zeroloss peak, and a list of elements to model. A routine for splicing parts of the spectrum together is also included. We hope that this work will make EELS quantification more practical for a broad community of potential users and will improve the consistency of their results. In the following sections, the essential and optional inputs and some of the theory for the fitting procedure are discussed. In particular, an approximation is given that allows one to accurately model plural scattering in the elemental edges (see Modeling the Core-Loss Edges section), while using a single linear parameter for the abundance of each element, and a model for the low-loss background is introduced that incorporates plural scattering (see Modeling the Background section). In the Results section the benefit gained from the background model from the Modeling the Background section is demonstrated and the user-interface is described. Future work is discussed in the Discussion section.
M ATERIALS AND M ETHODS Required Input The following are required from the user:
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3. treatment of near-edge structure (see Workflow for Elemental Analysis section); 4. whether to fit an additive constant.
Fitting Region Beginning with the full spectrum, the first decision to be made is where in the spectrum to start fitting. Ideally, one might like to model the entire spectrum, but constructing an accurate model for the low-loss region is not practical since it would involve a detailed application of solid state theory. If not specified by the user, a cutoff energy-loss is chosen such that at least one edge for each element of interest occurs at higher energy-loss (>90 eV). It is also necessary that fitting begin in a smooth region of the spectrum—otherwise it will be difficult to apply a background model. The program also avoids, when possible, a situation where there are large energy differences between edges since it is more difficult for a background model to be applied successfully over a wide region of energy-loss.
Modeling the Core-Loss Edges In the limit of a very thin specimen, the EELS spectrum is modeled as j
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where s(ν), l(ν), and k(ν) are the Fourier transforms of the quantities represented by the same letters in equation (1). In Fourier space, the plural-scattered, measured spectrum, j(ν) is related to the single scattering spectrum s(ν) as
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(Egerton, 1996: 247) where I0 is the integrated intensity of the zero-loss peak or instrumental response function z(ν). Substituting into equation (3) for plural scattering we get
There are a number of optional inputs that change aspects of the fitting procedure: 1. energy-loss loss at which to start fitting (see Fitting Region section); 2. model for pre-edge background (see Modeling the Background section);
jðνÞ ¼ zðνÞ exp½sðνÞ=I0 ;
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Separating the two terms in the exponent in equation (4), we have jðνÞ ¼ jL ðνÞ exp½ai ki ðνÞ=I0 ;
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where the plural-scattered, extrapolated low-loss spectrum jL(ν) is defined as jL ðνÞ zðνÞ exp½lðνÞ=I0 :
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Equation (5) provides a model that can be applied to an EELS spectrum if both the low-loss and core-loss regions are acquired. However, since there is much less intensity in the core-loss edges than in the low-loss it is possible to expand the exponential in equation (5) as a power series: " # ai ki ðνÞ 1 ai ki ðνÞ 2 1 ai ki ðνÞ 3 jðνÞ ¼ jL ðνÞ 1+ + + ¼ : I0 2! I0 3! I0 (7) Keeping the first two terms in the expansion, the parameters ai become linear parameters which is beneficial since linear fitting is faster and less error prone (Lawton & Sylvestre, 1971): jðνÞ jL ðνÞ+ai jL ðνÞki ðνÞ=I0 :
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Equation (8) corresponds to the familiar Fourier–ratio method (Egerton & Whelan, 1974). The inverse Fourier transform of equation (8) provides an approximate model that is linear in parameters ai that multiply the convolved core-loss edges. It is possible to maintain the quadratic term in equation (7) without sacrificing linearity in parameters ai. We introduce the following approximation, which makes use of the Fourier transform of the measured spectrum jexper(ν): jðνÞ jL ðνÞ + ai ki ðνÞ
jL ðνÞ + jexper ðνÞ : 2I0
