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Analytical Electron Microscopy

  • Gianluigi BottonEmail author
  • Sagar Prabhudev
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

Analytical electron microscopy ( ) refers to a collection of spectroscopic techniques that are capable of providing structural, compositional, and bonding information about samples probed by an electron beam, typically inside a transmission electron microscope ( ). Several AEM techniques are covered with particular attention given to the (energy-dispersive x-ray spectroscopy) microanalysis and (electron energy-loss spectroscopy) techniques. First, the different AEM techniques available in TEMs are surveyed and a parallel between EELS and EDXS is drawn. A fundamental description of the elastic and inelastic scattering events responsible for these signals is presented. The practical challenges related to electron optics and instrumentation capabilities are then discussed. Technical advances that have affected the performance of these AEM techniques are outlined, including successive generations and technologies of energy filters, monochromators, aberration correctors, and advanced energy-dispersive x-ray spectrometers. The different approaches of spectroscopic imaging with x-rays and energy-loss spectroscopy, the resolution limits, and the effects of electron-beam propagation are also described along with the types of information that can be extracted with electron-energy-loss near-edge structures. After a review of dielectric theory and low-loss spectroscopy, examples of plasmonic imaging are presented. The review also draws attention to the many efforts to extend the limits of spatial resolution and the atomic-level chemical analyses of materials. Some important progress in the statistical analysis of signals and associated numerical methods is mentioned. The review also presents some novel developments in image capture, such as the pixelated detectors. Finally, the realm of phonon spectroscopy made possible through the latest instrumentation is also discussed.

7.1 Overview

The term AEM refers to the collection of spectroscopic data in the TEM based on various signals generated by the inelastic interaction of the incident electron beam with the sample. These signals can be used to identify and quantify the concentration of the elements present in the area analyzed, map their distribution in the sample with high spatial resolution (down to \({\mathrm{1}}\,{\mathrm{nm}}\) or better), and even determine their chemical state. Although several signals are generated by the interaction of primary incident electrons with a sample (Fig. 7.1), the two main techniques at the core of AEM are based on the detection of x-ray signals generated in the sample by the primary incident electrons with the technique called energy-dispersive x-ray spectroscopy (EDXS) and the measurement of the energy lost by the incident electrons with EELS. These techniques are used within a transmission electron microscope (Fig. 7.2) equipped with EDXS, EELS detectors and other detectors to record TEM images, diffraction patterns, and other signals that can be combined to form new images containing additional information about the chemical nature and structure of the sample. Other techniques such as cathodoluminescence, Auger spectroscopy, electron-beam-induced current imaging are also part of the arsenal of techniques available in the transmission electron microscope but are less frequently implemented in commercial TEMs or have been used only as proof of concept. Convergent-beam electron diffraction has been considered traditionally as one of the AEM techniques but given the breadth of this topic, this chapter will focus on the principles of EDXS and EELS, instrumentation requirements, the quantitative aspects of microanalysis, and the respective advantages and limitations. Several examples of application of these techniques in practical problems will be given. A comparison of these AEM methods with other analytical techniques is presented in a summary. Since this topic has been extensively covered in earlier textbooks [7.1, 7.2, 7.3] and reviews by Sigle [7.4] and Egerton [7.5], an update on the current status of the techniques will be given based on recently published literature and practical examples.

Fig. 7.1

Signals generated when high-energy incident electrons interact with a thin sample

Fig. 7.2

Example of a vintage high-performance analytical TEM. Visible are the energy-dispersive x-ray detector (old generation Si(Li)), the energy-loss spectrometer, digital cameras, and other detectors. The sample is inserted inside the column via a sample holder. For more modern instruments, such detectors are usually not visible as they are incorporated in the housing of the microscope or are integrated within the column. For a superficial glance at more modern exterior appearances, the reader is referred to marketing materials from vendors

7.1.1 EDXS

As incident electrons generated by the electron gun in the transmission electron microscope interact with atoms and their electrons in the solid, various inelastic processes occur. When the primary electron excites bound electrons on a given shell (Fig. 7.3a-c), a core hole is created. This excited state is only temporarily created and the empty state is therefore filled by electrons from higher energy levels through a de-excitation process accompanied by generation of photons (of sufficient energy to be considered in the x-ray part of the spectrum) or Auger electrons. The energy of these x-rays, typically in the range of a few hundred eV to \(20{-}40\,{\mathrm{keV}}\), is also characteristic of the energy differences between the levels involved in the excitation and de-excitation process (Figs. 7.4 and 7.5). Because various energy shells can be excited, peaks in the spectra are labeled according to the corresponding quantum levels involved in the transitions based on the nomenclature illustrated in Fig. 7.4. The dependence on characteristic energy levels of the bound electrons makes it possible to identify the atomic number of the elements that have been involved in the excitation process. However, not all transitions between the energy levels are allowed: there must be a change in the angular momentum quantum numbers \(\ell\) and \(j\) for transitions to be observed according to the rule \(\Updelta\ell=\pm 1\), \(\Updelta j=1,0,+1\). The probability of these excitations and generation of x-rays or Auger electrons varies with atomic number based on cross sections and fluorescence yield. The efficiency of the detection of the x-rays also varies according to their energy because of absorption in the detector material. In addition, the x-rays generated can be absorbed in the sample itself before reaching the detector. The spectrum consists of characteristic x-ray peaks for the excited atoms present in the sample superimposed on a continuum noncharacteristic background (Fig. 7.6).

Fig. 7.3a-c

Energy-level diagrams showing the transitions required for the generation of x-rays and Auger electrons following the excitation of core electrons by the primary incident electrons. (a) Initial excitation of an inner shell electron in a ground state to an excited state creating a hole and an electron above the Fermi level. (b) De-excitation process to fill the hole accompanied by generation of photons. (c) De-excitation process to fill the generated hole accompanied by generation of Auger electrons. The energy levels A, B and C are used in the text to identify the energies involved in the Auger Process (see text vide infra)

Fig. 7.4

Detailed energy levels, their associated quantum number \(n\), \(\ell\), and \(j\) and associated families with the transitions respecting the selection rules described in the text

Fig. 7.5

Characteristic x-ray energy for different families of lines as a function of atomic number

Fig. 7.6

Simplified EDS spectrum demonstrating the characteristic lines (with energies presented in Fig. 7.5) and the continuum background

In order to relate the intensity detected in spectra to the concentration of elements, several effects must be considered: the intensity of the peaks in the spectra must be adjusted using cross sections and fluorescence corrections, absorption in the sample and in the detector. Figure 7.7 summarizes the process of signal generation, collection, display, and quantification in a flow chart containing various steps of signal correction. On the basis of the quantification of signals, it is possible to measure the local concentration to an accuracy limited by the statistic uncertainty of the spectrum and the errors in the cross sections.

Owing to the limited signal collection efficiency of the detector, the finite acquisition time and the presence of a noncharacteristic background under the element-specific peaks, the technique is generally not suitable for the detection of trace elements in samples but it can provide rapid quantitative data on elements to within a few percent accuracy, with detection limits typically of a few percent. Improvements in collection efficiency, particularly with multiple silicon-drift detectors ( s) surrounding the sample, and long acquisition times, however, have led to detection limits of fractions of \({\mathrm{1}}\%\) (Sect. 7.7 of this chapter).

Fig. 7.7

Schematic diagram showing the generation of x-rays in the sample, detection of the x-rays in the detector (by generation of electron and hole pairs), pulse analysis and quantification process of the spectrum to derive the composition of the sample. Detailed procedures of quantification are described in Sect. 7.4

Using software that can control the electron-beam position on the sample and the data collection in a sequential manner, signals can be collected over an area of the sample so that the intensity of characteristic signals, as a function of position, represents the local composition variations in the sample as displayed in an elemental map (Fig. 7.8). Further processing of spatially resolved spectra can also be carried out so that quantitative maps and statistical analysis of the concentration and element distribution can be displayed. Details of the performance and limitations of EDXS as well as the approaches used to quantify the data are given in the subsequent sections.

Fig. 7.8

Schematic diagram describing the processes necessary to record an elemental map with energy-dispersive x-ray spectroscopy. The synchronous scan of the beam is associated with a pixel position where the recording of the signal takes place. The peak intensity is measured for each element at each pixel position and is plotted in a two-dimensional () elemental map

7.1.2 Electron Energy-Loss Spectroscopy

Electron energy-loss spectroscopy (EELS) is based on the measurement of the energy that the primary incident electrons have lost while causing various inelastic processes in the sample. The excitation of electrons from core energy levels that precedes the generation of x-rays or Auger electrons is only one of the mechanisms by which the primary electrons can lose some of their energy. For this particular case, the energy-loss process gives rise to signals known as core edges with characteristic energies closely related to the binding energy of the excited electrons (Fig. 7.9) in the atoms. Excitation of valence electrons into the conduction band and collective excitation of weakly bound electrons are also potential energy-loss processes (Fig. 7.9) called plasmons. These losses contribute to the low-loss part of the spectrum (Fig. 7.10) from a few eV to about \(50{-}100\,{\mathrm{eV}}\). Although characteristic core edges can appear at relatively low energies, strong low-loss signals contain information about the dielectric properties of the material. With reference spectra and databases, it is therefore possible to identify particular compounds based on the shape of the low-loss spectrum. Identification of the chemical state and the nature of the compound is also possible through the analysis of fine modulations appearing in the first few eV from the core edge threshold. These modulations are known as electron energy-loss near-edge structure ( ) and contain information about the electronic structure and bonding environment of the excited atom. This information is now frequently used in the study of electronic structure and the chemical state of materials with applications ranging from semiconductor devices to the study of minerals whereas low-loss structures have been used to study structures as diverse as biological specimens, battery materials, and superconductors. More recently significant applications in the area of plasmonics, the field of study of surface plasmon resonances in nanostructures, have emerged with signals at energy losses typically below \({\mathrm{2}}\,{\mathrm{eV}}\).

Fig. 7.9

Schematic diagram of the associated features in a spectrum. The core edges arise from transitions from deep core levels to the first unoccupied states above the valence band ( ) (and Fermi energy) and the continuum. Excitation from defect states in the gap are also shown as well as collective excitations of valence electrons giving rise to broad features called plasmon peaks

Fig. 7.10

Full energy-loss spectrum recorded over a large energy range demonstrating the large dynamic range of the recorded intensities and relative intensity of core-losses as compared with the background. Spectrum courtesy of H. Sauer, Fritz-Haber Institut/MPG, Berlin

As in the case of EDXS microanalysis, the intensity of core edges is related to the probability of excitation and thus to cross-section values and the concentration of elements. The intensity of edges relative to the background, however, is strongly dependent on the thickness of the analyzed area and edges can remain simply undetected in the case of thick samples. This technique is again not ideal for routine detection of trace elements owing to the very intense background typically dominating the signal at the edges and to the overall small recorded signal of edges with respect to the total recorded signal (Fig. 7.10) although acquisition conditions can be optimized for the detection of minor constituents (discussed in Sect. 7.7.1).

EELS signals offer the advantage of being generated by a primary event: the loss of energy. As compared to EDXS, the intensity of recorded signals is therefore not linked to the secondary process of fluorescence resulting in the de-excitation via x-ray emission. For light elements such as O, N, C, B, this is a remarkable advantage because the fluorescence yield (the relative probability of x-ray and Auger electron generation, Sect. 7.4.1) decreases by orders of magnitude as compared to higher atomic number elements such as transition metals. Therefore, EELS analysis is generally considered to be more appropriate for the detection of light elements than EDXS analysis.

The core edges can be identified and labeled according to the energy levels of the ejected electron and the respective quantum numbers. K, L, M, N, O edges are related to the transitions involving \(n=1\), 2, 3, 4, 5 principal quantum numbers, respectively. The angular momentum quantum number \(\ell\) (s,p,d,f) and \(j\) leads to sublabels as indicated in Fig. 7.11. A summary of the information that can be retrieved from EELS spectra is shown in Table 7.1 updated from [7.6].

Fig. 7.11

Diagram demonstrating the origin of the spectroscopic labels of energy-loss spectra and the associated core levels. Adapted from [7.2]

The collection of EELS spectra is carried out with an energy-loss spectrometer either attached at the bottom of the TEM column (postcolumn filter) or within the projector lens system (in-column filter) (Sect. 7.2.4, Spectrometers). In both cases, the electron energy distribution is analyzed with one or a series of dispersing elements that separate the electrons according to their energy. The dispersion will result in the generation of a spectrum that will be recorded on a detector system. Depending on the electron-optical configuration of the filter and detector system, spectra, images and diffraction patterns corresponding to specific energy losses can be recorded as discussed in Sect. 7.2.4, Spectrometers. When images or diffraction patterns are obtained using electrons with specific energy losses or with electrons having lost no energy, the technique is called energy-filtered microscopy.

Table 7.1

Information from EELS spectra

Spectral region

Type of information

Application

Full spectrum

Thickness, inelastic mean free path

All analytical methods of quantification, volume fraction

Low-loss

Average electron density

Microanalysis of alloys, H content, identification of phases, Li mapping based on phase identification, ceramic phase distribution

Low-loss

Joint density of states

Optical properties of solids, electronic structure, correlation effects, bandgap measurement

Low-loss

Dielectric properties/interfaces

Relativistic effects, interface excitation effects/modes

Very low-loss

Plasmonic response

Phonon excitation

Surface plasmon resonance response of metallic nanostructures

Molecular fingerprinting

Phonon polariton excitation

Temperature measurements

Core-loss

Edge intensity

Quantification of concentration of elements

Core-loss

Near-edge structures

Chemical state, coordination, ionicity/valence, phase identification

Core-loss

Extended fine structure

Determination of radial distribution functions

Core-loss

White-lines: Density of holes in the d-band

Formal charge, charge transfer

7.1.3 Comparison With Other Spectroscopies

EDXS and EELS offer complementary information to other techniques that yield compositional or spectroscopic data typically available in surface analysis instruments such as x-ray photoelectron spectroscopy ( ), Auger spectroscopy, x-ray absorption spectroscopy, inverse photoemission etc. XPS provides information on the binding energy (\(E_{\mathrm{b}}\)) of electrons in atomic core levels as ejected by incident photons. These photoelectrons with kinetic energy \(E_{\mathrm{k}}\) are detected in vacuum as they escape the sample surfaces and the system workfunction (\(\phi\)). This process typically detects photoelectrons with very low kinetic energy as the incident x-ray photons (\(E_{\upnu}\)) are typically a few keV and \(E_{\mathrm{k}}=E_{\upnu}-(E_{\mathrm{b}}+{\phi})\). The technique thus provides essential information from the topmost few atomic layers and is used to analyze ultrathin layers, quantify composition of thin films deposited on surfaces, surface contaminants etc. Although the lateral resolution is typically a few tens of microns in commercial instruments, near-micron resolution can be achieved in synchrotron facilities and in imaging XPS instruments. Using ion beams to sputter the sample surface, depth profiling can be carried out with depth resolutions of \(2{-}5\,{\mathrm{nm}}\) due to the (energy dependent) escape depth of the electrons and, when sputtering is used, the ballistic mixing induced by the incident ions. Angular resolved XPS methods can reach a depth resolution of about \(1{-}2\,{\mathrm{nm}}\) as the escape angle can be tuned with the spectrometer. The technique can therefore provide information on the composition of surface layers and changes in the chemical state of atoms as reflected in the changes in binding energy.

In Auger spectroscopy, the energy of the Auger electrons typically ranges from a few tens of eV to \(1{-}2\,{\mathrm{keV}}\) and, as in the case of XPS, the escape depth from the sample surface is also limited to the topmost few nanometers. The Auger electron energy is determined by the differences between energy levels of the initial core level (A: \(E_{\mathrm{A}}\)) and secondary levels (B and C: \(E_{\mathrm{B}}\) and \(E_{\mathrm{B}}\) respectively) as shown in Fig. 7.3a-c. It is determined by the energy levels involved in the transition of states A, B and C as \(E_{\text{ABC}}=E_{\mathrm{A}}-E_{\mathrm{B}}-E_{\mathrm{C}}-\phi\). The changes in bonding due to changes in oxidation state or structure are therefore reflected in the energy of Auger peaks as the energy levels would be affected by the changes in bonding. The technique therefore provides both information on elemental composition and chemical state.

X-ray absorption spectroscopy provides information on the absorption process of incident photons caused by transitions from inner-shell energy levels to the unoccupied states just above the Fermi energy and the continuum free states. The technique is therefore complementary to EELS as unoccupied states are probed but it offers the advantage of giving access to higher energy edges and the possibility of analysis in controlled nonvacuum environments. With zone-plate focusing of incident photons in third-generation synchrotrons, it is possible to obtain spot sizes of \(15{-}30\,{\mathrm{nm}}\) and better spatial resolution using ptychography methods. The x-ray absorption process can be directly observed in transmission mode or via indirect yield of electrons generated via the absorption process (total electron yield or fluorescence yield). In the latter case, the technique becomes surface sensitive as the escape depth of detected electrons is limited by their energy. Detection of elements in ppm concentration is possible since the background of edges is lower than for EELS.

As compared with these spectroscopic techniques, EDXS, carried out with typical commercial detectors, can be considered a bulk analysis technique yielding elemental information through the thickness of the thin TEM foil with practically no content on the chemical state of the detected elements. Electron energy-loss spectroscopy in the analytical microscope also provides information on the chemical composition through the thin foil thickness but offers the clear advantage of providing spectroscopic information on chemical state.

Table 7.2 summarizes the general applications of the techniques, limitations, resolution etc. Characteristic details on resolution vary depending on the acquisition conditions, energy of the elements of interest, and efficiency of the detection system.

Table 7.2

Comparison of spectroscopy techniques

Technique

Lateral resolution limits

Depth resolution

Detection limit

Elemental information

Spectroscopic information

EDXS

\(1{-}2\,{\mathrm{nm}}\)

No

Minor

Yes

No

EELS

\(0.5{-}1\,{\mathrm{nm}}\)

No

Minor

Yes

Yes

Auger

\(10{-}50\,{\mathrm{nm}}\)

Yes (\(2{-}5\,{\mathrm{nm}}\))

Minor

Yes

Yes

XPS

\(1{-}10\,{\mathrm{\upmu{}m}}\)

Yes (\(1{-}5\,{\mathrm{nm}}\))

Minor

Yes

Yes

XAS

\(20{-}100\,{\mathrm{nm}}\)

No

Trace

Yes

Yes

7.2 Instrumentation

The ultimate aim of analytical electron microscopy is to analyze materials with high spatial resolution. This requires the use of electron sources and electron optic components capable of producing intense beam currents into small electron probes, and detectors to collect the various analytical signals generated from the interaction of incident electrons with atoms in the solid. These requirements are met in a transmission electron microscope configured for analytical work offering bright electron sources, flexible condenser optics, a clean vacuum, and a range of detectors for imaging and spectroscopy. Thin and very clean samples are a necessity for achieving the ultimate performance expected from the high spatial resolution techniques and these are as important as the quality of the microscope.

7.2.1 Electron Sources and Probes

Transmission electron microscopes for conventional TEM, high-resolution TEM, and analytical electron microscopes may be equipped with two types of electron sources—thermionic and field emission. Thermionic sources such as tungsten (W) hairpin filaments and refractory crystals such as lanthanum hexaboride (\(\mathrm{LaB_{6}}\)) operate at high temperature and emit electrons that are subsequently accelerated by the anode potential (\(100{-}200\,{\mathrm{kV}}\) or more). The thermionic sources are heated either by a flow of current through the emitting material itself (for the W hairpin cathode filament) or by thermal contact of a low-workfunction emitter material such as \(\mathrm{LaB_{6}}\) and resistive heating of a W wire supporting material. The field-emission gun ( ) source operates on the principle of electron tunneling from the tip to vacuum resulting from the application of a strong electric field (\(\approx{\mathrm{10^{9}}}\,{\mathrm{V/m}}\)) generating a very large electric field gradient at the tip of the cathode. This high field results in a very narrow potential barrier allowing tunneling of the electrons from a low-workfunction metallic tip to vacuum. This emission is generated from a very small area (in the order of \({\mathrm{10}}\,{\mathrm{nm}}\) or less) of a single crystal tip resulting in high current density. There are variants to this gun configuration. Cold-FEGs emit at room temperature and have the disadvantage of requiring ultrahigh vacuum (\(E-8{-}E-9\,{\mathrm{Pa}}\)) in order to prevent the adsorption of gas molecules on the surface of the tip. This would lead to a reduction of the emission current and to instabilities caused by the increase of the workfunction due to the surface contaminants. To reduce this sensitivity to adsorption, thermally assisted field emission and Schottky emission sources have been introduced. Thermally assisted FEG are based on the application of a high electric field to W single-crystal tips heated to about \({\mathrm{1600}}\,{\mathrm{K}}\). The emission also occurs through tunneling from a small area of the tip and the characteristics are similar to cold-field emission. The benefit of high-temperature operation is the increased stability due to a cleaner tip at the expense of a higher energy spread of the emitted electrons than for cold FEG. The last type of source also considered in the class of field-emission guns is the Schottky gun . These guns are based on the Schottky emission principle that causes a reduction of the energy barrier for simple thermionic emission with a combination of electron image forces and strong electric fields applied to the tip [7.7, 7.8]. Strictly speaking, the emission is therefore not due to tunneling as it combines thermal emission with high electric fields. Owing to the high coherence and brightness, however, Schottky sources are also considered in the class of field-emission sources. To further decrease the workfunction, the W tip is generally coated with ZrO to increase the emission current. Optimal current stability is obtained with W crystals oriented so as to expose (100) crystalline facets and ZrO coating. The high operating temperature reduces the sensitivity to adsorption but it has the drawback that the material is sensitive to reactions with any gases present in the gun area. Ultrahigh vacuum is therefore required but at less strict levels than is necessary for cold FEG. High current densities can be achieved with Schottky sources because of the very small emission area. High total emission currents can also be generated by controlling the extraction voltage and the gun lens operation parameters. In some implementations of the Schottky guns, total beam currents in the order of \(100{-}300\,{\mathrm{nA}}\) can be achieved.

The previous description has been very qualitative and further analysis is required to compare the performances of the various types of sources and understand the requirements for AEM. A key quantity characterizing the gun characteristics is the mean brightness of the source \(B\) defined as the current per unit area and solid angle
$$B=\dfrac{i_{\mathrm{e}}}{\uppi{}\left(\frac{d}{2}\right)^{2}\uppi\alpha^{2}}=\frac{4i_{\mathrm{e}}}{(\uppi d\alpha)^{2}}\;,$$
(7.1)
where \(i_{\mathrm{e}}\) is the emission current, \(d\) is the beam diameter, \(\alpha\) is the convergence angle of the cone containing the electrons and \(\uppi\alpha^{2}\) is the solid angle corresponding to the cone. The units of \(B\) are \(\mathrm{A/(m^{2}{\,}sr)}\) and the values scale with the accelerating voltage (\(B\) values are typically given at \({\mathrm{100}}\,{\mathrm{keV}}\)) and are maintained throughout the optical system (from the emission source to the sample).
The emission current density \(J\) for thermal sources is related to the material and the operating temperature \(T\) and follows Richardson's law
$$J=AT^{2}\exp\left(-\frac{\phi}{k_{\text{B}}T}\right),$$
(7.2)
where \(A\) is a constant of the material (the Richardson constant given in units of \(\mathrm{A/(cm^{2}{\,}K^{2})}\)), and \(\phi\) is the workfunction. The field-emission current is related to the tunneling process and thus to the electric field at the tip. For Schottky emission, the workfunction is effectively reduced by a factor \(\Updelta\phi\) by the strong electric field applied to the tip and becomes
$$\phi_{\text{eff}}=\phi-\Updelta\phi=\phi-e\sqrt{\frac{eE}{4\uppi\varepsilon_{0}}}\;,$$
where \(E\) is the electric field (in the order of \({\mathrm{10^{8}}}\,{\mathrm{V/m}}\)), \(\varepsilon_{0}\) is the vacuum permittivity, and \(e\) is the electron charge. This reduction of the workfunction arises from the well-established Schottky effect [7.8, Chap. 6] that effectively increases the emission current by a factor \(\exp(\Updelta\phi/(k_{\mathrm{B}}T))\) [7.9] arising from the negative term in the exponential factor in Richardson's law (7.2). The temperature of the source will affect the temporal coherence of the source and the energy spread of the electrons due to thermal energy. The source size and the angular spread \(\alpha\) will affect the spatial coherence. Considering the source size, temperature of the source, the vacuum requirements and the brightness, the sources can be compared (Table 7.3).
Table 7.3

Comparison of sources

Characteristic

Thermionic W hairpin

Thermionic \(\mathrm{LaB_{6}}\)

Schottky emission FEG (ZrO on W crystal)

Thermal FEG W(100) orientation

Cold-FEG (W single crystal)

\(\phi\) (eV)

\(\mathrm{4.5}\)

\(\mathrm{2.7}\)

\(\mathrm{2.8}\)

\(\mathrm{4.5}\)

\(\mathrm{4.5}\)

\(J_{\mathrm{c}}\) (\(\mathrm{A/m^{2}}\))

\(\approx{\mathrm{10^{4}}}\)

\(\approx{\mathrm{10^{6}}}\)

\(\approx{\mathrm{10^{7}}}\)

\(\approx E7{-}E8\)

\(\approx{\mathrm{10^{9}}}\)

\(\alpha\) (rad)

\(\approx{\mathrm{10^{-2}}}\)

\(\approx{\mathrm{10^{-2}}}\)

\(\approx{\mathrm{10^{-3}}}\)

\(\approx E-3{-}E-4\)

\(\approx{\mathrm{10^{-4}}}\)

\(d\) (\(\mathrm{\upmu{}m}\))\({}^{\mathrm{a}}\)

\(\approx{\mathrm{50}}\)

\(\approx{\mathrm{10}}\)

\(\approx 0.02{-}0.03\)

\(\approx 0.01{-}0.1\)

\(\approx{\mathrm{0.01}}\)

Brightness (\(\mathrm{A/(m^{2}{\,}sr)}\))\({}^{\mathrm{b}}\)

\(\approx{\mathrm{10^{9}}}\)

\(\approx{\mathrm{10^{10}}}\)

\(\approx E11{-}E12\)

\(\approx{\mathrm{10^{12}}}\)

\(\approx E12{-}E13\)

Temperature (K)

\(\mathrm{2700}\)

\(\mathrm{1800}\)

\(\mathrm{1800}\)

\(\mathrm{1600}\)

\(\mathrm{300}\)

Energy spread (eV)

\(1{-}3\) \({}^{\mathrm{c}}\)

\(0.6{-}2\) \({}^{\mathrm{c}}\)

\(0.6{-}1.5\) \({}^{\mathrm{c}}\)

\(0.6{-}0.8\)

\(\mathrm{0.3}\)

Vacuum (Pa)

\(\mathrm{10^{-2}}\)

\(\mathrm{10^{-4}}\)

\(\mathrm{10^{-7}}\)

\(\mathrm{10^{-7}}\)

\(\mathrm{10^{-8}}\)

\({}^{\mathrm{a}}\) Specific values depend on the exact tip configuration

\({}^{\mathrm{b}}\) The brightness values are given for \(\mathrm{100}\)-\(\mathrm{keV}\) electrons and scale linearly with accelerating voltage. The reduced brightness is given by \(B/V\) (where \(V\) is the accelerating voltage in volts)

\({}^{\mathrm{c}}\) The energy spread is dependent on the operation condition of the gun (the bias of the Wehnelt, filament heating current, or the extraction voltage of the Schottky FEG)

We now comment briefly on the merits of the various electron sources for AEM. Without accounting for the impact of aberration correctors of probe-forming lenses, the high brightness of the cold FEG makes it the ideal tool for analytical work requiring a small probe. Minimal source demagnification is required as the source size is in the order of \({\mathrm{10}}\,{\mathrm{nm}}\) although the ultimate probe size and shape will depend on electron-optical considerations related to the aberrations of the probe-forming optics (see below). An additional advantage of the cold FEG is the low energy spread of the electron source allowing, in principle, detailed analysis of the energy-loss near-edge structures in EELS spectra.

