Once you acquire a spectrum or a collection of spectra, you need to extract information from that data. Here we will discuss how to identify which elements are present, as well as best practices to extract and quantify elemental specific information to determine sample composition.
A typical analysis follows these steps:
Remove plural scattering (if necessary)
Proceed to the next page for step-by-step instructions for using Gatan Microscopy Suite^{®} (GMS) 3 software to quantify your data.
Once you select the element of interest, it is important to choose a suitable edge to analyze. Key variables to consider:
Is the edge energy suitable?
Low energies (<150 eV) – It may be difficult to extract a signal because of other low-loss features (e.g., plasmons) overlapping with the edge
High energies (≥2000 eV) – Edges tend to be noisier, but may be easier to remove background from
What is the accuracy of the edge?
K-, L-edges
Small cross-sections (→low signal-to-noise ratio)
Not suitable for high Z elements (ionization energy too high)
However, software can compute cross-sections relatively accurately
M-, N-, O-edges
Larger cross-sections
Higher Z elements only
Less reliable computed cross-sections
Presence of other edges in spectrum (e.g., overlapping edges) that may make it difficult to accurately analyze your edge
In general, using the highest energy edge that still give sufficient signal is recommended.
Leapman, R. D.; Rez, P.; Mayers, D. F. K, L and M shell generalized oscillator strengths and ionization cross-sections for fast electron collisions. J. Chem. Phys. 72:1232 – 1243; 1980.
Once you acquire the spectra, it is time to identify your edge of interest.
Common indicators you may use to identify your edge include:
Edge threshold energy (e.g., point of steepest rise)
Edge shape (e.g., hydrogenic, delayed, white lines)
Accompanying edges (e.g., Si L-edge at 99 eV, Si K-edge at 1839 eV)
Edge identification can be done three different ways in Gatan Microscopy Suite^{®} (GMS) 3 software.
When you select the AutoID button it will show candidate edges on the spectrum. You will need to verify these edges using the common identifiers defined above. To aid in this task the Mark Edges button can be used to show edge families. Once verified, you can then add them to the quantification list by pressing the → button to the left of the list.
By right clicking on the spectrum itself, you will see the Edge ID menu where candidate edges are shown. The most probable edge will be in bold. You can add that edge to the quantification by using Add to Quant menu choice.
To reach this option, click the Open Periodic Table icon on the Elemental Analysis window. This will in turn open a periodic table where you can directly select known elements to automatically add them to the quantification.
Delayed edge features, plural scattering and overlapped edges can sometimes make edge identification difficult. Compare the acquired spectra to reference data (e.g., from the EELS Atlas) if you suspect overlapped edges or see unusual edge shapes. If the element has multiple edges available, confirm the existence of these features in you experimental data for unambiguous identification.
Ahn, C. C.; Krivanek, O. L.; Disko. M. M. EELS atlas: a reference collection of electron energy loss spectra covering all stable elements. HREM Facility, Center for Solid State Science, Arizona State University; 1983.
There are a variety of algorithms you can use to determine relative sample thickness. However, the log-ratio (relative) method is most commonly used.
Following Poisson statistics, the ratio of zero-loss electrons to the total transmitted intensity gives a relative measure of the specimen thickness in units of the local inelastic mean free path λ.
\(t/\lambda = -ln(I_{o}/I_{t})\)
\(t/\lambda\) is the mean number of scattering events per incident electron. You can compute this easily from low-loss spectrum via the Thickness button located in the EELS Processing palette in the Techniques panel.
When you perform this analysis, you can determine if the region is thin enough for EELS and if plural scattering has significant impact.
Malis, T.; Cheng, S. C.; Egerton, R. F. EELS log ratio technique for specimen-thickness measurement in the TEM. J. Electron Microscope Technique. 8:193; 1988.
Plural scattering occurs when a significant fraction of incident electrons that pass through a sample are scattered inelastically more than once.
The inelastic mean free path represents the mean distance between inelastic scattering events for these electrons. When you regard inelastic scattering as a random event, the probability of n-fold inelastic scattering follows a Poisson distribution.
\(P_{n}=\frac{I_{n}}{I_{t}}=\frac{\left ( \frac{t}{\lambda } \right )^{n}}{n!}exp\left ( \frac{-t}{\lambda } \right )\)
where
\(I_{n}\) = intensity of n-fold scattering
\(I_{t}\) = total intensity
For \(n=0\), we get the simple relationship
\(t/\lambda = -ln( I_{o}/I_{t} )\)
where
This method is known as the log-ratio method. From a spectrum, the integral of the ZLP will give \(I_{o}\), while the integral of the entire spectrum will give \(I_{t}\).