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The core-loss edges are essentially being convolved with the average of the extrapolated low-loss and measured spectrum. Substituting equation (7) for jexper(ν) in equation (9), we get ai ki ðνÞ jðνÞ jL ðνÞ + 2I0 " ( )# ai ki ðνÞ 1 ai ki ðνÞ 2 jL ðνÞ + jL ðνÞ 1 + + + ¼ ; ð10Þ I0 2! I0 "
# ai ki ðνÞ 1 ai ki ðνÞ 2 1 ai ki ðνÞ 3 ¼ jL ðνÞ 1 + + + + ¼ : I0 2 I0 4 I0 (11) The difference between our estimate given by eqaution (9) and the spectrum including all orders of multiple scattering of the core-loss edges in equation (5) is quantified by comparing the estimated expansion [equation (11)] with the
correct expansion in equation (7). This deviation occurs at the cubic term, and is ΔjðνÞ 1 ai ki ðνÞ 3 : (12) jL ðνÞ 12 I0 The quality of this approximation can be evaluated for a given specimen thickness t by replacing in equation (12) the fitted edges aiki(ν) with their integrated intensity IK, and replacing j(ν) and jL(ν) with their integrated intensities I and IL. IK represents the intensity of the core-loss edges over the entire core-loss region, rather than over a small energy-loss window, since we are fitting the whole core-loss spectrum. Equation (12) becomes ΔI 1 IK 3 : (13) IL 12 I0 From Poisson statistics: IK 1 - exp½ - t=λK ; ¼ exp½ - t=λtotal I0
(14)
where λK and λtotal are the mean free paths of the core-loss edges and the total inelastic mean free path, respectively. Since t=λK 1 for many transmission elecron microscopic specimens, exp[ − t/λK] can be approximated using a power series, giving IK =I0 t=λK exp½t=λtotal : (15) Considering, for example, the total Si L-edge at an accelerating voltage of 200 keV, assuming all electrons are collected, and t = λtotal, equation (14) gives IK =I0 0:21 using a total inelastic mean free path from Chou & Libera (2003) and total cross-sections from the Hartree–Slater model (Leapman et al., 1980). Applying this value to equation (13) we get ΔI=IL 8 ´ 10 - 4 . For comparison, the integrated first- and second-order terms in equation (7) have values of 0.21 and 0.022 if ai ki ðνÞ ! IK . The approximation given by equation (9) captures the second term in equation (7) whereas the Fourier–ratio method [equation (8)] does not. This secondorder term corresponds to electrons that undergo two core-loss events, therefore equation (9) will yield a greater increase in accuracy when there are core-loss edges with greater cross-sections, such as Si-L23. It is then more important when fitting at lower energy-losses, since edges at lower binding energies generally have greater cross-sections. The inverse Fourier transform of equation (9) JL ðEÞ + Jexper ðEÞ JðEÞ JL ðEÞ + ai Ki ðEÞ ´ ; (16) 2I0 is the model that will be fitted to the measured spectrum. This is linear in parameters ai, though there are nonlinear parameters implicitly contained in the extrapolated low-loss JL(E). The nonlinear parameters in JL(E) are fitted using a Levenberg–Marquardt algorithm (Press et al., 2007), and ai are included using linear fitting.
Simulated Spectra The effectiveness of our procedure for including multiple scattering in the model for the low-loss background,
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described below in the Modeling the Background section, is evaluated using simulated spectra. Simulated boron nitride (BN) spectra are generated using a Monte Carlo procedure where each electron is tracked individually. Each of 108 incident electrons begins with an energy of 200 keV, and its energy is slightly altered according to a normal distribution to simulate the spread in energies that would be seen in the zero-loss peak. The number of inelastic collisions the electron will suffer is drawn randomly from a Poisson distribution, and this depends on the specimen thickness relative to the inelastic mean free path. The simulated low-loss spectrum is based on the plasmon-pole approximation (Ritchie & Howie, 1977; Raether, 1980), and the Boron and Nitrogen K-edges are the same Hartree–Slater differential crosssections used for fitting (Leapman et al., 1980). The simulated spectra do not include all the features seen in real EELS spectra, such as those caused by band structure, but this is not important for the present purpose. The effects of multiple scattering are readily apparent in the BN spectrum shown in Figure 1, where a specimen thickness of two inelastic mean free paths is simulated. The plasmon peak at 25 eV repeats at 50 and 75 eV for two and three plasmon excitations. Multiple scattering also has the effect of softening the onset of the B-K and N-K core-loss edge, which would have sawtooth-like profiles for a thin specimen.