Although the ultimate brightness of the Schottky sources is lower by about one order of magnitude, the practicalities of the less stringent vacuum and demonstrated reliability, have made the Schottky guns very common in the AEM field. Worth mentioning is the fact that the total current might be a more important parameter for some AEM analysis technique such as energy-filtered imaging when a large field of view needs to be illuminated (with enough current per individual pixel in a map). In that case, the thermionic emission sources (\(\mathrm{LaB_{6}}\) in particular) provide very large total currents. New research is in progress to develop alternative even brighter sources by exploiting low workfunction materials, the electronic properties of carbon nanotubes [7.10, 7.11], and semiconductor p-n junctions.

Once the electrons are accelerated by the anode stack to the high voltage potential of the TEM, the condenser optics demagnify (Fig. 7.12) the source image in the plane of the sample. In this chapter, we will not cover the description of the various aberrations and defects of lenses but refer the reader to other sections in this Handbook. We remind the reader, however, that all round electromagnetic lenses have positive spherical aberration coefficients \(C_{\mathrm{s}}\) (in the order of \(C_{\mathrm{s}}=0.5{-}1.0\,{\mathrm{mm}}\) for high-resolution microscopes offering limited sample tilt and a small polepiece gap and around \(C_{\mathrm{s}}=1.5{-}3\,{\mathrm{mm}}\) for large tilt and large polepiece gaps). Aberration correctors for the probe-forming lens make it possible to reduce the higher order aberration coefficients and therefore reduce the probe size as discussed further below. Electromagnetic lenses also suffer from chromatic aberrations \(C_{\mathrm{c}}\) in the order of \(C_{\mathrm{c}}=1{-}2\,{\mathrm{mm}}\) and the fact that, just as in light optics, the wave nature of electrons imposes a diffraction limit on the resolution as imposed by the Rayleigh criterion. Given these imperfections of the lenses and wave nature of electrons, the probe size at the specimen level \(d_{0}\) is limited not only by the demagnified image source size \(d_{\mathrm{s}}\) but also by the combination of the aberration of the illumination system (spherical and chromatic) and the diffraction limit of the probe-forming aperture all added in quadrature to a good approximation [7.12, 7.7]
$$d^{2}_{0}=d^{2}_{\mathrm{s}}+d^{2}_{\mathrm{d}}+d^{2}_{\text{sa}}+d^{2}_{\mathrm{c}}+d^{2}_{\mathrm{f}}\;.$$
(7.3)
Here \(d_{\mathrm{s}}\) is the demagnified source image, \(d_{\mathrm{d}}=0.6\lambda/\alpha\) is the diffraction limit broadening, \(d_{\text{sa}}=0.5C_{\mathrm{s}}\alpha^{3}\) is the spherical aberration broadening, \(d_{\mathrm{c}}=(\Updelta E/E_{0})C_{\mathrm{c}}\alpha\) is the chromatic aberration contribution of the probe-forming lens, \(d_{\mathrm{f}}=2\alpha\Updelta f\) is the defocus contribution to a small defocus error. \(\alpha\) is the convergence half-angle at the specimen determined by the condenser \(\mathrm{C_{2}}\) aperture limiting the probe, and \(\Updelta f\) is the defocus of the probe-forming lens. The size of the unaberrated probe source image \(d_{\mathrm{s}}\) is related to the definition of brightness that, as mentioned above, is constant through the illumination system (from the gun to the sample) and is determined as
$$d_{\mathrm{s}}=\frac{2}{\uppi\alpha}\sqrt{\frac{I_{\mathrm{b}}}{B}}\;,$$
where \(I_{\mathrm{b}}\) is the beam current. Calculations of the unaberrated source size therefore assume that the current at the sample is known (or can be measured).
Fig. 7.12

Schematic diagram of a field-emission electron gun with gun anodes (also called gun lens) and the accelerating anodes (accelerating the electrons up to the accelerating voltage of the TEM). The illumination system consists of condenser lenses (\(\mathrm{C_{1}}\), \(\mathrm{C_{2}}\), \(\mathrm{C_{\mathit{m}}}\)) and the objective lens ( ). See text for function details

Equation (7.3) assumes incoherent illumination i. e., all electrons at the aperture are not in phase and every point within the aperture can be considered as an independent source. Strictly speaking, this assumption is not valid for high-brightness field-emission sources (and for particular operating conditions of thermionic electron sources) and one must define the conditions where incoherence in illumination applies in order to determine the conditions of validity of (7.3) for the calculation of the probe size. Study of the various aberration terms in (7.3) shows that there is an optimum probe size as a function of the illuminating angle. Plots for two microscope configurations and sources are shown in Fig. 7.13 for a high-resolution microscope configuration—not dedicated to analytical work but designed for optimum imaging resolution (low tilt angle in the specimen area: \(\pm 20{-}24^{\circ}\))—and for a high-tilt analytical configuration (\(\pm 35{-}40^{\circ}\)). It is clearly visible that the use of FEG allows smaller probes. On the basis of the equations above and typical gun brightnesses for Schottky guns, it is possible to calculate the probe current dependence with probe size, based on the incoherent illumination condition, for an FEG and an \(\mathrm{LaB_{6}}\) source (Fig. 7.14). We also demonstrate the effect of reducing the spherical aberrations in the probe-forming lens (Fig. 7.15) using the recently developed correctors in a cold FEG-STEM operating at \({\mathrm{100}}\,{\mathrm{kV}}\) based on the data reported in the literature [7.13] and Chap.  13 by Hawkes and Krivanek (Fig. 7.15). For the latter case, (7.3) needs to be modified to allow for residual aberrations [7.13].

Fig. 7.13

Variation of the electron probe size for analytical TEM work as a function of the illumination aperture convergence for instruments equipped with different electron sources and lenses in nonaberration-corrected analytical TEM

Fig. 7.14

Variation of the electron-beam current with electron probe size for two instruments equipped with a thermionic source and Schottky field-emission gun

Fig. 7.15

Variation of the probe current as a function of probe size for a conventional cold field-emission-source dedicated scanning transmission electron microscope and an aberration-corrected cold-FEG microscope

As discussed above, the coherence of the source is a key concept that needs to be considered for high-brightness field-emission sources and analytical microscopy. In conditions of coherent illumination one can consider that all electrons are emitted by a single point source, they are all in phase in the illuminating aperture and on the electron wavefront at the sample. This condition implies that interference effects between electrons can occur. A detailed discussion on coherence and sources can be found in [7.12] and we summarize the key elements of coherence relevant to the discussion on the probe size for the purpose of discussing the ultimate limits of the instrumentation. Following the concepts developed in classical optics, one must consider the transverse coherence width \(X_{\mathrm{a}}\) concept related to the width of the electron beam in the plane of the condenser aperture \(X_{\mathrm{a}}=\lambda/(2\uppi\theta_{\mathrm{s}})=f_{\mathrm{C}_{2}}\lambda/(\uppi d_{\mathrm{s}})\) where \(\theta_{\mathrm{s}}\) is the angular width of the probe of size \(d_{\mathrm{s}}\) (i. e., not the convergence angle) as subtended at the condenser aperture (Fig. 7.16). This angle can be determined geometrically given a probe size \(d_{\mathrm{s}}\) and focal length \(f_{\mathrm{C}_{2}}\) of the probe-forming lens. Source points in the aperture plane closer than \(X_{\mathrm{a}}\) can interfere as in a slit experiment with coherent light. This transverse coherence term must be compared with the diameter of the probe-forming aperture \(D_{\mathrm{C}_{2}}\) to determine whether the illumination is coherent or not. If \(X_{\mathrm{a}}\ll D_{\mathrm{C}_{2}}\), the aperture is incoherently illuminated and all points in the illumination aperture can be considered as emitting independently. Still, for a given aperture of convergence angle \(\alpha\) incoherently filled, the coherence width will be \(X_{\mathrm{s}}=\lambda/(2\uppi\alpha)\). Object points at the sample separated by a distance smaller than \(X_{\mathrm{s}}\) will be illuminated coherently and will give rise to interference effects.

Fig. 7.16

Factors entering the description of the source coherence for illumination of samples. Adapted after [7.12]

In the other extreme, if \(X_{\mathrm{a}}> D_{\mathrm{C}_{2}}\), the aperture is coherently filled and the electron wavefront illuminating the sample will be in phase. For example, for the smallest probe in an FEG (say \({\mathrm{0.2}}\,{\mathrm{nm}}\)), a \(\mathrm{C_{2}}\) aperture of \({\mathrm{5}}\,{\mathrm{\upmu{}m}}\) would be coherently filled. At \({\mathrm{200}}\,{\mathrm{keV}}\), for a \(\mathrm{LaB_{6}}\) emitter and a probe size of about \({\mathrm{3}}\,{\mathrm{nm}}\), one would need an aperture of \({\mathrm{0.5}}\,{\mathrm{\upmu{}m}}\) for the coherence criterion to be fulfilled. By changing the illumination conditions (demagnification of the source and reducing the accelerating voltage) and the condenser aperture, the coherence illumination condition could be achieved even for thermionic sources as demonstrated by [7.14].

In coherent illumination conditions, the terms contributing to the determination of the probe size cannot be added incoherently as independent contributions (as was done in (7.3)) and detailed calculations based on the incident electron wavefunction and the transfer function of the objective lens must be carried out. It has been demonstrated that, with the judicious choice of defocus of the probe-forming lens, sub-angstrom electron probes can be obtained even without aberration correctors [7.15] to improve imaging resolution albeit with very small currents and tails to the intensity distribution containing up to \({\mathrm{50}}\%\) of the beam current. Such beams, although relevant for optimal imaging resolution, are therefore not suitable for analytical work. Optimum probe sizes [7.16] for analytical work are achieved with convergence \(\alpha_{\mathrm{C}}\) and defocus \(\Updelta f\) conditions determined by
$$\alpha_{\mathrm{C}}={\mathrm{1.27}}C_{\mathrm{s}}^{-\frac{1}{4}}\lambda^{\frac{1}{4}}$$
and
$$\Updelta f=-{\mathrm{0.75}}C_{\mathrm{s}}^{\frac{1}{2}}\lambda^{\frac{1}{2}}\;.$$
In these optimal conditions, the probe size containing \({\mathrm{80}}\%\) of the intensity is
$$d_{({\mathrm{80}}\%)}={\mathrm{0.4}}C_{\mathrm{s}}^{\frac{1}{4}}\lambda^{\frac{3}{4}}\;.$$
For an aberration-corrected instrument these terms need to be revised to treat the contributions of the residual aberrations correctly as discussed in [7.13] and in Chap.  2.

7.2.2 Electron-Optical Configuration

In the discussion so far we have disregarded the technical aspects necessary to achieve the demagnification of the source required for small probe analysis and the need to provide, with the same system, illumination at the sample for conventional imaging and energy-filtered microscopy. Modern instruments capable of conventional TEM and STEM are based on the double-condensing optical system (\(\mathrm{C_{1}}+\mathrm{C_{2}}\)) with the addition of a supplementary weaker condenser lens \(\mathrm{C_{\mathit{m}}}\) (called the condenser minilens or minicondenser) and a strong magnetic field before the sample (i. e., a prefield) generated by an objective lens ( ) composed of two parts: an upper lens and a lower lens surrounding the sample (Fig. 7.17a,b). In recent instruments equipped with aberration correctors of the probe-forming lens, a third condenser lens (\(\mathrm{C_{3}}\)) is added for additional flexibility and the correctors are placed below the \(\mathrm{C_{3}}\) lens prior to the \(\mathrm{C_{\mathit{m}}}\) system. Parallel illumination at the sample plane is achieved by a combination of \(\mathrm{C_{2}}\) and \(\mathrm{C_{\mathit{m}}}\) yielding, at the focal point of the objective lens a convergent beam that is then made parallel by the strong upper objective lens field (Fig. 7.17a,ba). A small probe required for analytical work or for STEM imaging and analysis in STEM mode is obtained by effectively optically switching the \(\mathrm{C_{\mathit{m}}}\) off (using various schemes depending on the exact location of the \(\mathrm{C_{\mathit{m}}}\) lens) and by making use of the strong OL prefield (of the upper objective lens) to achieve large convergence angles and large source demagnification (Fig. 7.17a,bb). Practically speaking, this mode of operation is called the nanoprobe or EDS mode. STEM operation for these microscopes is achieved in exactly the same electron-optical configuration. Field-emission guns (cold and Schottky) have an additional electrostatic gun lens (Fig. 7.12) leading to additional demagnification of the source and more flexibility in the gun operation. The strength of the \(\mathrm{C_{1}}\) lens determines the fraction of electrons (hence the current) that will enter the \(\mathrm{C_{2}}\) aperture and the demagnification of the source. A strongly excited \(\mathrm{C_{1}}\) lens will give rise to larger demagnification and, conversely, a weakly excited \(\mathrm{C_{1}}\) will result in a smaller demagnification and more current entering the \(\mathrm{C_{2}}\) aperture. In this fixed operation mode, changes in the convergence angle are achieved by changing the physical aperture size (using a strip aperture containing \(4{-}8\) apertures, including a top-hat Pt aperture for analytical work—Sects. 7.2.3, Geometry of the EDXS Detector in the AEM and 7.7.2 on instrumental contributions in EDXS analysis). Continuous change in convergence is achieved by controlling the strength of the \(\mathrm{C_{\mathit{m}}}\) lens and OL prefield and/or addition of a third condenser lens. The aberration correctors, in this case, using transfer lenses, make it possible to couple the upper electron-optical path to the objective probe-forming lens and compensating for its aberrations while leaving higher order aberrations uncorrected.

Fig. 7.17a,b

Electron-optical configuration of the illumination system of a double-objective lens system. Configuration (a) describes the formation of parallel illumination while configuration (b) describes the conditions to achieve a highly convergent and small focused electron beam

The electron-optical configuration required for STEM imaging (with or without aberration correctors) and analysis is achieved, in a TEM-STEM instrument capable of both operation modes and as discussed above, by switching off the \(\mathrm{C_{\mathit{m}}}\) lens and then focusing the nearly parallel beam into a small source image by the upper OL field. With an aberration-corrected probe-forming lens, the wavefront would enter the objective lens with the necessary fields to compensate the objective lens aberrations. The scanning operation of the beam over the sample is carried out by deflection coils located before the specimen (optically before the upper objective lens or as part of the corrector lens system) so that the beam is shifted (but not tilted) on the specimen plane (Fig. 7.18). As the beam is rastered pixel by pixel over the area of sample of interest, various signals (including analytical information) can be recorded sequentially at each position to form images and elemental maps based on analog or digital signals recorded synchronously (e. g., bright field, secondary electrons, backscattered electrons, annular dark-field and EELS, EDXS etc.). Dedicated commercial STEM instruments (offering no TEM operation mode) built by the company Vacuum Generator in the 1980s and 1990s have traditionally been equipped with cold FEG and, in the later models, operate with a gun lens, two condenser lenses, and an asymmetric objective lens (no imaging lenses are necessary). Their design is based on the gun located at the bottom of the microscope column for stability reasons with the detectors at the top. Instruments developed based on the same approach and equipped with aberration correctors have also been developed [7.17] (Chap.  13 by Hawkes and Krivanek). Dedicated STEM instruments based on the upper portion of a conventional TEM column (but in this case without imaging lenses) are also available. These instruments are designed for quick and simple operation particularly popular for routine analysis in device fabrication environments.

Fig. 7.18

Electron-optical configuration of STEM operation showing the role of the deflection coils to shift the electron beam over the sample (without tilt of the beam) and the detection of signals on the dark-field STEM detectors and bright-field or energy-loss spectrometer. (Adapted after [7.18])

As supplementary information, we should briefly note that the process of image formation and interpretation in STEM mode and TEM mode are closely linked by the reciprocity theorem further discussed in Chap.  2 and earlier references [7.19, 7.20, 7.21]. This principle links the electron source in STEM to a detector point in a TEM image (pixel on a digital camera) and the detector in STEM to the source of electrons in TEM. Effectively, the principle states that for identical optical components, sources and detectors, STEM and TEM images will show the same resolution and contrast. Given the respective strengths and weaknesses of these techniques (related to the field of view, recording time, dose, sequential recording, and analytical signals) the techniques should be considered as complementary and not competitive [7.22]. Practically, this implies that materials analysis, and AEM in particular, should involve all possible optimized techniques and instrumentation both for TEM and STEM approaches. As demonstrated in the case of EELS analysis, energy-filtered imaging carried out in the TEM mode is highly complementary to the scanning EELS imaging method with both approaches offering advantages and presenting limitations (Sect. 7.6).

7.2.3 EDXS Detector Systems

The EDXS Detector

In analytical electron microscopy experiments, photons are emitted in the sample following ionization of atoms by the primary incident electrons and subsequent de-excitation process. The energy of these photons is in the x-ray part of the electromagnetic spectrum (few hundred eV to few tens of keV) and we therefore refer to them as x-rays. In the analytical TEM, the most common and effective tool for the detection of x-rays is the energy-dispersive detector that is attached to the microscope column (Figs. 7.2 and 7.19a,b) with the active component detecting the x-rays located as close as possible to the sample. Alternative approaches to detect x-ray signals with wavelength-dispersive detectors have been attempted in prototype systems but have not been commercially implemented due to low efficiency (serial analysis of x-ray energies) and compatibility with the geometry of the microscope column (solid angle, vacuum etc.). The key component of the energy-dispersive x-ray detector is a semiconductor material (Si or Ge) that absorbs incident photons by generating electron–hole (e–h) pairs through the photoelectric effect. When this process occurs in an electric field, a current pulse is generated and subsequently measured by low-noise electronic components. Detectors based on Si are more commonly used in AEM, being less expensive than Ge-based detectors (described below). Because of the fact that very high-purity Si crystals are not available, Li additions in Si are used to compensate for residual impurity dopants that would create undesirable recombination sites for electron–hole pairs and uncharacteristic current related to the existence of impurity-generated acceptor levels (thus holes in the valence band). The role of Li in the so-called Si(Li) detectors is therefore to effectively create an intrinsic semiconductor region where e–h pairs generated by the incident photons produce a current that can be measured. Incident photons generate e–h pairs at the rate of \({\mathrm{1}}\,{\mathrm{pair}}/{\mathrm{3.8}}\,{\mathrm{eV}}\). The number of e–h pairs generated in the active layer of the detector is thus proportional to the incident photon energy (which is, in turn, related to the energy level involved in the transitions following ionization of the atom by the incident electron beam). For Ge detectors, high-purity crystals can be obtained and Li need not be added.

Fig. 7.19a,b

Schematics of a commercial (a) Si(Li) EDXS detector showing the detector front, the dewar system to cool the detector, and various components that are interfaced in the electron microscope. (b) A SDD detector. Courtesy of Oxford Instruments NanoAnalysis

The electric field in the detector necessary to cause the current flow is generated by metallic contacts (Au) on the front and back surfaces of the crystals (typically \({\mathrm{3}}\,{\mathrm{mm}}\) thick) and the application of a bias (\(0.5{-}1\,{\mathrm{kV}}\)) between the front and back of the detector with electrons traveling to the positive electrode and holes to the negative electrode. This type of configuration is essentially the same as a small capacitor. Some loss of charge occurs, however, due to recombination in the charges near the metallic contact (about \({\mathrm{200}}\,{\mathrm{nm}}\) in width). These regions are typically referred to as dead-layers in the literature as they do not contribute to the generation of signal and transfer of charge. The thickness of the crystal is normally large enough to convert all the photons to e–h pairs but for high-energy x-rays, transmission through the detector is possible. Measurement of the current is carried out from the back electrode which is connected to a field-effect transistor ( ) that acts as a first stage of amplification.

The old technology of Si(Li) detectors and FET are cooled to low temperature by a Cu rod connected to a liquid nitrogen reservoir (Fig. 7.19a,ba) to prevent the diffusion of Li in the electric field and to reduce the thermal noise of the e–h generation and the FET electronics. The mechanical design of the detector assembly is crucial to reduce the transfer of mechanical vibrations into noise in the spectra. For example, bubbling in the dewar arising from floating ice crystals and vibrations from the microscope frame or other components (such as fans) touching the detector can result in additional noise. The detector should therefore be supported by the same support mechanism as the frame of the microscope column and be isolated from the rest of the microscope components.

Most instruments available on the market today are equipped with the more modern silicon-drift detectors (Fig. 7.19a,bb) requiring no cooling with liquid nitrogen (just thermoelectric coolers), in which the electrons generated by the photons are channeled towards the anode by establishing both radial- and transverse electric fields. A schematic representation of a typical SDD is illustrated in Fig. 7.20a-ca. The most significant distinction between the conventional Si(Li) and SDDs is the size of the charge collecting anode used. In the case of Si(Li) detectors, the area of the anode is equal to the active area of the detector, whereas in the modern SDDs the anode area is independent of the active area of the detector, and smaller by at least \(1000{-}10000\) times [7.23]. Since the capacitance of these detectors is proportional to the area of the electrode, having a small anode in the SDDs leads to dramatically low device capacitances compared to the conventional Si(Li) detectors. The typical anode capacitances observed in the modern SDDs is about \(25{-}150\,{\mathrm{fF}}\). One direct consequence of having such low device capacitances is that the counting rates are dramatically improved. This is due to the much faster response of the detector to an electron pulse (thanks to the lower capacitance), and the resulting short-duration electron pulses translate into improved x-ray throughputs. Whereas for a typical Si(Li) detector the counting rates saturate at around \({\mathrm{5000}}\,{\mathrm{cps}}\) at its best resolution (\(\approx{\mathrm{132}}\,{\mathrm{eV}}\) full-width at half-maximum ( ) measured with the Mn \(\mathrm{K}_{\alpha}\) line), the best SDD detectors are capable of at least ten times the throughput at a much better resolution (\(\approx{\mathrm{125}}\,{\mathrm{eV}}\) FWHM) [7.23].