You can compute this easily from low-loss spectrum via the Thickness button located in the EELS processing palette in the Techniques panel. GMS will fit the ZLP to extract the \(I_{o}\) contribution. While the total spectrum is never truly measured, \(I_{t}\) can be approximated from a measured energy range since a majority of the signal is contained in the first 50 – 100 eV. GMS uses a power law tail beyond the measured spectrum to estimate its contribution to \(I_{t}\).
When you perform this analysis, you can determine if the region is thin enough for EELS and if plural scattering has significant impact.
Typically, just the ratio of \(t/\lambda\) is needed, but methods are available to estimate the value of the mfp for your material. Another method to determine the absolute thickness of the sample is the Kramers-Kronig Sum Rule.
Malis, T.; Cheng, S. C.; Egerton, R. F. EELS log ratio technique for specimen-thickness measurement in the TEM. J. Electron Microscope Technique. 8:193; 1988.
Iakoubovskii, K.; Mitsuishi, K.; Nakayama, Y.; Furuya, K. Thickness measurements with electron energy loss spectroscopy. Microscopy Research and Technique. 71(8):626 – 31; 2008.
Iakoubovskii, K.; Mitsuishi, K.; Nakayama, Y.; Furuya, K. Mean free path of inelastic electron scattering in elemental solids and oxides using transmission electron microscopy: Atomic number dependent oscillatory behavior. Physical Review B. 77(10):104102; 2008.
Egerton, R. F., Cheng, S. C. Measurement of local thickness by electron energy-loss spectroscopy. Ultramicroscopy. 21(3):231 – 244; 1987.
The next step in quantification requires you to extract edge intensities from spectrum, while you disregard the underlying background intensity.
To separate the edge intensity, you will need to fit, extrapolate, and then subtract a background model. It is important to consider the following items when you perform this critical extraction step:
Which background model should I use?
Where should I fit the background model to the data?
What are the optimal width and position for the signal integration window?
To extract the edge intensity, you must determine a model for the background of your spectrum. First identify a pre-edge fitting region that allows you to determine parameters of the fit. Then extrapolate this fit to estimate the background intensity below the edge signal. However, an accurate background subtraction may become difficult below 100 eV due to the large number of scattering processes in this region (e.g., plasmons tails, plural scattering).
Typically, the model is determined using linear least-squares methods using a single pre-edge region \(\Gamma\).
where
\(\Gamma\) = background fit window
\(\Delta\) = signal integration window
\(I_{b}\) = background intensity
\(I_{k}\) = signal intensity
A power law is the most common background model.
\(J\left ( E \right )=AE^{-r}\)
where
\(A\) = scaling constant
\(r\) = slope exponent (usually 2 – 6)
This model has the physical basis when interpreted as the long energy tails of the preceding energy loss events.
When you choose the optimal background placement, it is important to consider these parameters:
The high energy side \(E_{be}\) should be as close to but still preceding the edge (e.g., 5 eV) to avoid chemical shifts and broadening detector tails
To limit statistical error, fit region \(\Gamma\) should be as wide as possible
To limit systematic error, you need to limit the fit region to 10 – 30% \(E_{k}\)
Background window end should be 5 eV from edge onset
Background window width should be at most 30% edge energy
May need to limit window size to avoid preceding edges where necessary
Once background placement is made, it is important to review common errors.
Unphysical
Symptom – Obvious error where the background model crosses the spectrum and may cause the signal to become negative
Systematic errors
Symptom – Small changes in the background window width or position have large effects on the background model
Overlapping edges
Symptom – Background extrapolation is ineffective for instances where the pre-edge region is obscured by the preceding edge
Solution – Reduce the window size or placement as well as limit the signal window size and offset; a multiple linear least-squares (MLLS) fitting or model based approach may be necessary
When you choose the optimal signal integration window placement, it is important to consider:
Statistical error – The region \(\Delta\) should be as wide as possible and start at the steepest intensity increase of the spectrum
Hydrogenic edges (e.g., K-, some L-edges) – Place window at threshold
Delayed edges (e.g., L-, M-, N-, O-edges) – Offset by a few tens of eV
White lines – Best to avoid inclusion for quantitative evaluation as their intensity can vary with chemical state, and are not well modeled in cross-section calculations
With Gatan Microscopy Suite^{®} (GMS) 3 software, the process of signal extraction is highly automated. However, the guidelines and concepts above still must be considered. GMS 3 quantification utilizes a model based approach where the spectral background and the edge intensity are treated as single model. If there are overlapping edges present, they are also added to the model to allow separation of the overlap. Follow the below steps for EELS signal extraction in GMS 3 software.