Modeling the Background Modeling the low-loss, or pre-edge, background is arguably one of the most difficult aspects of model-based and conventional EELS elemental analysis. It is essential for the background to be fitted accurately, since the calculated composition is proportional to the intensity of the core edges, and this intensity is only known when the background intensity is accounted for. It should be noted that in model-based quantification, the models for the outer core-loss edges constitute part of the background for edges with higher binding energies. For example, in a specimen containing boron, carbon, and
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nitrogen, the boron and carbon K-edges contribute to the background for the nitrogen K-edge. This is not the case in the conventional approach, where a background model is fitted independently for each core-loss edge in an energy window preceding the onset of each edge. This can lead to a quite nonphysical model where there is no correlation between the independent background models. To model a measured spectrum, as opposed to its SSD, one might extrapolate the measured low-loss spectrum using an analytic model function such as a power-law (Egerton, 1996), which is known to approximately match the energy-loss dependence in the tail of ionization differential cross-sections. This can be unrealistic for thicker specimens, as both experimental observations and computational modeling show that multiple scattering of the low-loss excitations leads to significant changes in the background, in particular a deviation from the well-known inverse power-law. We propose treating plural scattering in the background model by extrapolating the SSD rather than the raw spectrum, and then reversing the deconvolution used to obtain the SSD. The rationale for this approach is that an analytic background model can be more successfully applied to the SSD, where there is ideally no plural scattering, than to the raw spectrum. As with Manoubi et al. (1990) and Verbeeck & Van Aert (2004), the parameters for the background model are fitted to the entire core-loss spectrum simultaneously with those for the elemental composition [ai in equation (16)]. Before introducing the procedure, it is helpful to recall that the plural-scattered, measured spectrum j(ν) is given by equation (3) in Fourier space. An approximate model for plural scattering is calculated by the following steps: 1. First, the SSD is obtained using Fourier–log deconvolution. The SSD is reconvolved with the zero-loss peak to suppress noise, as described by Egerton (1996: 248). In terms of Fourier transforms of the zero-loss peak z(ν) and measured spectrum j(ν), the smoothed SSD, s0 ðνÞ, is s0 ðνÞ ¼ zðνÞ ln½jðνÞ=zðνÞ:
(17)
2. Starting at the energy-loss where one chooses to begin fitting, the smoothed SSD is extrapolated using a powerlaw or more sophisticated model. The Fourier transform of the extrapolated SSD will be represented by l0 ðνÞ, since it represents the low-loss spectrum. l0 ðνÞ is our model for a single-scattered low-loss spectrum, or alternatively, an SSD without core-loss edges. 3. Reversing the deconvolution, an approximation for the Fourier transform of the extrapolated, plural-scattered low-loss spectrum j0L ðνÞ is obtained by replacing s(ν) in equation (3) with the extrapolated SSD l0 ðνÞ, giving j0L ðνÞ ¼ I0 exp½l0 ðνÞ=I0 :
(18)
This constitutes an approximation since the noisereducing function z(ν) is applied to the SSD [equation (17)] before reversing the deconvolution. In Figure 2 the relevance of the above procedure is illustrated, by comparing the above background model to
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Figure 2. Plural scattering background model versus an unconvolved power-law. The top panel shows a simulated boron nitride spectrum (see Simulated Spectra section) and the same spectrum with core-loss edges excluded (actual low-loss). The accelerating voltage is 200 keV, collection half angle 200 mrad, and thickness two inelastic mean free paths. The plural and raw background curves are the background models optimized along with core-loss edges to fit the spectrum. “Plural background” is due to the procedure descibed in Modeling the Background section, convolving a power-law, and “raw background” results from simply applying a power-law to the entire core-loss region. Residuals for both background models with respect to the actual low-loss spectrum are shown in the lower panel.
that obtained by simply applying a power-law to the entire core-loss region. Here the above procedure involves applying a power-law to the SSD rather than the measured spectrum. Both background models are fitted, along with core-loss edges, to a simulated BN spectrum (see Simulated Spectra section) shown in the top panel, which has a thickness of two inelastic mean free paths. Also shown in the top panel of Figure 2 is the same simulated spectrum excluding core edges (actual low-loss), which includes electrons that have undergone multiple low-loss events. This is obtained by excluding simulated electrons that suffer a core-loss event. For a measured spectrum, this low-loss component would not be available after the onset of the B-K-edge at 190 eV. The bottom panel shows the difference from the spectrum excluding core edges for our background model (plural background) and the unconvolved power-law (raw background). The success of our procedure is demonstrated by the fact that our background model tracks the actual low-loss much more closely.