Fig. 7.20a-c

Schematic illustrations of Si-drift detectors ( ). (a) Single-anode SDD. (b) Tear-drop or a droplet SDD. (c) Multiple-anode SDD. After [7.24]

Other sources of electronic noise in the SDDs include current leakage in the amplifier (referred to as the shot noise) and the thermal effects (which are proportional to capacitance squared and to the temperature) [7.25]. If the noise is small enough, the detector can even be operated at much higher temperatures (\(\approx{\mathrm{-20}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\)) using a simple Peltier device, and without needing to use the liquid-nitrogen cooling [7.24]. Additionally, the mechanical design of the detector can be a determining factor for limiting parasitic capacitance at material contacts (e. g., bonding pads) and the capacitance established between the amplifier and the charge-collecting anode. In modern SDDs, such effects can be minimized by integrating the field-emitting transistor (FET) of the amplifying electronics onto the detector chip, and also coupling it to the anode by means of a metal strip [7.25]. Even more importantly, the modern SDD design involves offsetting the integrated FET (i. e., amplifier and the anode) outside the active area of the detector. This is designed so as to avoid the electrostatic field effects due to irradiation to the incident x-rays. Figure 7.20a-cb provides a schematic illustration of this design, which is usually referred to in the literature as the tear drop or droplet SDD. Finally, the SDD design principles discussed above can be realized in different topologies—the single-anode device as in Fig. 7.20a-ca or the multiple-anode device as in Fig. 7.20a-cc. The typical x-ray throughput (i. e., the count rate) for multiple-anode systems is better than \({\mathrm{10^{7}}}\,{\mathrm{cps/cm^{2}}}\) detector area (based on \({\mathrm{5}}\,{\mathrm{mm^{2}}}\) cell and \(\mathrm{39}\) cells) [7.24]. While these count rates are not relevant in the realm of TEM thin samples where ultimate spatial resolution is desirable and therefore the interaction volume is very small, the improved solid angle afforded by the more flexible geometry (discussed further below) and larger detectors and the possibility to analyze thicker samples at much faster acquisition rates has dramatically changed the field of EDS spectrometry in the TEM. It is worth mentioning the significance of SDDs also in the context of SEM analysis and not only in TEM. In comparison to the wavelength-dispersive x-ray spectrometers ( ) that are considered the gold standard in microbeam analyses in the SEM, the EDS-SDDs can match or exceed the accuracy and precision of a typical WDS measurement [7.26]. The carbides, oxides and nitrides, which typically produce soft x-rays, can be quantified in a much better manner than with the WDS technique thanks to the improved energy resolution and spectra with much larger signals. Therefore, even higher level trace measurements (as small as \({\mathrm{1000}}\,{\mathrm{ppm}}\)) can be performed in a more accurate manner using SDDs than with WDS [7.26, 7.27] in SEM since x-ray collection with much larger solid angles is now possible. More importantly, this level of performance is possible even when there are peak interferences (e. g., Ba L–M and Ti K–L with separation \(\approx{\mathrm{40}}\,{\mathrm{eV}}\)) and at simultaneously large element ratios (e. g., \(\text{Ba}/\text{Ti}> {\mathrm{20}}\)) [7.26]. In the context of TEM, SSD have allowed the detection of trace element dopants in semiconductors devices.

In summary, modern silicon-drift detectors have revolutionized EDS detector technology both in the analytical transmission electron microscope and in the SEM. Their notable characteristics such as, low x-ray energies, improved resolution, larger detector areas, faster response times, and low electronic noise make them an ideal replacement for the conventional Si(Li) detectors.

Detector Windows

Because of the low operating temperature for Si(Li) devices (less so for SDD cooled with Peltier devices), the detector can act as a cold trap attracting contaminants to the detector front. For this reason, detector windows are used to isolate the Si(Li) detector from the vacuum of the microscope (Fig. 7.21). The window technology has evolved over recent decades with the initial technology based on Be windows (\(7{-}8\,{\mathrm{\upmu{}m}}\) in thickness) allowing x-rays for elements down to Na to be detected due to the absorption of lower energy photons in this material. This limitation for the detection of light elements has led to the development of polymer-based thin windows, and some implementation of systems without windows at all (called windowless detectors). Various polymer-based window materials are available as ultrathin windows ( ). These use proprietary technology combining light element composites (polymers, diamond, nitrides etc.) some of which are strong enough to withstand atmospheric pressure (labeled as atmospheric thin windows (ATW )). Because of the combination of various x-ray absorbing materials, the sensitivity of the detector system for light elements strongly varies at low energy due to the absorption of the soft x-rays by the window material. UTWs, for example, allow detection of elements down to B but with reduced sensitivity as compared to windowless detectors that are actually able to detect x-rays down to the Be \(\mathrm{K}_{\alpha}\) line. The latter technology, however, normally requires the use of ultrahigh vacuum in the microscope to prevent rapid contamination of the crystal by the microscope environment and the related absorption of soft x-rays by the ice contamination layer. Whether windowless or UTW, most Si(Li) detectors are equipped with crystal heaters that gently warm up the crystal surface causing desorption of the ice layer built up on the crystal surface. Ice contamination is revealed by a reduction of the intensity of soft x-ray lines as compared to higher energy lines such as demonstrated by the \(\text{Ni-L}/\text{Ni-K}\) ratios measured as a function of time. Since the Ni-L line at \({\mathrm{0.85}}\,{\mathrm{keV}}\) is much more strongly absorbed than the Ni-K (at about \({\mathrm{7}}\,{\mathrm{keV}}\)) set of lines the ratio is an effective means to appreciate the contamination effect [7.28].

Fig. 7.21

Schematics of the detector front of a TEM with components inserted into the TEM column detector window. Courtesy of Oxford Instruments NanoAnalysis

The detector efficiency (Fig. 7.22) accounts for all these effects and represents the fraction of x-rays that are transmitted through the window system relative to the incident intensity as a function of energy. Windowless Si(Li) detectors and those used in the SDDs still show an important drop in efficiency at low energy due to the presence of the dead-layer and metallization in front of the intrinsic active portion of the detector. This applies even for SDD due to the nonuniform metallization grid shown in Fig. 7.20a-ca. Even for UTW and ATW detectors, the efficiency drops significantly for low-energy x-rays resulting in difficulties for the analysis of light elements in low concentration. Discontinuities in the detector efficiency are visible at the energies corresponding to the absorption edges of the elements contained in the window material (C, O, B, N for example), the detector, and metallization. As discussed above, metallization is required for the application of a bias on the semiconductor crystal but it is also necessary on the window material to prevent light (generated by some samples by cathodoluminescence) entering the detector system and resetting the signal amplification system.

Fig. 7.22

Detector efficiency curves for various materials used as detector windows. Windowless (solid), polymer UTW (dashed), Be window (dotted), and atmospheric UTW (dashed-dotted). After [7.2]

Besides the fact that high-purity Ge (HPGe ) crystals can be produced and no Li additions are required, Ge-based detectors offer the advantage that the absorption of x-rays is stronger (1 e–h pair/\({\mathrm{2.9}}\,{\mathrm{eV}}\)) and higher energy lines can be analyzed as the related high-energy x-rays are not transmitted through the crystal. The width of the x-ray peaks is narrower for Ge detectors than Si(Li) detectors leading to better sensitivity and less overlap in measurements. Some AEM systems are therefore equipped with a combination of both Si(Li) and HPGe detectors for a more efficient analysis of x-rays from a larger range of elements.

One drawback of SDD is their thickness of about \(300{-}500\,{\mathrm{\upmu{}m}}\) which does not allow the high-energy lines to be fully absorbed in the detector. These provide, therefore, reduced efficiency for high atomic number elements, with a drop in efficiency above about \({\mathrm{10}}\,{\mathrm{keV}}\).

Signal Processing

The e–h pair-generated current is detected by the FET as a pulse signal that is subsequently fed into a main amplifier system as a voltage. The sequence of pulses, separated by a time interval, generates a staircase signal where each step represents a photon arrival and the height is linked to the energy of the photon. After the integrated signal reaches a threshold level, the FET must be reset to a base value by means of a light pulse generated by a light-emitting diode in a opto-electronic feedback system. This is necessary to avoid saturation of the signals. Each step rise lasts in the order of \({\mathrm{150}}\,{\mathrm{ns}}\). Pulses can be amplified and shaped for subsequent analysis (to determine exact height and thus photon energy) with analog technology. In Si(Li) detectors, analog systems give the user flexibility on the process time of the pulse and thus accuracy in the signal analysis. High processing speeds of pulses (in the order of a few \(\mathrm{\upmu{}s}\) process time per pulse) result in low energy resolution of the peaks owing to the uncertainty in the pulse height and thus the energy of the photon. Low processing speed (about \({\mathrm{50}}\,{\mathrm{\upmu{}s/pulse}}\)) results in more accurate determination of the pulse height and more accurate determination of the x-ray energy. During analysis of the pulses, the detector is effectively not able to process more photons entering the detector resulting in analysis dead-time, which represents the time the detector is not processing signals. Because of this limitation, the output count rate is not linear with input count rate at high x-ray fluxes. In Si(Li), count rates imposing detector dead-times in the order of \({\mathrm{60}}\%\) used to be typically accepted for modern systems and exhibit a nearly linear response. Above \({\mathrm{60}}\%\) dead-time, there is a drop in the output rate with an increase in the input rate. Therefore, with thick samples and thus large photon fluxes, high processing speeds are required to reduce the process time and the resulting dead-time. Recent developments have allowed much faster pulse processing with digital technology resulting in higher throughputs of signals and linear response of the system with respect to the input count rates. With digital technology, the voltage rise output from the FET is directly digitized and can be subsequently processed with numerical pulse processing techniques leading to reduced noise and better high-count rates response. Details of the various tests and procedures to determine linearity of the system response and examples are given in [7.2, 7.29]. For SDDs, high count rates in the TEM are rarely an issue but often users take advantage of the high throughput by using relatively thick samples (few hundreds of nm). In these conditions, elemental mapping can be done almost in real time. Accurate quantification of material characteristics with the SDD still requires careful understanding of the artifacts involved. When count rates are high (in the range of above \({\mathrm{10}}\,{\mathrm{kcount/s}}\) at low processing speed) broadening of the lines can occur. However, at these count rates, the resolution, defined as the FWHM of a peak, is still about the same as an Si(Li) operated at count rates of \(1{-}2\,{\mathrm{kcount/s}}\). Coincidence events can also occur and these refer to the condition where the charge pulse from two or more x-rays reaches the anode sufficiently close in time that the electronics cannot distinguish the resulting pulse from a single x-ray with an energy equal to the sum of the incident x-ray energies [7.23]. In practice, these events are very rare in very thin films but become more likely as users prepare thick samples when it is desirable to acquire very fast maps and the spatial resolution is not a concern. The coincidence peaks can occur at twice the energy of the most characteristic peak or at the sum energy of any pair of characteristic peaks. A more serious scenario is when the coincidence events are due to continuum x-rays with characteristic peaks or other continuum x-rays because such peaks can occur at any energy, making it difficult to discern discrete individual peaks.

The coincidence events can be avoided (and in some cases even eliminated) by adopting the following protocols. The best way is to employ pulse-processing electronics with the smallest pulse-pair resolving times, i. e., the time it takes for the electronics to distinguish sequential events. Typically, in the modern SDDs the resolving time is in the order of tens to hundreds of nanoseconds. Whenever a coincident event is detected by the electronics the live time is readjusted by discarding the event. The second method to avoid coincidence events is to reduce the input count rate since the probability of coincidence events is proportional to the input count rate. In the conventional Si(Li) detectors, a dead-time between 30 to \({\mathrm{50}}\%\) would work. However, it appears that for the SDD detectors an estimate of optimal dead-time needs to be set on a case-by-case basis [7.23]. For the fastest processing times, count rates up to \({\mathrm{400}}\,{\mathrm{kcount/s}}\) can be used while maintaining the linearity in the response of the detector (although with compromises in terms of resolution).

Peak Shapes

The intrinsic width of an x-ray emission line is in the order of \(1{-}2\,{\mathrm{eV}}\). The width of the x-ray peaks as processed by the detector system, however, depends on the generation of e-hole pairs and the noise introduced during the measurement process. A peak in the spectrum represents the distribution of x-rays detected at a given energy with each incident photon generating a variable number of e–h pairs due to statistical fluctuations in the generation process. The standard deviation of the number of e–h pairs produced is one of the factors affecting the width of the peaks. The noise of the detector system and detector collection artifacts, however, also contribute to the broadening. The broadening due to the statistical generation of e–h pairs is given by \(\Updelta E_{\mathrm{s}}={\mathrm{2.35}}\sqrt{\varepsilon FE}\) with \(\varepsilon\) representing the energy for e–h pair creation (\({\mathrm{3.8}}\,{\mathrm{eV}}\) in Si, \({\mathrm{2.9}}\,{\mathrm{eV}}\) in Ge), \(E\) the energy of the x-ray peak, and \(F\) is a parameter representing the statistical correlation in the e–h pair-generation process known as the Fano factor which varies between 0 and 1. \(F=1\) if there is no correlation at all between the e–h generation events and \(F=0\) if the processes are completely deterministic (the process is completely reproducible and yields the same result time after time). Noise and artifacts also contribute to the broadening yielding a total broadening of a Gaussian peak distribution \(\Updelta E=\text{FWHM}\) of the x-ray peaks as
$$\Updelta E^{2}=(\Updelta E_{\mathrm{s}})^{2}+(\Updelta E_{\mathrm{N}})^{2}=(2.35)^{2}\varepsilon FE+(\Updelta E_{\mathrm{N}})^{2}\;.$$
(7.4)
For Si(Li), \(F={\mathrm{0.12}}\) and the FWHM of the Mn \(\mathrm{K}_{\alpha}\) line (used as a reference for resolution because of availability of radioactive standards producing Mn \(\mathrm{K}_{\alpha}\) peaks) is around \(138{-}140\,{\mathrm{eV}}\). If the noise contributions were completely removed, the theoretical resolution would be solely based on the e–h-generation process and would be around \({\mathrm{110}}\,{\mathrm{eV}}\) for the Mn \(\mathrm{K}_{\alpha}\) peak. The energy width decreases at lower energy with FWHM below \({\mathrm{100}}\,{\mathrm{eV}}\) for light elements. Since the noise term is not linear with energy and depends on the processing time of the amplification system, a full prediction of the peak resolution depends on the operating conditions and energy. Reference values given for the Mn \(\mathrm{K}_{\alpha}\) lines are used to compare the electronics and detector performance and are usually given at optimum process time with typical values around \({\mathrm{140}}\,{\mathrm{eV}}\) FWHM at the Mn \(\mathrm{K}_{\alpha}\) line. Improved energy resolution is achieved even for high count rates on digital detectors and with HPGe detectors due to the lower \(\varepsilon\) values (FWHM for Mn \(\mathrm{K}_{\alpha}\) is around \({\mathrm{120}}\,{\mathrm{eV}}\)). Noise introduced by vibrations (e. g., mechanical coupling with environment and/or ice crystals floating in the dewar) can also contribute to peak broadening, hence lowering the spectral resolution, and should be minimized. For SDDs, the FWHM at the Mn \(\mathrm{K}_{\alpha}\) is in the range of \({\mathrm{130}}\,{\mathrm{eV}}\), again depending on the processing conditions.

Detector and Signal Processing Artifacts

A summary of detection artifacts is presented here with further details given in more extended reviews [7.2, 7.29]. In perfect detector conditions, the peak shape is expected to be Gaussian but small distortions can arise if the generation of e–h pairs is perturbed. For example, recombination of e–h pairs in the dead-layer or at lattice defects generated by high-energy incident electrons accidentally entering the crystal can give rise to a phenomenon known as incomplete charge collection. This effect gives rise to low energy tails in the peak distribution as not all the e–h pairs are collected.

Incident x-rays can cause fluorescence of the Si \(\mathrm{K}_{\alpha}\) line (or a Ge line). If Si \(\mathrm{K}_{\alpha}\) photons are not absorbed within the detector and exit the active area, incident photons will have lost a fraction of their energy in this process equivalent to the Si \(\mathrm{K}_{\alpha}\) ionization energy (\({\mathrm{1.74}}\,{\mathrm{keV}}\)). This will cause an escape peak in the spectrum at an energy \(E_{\text{es}}=E-E_{\text{Si}\,{\mathrm{K}}_{\alpha}}\). This effect is particularly important when small trace elements are investigated since there is potential overlap between \(E_{\text{es}}\) and x-ray lines (for example Fe overlaps with the escape peak of the Cu \(\mathrm{K}_{\alpha}\) line).

If the count rate is high, there is the possibility that two incident photons of energy \(E\) will be perceived by the pulse counter as one single photon of energy \(2E\). This effect is known as a sum peak visible when count rates are above the reliable limit of the system (which of course varies depending on the processing technology). High count rates, leading to dead-times greater than around \({\mathrm{60}}\%\), are likely to lead to sum peaks.

Internal fluorescence peaks can also be detected if the incident photons generate an Si (or Ge) \(\mathrm{K}_{\alpha}\) peak in the dead-layer of the detector which is subsequently detected in the active area of the detector. This effect is small but can, once again be significant for trace analysis.

High-energy incident electrons can also generate spurious signals and damage the semiconductor crystals. The location of the detector and the operation of the microscope should be such that these contributions are minimized (for example, objective apertures must be removed during acquisition).

Geometry of the EDXS Detector in the AEM

In order to optimize the solid angle and thus the collection of the x-ray radiation generated by the incident electron, the detector is placed as close as possible to the sample area (Fig. 7.23a-ca). The Si(Li) detector active area \(A\) is typically \({\mathrm{10}}\,{\mathrm{mm^{2}}}\) with some systems as large as \({\mathrm{30}}\,{\mathrm{mm^{2}}}\). For modern SDDs, areas as large as \(80{-}100\,{\mathrm{mm^{2}}}\) are commercially available. For a detector positioned at a distance \(R\) with respect to the optic axis (and the origin point of the emission) the solid angle \(\Omega=A/R^{2}\) (measured in steradians) is the key parameter determining how effective the system collects emitted x-rays. For an optimal solid angle, the detector normal is in direct line of sight to the emission point and not tilted away from it. Typical solid angles in vintage (i. e. pre-SDD detectors) AEM are \({\mathrm{0.13}}\,{\mathrm{sr}}\) (for \({\mathrm{10}}\,{\mathrm{mm^{2}}}\)) but with combinations of large detector areas and effective coupling with the microscope specimen area, solid angles in the order of \({\mathrm{0.3}}\,{\mathrm{sr}}\) have been achieved with single SSD. For \(\Omega={\mathrm{0.13}}\,{\mathrm{sr}}\) one can see that the fraction of collected x-rays with respect to the full emission solid angle is only \({\mathrm{1}}\%\)! The detection of x-rays is therefore a very inefficient process considering that the x-ray emission is fully isotropic. With multiple SDDs in modern AEMs, solid angles typically range between \(\mathrm{0.6}\) and \({\mathrm{1}}\,{\mathrm{sr}}\) (Fig. 7.23a-cb). Shadowing effects from the sample holder (Fig. 7.23a-cc) also have the potential to reduce even further the collection efficiency.

Fig. 7.23a-c

Interface of the detector with the microscope sample area. (a) For a single detector, after [7.30], and for (b) multiple detectors with related solid angles and shadowing regions of detectors 1 and 2 due to the tilt angle \(\theta_{\mathrm{T}}\) of the sample. (c) Shows the shadowing effects from the sample holder. After [7.31]

The elevation angle (also known as take-off angle in the literature) is an important parameter affecting the quantification of data through the absorption correction and the quality of the spectra. A large elevation angle minimizes the path length of x-rays into the sample (see Sect. 7.4.1, Quantification in EDXS) and also reduces the continuum background emission which is forward peaked. High detector elevation angles, however, are impractical in the TEM due to the fact that the detector would need to be above or within the objective lens at a large distance from the sample resulting in even lower collection efficiency. In addition, backscattered electrons have direct sight to the detector and can cause significant contributions and potential damage to the detector. Lower elevation angles (\(0{-}20^{\circ}\)), allow larger solid angles and lead to an effective shielding of the backscattered electrons by the objective lens magnetic fields. This shielding is not as effective for high elevation angles.

The interest in large solid angles and the proximity of the detector to the sample lead to significant drawbacks in terms of spurious signal collection. The field of view of the detector is much larger than the sample area (Fig. 7.23a-ca) and x-rays generated by backscattered electrons or by fluorescence of hard x-rays generated in upper parts of the illumination area of the microscope, easily enter into the detector (Sect. 7.7.2 on instrumental contributions). High-energy backscattered electrons can also enter the detector and generate additional secondary electrons/x-rays whilst low-energy electrons would spiral away from the detector due to the high magnetic field of the objective lens or the presence of a magnetic electron trap in the detector system (Fig. 7.19a,b). To reduce these effects, detectors are equipped with collimators that limit the field of view to the smallest possible area, thus preventing hard x-rays generated in the illumination system from directly hitting the detector, and contain baffles that reduce the effects of potential incident backscattered electrons that might enter the collimation system (Fig. 7.23a-ca).

Many other contributions arising from stray electrons hitting the microscope components such as apertures, cold traps, the polepieces etc. lead to increased noncharacteristic signals resulting in weak detection limits. As demonstrated in the work of Nicholson et al [7.32], many of these contributions can be reduced by improving the microscope and detector chamber using coatings to cover the microscope components with low atomic number materials and by improving the collimation system. These effects can be minimized in systems using the precautions discussed in Sect. 7.7.2.

For multiple detectors within the AEM, now possible using the SDD technology, the situation is more complex. Figure 7.23a-cb illustrates the modern EDS detector geometry based on an FEI Super-X design [7.31]. This design incorporates four windowless SDD detectors that are symmetrically placed around the optic axis, usually termed as the quad-array detector . Assuming a nominally rectangular-shaped active area for these individual detectors the collection solid angle (\(\Omega\)) can be obtained as follows [7.33]
$$\begin{aligned}\displaystyle\Omega&\displaystyle=(1-f_{\mathrm{s}})\,4\,\arcsin(\sin\alpha\,\sin\beta)\;,\\ \displaystyle\alpha&\displaystyle=\arctan(w/2d)\;,\\ \displaystyle\beta&\displaystyle=\arctan(h/2d)\;.\end{aligned}$$
Here \(w\) and \(h\) are the width and the height of the detector active area, and \(d\) is the distance from a point of interest on the sample. The maximum theoretical solid angle that is achievable is \(2\uppi\) sr. However, the edges of the detectors are normally rounded so as to accommodate for other physical design constraints, and the resulting decrease in the active area can be accounted for in the \(f_{\mathrm{s}}\) prefactor. The \(f_{\mathrm{s}}\) prefactor can also account for any physical shadowing of the detector area due to ancillary windows and support structures such as the collimator system. As illustrated in Fig. 7.23a-cb, the limits of heights of the shadow created on the detector are defined by \(\phi_{\mathrm{U}}\) and \(\phi_{\mathrm{L}}\) in the literature. Additionally, the specimen holder itself can shadow the detector over certain angular ranges (i. e., tilts and shifts) which cannot be accounted for from the calibration alone [7.31]. For example, a holder tilted to \(15^{\circ}\) from the optic axis as illustrated in Fig. 7.23a-cb creates a partial shadow on the detectors \(\mathrm{1}\) and \(\mathrm{2}\) in the quad-array detector setup. In addition to shadowing, the complexities of electron channeling in the specimen and the effect on x-ray throughput cannot be ignored. All these factors pose a major challenge to absolute EDS quantification, particularly at atomic levels, and some recent efforts to model these relationships seem promising [7.34]. Apart from the quad-array design described above, alternative combinations of more detectors and a few other variations of such arrays have also been proposed with the ultimate objective of improving the collection efficiency of the detector. However, the limited physical space available in the microscope column and the necessary collimation requirements can be major determining factors. The geometry is not only important for collection efficiency but as discussed further below, it will be essential for the quantification aspects of analysis.

7.2.4 EELS

Spectrometers

The measurement of energy losses suffered by the incident electrons as they exit the sample is carried out with energy-loss spectrometers. These devices also make it possible to select electrons with a particular energy loss (or no loss at all) with the use of energy-selecting slits. The capability of selecting electrons with a particular energy loss is called energy-filtered microscopy. The technique also allows the operator to obtain images and diffraction patterns where parts of the inelastically scattered electrons are filtered out so that information deriving from the elastically scattered electrons only is used.

These instruments are based on the use of a magnetic field that modifies the trajectory of the electron according to the electron energy. The radius of curvature \(R_{\mathrm{e}}\) of the electron trajectory is related to their velocity \(v\) and magnetic field strength \(B_{\mathrm{f}}\) as
$$R_{\mathrm{e}}=\frac{\gamma m_{0}}{eB_{\mathrm{f}}}v\;,$$
(7.5)
where
$$\gamma=\frac{1}{\sqrt{1-v^{2}c^{2}}} $$
is the relativistic factor and \(m_{0}\) is the rest mass of the electron. Slower electrons will follow trajectories with a smaller radius of curvature and will be dispersed on a detector plane located after the spectrometer. The dispersion refers to the separation of electron energies in space and is typically in the order of \(1{-}2\,{\mathrm{\upmu{}m/eV}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\). The exact location of this detector depends on the implementation of the spectrometer and its coupling with the microscope column (see below). The most common electron-optical component generating the magnetic field is a magnetic sector (used in various configurations, whether the spectrometer is implemented within the microscope column or after the viewing chamber of standard TEMs). Current flow in the prism generates the required field \(B_{\mathrm{f}}\) that disperses the electrons (Fig. 7.24). Other approaches to filtering have been implemented in prism-mirror spectrometers (see below) and Wien spectrometers which combine magnetic and electrostatic fields in different configuration [7.35].
Fig. 7.24

Prism spectrometer system showing the bending of electrons as they travel through the spectrometer. Dispersion of electrons according to their energy is achieved by the spectrometer in one direction and focusing is achieved in the other direction of travel

One should distinguish between two types of energy-filtering spectroscopy approaches (Fig. 7.25a,b). The so-called in-column filter/spectrometers are located within the projector lens system/postspecimen area of the microscope column. These spectrometers generate electron trajectories and dispersion that will result in the transfer of the electrons into the projector lens system and viewing chamber of the microscope. The alternative approach is realized with the postcolumn spectrometers/filters attached at the bottom of the microscope column. In the case of in-column filters, energy-loss spectra and/or energy-filtered images (obtained by selection of electrons of a particular value of energy loss using a slit) are realized. For postcolumn spectrometers, dedicated imaging lenses are required to generate energy-filtered images after selection of electrons of a particular energy loss. Two implementations of postcolumn spectrometers therefore exist. For acquisition of spectra only, the magnetic prism is followed by a series of optical components dedicated to increase/vary the dispersion at the detector system. For energy-filtered imaging, a more elaborated series of nonround lenses (i. e., based on multipoles) and a removable energy-selecting slit are used to provide both spectroscopy and imaging capabilities.