Joy, D.C.; Maher, D. M. The quantization of electron energy-loss spectra. J. Microsc. 124:37 – 48; 1981.
Egerton, R. F. A revised expression for signal/noise ratio in EELS. Ultramicroscopy. 9:387 – 390; 1982.
Leapman, R. D.; Swyt, C. R. Separation of overlapping core edges in electron energy loss spectra by multiple-least-squares fitting. Ultramicroscopy. 26:393 – 404; 1988.
Kothleitner, G.; Hofer F. Optimisation of the signal to noise ratio in EFTEM elemental maps with regard to different ionisation edge types. Micron. 29349 – 357; 1998.
Verbeeck, J., Van Aert, S. Model based quantification of EELS spectra. Ultramicroscopy. 101(2 – 4):207 – 24; 2004.
Riegler, K.; Kothleitner, G. EELS detection limits revisited: Ruby – a case study. Ultramicroscopy. 110(8); 2010.
Thomas, P.; Twesten, R. A Simple, Model Based Approach for Robust Quantification of EELS Spectra and Spectrum-Images. Microscopy and Microanalysis. 18(S2):968 – 969; 2012.
Jump to step-by-step instructions in Gatan Microscopy Suite 3
The probability \(P_{k}\) that a given incident beam electron will end up as a count in core-edge \(k\) of a particular element is directly proportional to the projected (areal) density, \(N\), of atoms of that element.
\(P{_{k}}=N\sigma _{k}\)
The constant of proportionality, \(\sigma _{k}\), is a factor that depends on the experimental conditions and on the limits over which you integrate the edge counts. It reflects the intrinsic scattering strength of shell \(k\) of the element.
\(P_{k}=\frac{I_{k}}{I_{t}}\)
The probability \(P_{k}\) can also be measured from the energy loss spectrum of interest by taking the extracted edge count integral \(I_{k}\) and dividing by the total number of counts in the spectrum.
Hence you can equate the two yielding the areal density of atoms present in terms of measurable quantities:
\(N = I_{k}/\sigma _{k}I_{t}\)
While quantification is simpler without plural scattering, it may occur and it is important to know how plural scattering can impact your quantification. Specifically, it can:
Redistribute the core-loss intensity – Due to a convolution of the edge shape and low-loss spectrum
Alter the edge shape and intensity
Invalidates the computed cross-section
Reduces the signal-to-background ratio (SBR) when it shifts the edge intensity to higher losses and increases the preceding background
In general, higher Z and thicker samples are more affected by plural scattering
It is useful to know the impact that plural scattering may have on your quantification. Samples have a finite thickness and you need to decide if the local thickness is low enough to assume plural scattering is negligible (e.g., \(t/\lambda =0.3\), plural scattering contribution is approximately 15%).
While most quantification equations assume no plural scattering, you can remove it using deconvolution (Fourier-based methods) or incorporate it into the quantification. Here are guidelines for the types of quantification routines you can use based on the level of plural scattering present in your sample.
Measurement type |
\(t/\lambda\) |
---|---|
Linear least squares (LLS) fitting (<1500 eV) |
0.0 – 1.0 |
LLS fitting (>1500 eV) |
0.0 – 2.0 |
LLS with deconvolution |
0.3 – 2.0 |
Thickness measurements |
0.15 – 6.0 |
When you safely ignore plural scattering, the areal density of atoms (atoms/nm^{2}) of type, \(k\), can be approximated by:
\(N = \frac{I_{k}^{1} (\beta , \Delta )}{I_{o}(\beta ) \sigma _{k} (\beta , \Delta )}\)
where
\(I_{k}^{1} (\beta , \Delta )\) = integral edge \(k\) with no plural scattering integrated over a window \(\Delta\)
\(I_{0}\) = zero-loss integral
\(\beta\) = effective collection angle
\(\Delta\) = signal integration width
\(\sigma _{k}\left ( \beta ,\Delta \right )\) = partial inelastic scattering cross-section
Here we assume most of the beam is still in the zero-loss peak and we can approximate the total beam integral by that of the zero-loss peak. Both the edge integral and the cross-section will scale with the signal integration width. The effect of the collection angle is included in the edge cross-section calculation. For small collection angles (<~5 mrad), the cross-section changes rapidly, but for larger angles, the cross-section changes more slowly. As a rule of thumb, you should know the collection angle to about 10% of the correct value.