Figure 3. Simulated boron nitride spectra for thickness of 1.55 inelastic mean free paths and collection half angles 200 (top) and 5 mrad (bottom). Fitted curves are shown for the case where the background model incorporates plural scattering.
Although the SSD is used to obtain the model for the low-loss background, the theoretical cross-sections are still being fitted to the measured spectrum rather than its SSD.
RESULTS Background Model In this section the fitting procedure described above is applied to simulated EELS spectra, and the benefit of the method from the Modeling the Background section for considering plural scattering in the background model is demonstrated. Spectra are simulated (see Simulated Spectra section) for values of specimen thickness up to three mean free paths, and for collection half angles of 200 and 5 mrad. A simulated spectrum is shown for both collection angles in Figure 3, along with the fitted model. The fitted boron to nitrogen (B/N hereafter) ratios are plotted in Figure 4. Also plotted are B/N ratios obtained by extrapolating the measured low-loss with a power-law (see Fig. 2) rather than including multiple scattering as described in the Modeling the Background section. In both cases, the power-law is fitted over the entire core-loss region. The plotted data points represent the average B/N ratio for ten simulated spectra corresponding to the same thickness. Error bars are equal to the standard deviation in the mean B/N ratio at each thickness: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u X σx 1 σ x ¼ pffiffiffiffi ¼ t (19) ðxi - xÞ2 ; NðN - 1Þ i¼1 N (Lyons, 1991) where x is the B/N ratio and N = 10.
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iii. Do nothing: include the channels containing near-edge structure as with the rest of the fitted core-loss region. This most simplistic option can introduce significant bias, since the Hartree–Slater cross-sections do not include near-edge structure (Manoubi et al., 1990).
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For both collection angles, the B/N ratio varies less with thickness when plural scattering is incorporated in the background model using the SSD. This is evidence of the importance of using a background model that accounts for the effects of plural scattering.
Workflow for Elemental Analysis In this section a sample workflow is given to illustrate the ease of use of this EELS quantification procedure. 1. After importing a CsI spectrum (Ahn et al., 1983) from a two-column text file, one arrives at the “setup” screen, shown in Figure 5. 2. Set the acceleration voltage and collection half angle (a and b from Fig. 5). 3. Select elements cesium and iodine from the periodic table (Fig. 5, c). Interactive edge onset markers appear (l). These can be adjusted to change the binding energies for the fitted theoretical edges. 4. (Optional) Choose how to treat the near-edge structure. There are three choices: i. Apply a nonparametric fit to the near-edge structure, using the measured spectrum to approximate plural scattering. This is similar to the treatment by Verbeeck et al. (2006). ii. Masking: exclude the channels in the near-edge region from the fit.
If (i) or (ii) is chosen, the range of channels to treat as nearedge structure is specified in controls e and f in Figure 5. 5. (Optional) Choose a power-law (default) or more detailed model for the low-loss background. 6. (Optional) Choose whether or not to incorporate plural scattering in the background model as described in the Modeling the Background section. 7. (Optional) Choose whether to include a uniform background intensity as a free parameter in the fitted model. By default, this decision is made automatically based on the negative energy-loss channels. 8. (Optional) The lowest energy-loss for modeling the spectrum is chosen automatically by default (j in Fig. 5), or the user can manually select the first channel (k). 9. Clicking on the quantify button (m in Fig. 5) runs the fitting algorithm, and displays the fitted model and elemental composition results as shown in Figure 6. Clicking on “Setup” (n) returns one to the setup screen (Fig. 5) where all of the above settings can be modified.