Fig. 7.25a,b

Schematic diagrams of energy-filtered electron microscopes. The postcolumn configuration (a) is based on the simple prism attached at the bottom of the microscope while the in-column configuration (b) is achieved by various components inserted in the projector lens system

There are various implementations of in-column filters. The earliest commercial applications were based on the electrostatic mirror-prism system initially proposed by Castaing and Henry [7.36] implemented in the Zeiss microscope (Fig. 7.26a-ca). These instruments were developed on \(\mathrm{80}\)-\(\mathrm{keV}\) microscopes and thus remained very popular for biological applications although excellent fundamental electron-scattering experiments were carried out on such instruments [7.37]. The first portion of the prism is used for an initial dispersion, the electrostatic mirror's function is to deflect the electrons back towards the second section of the prism and thence to the optic axis. The mirror voltage must be close to the accelerating voltage of the microscope. After the mirror-prism, electrons are dispersed and continue to travel down the optic axis of the microscope. Energy-filtered images are obtained by allowing electrons to pass through the slit and the projector lens system. Energy-loss spectra, angular-resolved energy-scattering diagrams (showing the energy-loss distribution as a function of scattering angle) and filtered images and diffraction patterns can be obtained by careful selection of the operating conditions of the microscope and cross-over points. This can be achieved by selecting, with the microscope postspecimen lenses (objective, intermediate), the object point entering the spectrometer and the transfer of the cross-over points on the viewing screen. Subsequent implementations of the in-column filters in higher voltage instruments (\(\mathrm{100}\), \({\mathrm{200}}\,{\mathrm{keV}}\) microscopes) are based on Omega-type spectrometers that use four magnetic sectors (Fig. 7.26a-cb) generating the dispersion and transfer of electrons back onto the optic axis of the microscope. Aberrations of the spectrometer that lead to loss of resolution in spectra and generate nonuniformities in the energy distribution of electrons in images are reduced by a combination of design of the magnetic sectors entrance and exit faces, the symmetry of the configuration (the fact that the aberration of the first two sectors is compensated by the aberrations in the opposite direction of the third and fourth sectors), and the use of series of multipoles within the path of the electrons. These filters are introduced within the projector lens system (Fig. 7.27). The last implementation of in-column spectrometers is the Mandoline filter (Fig. 7.26a-cc). This filter generates larger dispersions (a factor 3 larger than Omega) and is optimized for lower aberrations. This system is ideally suited for energy filtering with very narrow energy windows and large fields of view. Unfortunately, all these in-column implementations proposed by Zeiss are no longer commercially available. An in-column filter implementation of the Omega-type system has also been made available by the company JEOL.

Fig. 7.26a-c

Various in-column spectrometer configurations. (a) Mirror-prism spectrometer, (b) Omega filter, and (c) Mandolin filter. Diagrams courtesy of Zeiss

Fig. 7.27

Schematic diagram of the in-column Omega-type energy-filtered Zeiss Libra microscope. Courtesy of Zeiss

Images, diffraction patterns (filtered/nonfiltered), and spectra for in-column filters could be observed directly on the viewing screen of the microscope and recorded with analog techniques (on negatives), imaging plates, or a digital camera following conversion of the incident electrons to photons using a scintillator material (YAG, phosphor etc.).

For postcolumn spectrometers/imaging filters, the technology is based on the magnetic sector (Fig. 7.24). Aberrations of the prism can be minimized through design of the spectrometer entrance and exit faces and the use of multipole correcting elements before the prism. Only one prism generates the required dispersion to form a spectrum in the dispersion plane. The early spectrometers were used to generate energy-loss spectra by making use of a serial detection system where the spectrum is scanned, using an electrostatic field, in front of an energy slit. The number of electrons (or the current) entering through the slit is subsequently measured with a scintillator detector and photomultiplier with pulse counting or current measurement methods. This serial detection process (one energy recorded at a time) is extremely inefficient for recording large energy ranges (several seconds/minutes acquisition times) and therefore parallel detectors were developed in the late 1980s [7.38] to record a portion of the spectrum of \(\mathrm{1024}\) energy channels. For this early technology, the slit is replaced by a series of multipoles lenses (three quadrupoles) to focus and magnify the spectrum (Fig. 7.28). These optical elements change the prism dispersion (from about \({\mathrm{1.8}}\,{\mathrm{\upmu{}m/eV}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\)) up to \({\mathrm{1}}\,{\mathrm{mm/eV}}\) at the detector plane [7.38]. Incident electrons are converted to photons using a scintillator material (e. g., YAG) and a photodiode array (with \(\mathrm{1024}\) channels) is used as the recording system with acquisition times as short as \({\mathrm{25}}\,{\mathrm{ms}}\). Recent commercial spectrometers have more complex optics (better focusing capability and aberration correction of the magnification lenses), new scintillator materials with improved transfer function, and improved transfer of the generated light between the scintillator and detector. Fast-readout two-dimensional arrays (\({\mathrm{100}}\times{\mathrm{1200}}\) pixels, or twice this number) allow acquisition of more than one hundred spectra per second in commercial systems (the ENFINA/Enfinium spectrometers from Gatan). Noncommercial systems have achieved the same readout performance and used optical lenses to transfer the signal generated in the scintillator to two-dimensional detectors.

Fig. 7.28

Schematic diagram of the parallel EELS spectrometer. After [7.38]

For the acquisition of energy-filtered images using postcolumn filters, a series of multipole lenses (Fig. 7.29) is used to transform the spectrum at the slit plane back to an image (or diffraction pattern) at the detector plane and to correct the image distortions and aberration. This transformation is possible because both the in-column filter and the postcolumn prism are electron-optical components that produce, at the dispersion plane, spectra that contain information on the object that enters the spectrometer. The slit allows selection of the electrons with the energy loss of interest that is subsequently used to form an image at the detector plane. As for the in-column filters, both images and diffraction patterns can be obtained in the detector plane of postcolumn filters depending on whether an image or diffraction pattern is projected at the entrance plane of the spectrometer. Worth mentioning is the use of filters to remove inelastically scattered electrons from diffraction patterns (Fig. 7.30a,b) of thick samples to enhance the visibility of the dynamic structure in the convergent beam disks and to retrieve quantitative information on structure factors [7.40]. Further details of the optical function, aberration of the spectrometers and filters can be found in [7.35, 7.41, 7.42, 7.43] and Chap.  13 by Hawkes and Krivanek.

Fig. 7.29

(a) Postcolumn imaging filter schematic diagram of the electron optics components and detection system. (b) Actual spectrometer (Tridiem™ model). (c) The more recent Quantum™ spectrometer. Courtesy of Gatan, Inc. [7.39]

Fig. 7.30a,b

Energy-filtered electron diffraction pattern of Si(110) orientation. (a) Pattern recorded by selecting only the electrons that have lost no energy (zero-loss ( ) filtering) and (b) pattern recorded without energy filtering

Alignment in postcolumn filters is carried out using automated routines that tune the multipole lens functions so as to minimize distortions, aberrations, and the uniformity of the energy within the field of view so that every point in an image corresponds to the same energy loss (the isochromaticity). The detection of images in postcolumn filters can only be carried out using scintillators, CCD cameras and more recently direct electron detectors.

From a practical point of view, in-column energy filters offer the advantage that energy-filtered images can be directly observed on the microscope viewing screen while the postcolumn filters offer the advantage of being added to a microscope column as a optional attachment. As various implementations and models of in-column and postcolumns exist, a detailed comparison of the two filtering techniques should be carried out cautiously, based on the type of application of interest and thus the specific relevant parameters for that application.

Electron Monochromators

Based on the interest in energy-loss near-edge structures and for the purpose of increasing the temporal coherence terms of the imaging transfer function, the energy spread of the electron energy source must be decreased. One approach is to use electron guns with narrower energy distribution (e. g., Table 7.3 in Sect. 7.2.1). This approach is limited to about \({\mathrm{0.3}}\,{\mathrm{eV}}\) energy distribution with cold FEG as measured at the FWHM of the energy distribution. For cold FEG, the distribution is not symmetrical due to the nature of the tunneling process which follows a Fowler–Nordheim distribution . Although this energy width allows the acquisition of good quality energy-loss spectra where the intrinsic broadening of the features is in the order of \({\mathrm{0.3}}\,{\mathrm{eV}}\) or greater, there are cases when spectra obtained with electrons having a narrower energy distribution is of interest [7.44, 7.45, 7.46, 7.47]. This can be achieved with gun monochromators, and improvements in microscope stability and spectrometer resolution [7.48]. Improvements in energy resolution to \({\mathrm{0.1}}\,{\mathrm{eV}}\) have been obtained in commercial instruments based on various approaches (Fig. 7.31a-d). Four types of monochromators have been commercially developed based on:
  1. 1.

    The Wien spectrometer [7.49]

     
  2. 2.

    The double-focusing Wien spectrometer [7.50]

     
  3. 3.

    The Omega-type filter [7.51]

     
  4. 4.

    The Alpha-type systems [7.52].

     
Fig. 7.31a-d

Various implementations of monochromators in commercially available instruments. (a) The FEI monochromator, single Wien filter. (b) The JEOL monochromator double-Wien filter. (c) The electrostatic Omega filter implemented in the Libra Zeiss microscope. (d) The Alpha monochromator from NION

The single-Wien filter consists of an electron-optic device in which the incident electron trajectory is dispersed by the action of a magnetic field and an electrostatic field perpendicular to each other and the energy selection is achieved by a mechanical slit (used to mechanically select a portion of the dispersed electrons). The double-Wien filter is a variant of the single-Wien design where an additional Wien filter is placed after the energy-selecting slit. Here the use of a second filter eliminates the energy dispersion at the conjugate plane of the electron source. The cancelation of energy dispersion is of particular importance to ensure that the brightness is preserved in the beam incident on the specimen [7.52].

The Omega-type monochromator involves a more complicated arrangement where the use of four electrostatic toroidal deflectors defines an omega-shaped electron path. One common feature among these three monochromators (single-Wien, double-Wien, and Omega-type) is that they are all mounted between the electron gun and the acceleration tube. This feature provides some advantages in particular because the monochromated beam can be obtained at various acceleration voltages under the same monochromator conditions. However, one disadvantage is that the length of the electron path in the latter part of the column is long and involves multiple beam crossovers, which can lead to Coulomb repulsion (Boersch effect), often resulting in energy broadening. One recent development that is designed to avoid this issue is to mount the monochromators after the acceleration tube, as in the case of the Alpha-type magnetic monochromator design [7.52, 7.53]. It is argued in the literature that this allows for large beam currents to be sent through the monochromators without running into excessive energy and source-size broadening (note that the high-energy electrons are less susceptible to Coulomb repulsion than low-energy electrons). The use of a combination of uniform- and gradient-field deflectors on either side of the energy-selecting slit results in an Alpha-shaped electron trajectory. The Alpha-type monochromator has similarities to the Omega-type, in the sense that the energy-selecting slit and energy-dispersion plane of the monochromators lie on the mid plane of the electron path. Furthermore, the Alpha-shaped monochromators use quadrupole lenses (arranged into six separate groups) to:
  1. 1.

    Magnify the energy dispersion at the slit

     
  2. 2.

    Cancel the dispersion for the beam re-entering the microscope column

     
  3. 3.

    Correct for the aberrations at the slit and in the re-entrant beam.

     
Typically, the energy resolution obtained with the use of such Alpha-shaped magnetic monochromators is just below \({\mathrm{10}}\,{\mathrm{meV}}\) [7.54], which is the highest attainable today. However, more systematic experiments are needed to quantify the ultimate brightness at a given energy resolution.

The effect of monochromation on the energy distribution of the incident electrons is shown in Fig. 7.32 and on core-loss spectra in Fig. 7.33a. Investigations of very low energy losses, in the realm of phonon excitation and molecular fingerprinting have been recently demonstrated as discussed in Sect. 7.8.2 of this chapter.

Fig. 7.32

Zero-loss peaks obtained with a monochromator switched on using a Schottky FEG source (with stabilized electronics) (solid line), the same instrument with the monochromator switched off (dotted line) and with a cold field-emission source (dashed line)

Fig. 7.33

(a) Effects of a monochromator (solid line) on the visibility of peaks of the Ti \(\mathrm{L_{23}}\) edge in \(\mathrm{CaTiO_{3}}\). Clearly resolved are the small triplet states in front of the first strong peak. (b) Comparison of spectra of the Ti \(\mathrm{L_{23}}\) edge in \(\mathrm{SrTiO_{3}}\) acquired with a regular CCD (dashed) and a direct detection camera (solid line) in the same microscope and spectrometer setting conditions optimizing the counts/pixel with the same energy resolution. (c) Schematic illustration of the modern Gatan K2 summit (DDC) which is also coupled with the Gatan US1000FTXP (IDC). Courtesy of Gatan, Inc

Direct Electron Detectors for EELS

Conventionally, the EELS spectrometers have employed indirect detection cameras ( ) for electron imaging. In most cases, the typical IDC consists of a scintillator and a charge-coupled device (CCD) that relies on fiber-optically coupled photons as the intermediate signal carriers separating the complementary metal-oxide semiconductor technology detector from the beam electrons. This approach is discussed in detail in the earlier sections and is available in most systems today because of the robust performance given the highly intense features in the spectra impinging on the detectors. Alternatively, the latest direct detection cameras ( ) count individual primary electrons (generating a large number of electrons within the pixel) directly impinging on the CMOS structure without the need of a scintillator, and fiber optics or lens coupling [7.55]. These cameras have had a significant impact in the area of cryo-electron microscopy but have not been used until recently for the measurement of EELS spectra where there are equally important benefits. In comparison to the conventional CCD cameras the DDCs offer a much narrower point-spread function ( ), low read-out noise, and the potential for higher frame rates. For example, Hart et al compared the FWHM of the zero loss peaks ( s) with a Gatan K2 summit (DDC) and a Gatan US1000FTXP (IDC) [7.56]. The ZLP detected using the DDC is much narrower (FWHM of about \({\mathrm{1.0}}\,{\mathrm{eV}}\)) than that detected by the conventional IDC (FWHM of about \({\mathrm{3.0}}\,{\mathrm{eV}}\)) using the same dispersion settings. Hart et al [7.56] further compared the Ti-L edge spectra acquired with these cameras. The features in the spectra acquired with a DDC are better resolved than with an IDC, which is apparent from the Ti-L\({}_{2,3}\) splitting visible in the former but absent in the latter (Fig. 7.33b).

Paying heed to the acquisition time is paramount in comparing the performance of DDC and IDC particularly for reducing the dose inflicted on the sample. For example, whereas the DDC yields a better signal-to-noise-ratio ( ) than IDC at very short acquisition times, this may not be true at longer acquisition times (e. g., \(\mathrm{0.1}\) versus \({\mathrm{10}}\,{\mathrm{s}}\)). This effect is attributed to the interplay between factors such as read-out noise, shot noise, and the electron dose [7.57]. At short acquisition times the read-out noise is said to be the dominant noise component. Hence, the DDC can provide highest SNR with its low read-out noise and correspondingly, higher detector quantum efficiency ( ; evident from the narrow PSF). In contrast, at longer acquisition times the shot noise is said to be the limiting factor, and since it is inherent in the predetected signal, the PSF is convoluted by the shot noise.

For EELS applications, the commercially available Gatan K2 DDC (\({\mathrm{3838}}\times{\mathrm{3710}}\) pixels in \({\mathrm{5}}\,{\mathrm{\upmu{}m}}\) pixel size) (Fig. 7.33c) can be operated under three different modes, namely the counting, super-resolution, and linear modes [7.56]. In the counting mode, the individual electrons are identified and digitized as a discrete count at a particular pixel. One issue at high-dose rates is that two or more electrons can hit an individual pixel, resulting in coincident losses, a condition where the camera is unable to distinguish between such single and multiple electron events. Thus, in order to minimize coincident losses high frame rates would be required for efficient operation in the counting mode. In the latest K2 DDC, the frame rate is about 10 times higher than that provided by regular cameras. Additionally, the DQE is also high, as much as \({\mathrm{80}}\%\) (at small exposure times). The operation of K2 in the super-resolution mode shares similar principles as the counting mode, except that by using a special algorithm the position of incident electrons is located to subpixel accuracy. One negative feature is that the size of the individual images can be extremely large and requires additional software setup to carry out routine handling and data processing.

Finally, the linear mode in K2 is similar to conventional IDCs where the charge carriers generated by the incident electrons are read out during a predefined exposure time to form an image. The main improvement here is that the read-out time in K2 DDC can be much shorter than IDC.

7.2.5 Sample Preparation Requirements

Although the quality of the microscope and vacuum are key elements of good AEM work, sample preparation and cleanliness are certainly factors affecting the quality of the data. In order to avoid sample contamination build up under the electron beam, plasma cleaning of the samples with dedicated systems prior to insertion into the TEM column has become a routine practice. This technique gently burns off all hydrocarbons built up on the surface of the sample so that diffusion of the species during analysis in the TEM is no longer possible. An alternative to this approach is to gently heat the samples prior to insertion on the TEM with a halogen lamp in vacuum (it is possible to build such a system with off-the shelf vacuum components) at temperatures of about \(70{-}80\,{\mathrm{{}^{\circ}\mathrm{C}}}\). Although this technique does not replace the plasma cleaning approach, it is a solution for samples that might be sensitive to the reactive gases used in the plasma process and often solves contamination problems due to the sample. Samples permitting, alternatives are also low-energy milling (few hundred eV ions) and prior sample insertion (with regular mills or dedicated low-energy systems). Cooling samples in a dedicated cryogenic analytical holder in the TEM also reduces the contamination buildup during analysis but at the expense of increased drift rate during very long analyses.

Removal of the amorphized layer produced by ion milling has become an increasingly used approach as ever thinner samples are used to achieve the ultimate spatial resolution in analysis. A combination of mechanical polishing and low-energy milling has proved excellent for obtaining very clean samples and minimal amorphous damage. Several manufacturers offer these new approaches to improve the quality of the samples. Often these methods make the difference between successful work, major scientific breakthrough, and plain disaster!

7.2.6 New Developments in Electron Optics

Since around 2002, there have been significant developments in commercially available electron optics components that have led to a new generation of microscopes with much finer probe-forming capabilities making use of aberration correctors (Sect. 7.2.1 and Chaps.  2 and  13). The early literature focused on point spectra acquisition with EELS [7.58, 7.59] with atomic-resolved maps demonstrated (initially without corrector by Kimoto et al [7.60]) in [7.61, 7.62, 7.63]. High-resolution EDXS has also been shown since the implementation of aberration correctors with the first results presented by [7.64, 7.65] and atomic-scale maps published by Chu et al [7.66]. These capabilities, in both EDXS and EELS, as well as the resolution limits and interpretation, are discussed in Sect. 7.9 of this chapter. While atomic-scale EDXS or EELS can be useful in specific experiments and can be quantitative [7.67], the main advantage of the aberration corrector of the probe-forming lens for AEM purposes is not just the ability to form smaller probes (although this is a distinct advantage) but rather the significant increase in electron-beam current for a given spot size (Fig. 7.15). It is expected that the widespread use of instruments equipped with such correctors will result in a dramatic improvement in the analytical performance of new AEMs as demonstrated in Sect. 7.9 of this chapter. However, users will be faced with new limitations as electron-beam damage will undoubtedly be the ultimate barrier in the analysis of the most interesting materials even though new approaches can perhaps be used to circumvent some of the problems or beam damage [7.68].

7.3 Fundamentals

7.3.1 Elastic and Inelastic Scattering

The interaction of primary electrons with electrons and nuclei in the solid results in various scattering processes and the generation of signals that can be detected in different analytical tools such as the transmission electron microscope, the scanning electron microscopes, or surface analysis instruments. We can classify these processes as elastic and inelastic based on the energy changes of the primary incident electrons following the scattering event. Elastic scattering caused no detectable change in the energy of the primary electrons within the resolution of the measurement system typically available in the TEM. These processes do not give rise to analytical signals in the strict sense of the term but are nevertheless very relevant to the understanding of signal generation, imaging, and all discussions on spatial resolution as they significantly affect the angular distribution of the incident electrons, their propagation in the solid and consequently the spatial spread of these electrons as they travel through the sample. Elastic scattering is at the basis of contrast mechanisms in TEM and STEM imaging and can directly affect the interpretation of energy-filtered images. Furthermore, the powerful technique of \(Z\)-contrast imaging (Chap.  2) is based on signals generated by electrons elastically scattered at high angles. The technique is commonly used in combination with analytical measurements using electron energy-loss spectroscopy or x-ray microanalysis and provides indirect information on changes in average atomic number within the area illuminated by the electron beam. Although traditionally this is not considered as an analytical technique, this imaging method, combined with spectroscopic measurements such as EDXS and EELS, provides the most powerful capability for the analysis of materials with the highest spatial resolution.

Inelastic processes are the key to all analytical measurements as they directly and indirectly give rise to the signals detected. Figure 7.34 summarizes the various energy-loss processes generating the excitation and the subsequent signal generation processes by de-excitation. Analysis techniques focus, on one hand, on the detection of the primary event of energy loss where excitation of single electrons from strongly bound core energy levels (inner-shell ionization) and weakly bound electrons (from the valence band) occurs. These excitations cause transitions of electrons from the deep bound states to levels just above the Fermi energy and the continuum. Energy losses for the primary electrons can also occur through collective excitation of weakly bound valence electrons in the solid behaving as an electron gas (known as plasmon excitations; Fig. 7.34) or from defect states within the gap of a material. In the case of EELS, the energy losses of the primary electrons range from a few eV to tens of eV (typically up to \(50{-}60\,{\mathrm{eV}}\)) in the case of single electron valence excitation and plasmon losses while core-losses range from a few tens of eV (typically down to \({\mathrm{30}}\,{\mathrm{eV}}\)) up to a few keV for the core-losses with significant overlap between these regions. Other processes of inelastic scattering leading to detectable signals in the spectra include the excitation of quasi-free single electrons (also known as Compton scattering ) and losses due to radiative phenomena (Cerenkov effect) .

Fig. 7.34

Various processes of inelastic scattering events from core levels, valence states, and defect states. Inelastic processes via collective excitation of valence electrons is also possible. Measurement of the energy losses of the primary electrons is possible with EELS. The de-excitation processes give rise to x-ray and Auger signals

On the other hand, secondary processes give rise to other signals used in analytical techniques. The de-excitation process leads to the generation of photons detected with EDXS or Auger electrons with related characteristic energies.

7.3.2 Elastic Cross Sections

As discussed above, some theoretical background on elastic scattering is useful to understand electron propagation in the sample and the related contributions on inelastic signal generation. A good understanding of elastic scattering is particularly useful to simulate electron trajectories, estimate the electron-beam broadening and thus the spatial resolution in analytical measurements. We will follow the description given in detail by Reimer [7.35, 7.69] using the terminology adopted by R.F. Egerton.

The differential elastic cross-section \(\mathrm{d}\sigma/\mathrm{d}\Omega\) representing the probability of scattering per unit solid angle \(\mathrm{d}\Omega\) is given by
$$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=|f|^{2}\;,$$
(7.6)
where the amplitude of the scattering factor \(f\) is directly proportional to the Fourier transform of the potential of the atom. Various models and approximations of the potential can be used. In its simplest form, the potential considers the unscreened electrostatic Coulomb potential of the type
$$V(r)=\frac{Ze}{4\uppi\varepsilon_{0}r}$$
(7.7)
for a free atom yielding the Rutherford cross section
$$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{4\gamma^{2}Z^{2}}{a^{2}_{0}q^{2}}\;,$$
(7.8)
where \(a_{0}\) is the Bohr radius, \(\gamma\) is the relativistic factor (\(\gamma^{2}=(1-v^{2}/c^{2})^{-1}\)), \(q=2k_{0}\sin(\theta/2)\) represents the amplitude of the scattering vector \(\boldsymbol{q}\) shown in Fig. 7.35, and \(\hbar\boldsymbol{k}_{0}=\gamma m_{0}\boldsymbol{v}\) represents the momentum of the incident electron; \(\boldsymbol{q}\) is the scattering vector defined in Fig. 7.35 and \(\hbar\boldsymbol{q}_{0}\) is the momentum given to the nucleus.
Fig. 7.35

Diagram for an elastic scattering process

This first approximation of the unscreened electrostatic Coulomb potential satisfactorily describes scattering for light atoms, for large scattering angles, and high incident electron energy (typical of TEM). The model, however, needs refinements for elements of high atomic number, for small scattering angles and low incident energies (in low voltage SEM for example) because of a singularity in the cross sections at \(\theta=0\) arising from the neglect of screening. The Rutherford model can be subsequently refined by incorporating the exponential screening factor used in the Wentzel atomic model
$$V(r)=\frac{Ze}{4\uppi\varepsilon_{0}r}\exp\left(-\frac{r}{r_{0}}\right),$$
(7.9)
where \(r_{0}\) is the screening radius that can be estimated by \(r_{0}=a_{0}Z^{-1/3}\).
This potential and the screening radius lead to a Lenz model of differential cross section
$$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{4\gamma^{2}}{a_{0}^{2}}\left(\frac{Z}{q^{2}+r_{0}^{-2}}\right)\approx\frac{4\gamma^{2}Z^{2}}{a^{2}_{0}k^{4}_{0}}\frac{1}{\left(\theta^{2}+\theta^{2}_{0}\right)^{2}}\;,$$
(7.10)
where \(\theta_{0}\) is the characteristic elastic scattering angle representing the width of the angular distribution of elastic scattering.
The total scattering cross section is obtained by integrating (7.10) over all scattering angles
$$\sigma_{\mathrm{e}}=\int^{\uppi}_{0}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}2\uppi\sin\theta\mathrm{d}\theta=\frac{4\uppi\gamma^{2}}{k^{2}_{0}}Z^{4/3}\;.$$
(7.11)
A more accurate model of the potential considering electron spin and relativistic effects leads to the Mott cross section applicable to a large range of atomic numbers, low voltages, and scattering angles.
On the basis of elastic cross sections and the number of atoms per unit volume \(n_{\mathrm{a}}\) we can define a quantity called the elastic mean free path
$$\lambda_{\mathrm{e}}=(\sigma_{\mathrm{e}}n_{\mathrm{a}})^{-1}$$
(7.12)
which represents the mean distance between elastic scattering events in a solid assumed, for simplicity, to be amorphous.