When you do not remove plural scattering, you can apply an approximate correction:
\(N\approx \frac{I_{k}(\beta ,\Delta )}{I_{low}(\beta ,\Delta )\sigma _{k}(\beta ,\Delta )}\)
where
\(I_{low}( \beta,\Delta)\) = low-loss intensity integrated up to energy loss \(\Delta\)
When you divide by \(I_{low}( \beta,\Delta)\) instead of \(I_{0}\), this approximately compensates for extra signal from plural scattering. If the low-loss signal is available (in same spectrum or a DualEELS™ pair), DigitalMicrograph^{®} EELS analysis will automatically include this correction.
Plural scattering is a collection of independent events and follows predictable statistics. We can exploit this to deconvolve the effect of plural scattering from the spectrum and recover the single scattering distribution. The removal of plural scattering will improve background fit and cross-section agreement, but not the signal-to-noise ratio (SNR) for microanalysis.
Starting with Poisson statistics for the scattering events, it can be shown that the single scattering distribution \(j^{1}\left ( E \right )\) can be derived from the recorded spectrum \(j\left ( E \right )\) using:
\(j^{1}\left ( \nu \right )=g\left ( \nu \right )ln\left [ \frac{j\left ( \nu \right )}{z\left ( \nu \right )} \right ]\)
where
\(\nu\) = Fourier-frequency
\(g(\nu)\) = reconvolution function to remove noise amplification (e.g., Fourier transform (FT) of ZLP or a Gaussian of the same full width at half maximum (FWHM))
\(z(\nu)\) = FT of the ZLP
The inverse FT of \(j^{1}(v)\) will give the signal scattering distribution. This formula needs a spectrum from zero-loss up to edge of interest, and is therefore not suitable for higher energy edges. However, it is ideal for low-loss analysis such as plasmon fingerprinting and dielectric function analysis.
For an isolated core-loss edge, we can treat the low-loss spectrum as an instrument broadening function and employ classical deconvolution techniques. The single scattering distribution \(j_{k}^{1}(E)\) is derived from the core-loss spectrum \(j_{k}(E)\) and corresponding low-loss spectrum \(j_{low}(E)\) using:
\(j_{k}^{1}(\nu )=\frac{g(\nu )j_{k}(\nu )}{j_{low}(\nu )}\)
where
\(\nu\) = Fourier-frequency
\(g(\nu )\) = reconvolution function (e.g., FT of ZLP/Gaussian)
This formula is useful for high-loss edges (but requires corresponding low-loss spectrum). When you use this formula, you must remove the leading edge background prior to deconvolution for the Fourier analysis to be valid.
It may not always be possible (or desirable) to acquire the low-loss distribution along with every core-loss spectrum. In this case, relative quantification is still possible.
You can obtain an atomic ratio of species \(A\) and \(B\) following the formula below where the zero-loss integral \(I_{0}\) divides out of the equation:
\(\frac{N_{A}}{N_{B}}=\frac{conc. A}{conc. B}\approx \frac{I_{A}(\beta ,\Delta )\sigma _{B}(\beta ,\Delta )}{I_{B}(\beta ,\Delta )\sigma_{A}(\beta ,\Delta )}\)
Where possible, use the same edge type to cancel computed cross-sections errors. Likewise, the use of the same signal windows widths will partially correct for plural scattering if \(A\) and \(B\) are edges of similar shape. This also corrects for some artifacts, including thickness and diffraction contrast.
You can describe inelastic scattering process by:
\(\frac{d^{2}\sigma }{d\Omega dE}=\frac{8a_{0}^{2}R^{2}}{Em_{0}\nu ^{2}}\left ( \frac{1}{\theta ^{2}+\theta_{E} ^{2}} \right )\frac{df_{n}}{dE}\)
where
\(\Omega\) = solid angle of scattering
\(E\) = energy loss
\(R\) = Rydberg energy (13.6 eV)
\(\theta\) = scattering angle
\(\theta_{E}\) = characteristic inelastic scattering angle (~= E/2E_{0}) where E_{0} is the primary beam energy
\(a_{0}^{}\) = Bohr radius (0.529 Å)
\(\nu\) = electron velocity
\(m_{0}\) = electron rest mass
In this scenario
Cross-section decreases with higher energy loss and accelerating voltage
Angular distribution of cross-section follows Lorentzian distribution (approx.)