DISCUSSION The procedure for model-based EELS elemental analysis described here is meant to be appropriate for both nonexpert and advanced users. The utility of the method described in the Modeling the Background section is demonstrated in the Results section. Regarding the user-interface described in the Workflow for Elemental Analysis section, it is important to note that one can obtain elemental analysis results by supplying only the measured spectrum, accelerating voltage, collection angle, and elements to be fitted. In the remainder of this section, future improvements are discussed. These fall in two categories—adding new functionality to more accurately model difficult spectra, and further automating the user-interface. In the first category, the most obvious addition will be to support the use of core-loss edges measured from standards in place of the theoretical Hartree–Stater differential crosssections (Leapman et al., 1980). This will allow the fitting algorithm to more accurately handle spectra with overlapping edges due to different elements, when neither element has another (nonoverlapping) edge that can be modeled (Leapman & Swyt, 1988). An example of such a spectrum is shown in Figure 7. When the edge onsets are too close together, and one chooses to either mask or use nonparametric fitting for the near-edge region, the total model can become degenerate for concentrations of the two elements, since it is not clear without fitting the edge onsets how much of the intensity is due to either element. A standard specimen with band structure similar to that seen in the spectrum could allow
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Figure 5. Electron energy-loss spectroscopy (EELS) quantification “setup” screen: (a) accelerating voltage, (b) collection half angle, (c) element selection periodic table, (d) near-edge modeling option, (e, f) energy range over which to apply the near-edge model, (g) background model, (h) whether or not to use single scattering distribution to incorporate plural scattering in the background model (see Modeling the Background section), (i) whether to apply an additive constant level to the spectrum fit, (j) whether to automatically determine energy-loss channel at which to start fitting, (k) slider for manually choosing first channel to fit, (l) atomic edge onset interactive markers, (m) setup button, (n) quantify button.
one to adequately model the near-edge structure without adding extra parameters. To effectively use edges measured from standards, one would have to correct for differences in collection angle and specimen thickness between the standard spectrum and that which is being modeled. The fitting procedure might also be expanded to support quantitative elemental mapping and tomography, which might require further optimization for speed. Another important addition will be to include the beam convergence angle in the calculation of core-loss edges, as in Kohl (1985). With regard to automation, the general mindset behind the procedure described in the Workflow for Elemental Analysis section is to remove, or at least make optional, unnecessary manual settings that affect the outcome of elemental analysis. In this spirit, an algorithm could be used to automatically identify which elements are present in the spectrum. One might step through the spectrum, looking for increases in intensity that satisfy some calculated threshold. It could also
be helpful to automatically search for and apply saved standard edges. As mentioned in the Workflow for Elemental Analysis section, edge onset energies have been left to the user to change manually, and these could be added as free parameters, which might be particularly helpful if standard edges are used.
SUMMARY This paper adds to prior work on model-based EELS analysis an approximation to allow for more accurate modeling of plural scattering in the core-loss edges while maintaining linearity in the elemental composition parameters (see Modeling the Core-Loss Edges section), a method for producing a low-loss background model that incorporates the effects of plural scattering (see Modeling the Background section), and an intuitive user-interface that allows one to arrive at elemental analysis results with very little effort (see Workflow for
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Figure 6. Electron energy-loss spectroscopy (EELS) quantification “quantify” screen. The controls on the left side toggle on and of the measured spectrum and curves corresponding to final and initial models, the single scattering distribution, residuals (data minus fitted), and the background model. Fractional composition results are shown as a pie chart and in a table at the bottom of the screen.
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Elemental Analysis section). It is our hope that this work makes elemental analysis with EELS accessible to a broader community of scientists and technicians.
Ti−L2,3
Counts
6
ACKNOWLEDGMENTS We gratefully thank Professor Jo Verbeeck for taking his time to have a number of interesting and productive discussions with us on model-based EELS analysis.
4 O−K
Ti−L1
2
REFERENCES 0
450
500
550 eV
600
650
Figure 7. Edge overlap is illustrated in this TiO2 spectrum from the EELS Atlas (Ahn et al., 1983). Owing to near-edge structure, there is no smooth region between either the Ti-L23 and O-K, or the O-K and Ti-L1 edges where an atomic model for the differential cross-sections could be applied.
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