In crystals, the elastic scattering must account for diffraction effects and the treatment of the electron distribution in the solid follows the diffraction theory developed elsewhere in this handbook (see Chaps.  1 and  18). With the development of \(Z\)-contrast imaging as one of the tools available in the analytical electron microscope and use of this technique with electron probes of diameter smaller than the unit cell, one should be aware of the characteristics of the electron-beam propagation in the sample. Detailed calculations with the multislice technique have shown that when a small probe is positioned on an atomic column, electrons are channeled along the column for samples of thicknesses up to a few nm. When the beam is positioned in between the columns it strongly disperses and broadens while propagating in the sample [7.70] (Fig. 7.36a,b). This effect is probe size-dependent with the channeling conditions being more relaxed with larger probes of \({\mathrm{0.2}}\,{\mathrm{nm}}\). Calculations show that the electron intensity builds up in the adjacent columns as the electron beam propagates (Fig. 7.36a,bb) [7.71]. The effect is demonstrated in other calculations [7.71, 7.72] showing that the beam can channel along an adjacent column and give rise to a signal even if not nominally on the initial position (Figs. 7.36a,bb and 7.37a,bb). Although these effects are not part of the traditional AEM literature, they are becoming increasingly relevant when the very smallest probes produced by aberration-corrected instruments are used to achieve the ultimate spatial resolution in very thin samples (Sect. 7.5.1) and when the sample is oriented along a zone axis to obtain apparent atomic-scale maps. While it is clear that such maps contain atomic resolution information on very thin samples, where the beam propagation occurs in one single atomic column, this is not the case for moderately thick samples above a few tens of nm because of the convergence and channeling into neighboring columns. Such effects, including contributions from thermal diffuse scattering on the propagation and apparent resolution are discussed in the application section (Sect. 7.9).

Fig. 7.36a,b

Real-space intensity distribution of the probe electron density in the sample as it propagates through the thickness of the foil. (a) Plots of the intensity distribution at two depths (\(\mathrm{100}\) and \({\mathrm{500}}\,{\mathrm{\AA{}}}\)) are shown for incident probe sizes of \(\mathrm{2}\), \(\mathrm{1.4}\), and \({\mathrm{0.7}}\,{\mathrm{\AA{}}}\) (as obtained with an aberration-corrected microscope) when the electron beam is positioned on the atomic column down the (110) orientation of the crystal (each square panel lateral length is \({\mathrm{16.3}}\,{\mathrm{\AA{}}}\)). The electron beam is channeled onto the atomic column but the intensity maxima moves from one atomic column to the adjacent one. Reprinted from [7.70], with permission from Elsevier. (b) Alternate visualization of the process of channeling viewed as a function of thickness. The beam intensity clearly channels from one atomic column to the adjacent one as the electrons propagate in the sample. From [7.71] reproduced with permission

Fig. 7.37a,b

Real-space intensity plots demonstrating the dispersion of the electron intensity when the electron beam is located on top of the atomic column and when it is located just in-between two atomic columns. The bright empty circles indicate the position of the atoms in the cell closest to the point of impact of the electron beam. Channeling is observed when the beam is positioned on the atomic column (a) while much stronger dispersion is observed when the electron beam is not on the atomic column (b). Reprinted from [7.70]. Copyright Elsevier (2003)

7.3.3 Inelastic Scattering Cross Sections

Before giving a description of the cross sections of various inelastic processes, it is useful to review the concept of the total inelastic cross sections so that a comparison of elastic and inelastic scattering distributions can be made. The total inelastic cross section is also relevant to understand the process of energy losses that slow down the electrons in thin foils or in bulk samples. This concept is useful in the modeling of electron propagation in samples using Monte Carlo methods.

The differential inelastic cross section can be described by
$$\frac{\mathrm{d}\sigma_{1}}{\mathrm{d}\Omega}=\frac{4\gamma^{2}Z}{a^{2}_{0}q^{4}}\left(1-\frac{1}{[1+(qr_{0})^{2}]^{2}}\right),$$
(7.13)
where \(\gamma\), \(a_{0}\), and \(r_{0}\) have been described in Sect. 7.3.2. As the scattering vector \(\boldsymbol{q}\) is energy-loss-dependent (Sect. 7.3.4), it is approximated for the purpose of the evaluation of the total cross sections by
$$\boldsymbol{q}^{2}\approx k^{2}_{0}\left(\theta^{2}+\bar{\theta}^{2}_{\mathrm{E}}\right),$$
(7.14)
where \(k_{0}=2\uppi/\lambda=\gamma m_{0}v/\hbar\) is the magnitude of incident wavevector, \(\theta\) is the scattering angle, and \(\bar{\theta}_{\mathrm{E}}=\bar{E}/(\gamma m_{0}v^{2})\) is the characteristic scattering angle corresponding to the mean energy-loss \(\bar{E}\).

As pointed out in [7.35], the first term of the expression for the total inelastic cross section (7.13) is similar to the Rutherford elastic cross section with a second term being described by the inelastic form factor.

Replacement of the scattering vector by the scattering angle leads to the angular dependence of total inelastic scattering [7.73]
$$\frac{\mathrm{d}\sigma_{\mathrm{i}}}{\mathrm{d}\Omega}=\frac{4\gamma^{2}Z}{a^{2}_{0}k^{4}_{0}}\frac{1}{\left(\theta^{2}+\bar{\theta}^{2}_{\mathrm{E}}\right)^{2}}\left[1-\frac{\theta^{4}_{0}}{\left(\theta^{2}+\bar{\theta}^{2}_{\mathrm{E}}+\theta^{2}_{0}\right)^{2}}\right],$$
(7.15)
where \(\theta_{0}=(k_{0}r_{0})^{-1}\) is related to the elastic scattering distribution defined in Sect. 7.3.2. As demonstrated in Fig. 7.38, the expressions for the total inelastic and elastic scattering cross sections make it possible to compare the width of the respective angular distributions. The angular distributions are Lorentzian with an angular width of \(\bar{\theta}_{\mathrm{E}}\) for the inelastic distributions and \(\theta_{0}\) for the elastic distribution. It can also be seen that the angular width of the elastic distribution (for scattering from free atoms) is larger than the inelastic distribution.
Fig. 7.38

Comparison of the elastic and total inelastic angular scattering distributions for carbon atoms at \({\mathrm{100}}\,{\mathrm{keV}}\). The characteristic angles for the inelastic and elastic distributions (\(\theta_{\mathrm{E}}\) and \(\theta_{0}\)), the mean angle \(\bar{\theta}\), the median angle \(\tilde{\theta}\), and the root-mean-square angle \(\theta_{\text{rms}}\) are shown. (After [7.35])

The total cross section integrated up to a scattering angle \(\beta\) is relevant when calculating the inelastic free path or the stopping power (see below)
$$\sigma_{\mathrm{i}}(\beta)\approx\frac{8\uppi\gamma^{2}Z^{\frac{1}{3}}}{k_{0}^{2}}\text{ln}\,\left[\frac{\left(\beta^{2}+\theta^{2}_{\mathrm{E}}\right)\left(\theta^{2}_{0}+\theta^{2}_{\mathrm{E}}\right)}{\theta^{2}_{\mathrm{E}}\left(\beta^{2}+\theta^{2}_{0}+\theta^{2}_{\mathrm{E}}\right)}\right].$$
(7.16)
The total inelastic scattering cross section integrated over all scattering angles is approximated by
$$\sigma_{\mathrm{i}}\approx 16\uppi\gamma^{2}Z^{\frac{1}{3}}\text{ln}\,\left(\frac{\theta_{0}}{\theta_{\mathrm{E}}}\right)\approx 8\uppi\gamma^{2}Z^{1/3}\text{ln}\,\left(\frac{2}{\theta_{\mathrm{E}}}\right) $$
(7.17)
by identifying the cutoff angle \(\theta_{0}\) with the Bethe-ridge angle \((2\theta_{\mathrm{E}})^{1/2}\). This expression leads to a comparison of the relative magnitude of the elastic and inelastic cross sections as described by
$$\frac{\sigma_{\mathrm{i}}}{\sigma_{\mathrm{e}}}\approx 2\dfrac{\text{ln}\,\left(\frac{2}{\bar{\theta_{\mathrm{E}}}}\right)}{Z}=\frac{C}{Z}\;,$$
(7.18)
where the coefficient \(C\) (around \(\mathrm{20}\)) does not vary significantly with atomic number and incident electron energy.

This expression can be used to calculate the scattering contrast as defined in Reimer [7.69] and to interpret the contrast in STEM images obtained by calculating the ratio of inelastic and elastic signals.

These calculations are a good first approximation of the behavior of the inelastic cross sections with further refinements, accounting for the outer-shell electrons, leading to systematic variations in the total cross sections related to the filling of the periodic table. Minima in the inelastic cross sections occur for atoms with a closed shell whilst maxima occur for atoms filling the s shell due to the strong effects of valence excitations [7.35] (Fig. 7.39).

Fig. 7.39

Total inelastic cross sections as a function of atomic number (\({\mathrm{80}}\,{\mathrm{keV}}\) electrons). Open circles are calculations based on the Hartree–Slater ( ) models, the full squares are based on calculations accounting for plasmon losses, and the full circles are experimental data. (After [7.35])

The importance of the total inelastic cross section becomes apparent in AEM as it is at the basis of calculations of the stopping power of the electrons in the solid. This quantity is therefore relevant to the understanding of propagation of the electrons and simulation of electron trajectories in Monte Carlo calculations (Sect. 7.5.1)
$$S=\frac{\mathrm{d}E}{\mathrm{d}z}=n_{\mathrm{a}}\bar{E}\sigma_{\mathrm{i}}\;,$$
(7.19)
where \(E\) is the energy loss, \(z\) is the distance traveled in the sample, \(\bar{E}\) is the mean energy loss for the inelastic event, and \(n_{\mathrm{a}}\) is the number of atoms in the solid per unit volume.
The total inelastic cross section considers all possible events giving rise to energy losses represented by an average energy-loss \(\bar{E}\) and does not consider the individual interactions of the incident electrons with the inner-shell or outer-shell atomic electrons. Predictions of the details of a spectrum and the intensity at a given energy loss, however, must take into account these various inelastic processes, their energy dependence, and the angular distribution of scattering. In order to do so, we must consider Bethe's theory to predict the probability of transitions of electrons from an initial state wavefunction \(\psi_{0}\) to a final state wavefunction \(\psi_{n}\) following interaction with incident fast electrons and the related cross section. If the energy losses and the momentum transfer are small compared to the momentum of the incident electron and there is only one scattering event during the interaction (the first Born approximation) the cross section can be described by
$$\frac{\mathrm{d}\sigma_{n}}{\mathrm{d}\Omega}=\left(\frac{4\gamma^{2}}{a^{2}_{0}\boldsymbol{q}^{4}}\right)\frac{k_{1}}{k_{0}}|\varepsilon_{n}(\boldsymbol{q})|^{2}\;,$$
(7.20)
where \(k_{0}\) and \(k_{1}\) are the magnitudes of the wave vectors of the incident electron before and after scattering respectively, and \(\boldsymbol{q}\) is the scattering vector related to the momentum transfer \(\hbar\boldsymbol{q}=\hbar(\boldsymbol{k}_{0}-\boldsymbol{k}_{1})\) (Fig. 7.40). The first term has already been encountered in the description of Rutherford scattering for a single charge (7.8) and constitutes the amplitude factor in the cross section. This term is modified by the inelastic form factor related to the transition matrix element defined as
$$\begin{aligned}\displaystyle\varepsilon_{n}&\displaystyle=\int\psi^{\ast}_{n}\sum_{j}\exp(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}_{\mathrm{j}})\psi_{0}\mathrm{d}\tau\\ \displaystyle&\displaystyle=\left\langle\psi_{n}\left|\sum_{j}\exp(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}_{j})\right|\psi_{0}\right\rangle\;,\end{aligned}$$
(7.21)
which expresses the interaction of the incident electron and the atomic electron via an operator \(\exp(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}_{\mathrm{j}})\) where \(\boldsymbol{r}_{j}\) is the coordinate position of the fast electron treated as a plane wave and the sum is carried out over the atomic electrons from \(j=1\) to \(Z\). This form factor contains the information related to the properties of the material through the wavefunctions of the electrons in the solid (Sect. 7.8.1). \(|\varepsilon_{\mathrm{n}}(q)|^{2}\) is independent of the electron energy and solely dependent on the atom and its environment.
Fig. 7.40

Inelastic scattering diagram showing the scattering vectors, the energy loss, and the effect of the classical impact parameter \(b\) on the scattering angle (also refer to discussions on spatial resolution in Sect. 7.6)

From the form factor, one can define the generalized oscillator strength ( )
$$f_{n}(q)=\frac{E_{n}}{R}\frac{|\varepsilon_{n}(q)|^{2}}{(qa_{0})^{2}}\;,$$
(7.22)
where \(R\) is the Rydberg energy (\({\mathrm{13.6}}\,{\mathrm{eV}}\)) and \(E_{n}\) is the energy loss of the transition. This term also contains information on the probability of the transition from an initial state wavefunction \({\psi}_{0}\) to a final state wavefunction \({\psi}_{n}\).
Using the concept of the generalized oscillator strength [7.35], the cross section can be expressed as
$$\frac{\mathrm{d}\sigma_{n}}{\mathrm{d}\Omega}=\frac{4\gamma^{2}R}{E_{n}q^{2}}\frac{k_{1}}{k_{0}}f_{n}(q)\;.$$
(7.23)
The full angular and energy dependence of the scattering can be described by the double differential cross section
$$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}=\frac{4\gamma^{2}R}{Eq^{2}}\frac{k_{1}}{k_{0}}\frac{\mathrm{d}f}{\mathrm{d}E}(q,E)\;,$$
(7.24)
where the scattering vector \(\boldsymbol{q}\) is expressed in terms of scattering angle \(\theta\) (Fig. 7.40) and the initial and final wavevectors \(\boldsymbol{k}_{0}\) and \(\boldsymbol{k}_{1}\). For small scattering angles and small energy losses relative to the incident energy (as in typical TEM experiments) \(\boldsymbol{k}_{1}/\boldsymbol{k}_{0}\approx 1\) and
$$q^{2}\cong k^{2}_{0}\left(\theta^{2}+\theta^{2}_{\mathrm{E}}\right)$$
(7.25)
with \(\theta_{\mathrm{E}}\) the relativistically corrected characteristic scattering angle
$$\begin{aligned}\displaystyle\theta_{\mathrm{E}}&\displaystyle=\frac{E}{\gamma m_{0}v^{2}}=\frac{E}{\left(E_{0}+m_{0}c^{2}\right)\left(\frac{v}{c}\right)^{2}}\\ \displaystyle&\displaystyle=\frac{E}{E_{0}}\left(\frac{E_{0}+mc^{2}}{E_{0}+2mc^{2}}\right)\end{aligned}$$
(7.26)
and the cross section becomes
$$\begin{aligned}\displaystyle\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}&\displaystyle\approx\frac{4\gamma^{2}R}{Ek^{2}_{0}}\left(\frac{1}{\theta^{2}+\theta^{2}_{\mathrm{E}}}\right)\frac{\mathrm{d}f}{\mathrm{d}E}\\ \displaystyle&\displaystyle=\frac{8a^{2}_{0}R^{2}}{Em_{0}v^{2}}\left(\frac{1}{\theta^{2}+\theta^{2}_{\mathrm{E}}}\right)\frac{\mathrm{d}f}{\mathrm{d}E}\;.\end{aligned}$$
(7.27)
The first two terms of (7.27) represent a kinematic term of the scattering while the \(\mathrm{d}f/\mathrm{d}E\) term provides the information on the initial and final wavefunctions of the electrons (including changes in the bonding state) either via inner-shell or valence excitations. The second term imposes a Lorentzian angular distribution of scattering at small scattering angles as it contains the term \(\theta_{\mathrm{E}}\) representing the half-width of the distribution. At low energy losses and small scattering angles (i. e., \(q\rightarrow 0\)), \(\mathrm{d}f/\mathrm{d}E\) does not vary with \(\theta\) and it can therefore be considered as a dipole oscillator strength. In this region of scattering (achieved by limiting the maximum scattering angle \(\theta\) with a small aperture \(\beta\)) the response to the excitation from an electron would be equivalent to the excitation by a photon.

Outer-Shell Excitations

The expression for \(\mathrm{d}f/\mathrm{d}E\) has been derived in Bethe's theory using atomic models. In order to treat inner-shell excitations, the initial and final wavefunctions of the electrons must be known. These wavefunctions vary with the chemical state in a solid but the atomic models are still a good approximation for the purpose of quantification of the atomic concentration based on intensity of edges. In the case of a solid, however, wavefunctions related to outer-shell excitations are much more strongly modified by the interaction between atoms and the collective behavior of the electrons. Similarly, when one wants to model fine modulations in the edges simple atomic models are not sufficient (Sect. 7.8.1). In order to consider these effects it is more convenient to consider the dielectric response of the medium \(\varepsilon(q,E)\) to the incident electron treated as a point charge perturbing the solid.

The dielectric formulation of scattering relates the double differential scattering cross section to
$$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}\approx\dfrac{\Im\left\{\frac{-1}{\varepsilon(q,E)}\right\}}{\uppi^{2}a_{0}m_{0}v^{2}n_{\mathrm{a}}}\left(\frac{1}{\theta^{2}+\theta^{2}_{\mathrm{E}}}\right),$$
(7.28)
thus the GOS relates to the dielectric function as
$$\frac{\mathrm{d}f}{\mathrm{d}E}(q,E)=\frac{2E}{\uppi E^{2}_{\mathrm{a}}}\Im\left\{\frac{-1}{\varepsilon(q,E)}\right\},$$
(7.29)
where \(E^{2}_{\mathrm{a}}=\hbar^{2}n_{\mathrm{a}}{e}^{2}/(\varepsilon_{0}m_{0})\), \(n_{\mathrm{a}}\) is the number of electrons per unit volume and \(\Im\{-1/\varepsilon(q,E)\}\) is the loss function that contains all the material dependence via the complex dielectric response of the solid \(\varepsilon=\varepsilon_{1}+\mathrm{i}\varepsilon_{2}\) to electromagnetic radiation. The real part of the dielectric function \(\Re\{\varepsilon(q,E)-1\}=\varepsilon_{1}(q,E)-1\) is related to the polarizability of the medium and the imaginary part \(\Im\{\varepsilon(q,E)\}=\varepsilon_{2}\) is related to the absorption. Maxima in the loss function \(\Im\{-1/\varepsilon\}=\varepsilon_{2}/(\varepsilon^{2}_{1}+\varepsilon^{2}_{2})\) that result in strong peaks in the energy-loss spectra (see Sect. 7.8.2) occur when \(\varepsilon_{1}=0\) and \(\varepsilon_{2}\) is small (the damping is weak). The condition \(\varepsilon_{1}=0\) suggests that a condition of resonance is met and that the medium in unstable. This instability corresponds to creation of a quasiparticle called a plasmon of energy \(E_{\mathrm{a}}\). Further discussion on the derivation of the loss function and applications can be found in Sect. 7.8.2.

7.3.4 Calculations of Cross Sections

Quantification of EELS and EDXS spectra based on signals recorded from edges and x-ray peaks (Sect. 7.4) is based on the knowledge of cross sections. For inner-shell excitations in EELS (and the related EDXS peaks) it is assumed, as a first approximation, that the initial and final wavefunctions are not affected by collective electron behavior and the cross sections can be calculated using various models based either on the simple hydrogenic description of the atomic electrons or the more accurate Hartree–Slater method. For an atom of atomic number \(Z\), the hydrogenic model uses the simplification of the electrostatic potential arising from the treatment of the nuclear charge \(Ze\) and the screening due to remaining inner nonexcited electrons. Different expressions for the effective charges are used for K and L shells and consideration is given to the presence of outer electrons in higher energy levels that modify the binding energy of the inner-shell electrons. Solving the Schroedinger equation in the revised simplified potential leads to analytical solutions which can be easily calculated for K shells using programs developed by Egerton [7.35]. The treatment of L shells, although initially unsuccessful due to the simplifications of the hydrogenic model, was revised by considering experimental optical and energy-loss data with built-in corrections. These modifications have led to improved accuracy in the treatment of L shells for transition metals; programs and models for L shells are also available [7.35] in the literature and in commercial EELS analysis programs (the Digital Micrograph™ software from Gatan). Empirical modifications of the hydrogenic models for M edges have also been developed [7.74]. The Hartree–Slater approach requires iterative solutions, which lead to more accurate cross sections that consider a potential calculated based on the charge density of the electrons in a self-consistent manner. Cross-section tabulations have been developed by Leapman et al [7.75] and Rez [7.76] and the results have been implemented within commercial EELS analysis software for K, L, and M shells (Digital Micrograph™). The HS model predicts more realistic shapes of edges and includes corrections for sharp features related to unoccupied bound states present on transition-metal L edges. Recent developments allow the combination of solid-state effects near the edge threshold, calculated from band structure techniques, with the atomic models at higher energy losses into more accurate cross-section models [7.77].

Angular and Energy Dependence of the GOS

Some physical insight into the scattering process is given by the analysis of the angular and energy dependence of the GOS \(\mathrm{d}f(q,E)/\mathrm{d}E\), known as the Bethe surface, as plotted for the C K edge in Fig. 7.41a. This plot also provides information related to the optimization of the acquisition conditions as it shows the dependence of the differential cross section on angle and energy loss. There are two regions of key importance on the Bethe surface. At energy losses just above the threshold, the distribution is peaked at small scattering angles (\(\theta=0\), \(q\rightarrow 0\)) and the angular dependence in the cross-section \(\mathrm{d}^{2}\sigma/\mathrm{d}\Omega\mathrm{d}E\) is controlled by the kinematic term of the cross section since the GOS does not vary strongly with \(q\) at small scattering angles. This region of the GOS is known as the dipole region and represents collisions with large impact parameter \(b\) (Fig. 7.40 and Sect. 7.5.2) and little momentum transfer. Spectra acquired with small scattering angles in electron energy-loss experiments are thus equivalent to optical (x-ray) absorption spectra where only dipole transitions are allowed. On the basis of this equivalence, photoabsorption spectra can be used to improve GOS models for use in cross-section calculations [7.78, 7.79]. The dipole transitions also imply that the change of angular momentum quantum number \(\ell\) when the electron is initially excited from a core level \(i\) to a final level \(f\) is \(\Updelta\ell\pm 1\).

Fig. 7.41

(a) Two-dimensional angular and energy distribution of the general oscillator strength per unit energy loss (\(\mathrm{d}f/\mathrm{d}E\)) for the C K-edge (also described as the Bethe surface of scattering). (After [7.35]). The hump at high energy from the edge onset and large scattering vector \(\boldsymbol{q}\) is the Bethe ridge . (b) Two-dimensional schematic plot of the full energy and scattering angle distribution of inelastic losses including plasmon losses, core-losses as well as the Bethe ridge and the plasmon dispersions. The diagram shows the elastic scattering as part of Bragg peaks (as in a crystalline sample). Scattering from atoms or amorphous samples would give rise to a broad peak centered at zero scattering angle. (Adapted after [7.80]). The angle (\(\beta\)) and energy (\(\Delta\)) integration windows for the calculations of the partial cross sections are identified

At larger energy losses from threshold, corresponding to collision with nearly free electrons, the GOS peaks at large scattering angles \(\theta_{\mathrm{C}}=(E/E_{0})^{2}\) corresponding to what is known as the Bethe ridge . Experimentally, this peak can be readily visualized by collecting energy-loss spectra in thin samples with a spectrometer aperture centered a few tens of mrad off the transmitted beam collection angle (thus away from the forward scattered direction). This acquisition condition is equivalent to Compton scattering experiments between an incident electron and a free electron. As discussed in a review by Schattschneider and Exner [7.81], a quantitative analysis of the width of this experimental peak provides information on the momentum distribution of atomic electrons. Similarly, by recording diffraction patterns of thin specimens with an energy filter (Sect. 7.2.4, Spectrometers) at high energy loss (a few \({\mathrm{100}}\,{\mathrm{eV}}\)) it is possible to see a ring corresponding to the Compton scattering peak [7.35, 7.37]. The full energy and scattering angle dependence of the entire spectrum includes therefore the various processes of low-loss scattering (including the dispersion of plasmon losses, the elastic Bragg scattering (in crystalline specimens) and the core-losses (with the Bethe ridge visible) (Fig. 7.41b.