Since \(\theta_{E}\) increases with energy loss, angular distribution broadens with \(E\)
The quantity \(df/dE\) is known as the generalized oscillator strength (GOS). It is through this term that the atom specific physics enters into the cross-section by means of a transition matrix of the form.
\(\left \langle \Psi _{f}\left | exp(iq.r) \right | \Psi _{i}\right \rangle\)
where
\(\Psi _{i}\) = initial (core) electron state
\(\Psi _{f}\) = final (ionized) electron state
\(q\) = momentum transfer
In practice, there are two approaches you can take to determine the partial cross-section \(\sigma (\beta ,\Delta )\). You can evaluate values for GOS (\(q,E\)) in analytical form using hydrogenic approximations. This is incorporated into the SIGMAK and SIGMAL programs by Egerton; however, you can only compute K- and L-edge cross-sections with these routines. Alternatively, you can numerically compute these values utilizing more sophisticated models of the atoms such as Hartree Slater atomic wave functions. Both these approaches are available in the DigitalMicrograph EELS analysis software. Keep in mind these approaches assume isolated, spherical atoms. The effects of bonding and crystallinity are not incorporated in these models.
Alternatively, you can measure edge intensities from known standard samples when you use a fixed set of experimental conditions and \(\sigma (\Delta ,\beta )\) that is derived from them. Caution, such measurements can be difficult and each reference covers only a limited range of \(\Delta\) and \(\beta\).
Knowledge of the effective collection angle \(\beta\)* is essential for accurate determination of the partial cross-section \(\sigma (\beta ^{*},\Delta )\). In the majority of cases in the conventional TEM you define the effective collection angle by the actual collection angle \(\beta ^{*}\cong \beta\). However, a convergence angle a of similar or greater magnitude to the collection angle will alter the partial cross-section. If this occurs, you should evaluate the effective partial cross-section \(\sigma (\alpha ,\beta,\Delta)\). If both angles are entered in the DigitalMicrograph EELS analysis software, the correct cross-section will be calculated. It is advisable to work in the regime where \(\beta\geq \alpha\) otherwise significant signal will be excluded from the detector.
When you increase \(\beta\), you will increase the total signal entering the spectrometer. However, beyond a certain angle (dependent on the edge energy) the background under the edge will start increasing faster than the edge signal. This crossover effect will lead to a maximum in the SNR for the edge.
As you optimize the collection angle \(\beta\) for EELS, it is important to keep in mind the below parameters.
To collect the maximum inelastic signal and minimum background, make the collection angle 2 – 3x the characteristic scattering angle \(\theta _{E}\) for the energy loss. The characteristic scattering angle \(\theta _{E}\) is the half width of the scattering distribution and is related is to the energy loss \(E\) and beam energy \(E_{0}\) via:
\(\theta _{E}\approx \frac{E}{2E_{0}}\)
e.g., Si K (1839 eV loss) at 200 kV, require 10 – 15 mrad collection angleTo ensure you attain the most robust results possible, it is important to pay close attention to the errors that may occur during analysis. Common errors include:
Systematic errors
Uncertainty in background model
Redistribution of signal from plural scattering; e.g., thicker regions when \(t/\lambda > 0.3\)
Uncertainty in experimental parameters; e.g., collection or convergence angle
Computation of cross-section
5 – 10% for K-edges
10 – 20% for L-edges
Much worse for M-, N-, O-edges
Statistical errors
Uncertainty in background fit (e.g., \(h\) parameter)
Noise in signal integral (shot noise sqrt(recorded intensity) shot noise = \(N{_{E}}^{1/2}\) )
In Gatan Microscopy Suite^{®} (GMS) 3 software, the extraction of the edge signals is approached somewhat differently. Rather than treating the edges and the background a two separate entities, they are treated as a single function. The background is treated as a power law while the edges present are modeled by the theoretical cross-sections. This allows overlapping edges to be handled automatically. The effect of plural scattering is handled not by deconvolution of the extracted signal but by forward convolution of the model cross-section prior to fitting the model to the data. This results in a much better fit to the data and reliable extraction of overlapping edges. The method was inspired by the work of Veerbeeck and Van Aert (Verbeeck, J.; Van Aert, S. Model based quantification of EELS spectra. Ultramicroscopy. 101(2 – 4):207 – 24; 2004).
Leapman, R. D.; Rez, P.; Mayers, D. F. K, L and M shell generalized oscillator strengths and ionization cross-sections for fast electron collisions. J. Chem. Phys. 72:1232 – 1243; 1980.
Rez, P. Cross-sections for energy-loss spectrometry. Ultramicroscopy. 9:283 – 288; 1982.