Partial and Total Ionization Cross Sections

Whether we have to compute cross sections for quantification of EDXS or energy-loss spectra, the differential cross sections have to be integrated with respect to angle and energy loss to yield partial or total cross sections. In the case of energy-loss experiments, the spectra are recorded with a fixed collection angle \(\beta\) and the partial cross section is therefore related to the double differential cross section through integration up to an angle \(\beta\)
$$\frac{\mathrm{d}\sigma}{\mathrm{d}E}\cong\frac{4R\hbar^{2}}{Em^{2}_{0}v^{2}}\int^{\beta}_{0}\frac{\mathrm{d}f(q,E)}{\mathrm{d}E}2\uppi\frac{\theta}{\left(\theta^{2}+\theta^{2}_{\mathrm{E}}\right)}\mathrm{d}\theta\;,$$
(7.30)
where the GOS is expressed in terms of the scattering vector \(\boldsymbol{q}\) and the scattering angle is \(\theta\). By considering the kinematics of scattering (Fig. 7.40), \(q^{2}\cong k^{2}_{0}(\theta^{2}+\theta^{2}_{\mathrm{E}})\) and the integration can be either carried out over \(\theta\) or, after transformation of variables and limits of integration, over \(q\) [7.35]. The partial cross section follows a simple trend as \(\mathrm{d}\sigma/\mathrm{d}E\propto E^{-\mathrm{s}}\) where \(s\) is the slope of the function.
For the purpose of quantification in energy-loss spectroscopy (Sect. 7.4.2), core shell edge spectra acquired with a collection angle \(\beta\) are integrated over an energy window \(\Delta\). The integrated signal for a given collection angle \(\beta\) and energy window is thus related to the partial cross section as
$$I^{1}_{k}(\beta,\Delta)=NI_{0}\sigma_{k}(\beta,\Delta)\;,$$
(7.31)
where \(N\) is the number of atoms per unit area, \(I_{0}\) is the incident intensity (approximated by the number of counts under the zero-loss peak) and
$$\sigma_{k}(\beta,\Delta)=\int^{E_{k+\Delta}}_{E_{k}}\frac{\mathrm{d}\sigma}{\mathrm{d}E}\mathrm{d}E\;.$$
(7.32)

Examples of the values of partial cross section for light elements as a function of collection angle \(\beta\) are shown in Fig. 7.42. These plots show the increase of expected signal with increases of collection angle and the variation with atomic number. One can note the asymptotic behavior of the cross sections for large scattering angles approaching \({\mathrm{100}}\,{\mathrm{mrad}}\). This effect suggests that, by using large collection angles above a few tens of mrad, the collected intensity does not increase significantly. This asymptotic behavior affects the optimization of signals and signal-to-background ratios as discussed in relation to the detection limits (Sect. 7.7). This integration over scattering angles and energy window is demonstrated in the hatched area in Fig. 7.41b.

Fig. 7.42

Angular dependence of the partial cross sections for lighter elements. The energy integration is carried out over an energy window \(\Delta\) equivalent to the \(\mathrm{0.2}\) \(E_{k}\) where \(E_{k}\) is the threshold energy. (After [7.35])

For the calculation of the probability of x-ray signal generation following ionization, the collection angle of the primary electrons is irrelevant and one must consider all possible scattering angles of the incident electron. The integration must be carried out up to \(\beta=\uppi\) which includes possible backscattering. This integration leads to the total inelastic cross section for core shell ionization that is used in x-ray quantification (Sect. 7.4.1, Quantification in EDXS).
$$\sigma_{k}\cong 4\uppi a^{2}_{0}N_{k}b_{k}\left(\frac{R}{T}\right)\left(\frac{R}{E_{k}}\right)\text{ln}\left(\frac{c_{k}T}{E_{k}}\right),$$
(7.33)
where \(N_{k}\) is a number related to the occupancy in the particular shell (2, 8, 18 for K, L, M shells, respectively), \(b_{k}\) and \(c_{k}\) are factors deduced from theory or experiments [7.82, 7.83, 7.84], also tabulated in [7.85].
It is useful to compare the magnitude of the cross sections for elastic and various inelastic events such as K and L shell excitation, outer shell (including excitations of plasmons) and generation of secondary electrons (Fig. 7.43) as a function of energy. On the basis of the inelastic cross sections it is possible to define an inelastic mean free path (mean distance between inelastic scattering of events) in a similar fashion to the elastic mean free path (Sect. 7.3.1). For an inelastic scattering event \(i\),
$$\lambda_{\mathrm{i}}=(n_{\mathrm{a}}\sigma_{\mathrm{i}})^{-1}\;,$$
(7.34)
where \(n_{\mathrm{a}}\) is the number of atoms per unit volume and \(\sigma_{\mathrm{i}}\) is the cross section for an inelastic event \(i\) (either total inelastic scattering or K, L shell excitation).
Fig. 7.43

Comparison of cross section for various scattering processes for Al as a function of incident electron energy. The comparison shows the strongest inelastic scattering (dominated by plasmons losses), followed by elastic scattering, \(\mathrm{L}\) shell ionization, \(\mathrm{K}\) shell ionization, fast secondary electrons, and secondary electrons. The latter two are not discussed in this chapter. (After [7.1])

7.3.5 Shape of EELS Edges

There are two factors that contribute to the shape of inner-shell absorption edges observed in energy-loss spectra. The atomic calculations of the cross sections (hydrogenic of Hartree–Slater) give, to a good first approximation, the expected shape of the edge related to transitions from a core state to the free continuum states. Although the general shape is reproduced particularly well at high energies from the threshold, details arising from transitions to discrete unoccupied bound states can be important. These effects are visible both in terms of the integrated edge intensity (particularly for transition-metal L edges) and in terms of the fine details arising from electronic structure and the bonding environment in the solid state. These latter modulations are called energy-loss near-edge structures ( ). The dipole selection rules for the transitions from an initial core level to a final state (Sect. 7.8.1) imply that there must be a change in angular momentum quantum number \(\Updelta\ell=\pm 1\). This implies that the angular character of the final states (i. e., the symmetry) is determined by the core state quantum numbers. Making use of the nomenclature presented in Fig. 7.11, the following rules can be summarized. For K edges, when the principal quantum number is \(n=1\) and the angular momentum quantum number of the core state is s (i. e., \(\ell=0\)) then the final state character is p (\(\ell=1\)). For \(\mathrm{L_{23}}\) edges, the core level is 2p and the final states probed are s (\(\ell=0\)) and d (\(\ell=2\)). For \(\mathrm{M_{45}}\) edges, the core level is 3d character and the final states will be f or p character. Spectroscopically the K, L, M, N, and O edges therefore reflect the main quantum number of the core level \(n=1\), 2, 3, 4, 5, respectively and each edge presents sublevels due to the angular momentum quantum number \(\ell\) and spin-orbit coupling due to the \(j\) quantum number (Figs. 7.11 and 7.4). L edges thus present transitions to three sublevels for 2s, 2p\({}_{3/2}\), and 2p\({}_{1/2}\) giving rise to the L\({}_{1}\), L\({}_{3}\), and L\({}_{2}\) edges, respectively, while M edges have five sublevels and so forth as summarized in Fig. 7.11.

A summary of the types of shapes is reflected in Figs. 7.44a-d and 7.45 along with a table (Table 7.4) showing the expected edges visible for energy losses up to \({\mathrm{2000}}\,{\mathrm{eV}}\). These edges can be detected in typical TEM experiments. K edges are observed for light elements up to Si (\({\mathrm{1800}}\,{\mathrm{eV}}\)) and generally present a sawtooth shape. The overall edge shapes are relatively well predicted by the hydrogenic and Hartree–Slater cross-section models although, within the first \({\mathrm{30}}\,{\mathrm{eV}}\) from the edge threshold, fine variations due to the bound final states of p-character in the ELNES are present. L edges are observed for a larger range of elements and the shape of these varies with the atomic number. The L\({}_{2}\) and L\({}_{3}\) edges (also labeled as L\({}_{23}\) edges) are the strongest in the series due to the large cross sections and the selection rules imposing transitions to unoccupied levels with large density of states above the Fermi energy (if there are 3d empty states). The L\({}_{1}\) edges are hardly visible in typical acquisition conditions favoring the dipole transitions (i. e., small collection angle). The edges of elements Na to Ar present the so-called delayed edges (Fig. 7.44a-db) with a broad maximum of the edge intensity \(10{-}20\,{\mathrm{eV}}\) above the edge threshold. This delayed shape is predicted by the cross-section calculations and is due to a maximum in the effective atomic potential resulting in a centrifugal barrier that prevents the overlap between the core and the broad final state wavefunctions at low energy from the threshold. This poor overlap results in low values of the matrix element near the edge threshold (Sect. 7.8.1). When the final state wavefunctions are very localized, as in the case of the 3d electrons, they are contained within the barrier and there is therefore a strong overlap with the core wavefunctions. This results in strong features at the edge threshold for the L\({}_{23}\) edges in the series K to Ni where the 3d band is being filled and transitions to empty 3d electron states are possible (Fig. 7.34c). In these cases, sharp features, known as white-lines , are clearly visible leading to noticeable trends in the transition-metal series (Fig. 7.45). For the Cu L\({}_{23}\) edge, where the 3d band is full with 10 d electrons, the delayed maximum is again observed without strong peaks at the edge threshold. When solid-state effects result in changes in the electronic configuration, however, for example when Cu loses 3d electrons when bonding to O (such as CuO or \(\mathrm{CuO_{2}}\)), white-lines do appear in the spectra (Fig. 7.46). In Zn and its oxide ZnO, however, the electrons affected by bonding are of \(\mathrm{s}\) character and the changes with bonding are not visible in the L edge of Zn. White-line features in L\({}_{23}\) edges are observed for the 4d transition metals (the series Rb to Pd) at higher energy losses.

Table 7.4

Summary of expected edges from [7.86]

Edge

Level

Elements

Shapes

K

1s

Li–Si

a

L\({}_{23}\)

2p\({}_{1/2}\), 2p\({}_{3/2}\)

Mg–Ar

K–Ni

Cu–Br

b

c

b

M\({}_{23}\)

3p\({}_{1/2}\), 3p\({}_{3/2}\)

K–Cu

d

M\({}_{45}\)

3d\({}_{3/2}\), 3d\({}_{5/2}\)

Se–Kr

Rb–I

Cs–Yb

Lu–Au

d

b

c

b

N\({}_{45}\)

4d\({}_{3/2}\), 4d\({}_{5/2}\)

Cs–Yb

d

O\({}_{45}\)

5d\({}_{3/2}\), 5d\({}_{5/2}\)

U

Th

d

d

Fig. 7.44a-d

Schematic diagram of edge shapes from [7.86]. Panels (ad) refer to the types of edges presented in Table 7.4

Fig. 7.45

(a) Trends in K edges for light elements and (b) sp electron metals. (c) Trends in \(\mathrm{L_{23}}\) edge intensities (called white-lines) for the transition-metal series. It is clearly visible that the strong peaks at the edge threshold decrease in intensity with respect to the continuum after the edge threshold. This effect reflects the filling of the 3d transition-metal band with electrons. (d) Shape of delayed M edges. Adapted from [7.87], with permission from Elsevier

Fig. 7.46

Cu \(\mathrm{L_{23}}\) edge in metallic Cu, CuO, \(\mathrm{Cu_{2}O}\). The presence of the white-line in CuO and \(\mathrm{Cu_{2}O}\) (as compared to metallic Cu) reflects the transfer of electrons from the 3d band to O atoms in the oxidation

Fig. 7.47

Nb \(\mathrm{M_{45}}\) and Ba \(\mathrm{M_{45}}\) edges. While the Ba edge shows strong peaks and a lower continuum past the edge onset, the Nb \(\mathrm{M_{45}}\) edge shows a strong rising intensity post the edge threshold

The M\({}_{45}\) and M\({}_{23}\) edges are the strongest in the M series with large cross sections. These edges exhibit the same trends with delayed maxima and strong white-lines depending on the atomic number (Fig. 7.47). For the series Rb to In, a delayed maximum is observed due to transitions to delocalized empty \(\mathrm{f}\) states well above the Fermi level whilst strong white-lines are visible for Cs to Yb since the \(\mathrm{f}\) states are bound and are being filled in the series. The very strong white-lines make these edges very suitable for detection of trace quantities of elements in samples. For Lu to Au, as the \(\mathrm{f}\) band is completely filled, very broad delayed edges are present and their detection, due to the background in the spectra, is often difficult even for elements present in samples in large concentrations. M\({}_{23}\) edges are detected for the K to Zn series in the energy range between \(\mathrm{30}\) and \({\mathrm{100}}\,{\mathrm{eV}}\) with plasmon-like features represented in Fig. 7.44a-dd. These edges overlap with the strongly varying background in the low portion of the energy-loss spectrum and are thus difficult to extract from the spectra. Because of the large cross sections, however, elemental maps can be easily obtained with very high spatial resolution [7.88].

7.3.6 The Background in EELS and EDXS

The background in core-level energy-loss spectra (Fig. 7.8) is dominated by lower energy-loss processes such as lower core-losses and plasmon/outer shell losses and the characteristic signal of edges must be extracted with procedures described in Sect. 7.4.2, Quantification Procedures. In the case of energy-dispersive x-ray spectra, the continuum background depicted in Fig. 7.4c arises from the bremsstrahlung radiation (or braking radiation) as the electrons are subjected to centripetal acceleration forces in the Coulomb potential of the atoms in the solid as their trajectories are modified. The x-rays are generated over a continuous energy range from \({\mathrm{0}}\,{\mathrm{eV}}\) up to the energy of the incident electrons. The angular distribution of the emitted x-rays due to the bremsstrahlung radiation is not isotropic as in the case of the characteristic radiation emitted following ionization and de-excitation. The distribution is almost forward-peaked with maxima in intensity a few degrees from the forward direction and very little intensity emitted in the backscattered direction. Hence the peak-over-background ratio is dependent on the location of the EDXS detector (lowest peak-over-background with higher elevation angles) although additional difficulties in the acquisition of spectra arise from the solid angle limitations and the instrumental contribution to the spectra (Sect. 7.7.2). Expressions describing the continuum spectrum intensity (\(I_{\text{cm}}\)) demonstrate the dependence on the incident current and the average atomic number \(\overline{Z}\) of the sample irradiated by the electron beam [7.89]
$$I_{\text{cm}}(E_{\nu})\propto i_{\mathrm{b}}\bar{Z}\left(\frac{E_{0}-E_{\nu}}{E_{\nu}}\right),$$
(7.35)
where \(i_{\mathrm{b}}\) is the beam current, \(E_{0}\) is the incident energy of the electrons, and \(E_{\nu}\) is the energy of the emitted radiation. At lower energy, the recorded spectrum does not show the increase in the intensity expected from (7.35) due to the absorption in the sample and in the detector. Although the generated continuum spectrum does not show sharp features, in crystalline solids well-defined peaks can occur due to coherent bremsstrahlung reported by Reese et al [7.90] when electrons travel close to a zone axis. The coherent bremsstrahlung energy peak positions \(E_{\text{CB}}\) are related to the spacing of the atomic planes in the direction of the electron beam \((L)\), the ratio of the velocity of the electrons/velocity of light \((v/c)\), and the elevation angle \(\alpha_{\mathrm{E}}\) (Fig. 7.23a-c) as described by [7.90, 7.91]
$$E_{\text{CB}}\,(\text{keV})=\dfrac{12.4\left(\frac{v}{c}\right)}{L\left[1-\left(\frac{v}{c}\right)\cos(90+\alpha_{\mathrm{E}})\right]}\;.$$
(7.36)
These peaks can be mistaken for minor constituent phases when they overlap with characteristic peaks and can be minimized by carrying out analyses when the samples are oriented far from major zone axes.

7.4 Quantification

7.4.1 EDXS Microanalysis

Quantification in EDXS

As indicated in Fig. 7.7, there are several processes that need to be considered in the measurements of concentration based on the detected signals. First of all, x-rays generated by the excited atoms must travel through the specimen and are therefore subject to absorption. These x-rays can also cause fluorescence of secondary x-ray radiation within the sample, which generates additional signals further away from the site irradiated by the primary electrons. The subsequent detection process includes transmission through a detector window, various detector layers and generation of electron–hole pairs in the solid-state detector device. All these effects need to be considered in the quantification procedure. We shall consider the different processes starting with the generation of x-rays in the sample assuming that the characteristic signals in the spectrum can be extracted from the continuum background by simple interpolation of the background intensity under the characteristic peak from the background on the low and high energy sides of the characteristic peak.

As electrons travel through a sample of thickness \(t\) and are scattered at various angles according to the elastic cross section (Sects. 7.3.1 and 7.3.2) losing energy as dictated by the inelastic cross sections and the stopping power (Sects. 7.3.1, 7.3.3 and 7.3.4) they generate photons in the x-ray portion of the electromagnetic spectrum. The generated x-ray intensity for element \(A\) is
$$I^{\ast}_{A}\propto\frac{(C_{\mathrm{A}}\omega_{\mathrm{A}}\sigma_{\mathrm{A}}a_{\mathrm{A}}t)}{A_{\mathrm{A}}}\;,$$
(7.37)
where \(C_{\mathrm{A}}\) is the weight fraction of element \(A\), \(\omega_{\mathrm{A}}\) is the fluorescence yield for the K, L or M line, \(a_{\mathrm{A}}\) is the fraction of the total K, L or M line intensity that is measured, \(A_{\mathrm{A}}\) is the atomic weight of element \(A\), \(\sigma_{\mathrm{A}}\) is the total ionization cross section for a shell K, L or M for atom \(A\) in the specimen, and \(t\) is the sample thickness.
The total ionization cross section derived in Sect. 7.3.4 can be expressed in terms of the overvoltage \(U=(E_{0}/E_{\mathrm{C}})\) relative to the ionization energy \(E_{\mathrm{C}}\) and incident energy \(E_{0}\) commonly used in the AEM literature where the constants have been included into one single term and \(N,b,c\) are the same as the constants found in (7.33) with subscripts related to the shell \(\mathrm{s}\) i. e.,
$$\sigma=Q=\frac{{\mathrm{6.51\times 10^{-20}}}}{E^{2}_{\mathrm{C}}}N_{\mathrm{s}}b_{\mathrm{s}}\text{ln}\,(c_{\mathrm{s}}U)\;.$$
(7.38)
When the thickness of the sample satisfies the thin-film criterion (Sect. 7.4.1, Absorption), the absorption and fluorescence in the sample can be neglected. In this condition, the x-ray intensity still remains to be corrected for the detection process that includes absorption in the detector's various layers and for the efficiency of the detector active layer in generating electron–hole pairs from the impinging x-rays (Sect. 7.2.3, The EDXS Detector, Sect. 7.2.3, Detector Windows). For each particular x-ray peak, the intensity detected must therefore include a detector efficiency term that describes how the intensity is absorbed
$$I_{\mathrm{A}}=I^{\ast}_{\mathrm{A}}\varepsilon_{\mathrm{A}}\;,$$
(7.39)
where
$$\begin{aligned}\displaystyle\varepsilon_{\mathrm{A}}&\displaystyle=\left(\exp\left[-\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\mathrm{W}}\rho_{\mathrm{W}}X_{\mathrm{W}}\right.\right.\\ \displaystyle&\displaystyle\qquad\qquad\left.\left.-\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\text{Au}}\rho_{\text{Au}}X_{\text{Au}}-\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\text{Si}}\rho_{\text{Si}}X_{\text{Si}}\right]\right)\\ \displaystyle&\displaystyle\quad\,\,\times\left(1-\exp\left[-\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\text{Si}}\rho_{\text{Si}}Y_{\text{Si}}\right]\right)\end{aligned}$$
(7.40)
and \((\mu/\rho)\) is the mass absorption coefficients for elements A in the window material W, the Au contact layer and Si dead layer, \(\rho\) is the densities of the window material W, Au surface contact layer, and Si dead layer, \(X\) is the thickness of the window W, the Au layer and the Si dead layer, and \(Y_{\text{Si}}\) is the EDXS detector active layer thickness.

The second term accounts for the detector thickness that generates the signal based on the electron–hole pairs created by the incident photons (the active layer thickness).

Although these equations can, in principle, be easily calculated, exact values of the various constants, sample thickness \(t\), and detector layer thicknesses are not known to sufficient accuracy and alternative approaches have been suggested to eliminate the influence of these parameters on quantification. The calculation of the relative concentration of elements \((C_{\mathrm{A}}/C_{\mathrm{B}})\) makes it possible to include all the constant terms both in the cross sections and detector response into one single term and thus eliminate the necessity to know the sample thickness
$$\frac{C_{\mathrm{A}}}{C_{\mathrm{B}}}=\left(\dfrac{\left(\frac{\sigma\omega a}{A}\right)_{\mathrm{B}}\varepsilon_{\mathrm{B}}}{\left(\frac{\sigma\omega a}{A}\right)_{\mathrm{A}}\varepsilon_{\mathrm{A}}}\right)\frac{I_{\mathrm{A}}}{I_{\mathrm{B}}}$$
(7.41)
The term within brackets is a constant for a given set of x-ray peaks for elements A and B, detector and accelerating voltage. This term allows the user to relate the intensity ratio for two peaks to the relative concentration of the elements in the sample. The \(k_{\text{AB}}\) factor approach suggested by Cliff and Lorimer [7.92] allows us to simplify the calculation of relative concentration of elements provided that standards can be obtained so that
$$\frac{C_{\mathrm{A}}}{C_{\mathrm{B}}}=k_{\text{AB}}\frac{I_{\mathrm{A}}}{I_{\mathrm{B}}}\;.$$
(7.42)
For binary systems, the concentration can be retrieved assuming \(C_{\mathrm{A}}+C_{\mathrm{B}}=1\) and for ternary systems a similar assumption can be made. In principle, the \(k_{\text{AB}}\) term in (7.41) can be calculated based on knowledge of the terms within the bracket but the significant advantage of the Cliff–Lorimer approach resides in the fact that the \(k_{\text{AB}}\) factors can be obtained from standards of known composition \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) related to one reference element. As a convention, \(k\)-factors can be expressed in terms of a standard with respect to Si and Fe (i. e., \(k_{\text{ASi}}\) \(k_{\text{BSi}}\) or \(k_{\text{AFe}}\), \(k_{\text{BFe}}\) etc.) so that any set of \(k_{\text{AB}}\) factors can be retrieved from
$$k_{\text{AB}}=\frac{k_{\text{ASi}}}{k_{\text{BSi}}}\;.$$
(7.43)
Reference \(k\)-factors can be developed with respect to any elements but reference to Si and Fe were historically developed for users in the geological science and metallurgy communities. Examples of \(k_{\text{AFe}}\) factors are shown in Fig. 7.48a,b and can be used as a good approximation to quantify measurements although the exact values are dependent on a particular detector (and in particular detector efficiency) and accelerating voltage. Some tabulated values for a range of elements can be found in [7.2]. Users of EDXS in AEM typically develop their own standards based on samples of known composition \(C_{\mathrm{A}}\), \(C_{\mathrm{B}}\) to quantify unknown samples containing the same elements. These \(k_{\text{AB}}\) factors would therefore account for the specific features of the microscope-detector system. Standardless \(k_{\text{AB}}\) factors can be calculated based on the knowledge of all the terms expressed in (7.41) making use of constants tabulated in the literature.
Fig. 7.48a,b

Experimental \(k_{\text{AFe}}\) factors for a range of elements (a) for \(\mathrm{K}_{\alpha}\) lines and (b) for \(\mathrm{L}_{\alpha}\) lines (data from [7.2]) as a function of the characteristic x-ray energy from element A. The limits reflect the range of theoretical values obtained with different cross-section parametrizations

The fluorescence term can be derived on the basis of the parametrization
$$\left[\frac{\omega}{(1-\omega)}\right]^{\frac{1}{4}}=A+BZ+CZ^{3}\;,$$
(7.44)
where the constants for K, L, and M edges have been summarized in [7.85] and they give rise to the dependence shown in Fig. 7.49.
Fig. 7.49

Fluorescence coefficient for families of lines as a function of atomic number. (After [7.35])

Parametrization of the cross sections and the constants found in (7.37) and (7.41) have been discussed in detail in [7.85] and show that relative errors in \(k_{\text{AB}}\) factors in the range of \(1{-}5\%\) for K lines and higher for L lines (\(10{-}15\%\)) can be expected based on the various sources of data and estimates of detector parameters as compared with experimental data. Quantification packages in modern AEM systems provide accurate quantification based on more precise knowledge of the internal parameters for the detector often not accessible to the end user. Verification of the quantification procedure is, however, suggested based on standards of known composition to validate the built-in procedures of the EDXS software.

Absorption

The simple expression for the quantification of EDXS spectra with \(k_{\text{AB}}\)-factors based on standards, or with the standardless approach discussed above, assumes that the sample is thin enough that the absorption and fluorescence terms can be neglected—i. e., the thin-film criterion is applicable. Photons generated by the incident electrons, however, are absorbed by atoms in the sample via electron excitation processes while traveling from their origin point to the detector. As in the case of absorption of x-rays occurring in the EDXS detector, the strength of the absorption process is related to the x-ray absorption coefficient (tabulated in [7.93]) for a given x-ray line and a specific absorbing element. The intensity of the incident radiation (and not the energy) will decrease with an exponential decay (Fig. 7.50) while it travels through a sample. Large absorption coefficients occur when low-energy x-ray lines are just above the absorption thresholds of the absorbing element—i. e., when the photoabsorption cross sections are high. On the other extreme, small absorption coefficients occur for high-energy lines far away from the absorption threshold of the absorbing element. Considering these effects, the thin-film criterion stipulates that the \(k_{\text{AB}}\) factors should not change more than the arbitrary value of \({\mathrm{3}}\%\) (or less stringent \({\mathrm{10}}\%\)) due to absorption.

Fig. 7.50

Absorption process of x-rays and variation of the absorption coefficient with energy

If the generation of x-rays is uniform throughout the sample thickness, the \(k_{\text{AB}}\) factor is modified by the absorption correction factor ( ) term
$$\begin{aligned}\displaystyle k_{\text{AB}}&\displaystyle=k_{\text{AB}}^{\text{Thin film}}(\text{ACF})\\ \displaystyle k_{\text{AB}}&\displaystyle=k_{\text{AB}}^{\text{Thin film}}\left(\dfrac{\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\text{spec}}}{\left(\frac{\mu}{\rho}\right)^{\mathrm{B}}_{\text{spec}}}\right)\\ \displaystyle&\displaystyle\quad\,\times\left(\dfrac{1-\exp-\left[\left(\frac{\upmu}{\rho}\right)^{\mathrm{B}}_{\text{spec}}\text{cosec}\,\alpha(\rho t)\right]}{1-\exp-\left[\left(\frac{\upmu}{\rho}\right)^{\mathrm{A}}_{\text{spec}}\text{cosec}\,\alpha(\rho t)\right]}\right)\end{aligned}$$
(7.45)
where \(k_{\text{AB}}^{\text{Thin film}}\) is the \(k_{\text{AB}}\) factor for the thin film without the absorption term, \(\mu/\rho\) is the absorption coefficient for element A or B in the sample, and \(\alpha\) is the x-ray elevation (or take-off) angle required to calculate the path length of x-rays in the sample (Fig. 7.51).
Fig. 7.51

Absorption length dependence on elevation angle and foil thickness

In practical cases, it is clear that the \({\mathrm{3}}\%\) limit to the absorption correction is very stringent and severely limits the practical applications of the Cliff and Lorimer approach thus forcing the user to include the correction term and measure the sample thickness. Examples of calculations of the thickness satisfying the thin-film criterion (7.45) are given in Table 7.5 for the 10 and \({\mathrm{3}}\%\) criteria [7.85]. By inspection, it is clear that, in order to avoid absorption corrections, extremely thin samples are required when low-energy x-ray lines are generated in the sample and when there are elements with absorption thresholds close to the x-ray lines. For higher energy photons (and larger separation between the photon energy and the absorption threshold), absorption can be neglected in typical AEM samples. To illustrate this effect, Table 7.5 indicates that an NiAl specimen of \({\mathrm{9}}\,{\mathrm{nm}}\) thickness would satisfy the \({\mathrm{3}}\%\) criterion, while an Fe-\({\mathrm{5}}\%\) Ni alloy would require a \(\approx\)\(\mathrm{90}\)-\(\mathrm{nm}\) thick foil.

Table 7.5

Thickness criteria for 10 and \({\mathrm{3}}\%\) absorption limit. (From  [7.1])

Material

Thickness (nm)

Absorbed x-ray line

 

\({\mathrm{10}}\%\) limit

\({\mathrm{3}}\%\) limit

 

Fe-\({\mathrm{5}}\%\)Ni

\(\mathrm{322}\)

\(\mathrm{89}\)

Ni K\({}_{\alpha}\)

MgO

\(\mathrm{304}\)

\(\mathrm{25}\)

Mg K\({}_{\alpha}\), O K\({}_{\alpha}\)

NiAl

\(\mathrm{32}\)

\(\mathrm{9}\)

Al K\({}_{\alpha}\)

SiC

\(\mathrm{13}\)

\(\mathrm{3}\)

Si K\({}_{\alpha}\), C K\({}_{\alpha}\)

\(\mathrm{Si_{3}N_{4}}\)

\(\mathrm{413}\)

\(\mathrm{6}\)

Si K\({}_{\alpha}\), N K\({}_{\alpha}\)

The calculation of the absorption correction factor implies that absorption coefficients are known, the density of the sample is known, and that the thickness of the sample can be measured at the point of analysis so that the path length of the x-rays can be determined. Although tabulated values for the mass absorption coefficients for pure elements can be easily found in the literature [7.85], the knowledge of the actual sample mass absorption coefficient and sample density means that the composition of the sample should be calculated in an iterative process where initial input of composition, and thus density and mass absorption coefficient values, are refined at each iteration step. An additional complication resides in the requirement to calculate the mass absorption coefficients based on all elements present in the sample even if only relative concentrations are of interest. The mass absorption coefficient of the sample is determined from
$$\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{\text{spec}}=\sum_{i}C_{i}\left(\frac{\mu}{\rho}\right)^{\mathrm{A}}_{i},$$
(7.46)
where the composition \(C_{i}\) for each element is used. All elements where absorption is potentially significant should be considered, particularly for low-energy peaks in the presence of light elements where absorption is important. Similar difficulties arise with the determination of the sample density that must be deduced following a weighed average based on the concentration of the pure elements (whether the elements are detected/analyzed or not). Given the complexity of novel synthetic structures produced by chemists and physicists and materials scientists, extrapolating the density of materials is not a trivial task for some samples and the best approach to quantification is to avoid thick regions where absorption correction is required.

An additional parameter required for absorption correction is the sample thickness that must be determined accurately. Various approaches based on convergent beam electron diffraction [7.2, 7.94], contamination spot measurements [7.2, 7.85], and electron energy-loss spectroscopy (Sect. 7.8.2 in this chapter) exist to retrieve the sample thickness. Their accuracy, however, is approximately \(5{-}10\%\) at best. An additional difficulty in the evaluation of the absorption path length, from the point of origin to the exit point of the sample, is the knowledge of the sample geometry and the position of the detector relative to the point of analysis. This is demonstrated in Fig. 7.52 where the sample is tilted for analysis or when the area of analysis is not suitably positioned with respect to the detector (this might be necessary to achieve a suitable contrast condition in the image or to align an interface so that the electron beam is parallel to the interface plane). Significant differences in absorption corrections, and therefore relative concentration, can occur if careful consideration is not given to these issues during experiments [7.85, 7.95]. Errors in concentration up to a factor of two have been demonstrated if the detector position is not known during the analysis [7.95]. Similarly, quantitative analysis of interfaces and grain boundaries should be carried out so that the x-ray path from the analysis point to the detector and the electron beam are contained in the same plane. This approach simplifies all correction procedures since the absorption is considered within the boundary itself and the x-rays are generated by the electron beam traveling within the same material (Fig. 7.53).

Fig. 7.52

Absorption path for two configurations of sample with respect to the detector(s). When the sample is rotated so that the detector is in position A (relative to the sample), the absorption length is significantly longer than when the detector is in position B (relative to the sample). Adapted after [7.85]. This also applies for two-detector configurations in dual SDD systems (and even for four SDD systems) where each one would be affected by different absorption paths when quantification needs to be considered

Fig. 7.53

Ideal geometry for analysis of interfaces and grain boundaries. The electron beam and the absorption path of x-rays from the generated area to the detector are all contained within the plane of the boundary

These difficulties in accurately determining the correction coefficients, density, exact geometry of analysis and the detector efficiency particularly at low energies, suggest that quantitative work on carbides, nitrides, and oxides is particularly prone to inaccuracies and error propagation so that complementary quantitative measurements with EELS should be carried out whenever possible. For higher energy peaks (such as transition-metal K lines) absorption is significantly lower and can more easily be neglected in TEM measurements on typical thin foils.

Alternative approaches to retrieve composition of TEM samples while considering the absorption effects are given by the extrapolation techniques. These approaches involve sequences of measurements at increasing thicknesses as discussed below (Sect. 7.4.1, Correction Techniques).

In the case of multiple detectors, the detector configuration also has significant effects on the quantification. All the equations developed for the absorption and fluorescence corrections rely on a well-defined geometry with a single detector in consideration so that the path of a photon reaching the detector can be identified (and the appropriate angles for the calculations can be identified). Since there are multiple detectors and the detector areas are larger, the quantification will depend on the specific detector collecting the signal and, if the size is now larger than the regular Si(Li) detectors, simple approximations using a single average angle will no longer apply. Therefore, it is of utmost importance that the signal being integrated is quantified for the specific detector that has recorded this signal. For example, in a dual-detector geometry (and even more significant in a quad-detector configuration), the absorption such as considered in Fig. 7.52, will be minimal for detector B and maximal for detector A. Therefore, while absorption might be negligible for one detector, it might not be for another detector (or for three other detectors). It could also be possible that no signal is detected on three detectors if the sample is attached to a Cu grid used to support focused ion beam samples. The user must then be extremely vigilant when considering the sample geometry, and the mounting process of the sample in the sample holder. In the study of grain boundaries and particularly on interfaces, this issue is significantly problematic as the absorption would be on two different materials and the interface needs to be oriented with respect to one detector (or one set of detectors) to consider the geometry shown in Fig. 7.53).

Fluorescence

Fluorescence occurs when x-rays have sufficient energy to ionize atoms and new photons are generated by the de-excitation process. For example, the Fe-\(\mathrm{K}_{\alpha}\) (\({\mathrm{6.4}}\,{\mathrm{keV}}\)) can generate fluorescence of Cr (\({\mathrm{5.9}}\,{\mathrm{keV}}\)) x-rays but cannot excite Mn-K (\({\mathrm{6.5}}\,{\mathrm{keV}}\)) lines. However, the Fe-\(\mathrm{K}_{\beta}\) (\({\mathrm{7.0}}\,{\mathrm{keV}}\)) line can generate the Mn-K x-rays (although the contribution would be small since the intensity of the \(\mathrm{K}_{\beta}\) is about \({\mathrm{14}}\%\) of the \(\mathrm{K}_{\alpha}\)). At low energies, fluorescence can be important given the proximity of several lines. Although it is possible to generate fluorescence far away from the point of origin, fluorescence in thin specimens is not as important as in bulk samples for the simple reason that there is less material to fluoresce. The original developments required to correct for fluorescence effects are due to Philibert and Tixier [7.96]. Their model considered that the x-rays are generated from the middle of the thin foil and predicted small changes in intensities due to fluorescence. Improvements were later proposed by Nockolds et al [7.97] to consider uniform emission of x-rays in the sample including tilted sample geometries when element B causes fluorescence of element A
$$\begin{aligned}\displaystyle\frac{I^{\mathrm{A}}}{I_{\mathrm{A}}}&\displaystyle=C_{\mathrm{B}}\omega_{\mathrm{B}}\left[\frac{(r_{\mathrm{A}}-1)}{r_{\mathrm{A}}}\right]\frac{A_{\mathrm{A}}}{A_{\mathrm{B}}}\left(\frac{\mu}{\rho}\right)^{\mathrm{B}}_{\mathrm{A}}\frac{E_{C_{\mathrm{A}}}}{E_{C_{\mathrm{B}}}}\left(\dfrac{\text{ln}\frac{E_{0}}{E_{C_{\mathrm{B}}}}}{\text{ln}\frac{E_{0}}{E_{C_{\mathrm{A}}}}}\right)\\ \displaystyle&\displaystyle\quad\times\frac{\rho t}{2}\left(0.932-\text{ln}\left[\left(\frac{\mu}{\rho}\right)^{\mathrm{B}}_{\text{spec}}\rho t\right]\right)\sec\alpha\;,\end{aligned}$$
(7.47)
where \(I^{\mathrm{A}}/I_{\mathrm{A}}\) is the ratio of fluorescence intensity to primary intensity, \(\omega_{\mathrm{B}}\) is the fluorescence yield of element B, \(r_{\mathrm{A}}\) is the absorption edge jump-ratio of element A, \((\mu/\rho)^{\mathrm{B}}_{\mathrm{A}}\), \((\mu/\rho)^{\mathrm{B}}_{\text{spec}}\) are the mass absorption coefficients of x-rays from element B in element A, and the specimen, \(A_{\mathrm{A}}\), \(A_{\mathrm{B}}\) are the atomic weights of elements A and B, and \(E_{C_{\mathrm{A}}}\), \(E_{C_{\mathrm{B}}}\) is the critical excitation energy for the characteristic radiation of A and B.

As in the case of the absorption corrections, prior knowledge of the sample composition (in this case of element B) is required for the calculations thus resulting in an iterative process to estimate the correction term. This expression proved effective in correcting for fluorescence in very thick samples by modern AEM standards (Fig. 7.54).

Fig. 7.54

Effect of fluorescence correction on the quantification of spectra as a function of thickness. The raw data show strong thickness dependence while corrected data is independent of thickness following corrections using (7.47)

Additional corrections for the fluorescence caused by the continuum radiation are presently considered to be negligible.

Correction Techniques

The \(k\)-factor expression (7.42) assumes that the sample is infinitesimally thin and corrections are not required. The thin-film criterion discussed above states that corrections are not required if the \(k_{\text{AB}}\) factors do not change by more than about \({\mathrm{3}}\%\). This implies that estimates of the absorption correction factors using (7.45) are necessary if the thickness does not satisfy this criterion and that estimates of the sample thickness for the points of analysis are required. If the ACF is greater than \(\mathrm{1.03}\) or smaller that \(\mathrm{0.97}\), then corrections must be carried out and exact measurements of the thickness necessary.

Similarly, fluorescence corrections are required if the x-rays of one of the elements generate significant x-ray emission from other elements present in the sample. The fluorescence correction term proposed by [7.97] with estimates of the thickness is used to calculate the ratio of fluoresced intensity to primary intensity \(I^{\mathrm{A}}/I_{\mathrm{A}}\) for element A (and for the other elements present). When the correction term is greater than \({\mathrm{5}}\%\), the thin-film criterion is not met and the correction is required. If the thin-film criteria for absorption and fluorescence are both not satisfied, quantification requires the related corrections
$$\frac{C_{\mathrm{A}}}{C_{\mathrm{B}}}=k_{\text{AB}}\frac{I_{\mathrm{A}}}{I_{\mathrm{B}}}\,\text{(ACF)}\,\dfrac{1}{\left(1+\frac{I^{\mathrm{A}}}{I_{\mathrm{A}}}\right)},$$
(7.48)
where it is assumed that element A is fluoresced by element B and the thickness is calculated at each analysis point. As discussed in the absorption and fluorescence correction sections (Sects. 7.4.1, Absorption and Fluorescence), the calculations require prior knowledge of the sample composition, thus leading to the use of iterative techniques with initial assumptions of the composition and refinement of the compositions until there is agreement between the input and output concentrations.
Fig. 7.55

Extrapolation procedure using the method proposed by Horita et al [7.98]. The \(k_{\text{AB}}\) factor is plotted as a function of the integrated number of counts of an x-ray line (not absorbed in the sample) as an indirect indicator of the thickness of the sample. Extrapolation to zero counts, leads to the determination of the \(k_{\text{AB}}\) factor of a thin sample. The nonabsorbed x-ray line must be of sufficiently high energy and far from any absorption edge

Various approaches have been proposed over the last few years to simplify the quantitative analysis of samples without prior knowledge of the sample thickness. These methods rely on the extrapolation of the composition for a zero-thickness sample based on several measurements obtained from areas of different (but unknown) thicknesses. These measurements would require a constant acquisition time and electron-beam current during the series of measurements and a uniform sample concentration in the areas of analysis. The original approach was proposed by Horita et al [7.98] who suggested the use of measurements of \(k_{\text{AB}}\) factors at different sample thickness. By plotting the \(k_{\text{AB}}\) factors as a function of the intensity of one high-energy x-ray line in the spectrum and extrapolating the curve to zero intensity, absorption and fluorescence-free \(k_{\text{AB}}\) factors can be obtained (Fig. 7.55). This approach is based on the assumption that the high-energy x-ray line is not strongly absorbed and that its intensity is therefore directly related to the sample thickness (7.37). This method can be used to determine zero-thickness \(k_{\text{AB}}\) factors of standard samples and the composition of unknown samples following the extrapolation of x-ray line intensity ratios to zero thickness. Although extremely useful for the quantification of homogeneous samples, the extrapolation techniques are not suitable for the determination of composition of structures where the variations in thickness cause overlap of phases throughout the analyzed area. For example, when the composition of precipitates in a thin foil is required, only areas where the precipitates do not overlap with the matrix can be analyzed.

Various modifications to the Horita technique were proposed later by Van Cappellen [7.99] for work on alloys, Eibl and Van Cappellen and Doukhan [7.100, 7.101] for work on oxides using the charge neutrality concept, and in light element analysis demonstrated by Westwood et al [7.102]. An alternative quantification method makes use of the \(\zeta\) factor approach [7.103] and is discussed in the mapping Sect. 7.6.1 as it allows the thickness variations to be accounted for very effectively and can allow the determination of concentration and thickness at every single pixel of a map.

7.4.2 Quantification in EELS

Quantification Procedures

The atomic cross sections (Sect. 7.3) describe the probability of excitation of an atom by a fast incident electron. The energy dependence of these cross sections provide a basic description of the shape of the edges (without any solid state and bonding effects) while the partial cross section integrated for a given collection angle and energy window allows the intensity recorded in the spectrum to be related to the incident beam current and the number of atoms excited by the primary electron beam. The recorded signal for a particular edge, however, is superimposed on a large background due to lower energy excitations (including collective excitations and lower energy ionization edges). Therefore, even for elements present in the sample in large concentrations, the background can constitute the major contribution to the intensity at a given energy loss (Fig. 7.10) and must be removed, prior to the integration of the signal of the characteristic edge, using an extrapolation technique (Fig. 7.56). The extrapolation is based on a power-law dependence of the background derived from atomic physics (Sect. 7.3.4, Partial and Total Ionization Cross Sections)
$$I_{\mathrm{b}}(E)=AE^{-\mathrm{r}}\;,$$
(7.49)
where \(A\) and \(r\) can be considered as constants within a small region of the spectrum and their values can be retrieved based on a fit of the spectrum over an energy window prior to the edge of interest. The \(A\) parameter is strongly dependent on the intensity of the spectrum (hence, the current, recording time, collection efficiency, etc.) whereas the \(r\) parameter (related to the slope of the spectrum) varies between the values of \(2{-}6\) and is mostly dependent on energy loss and angular collection conditions. The fit and extrapolation of the background are carried out over energy windows (Fig. 7.56) using least-square fitting after transformation of the intensity and energy to a logarithmic scale so that the intercept of the fit relates to the \(A\) parameter and the slope to the \(r\) parameter. Alternatively, nonlinear least squares can also be used. Once the \(A\) and \(r\) parameters are found in the fitting region, the extrapolation allows subtraction of the background and extraction of the characteristic edge (Fig. 7.56). This power-law extrapolation assumes that the contributions from multiple inelastic losses to the spectrum in the region of interest are small. For typical samples having thicknesses in the order of \(50{-}100\,{\mathrm{nm}}\), this requirement is relatively easy to meet at high energy losses (a few \({\mathrm{100}}\,{\mathrm{eV}}\) and higher) but low-energy edges (\(<{\mathrm{80}}\,{\mathrm{eV}}\)) such as Al \(\mathrm{L_{23}}\), Li K are particularly difficult to analyze due to the rapid variation of \(r\) and the contributions of multiple scattering (Sect. 7.8.2). Very thin samples are thus required and more advanced approaches able to deal with the steep background and rapid variation of the \(r\) parameter for extrapolation are necessary.
Fig. 7.56

Background extrapolation of EELS edges. Fitting of the background is carried out over a window \(\Gamma\) and integration is carried out over an energy window \(\Delta\)

Polynomial-based extrapolation and/or constrained power law (forcing the background to cross the spectrum at a given energy) can be used to deal with numerical instabilities arising from the extrapolation. In order to remove the complications due to multiple scattering when extracting low-energy edges, deconvolution of the spectrum using the Fourier-log technique is necessary (Sect. 7.8.2). Fitting approaches using reference spectra and multiple scattering convolution with low-loss spectra have been developed recently [7.104] to deal very successfully with weak and/or overlapping edges (Fig. 7.57a,b). Signal extraction using difference techniques [7.78] has also been used to analyze very weak signals for elements present in trace concentrations [7.105, 7.106]. A variant of this latter method uses the numerical difference technique [7.107]. This technique is relatively insensitive to the sample thickness. Following the extraction of the background, integration of the signal over an energy window \(\Delta\) must be carried out (Fig. 7.56) to relate the signal to the number of atoms present under the electron beam.

Fig. 7.57a,b

Model-based quantification of EELS edges using maximum likelihood estimators of parameters of spectra and variation of quantification with respect to thickness (a). Models include thickness effects and cross sections. (b) This panel shows that quantification with this technique appears to be less sensitive to thickness variations based on a series of measurements with increasing thickness (relative to an inelastic mean free path) on a wedge sample. Reprinted from [7.104], with permission from Elsevier

Assuming that single scattering is dominant (i. e., one single loss event occurs when the electron travels through the sample), the integrated edge intensity, obtained for a given collection angle \(\beta\) and integration window \(\Delta\), is related to the number of atoms per unit area \(N\), the unscattered intensity \(I_{0}\) (the zero-loss peak is assumed to be nearly equal to the incident beam in a single scattering first approximation), and the partial cross-section \(\sigma_{k}\) for the same collection and integration conditions
$$I^{1}_{\mathrm{k}}(\beta,\Delta)=N\,I_{0}\sigma_{\mathrm{k}}(\beta,\Delta)\;,$$
(7.50)
where the index refers to the single scattering assumption. More realistically, we must consider that plasmon interactions redistribute the intensity in the spectrum causing multiple inelastic processes visible both at low energy and at the core edges (i. e., core-loss and plasmon scattering occurs as discussed in Sect. 7.8.2; Fig. 7.58). Furthermore, elastic scattering reduces the collected intensity by scattering electrons outside the collection aperture both for the apparent zero-loss intensity, relative to the incident beam intensity, and for the edge. In these conditions, we must integrate the spectrum intensity above the zero-loss peak so as to include inelastic scattering up to an energy-loss \(\Delta\) (Fig. 7.58).
$$I_{\mathrm{k}}(\beta,\Delta)=N\,I(\beta,\Delta)\sigma_{\mathrm{k}}(\beta,\Delta)$$
(7.51)
Although this equation allows us to measure the absolute number of atoms per unit area, the more typical approach in EELS quantification is to use intensity ratios to deduce the relative concentration using
$$\frac{N_{\mathrm{a}}}{N_{\mathrm{b}}}=\frac{I_{ka}(\beta,\Delta)}{I_{jb}(\beta,\Delta)}\frac{\sigma_{jb}(\beta,\Delta)}{\sigma_{ka}(\beta,\Delta)}\;.$$
(7.52)
In this case, the indices \(k\) and \(j\) relate to the fact that different types of edges can be used for quantification (K, L, or M). When multiple scattering is removed from the spectrum through the use of deconvolution techniques (Sect. 7.8.2) the expression remains the same but the extracted intensity is obtained from extrapolation of the deconvoluted spectrum.
Fig. 7.58

Extrapolation of edge signals and the integration windows used for quantification. The integration of the signal over the window \(\Delta\) is carried out for the signal at the edge and for the low-loss part of the spectrum (based on (7.51)). \(G\) is the value of the gain change (scale change factor) to visualize the high-energy part of the spectrum

Correction for Convergence

Equation 7.52 assumes that the incident beam is parallel and the angular distribution of scattering (represented by the Bethe surface in Fig. 7.41) is not altered by any convergence of the incident beam. Corrections to account for convergence of the incident beam on to the sample, however, become necessary when the electron beam is convergent \((\alpha\geq{\mathrm{0.3}}\beta)\) as in the case of analysis in STEM mode. In STEM configuration the convergence-angle \(\alpha\) typically exceeds this criterion (Fig. 7.59). The relative concentration must be calculated by including convergence correction factors into (7.52) that can be deduced analytically [7.35] for each particular edge. The relative concentration becomes
$$\frac{N_{\mathrm{a}}}{N_{\mathrm{b}}}=\frac{I_{ka}(\beta,\Delta)}{I_{jb}(\beta,\Delta)}\frac{\sigma_{jb}(\beta,\Delta)}{\sigma_{ka}(\beta,\Delta)}\frac{F_{\text{1b}}}{F_{\text{1a}}}\;,$$
(7.53)
where \(F_{1a,b}\) are the convergence correction factors that are dependent on the angular scattering distribution for a particular edge via the characteristic scattering angle \(\theta_{\mathrm{E}}\). Correction factors must therefore be calculated for each edge of interest in the quantification and can be deduced from Fig. 7.60 (from [7.35]) or from simple programs [7.35]. It is seen that for edges of very similar intrinsic angular distributions (i. e., two edges close in energy will have very similar \(\theta_{\mathrm{E}}\)) the ratio \(F_{\text{1b}}/F_{\text{1a}}\) is close to one and no effect of convergence is visible irrespective of the convergence angle.
Fig. 7.59

Relation between the convergence and collection angles for quantification of spectra accounting for the incident beam convergence

For absolute quantification, the correction of the expression relating the number of atoms and the recorded intensity under the edge (7.51) becomes necessary irrespective of the intrinsic angular distribution
$$I_{k}(\beta,\Delta,\alpha)\approx F_{2}NI(\beta,\Delta)\sigma_{\mathrm{k}}(\beta,\Delta,\alpha)\;,$$
(7.54)
where the correction factor \(F_{2}\approx F_{1}\) for \(\alpha\leq\beta\), \(F_{2}=(\alpha/\beta)^{2}F_{1}\) for \(\alpha\geq\beta\) and can be deduced from Fig. 7.60 [7.35].
Fig. 7.60

Correction factors \(F_{1}\) useful to quantify spectra accounting for convergence of the incident beam \(\alpha\) from [7.35]. The convergence factors \(F_{1}\) must be determined for each edge based on the convergence angle \(\alpha\) relative to the collection angle \(\beta\). For each edge, the characteristic angle \(\theta_{\mathrm{E}}\) relative to the collection angle \(\beta\) must be used to identify the appropriate curve to be used in the quantification. The \(F_{2}\) curves are used only for absolute quantification (not discussed in this section)

\(k\)-Factor Approach

All the expressions for relative quantification in EELS are similar to the \(k_{\text{AB}}\) factor equations in EDXS microanalysis. In a similar manner, a \(k\)-factor approach has been proposed [7.108] to quantify EELS spectra although the experimental and sample conditions (convergence, collection, sample thickness) must be very strictly controlled and reproducible. Once the \(k\)-factor has been determined on the reference sample accounting for the cross-section ratios and possible convergence effects, the same \(k\)-factor can be used to deduce the composition of the unknown samples as in EDXS microanalysis. Cross-section calculations using the currently available models (Hartree–Slater or Hydrogenic, Sect. 7.3.4) and convergence corrections are already built-in in commercial programs that control the acquisition functions of the spectrometers. An additional calculation approach to deduce the cross sections necessary for the quantification makes use of tabulations of the oscillator strengths obtained from optical data (Sect. 7.3.4, Angular and Energy Dependence of the GOS). These methods require the acquisition of spectra in dipole conditions (i. e., with small collection angles) and within the energy range where the angular distribution of scattering is still dominated by the Lorentzian term of (7.22), (i. e., near the edge threshold). The methodology is described in [7.79] and summarized in [7.78].

Limitations in Analysis and Quantification

Although EELS quantification is not affected by terms such as absorption or fluorescence, there are major drawbacks in quantitative and even qualitative EELS analysis. The major difficulty is the strong effect of the sample thickness on the detection of EELS edges. For example, spectra from specimens of very simple composition a few tens of nm in thickness can reveal well-resolved edges while spectra from thicker areas (as thin as \({\mathrm{100}}\,{\mathrm{nm}}\)) might show no edges at all (Fig. 7.61). This severe limitation to the visibility of edges is due to the contribution of multiple inelastic scattering that increases the background under the edge. The contribution of multiple scattering, however, is not uniform as a function of energy loss and the quantification of spectra for an increasingly thick sample demonstrates a variation of the apparent concentration with thickness. Systematic measurements of the ratio of two elements, for example, show that for samples with thickness relative to the mean inelastic free path (Sects. 7.3.4 and 7.8.2) \(t/\lambda> {\mathrm{0.5}}\) are unreliable (Fig. 7.62). Even when multiple scattering effects are removed with deconvolution techniques (Sect. 7.8.2), the effects persist [7.35] suggesting that additional contributions due to the angular distributions of losses are present and cannot be neglected in correction approaches. Calculations that include the contributions of elastic scattering [7.109] and convolutions of the energy and angular distributions of scattering angle [7.110] demonstrate trends that are consistent with the experimental variation of the composition with thickness. Correction programs have been developed to account for these effects [7.111]. The lack of visibility of edges in spectra, even for pure elements, implies that users should be particularly cautious about drawing a conclusion on the absence of elements during the analysis of point spectra and particularly with the application of energy-filtered imaging techniques. In the latter condition, since spectra are usually disregarded and the acquisition is done in an automatic procedure, erroneous results can often be obtained if care is not taken to check the sample thickness prior to the analysis of spectra. Relative thickness maps should be used therefore as a routine check prior to elemental imaging to verify the applicability of the \(t/\lambda<{\mathrm{0.5}}\) condition. The procedure for the measurement of the relative thickness is described in detail in Sect. 7.8.2.

Fig. 7.61

Variation of the visibility of spectra with increasing thickness of the sample. Thick samples will not necessarily show EELS edges even for major constituents

Fig. 7.62

Variation in the quantification of two elements as a function of thickness relative to the total inelastic mean free path

7.5 Resolution in Microanalysis

7.5.1 EDXS Microanalysis

There are two contributions determining the spatial resolution in x-ray microanalysis. The first contribution arises from the electron-beam diameter and the second from the electron-beam broadening generated when electrons travel in the sample. The early analytical TEMs provided beam diameters of the order of tens to a hundred nanometers and the beam broadening within the sample was not a significant issue. Measurements using contamination spots demonstrated that the illuminated area on the top of the sample was essentially the same as the exit surface of the electron beam [7.112]. The situation changed radically with the development of analytical instruments equipped with field-emission sources capable of achieving probe sizes in the order of a nanometer or smaller (Sect. 7.2.1) whereby the ultimate limits in spatial resolution due to the intrinsic beam broadening could be probed.

To determine the electron-beam broadening, we must consider that the trajectory of the incident beam is controlled by the elastic scattering process that causes the deviation of the incident electrons as they travel through the sample. Transport equations [7.113] and detailed multislice calculations [7.114, 7.70, 7.71, 7.72] (Sect. 7.3.2) have been developed to describe the electron-beam propagation (including the impact of sub-Angstrom beams [7.70]) but more extensive work has been carried out in the field of AEM using Monte Carlo simulations that consider the individual trajectories of the incident electrons, the elastic scattering cross sections that modify the electrons trajectories, and the inelastic scattering that causes the slow-down of the electrons (Fig. 7.63). Monte Carlo approaches are easily applicable in complex geometries of samples (for example interfaces and particles) although they neglect the effect of channeling of electrons in a crystal and assume therefore the sample is amorphous or tilted away from a zone axis. The Monte Carlo technique is based on the generation of random numbers that are used to calculate the scattering angles (via equations of the elastic cross-sections (7.11)), the path length between scattering events (using the elastic mean-free path of (7.12)), and the energy loss between the scattering events (using the stopping power derived from the inelastic cross-sections (7.19)). Electron trajectories simulated with the Monte Carlo method for thin films show the dependence of beam broadening on the accelerating voltage (compare Fig. 7.64 with Fig. 7.65). The effect of the average atomic number and sample geometry can also be easily determined. With these simulations, it is possible, in principle, to evaluate the exit area and the volume containing an arbitrary fraction of electrons that will contribute to the generation of the x-ray signal and thus the spatial resolution. For example, the interaction volume containing \({\mathrm{90}}\%\) of scattered electrons is typically used as a reference in the AEM literature to determine the resolution although more stringent criteria (with \({\mathrm{95}}\%\) of the electrons [7.117] and \({\mathrm{99}}\%\) of the electrons [7.118]) have also been proposed. Although broadening values can be retrieved from the simulated trajectories in a few minutes of computation even on laptop computers or through web-based programs [7.115], quick estimates of electron-beam broadening are necessary to evaluate the approximate loss of spatial resolution in microanalysis. Goldstein et al [7.119] developed a simple analytical model assuming a single scattering event in the middle of the foil thickness (Fig. 7.65) to calculate the broadening of electron-beam \(b\) that contains \({\mathrm{90}}\%\) of the scattered electrons
$$b={\mathrm{7.21\times 10^{5}}}\frac{Z}{E_{0}}\left(\frac{\rho}{A}\right)^{\frac{1}{2}}t^{\frac{3}{2}}\;,$$
(7.55)
where \(b\) is the broadening in cm units, \(Z\) is the atomic number, \(A\) is the atomic weight, \(E_{0}\) is the accelerating voltage (eV), \(\rho\) is the density (\(\mathrm{g/cm^{3}}\)), and \(t\) is the specimen thickness (in cm). This simple equation is the most common method to estimate broadening and is most suitable for foils of thickness approximately equal to the elastic mean free path (Sect. 7.3.2) although refinements were proposed to account for multiple scattering in thicker foils [7.120, 7.85].
Fig. 7.63

Simulation of the electron trajectories in a thin sample of Au (\({\mathrm{100}}\,{\mathrm{nm}}\) thickness) using the Monte Carlo method for \(\mathrm{100}\)-\(\mathrm{keV}\) incident electrons. Calculations of electron trajectories are carried out using the public-domain code CASINO developed in the Gauvin group [7.115] available on the web [7.116]

Fig. 7.64

Electron trajectories in a \(\mathrm{100}\)-\(\mathrm{nm}\) thick Au with Monte Carlo calculations for \(\mathrm{300}\)-\(\mathrm{keV}\) incident electrons

Fig. 7.65

Simplified diagram describing the broadening of electrons in samples. Parameter \(b\) describes broadening at the bottom of the foil using (7.55) and the single scattering assumption

The spatial resolution can be determined based on the contribution of both the electron-beam size \((d)\) and the electron-beam broadening \((b)\). Early models proposed the addition of the two terms in quadrature
$$R=(b^{2}+d^{2})^{\frac{1}{2}}$$
(7.56)
to yield the resolution \(R\) but the currently adopted definition is based on experimental evidence suggesting the use of a less stringent, but still arbitrary, average sum of the two terms leading to a definition of spatial resolution based on the broadening in the middle of the foil thickness as suggested in Fig. 7.66
$$R=\frac{d+R_{\text{max}}}{2}\;,$$
(7.57)
where \(R_{\text{max}}\) is determined from (7.56) as the maximum broadening at the bottom of the foil. If one considers \({\mathrm{90}}\%\) of the beam current distribution, the resolution becomes [7.121]
$$R=\sqrt{\frac{d^{2}+bd+b^{2}}{3}}\;.$$
(7.58)
Recent measurements with sample geometries that probe the electron-beam broadening at the bottom of the foil have been published and have led to the conclusion that, although the single scattering models predict the right beam broadening magnitude these quantitatively overestimate the broadening observed experimentally [7.122]. More accurate models [7.121, 7.123] consider a Gaussian probe distribution of standard deviation \(\sigma\) and hence a probe with full-width at half-maximum of \(\mathrm{4.29}\)\(\sigma\) propagating the foil with a distribution accounting for beam spreading and the incident probe distribution  [7.121]
$$I(x,y,t)=\frac{i_{\mathrm{b}}}{\uppi\left(2\sigma^{2}+\beta t^{3}\right)}\exp\left(\frac{-(x^{2}+y^{2})}{2\sigma^{2}+\beta t^{3}}\right),$$
(7.59)
where \(\beta={\mathrm{500}}(4Z/E_{0})^{2}(\rho/A)\) and \(i_{\mathrm{b}}\) is the incident beam current.
Using this more detailed electron distribution in the probe (as well as simpler models) as it travels through the sample [7.121, 7.124] determined the equilibrium segregation profile in grain boundaries based on composition line scans and two-dimensional map measurements (Fig. 7.67). On the basis of Gaussian intensity distribution of incident electrons one can define the resolution, for a given fraction of electrons \(Q\), as a function of thickness as
$$Q=\frac{1}{h}\int^{h}_{0}\left[1-\exp\left(\frac{-R^{2}}{4\left(2\sigma^{2}+\beta t^{3}\right)}\right)\right]\mathrm{d}t\;,$$
(7.60)
where the integration is carried out over the thickness up to \(t=h\) where \(R\) is the diameter of the cone defining the resolution. The distribution is assumed to be circularly symmetric (since there is no dependence on the azimuth angle in the equation) and applicable to a two-dimensional case. By defining the resolution criteria based on the fraction of electrons within a given radius, it is possible to plot the expected resolution (Fig. 7.68) for a diameter containing 50 and \({\mathrm{90}}\%\) of electrons. A comparison of the predictions with experimental profiles obtained on equilibrium segregation in Cu (for FWHM and at \({\mathrm{10}}\%\) maximum) [7.121] shows excellent agreement for thicknesses up to the elastic mean free path.
Fig. 7.66

Definition of parameters related to the calculation of the resolution with (7.57) and (7.56) (where \(R_{\text{max}}\) is estimated with (7.56) as the maximum broadening at the bottom of the foil) and \(R\) is the broadening in the middle of the foil. \(d\) is the diameter of the incident electron beam

Fig. 7.67

Composition profile of S segregation at an Ni grain boundary. The profile is based on the modeling of the beam profile using a Gaussian probe distribution and (7.59). Profile courtesy of V. Keast, U. Newcastle, Australia. After [7.121]

Fig. 7.68

Resolution variation as a function thickness for 50 and \({\mathrm{90}}\%\) fraction of electrons within the beam. (Adapted after [7.121])

With a dedicated scanning transmission electron microscope, work on grain boundary segregation and multilayer composite materials has shown that it is possible to detect submonolayer segregation at grain boundaries [7.121, 7.124, 7.125] and map the composition of quantum wells in semiconductor materials (Fig. 7.69). Early EDXS analysis with aberration-corrected instruments was reported by [7.126, 7.65] and detailed calculations accounting for beam propagation and quantification with multiple detector geometries (including a comparison with EELS) [7.67] will be discussed in more detail in Sect. 7.9.

Fig. 7.69

(a) Quantitative EDXS elemental maps of In-rich quantum wells in semiconductors and (b) related intensity profiles across the quantum wells. Courtesy of V. Keast, University of Newcastle

7.5.2 Energy-Filtered Microscopy and EELS Microanalysis

The spatial resolution in energy-loss spectroscopy measurements is strongly dependent on the operating mode of the microscope during EELS analysis due to the illumination conditions of the area of interest, the effects of lens aberration, and the spectrometer coupling conditions. The contribution from the microscope lens aberrations comes essentially from the objective lens and can be calculated as follows. With respect to an incident beam of primary energy \(E_{0}\) propagating along the optic axis, the inelastic interactions that induce an energy-loss \(E\) and scattering at an angle \(\theta\) will cause a blur of an object point (and the image on the viewing screen of the microscope) due to chromatic aberrations \(R=M_{0}\theta\Updelta f\) where \(M_{0}\) is the magnification of the objective lens and \(\Updelta f=C_{\mathrm{C}}(E/E_{0})\). According to the Rayleigh criterion, two object points with blurred independent distributions can be resolved if they are separated by a distance giving rise to an intensity drop of \({\mathrm{20}}\%\) between the maxima of the summed distributions (Fig. 7.70).

Fig. 7.70

Rayleigh definition of resolution based on the overlap of two diffraction-limited functions placed in close proximity to each other. If the sum of the two intensity profiles shows a dip of \({\mathrm{20}}\%\) of the maximum intensity, the peaks are considered to be resolvable

The distance between these two points is defined as the resolution and is obtained as [7.35]
$$r_{i}\approx 2\theta_{\mathrm{E}}\Updelta f\approx C_{\mathrm{c}}\left(\frac{E}{E_{0}}\right)^{2}.$$
(7.61)
This resolution is expected for energy-filtered images acquired at an energy-loss \(E\) when the incident electron energy is kept constant, the image is initially focused for electrons that have lost no energy (also known as a zero-loss image, Sect. 7.2.4, Spectrometers), and the scattering angles are only limited by the intrinsic scattering distribution at the energy-loss \(E\) (i. e., no limiting objective aperture is used). For images of thick samples, it should be pointed out that the inelastic losses can constitute the most significant part of the signal resulting in significant blurring of images. These contributions can be removed by energy filtering so that the inelastically scattered electrons are removed with the energy-selecting slit and the images appear sharp even for thick samples (Fig. 7.71). In typical conditions for optimum energy-loss imaging at an energy-loss \(E\), however, the incident electron energy is readjusted so that the spectrometer is focused at the energy-loss \(E\) (i. e., by raising the high-tension of the microscope so that the primary energy is now \(E_{0}+E\)) and the electrons of the related loss are in focus. If an energy-selecting slit \(\Updelta E\) wide is used, the resolution becomes [7.35]
$$r_{i}\approx 2\theta_{\mathrm{E}}\Updelta f\approx\left(\frac{C_{\mathrm{C}}}{4}\right)\left(\frac{\Updelta E}{E_{0}}\right)^{2}$$
(7.62)
When the angular distribution of scattering is limited by an objective aperture \(\beta\) smaller than the angular distribution of scattering—namely \(\theta_{\mathrm{E}}\), other contributions become dominant (diffraction limit, spherical aberration, delocalization of scattering) and the spatial resolution in elemental maps is derived from a combination of all factors as described in the energy-filtered imaging section (Sect. 7.6.2, Quantitative EELS Imaging).
Fig. 7.71

(a) Energy-filtered image of an Al alloy with strengthening precipitates. The image has been obtained by selecting only the electrons that have not lost significant amounts of energy (i. e., zero-loss image). (b) Unfiltered image containing all the inelastically scattered electrons. Although the total intensity is reduced, the precipitates are more clearly defined with sharper edges in the zero-loss image

For point analyses carried out in TEM image mode, the spectrometer entrance aperture is used to select electrons from the regions of interest so that they enter the spectrometer for analysis. Under these conditions, the chromatic aberration will also cause contributions from areas further away from the spectrometer aperture-delimited area due to the significant broadening of the angular distributions and high energy losses. If no angular-selecting aperture (the objective aperture in this case) is used, the total broadening due to angular scattering up to the Bethe ridge (\(\theta_{\mathrm{r}}=\theta_{\mathrm{C}}\) which includes most of the scattering intensity, see Sect. 7.3.4, Angular and Energy Dependence of the GOS) is
$$r_{\mathrm{c}}\approx\theta_{\mathrm{r}}\Updelta f$$
(7.63)
while the use of an angular-selecting aperture (in the objective back-focal plane when the TEM is operated in image mode) will limit the broadening to \(r_{\mathrm{c}}(\beta)\approx\beta\Updelta f\). Once again, one can distinguish two cases: \(\Updelta f=C_{\mathrm{c}}(E/E_{0})\) if the image is focused at the primary energy \(E_{0}\) and \(\Updelta f\approx C_{\mathrm{c}}(\Updelta E/4E_{0})\) if the electrons are focused at an energy loss of interest and an energy-selecting slit \(\Updelta E\) wide is used [7.35].

The broadening due to aberrations is a significant contribution that cannot be reduced in current instrumentation although chromatic aberration correctors have been developed and have demonstrated significant resolution improvements in energy filtered images. However, at the present time, the ultimate resolution is achieved by limiting the illuminated area with the electron beam. Current commercial Schottky-type field-emission gun instruments allow us to achieve a probe size of about \({\mathrm{0.2}}\,{\mathrm{nm}}\) with typically \({\mathrm{10}}\,{\mathrm{pA}}\) of current (more with a cold field-emission source). Spherical aberration correctors can be used to improve the probe-forming capability. These instruments have been implemented by major manufacturers on both dedicated scanning transmission microscopes and STEM-TEM instruments yielding sub-Angstrom probes with \(10{-}20\,{\mathrm{pA}}\) current or about a few hundred pA for \(\mathrm{0.2}\)-\(\mathrm{nm}\) diameter probes [7.127, 7.128, 7.13, 7.52] (refer also to Chap.  2 by Nellist in this volume). With such small probes, however, it becomes increasingly important to be aware of the detailed electron propagation within the sample as discussed in Sect. 7.3.2.

Since the electron-propagation process in the sample is the same irrespective of the microanalysis technique probing various signals, elastic scattering does affect the broadening of the electron beam not only in EDXS but also in EELS measurements. The impact of this broadening on the degradation of the spatial resolution can, however, be somewhat controlled with the use of a collection aperture that limits the scattering angles entering the spectrometer. As shown in Fig. 7.72a-ca, electrons scattered at high angles (thus away from the forward direction and the incident probe distribution) can be eliminated with the use of an angle-limiting collection aperture (either the objective aperture if the spectra are acquired in image mode or the spectrometer aperture if the spectra are acquired in diffraction and STEM mode). Assuming that the elastic scattering distribution is large compared to the collection angle and that there is no strong Bragg scattering (essentially an amorphous sample) the fraction of electrons contained within a radius \(r\) for a given sample thickness \(t\) and collection aperture \(\beta\) can be estimated geometrically (Fig. 7.72a-cb,c). On the basis of Fig. 7.72a-cb we can estimate this geometrical broadening contribution for a parallel incident electron beam. For a sample thickness of \({\mathrm{100}}\,{\mathrm{nm}}\) and a collection angle of \({\mathrm{10}}\,{\mathrm{mrad}}\), the fraction of electrons contained within \({\mathrm{0.5}}\,{\mathrm{nm}}\) would be 85 and nearly \({\mathrm{100}}\%\) for \({\mathrm{1}}\,{\mathrm{nm}}\). These contributions can be small indeed as compared to the intrinsic broadening due to the convergence of the electron beam required for STEM imaging and optimal probe size (in a range from a few mrad to a few tens of mrad) in the case of aberration-corrected STEM instruments (Sect. 7.2.1). The latter geometrical broadening contribution for a \(\mathrm{100}\)-\(\mathrm{nm}\) thick sample assuming optimal probe size with \(\mathrm{30}\)-\(\mathrm{mrad}\) convergence in an aberration-corrected STEM would be \({\mathrm{3}}\,{\mathrm{nm}}\)! Although these numbers represent an extreme effect due to the sample thickness, typical samples used to demonstrate the ultimate resolution tend to be in the order of \(5{-}10\,{\mathrm{nm}}\) thickness as required to limit the propagation of the electron beam within a single atomic column (Sect. 7.3.2). In these cases, the geometrical broadening would be of the order of \(0.1{-}0.3\,{\mathrm{nm}}\).

Fig. 7.72a-c

Geometrical contribution to the broadening of the electron beam [7.35]. (a) The effective broadening can be reduced by using an angle-limiting aperture even if the real broadening still occurs in the sample as larger scattering angles (causing the broadening) are cut-off by the aperture. (b) Diagram defining the broadening terms necessary to calculate the fraction of electrons scattered outside a radius \(r\) for a given combination of thickness and collection aperture. (c) Geometrical evaluation of the fraction of electrons contained within a radius \(r\), from [7.35]

When the electron-beam size is in the order of a fraction of a nanometer, a significant contribution affecting the spatial resolution is the delocalization of inelastic scattering—i. e., the excitation and energy loss can occur even for electrons traveling at a finite distance \(b\) from the target atoms and not only when the incident electron is directly on the atom in the classical particle point of view. This factor ultimately limits the spatial resolution in EELS analysis with aberration-corrected electron microscopes capable of achieving sub-\(\mathrm{\AA{}}\) beam sizes.

As expected from the wave-mechanical perspective, the origins of the delocalization effect are quantum mechanical in nature and relate to the uncertainty relation but a simple classical treatment already allows one to identify the significant elements related to the spatial resolution. On the basis of classical scattering, the impact parameter \(b\) is related to the scattering angle of electrons (Fig. 7.40): large scattering angles are indicative of a smaller impact parameter and thus interactions closer to the atom. Hence, measurement of energy losses carried out on electrons scattered at large scattering angle would imply higher apparent resolution. The simplest treatment of the spatial resolution involving the uncertainty principle [7.129] invokes the time of interaction (\(\Updelta\tau\)) of an electron traveling at a speed \(v\) and at a distance \(b\) from the atom. The interaction time is \(\Updelta\tau=b/v\). Applying the uncertainty relation \(\Updelta E\Updelta\tau\leq h\) yields \(b_{\text{max}}=hv/\Updelta E\) with \(b\) considered as the ultimate spatial resolution and \(\Updelta E\) here is the energy loss. For energy losses of about \({\mathrm{20}}\,{\mathrm{eV}}\) (at \({\mathrm{100}}\,{\mathrm{KeV}}\)) the delocalization is of the order of \({\mathrm{2}}\,{\mathrm{nm}}\) while for energy losses of \({\mathrm{200}}\,{\mathrm{eV}}\) it would be \({\mathrm{0.2}}\,{\mathrm{nm}}\). A more detailed analysis [7.130, 7.131] based on the same principles yields a delocalization with the form
$$b=\frac{\hbar v\beta}{\Updelta E}\left[\left(\beta^{2}+\theta^{2}_{\mathrm{E}}\right)\text{ln}\left(1+\frac{\beta^{2}}{\theta^{2}_{\mathrm{E}}}\right)\right]^{-\frac{1}{2}},$$
(7.64)
where \(\beta\) is the limiting collection angle (the maximum scattering angle entering the spectrometer) and \(\theta_{\mathrm{E}}\) is the characteristic scattering angle at the energy-loss \(\Updelta E(\theta_{\mathrm{E}}=\Updelta E/2E_{0})\).

The detailed quantum mechanical treatments by Muller and Silcox [7.132] and Kohl and Rose [7.133] as well as the work of Pennycook [7.130] lead to results (in terms of trends and order of magnitude) similar to those given by the simpler analysis presented by Egerton [7.134, 7.35] who considered a spatial resolution related to the diffraction limit imposed by the width of the energy-dependent scattering distribution and the collection aperture. This approach can be explained as follows. For high-energy losses, the angular distribution can be broader than the collection aperture and thus the diffraction limit imposes a resolution determined by the collection aperture \(\beta\) similar to the Rayleigh criterion i. e., \({\mathrm{0.6}}\lambda/\beta\).

When the angular scattering distribution is not limited by the collection aperture (typically for low energy losses), the scattering distribution is limited by a cut-off angle \(\theta_{\mathrm{c}}=(2\theta_{E})^{1/2}\) related to the Bethe-ridge maximum scattering angle (Sects. 7.3.3 and 7.3.4, Angular and Energy Dependence of the GOS). In this case, the median scattering \(\tilde{\theta}\) angle containing \({\mathrm{50}}\%\) of the electrons is \(\tilde{\theta}\approx(\theta_{\mathrm{E}}\theta_{\mathrm{c}})^{1/2}\) and thus \(\tilde{\theta}\approx{\mathrm{1.2}}(\theta_{\mathrm{E}})^{3/4}\). By considering that the limiting aperture containing \({\mathrm{50}}\%\) of the electrons is effectively given by \(\tilde{\theta}\), this diffraction limit contribution will be \({\mathrm{0.6}}\lambda/\tilde{\theta}\) i. e., \({\mathrm{0.5}}\lambda/(\theta_{\mathrm{E}})^{3/4}\). Combining the two limiting terms in quadrature, the delocalization contribution to the resolution, related to inelastic scattering of \({\mathrm{50}}\%\) of the intensity, is
$$d_{50}\approx\sqrt{\left[\frac{{\mathrm{0.5}}\lambda}{(\theta_{\mathrm{E}})^{3/4}}\right]^{2}+\left(\frac{{\mathrm{0.6}}\lambda}{\beta}\right)^{2}}\;. $$
(7.65)
Experimental measurements roughly agree with these estimates and more detailed quantum mechanical treatments [7.132, 7.133] (Fig. 7.73). More recent work deals with the impact of aberration correction and ultimate spatial resolution by combining the effects of inelastic scattering and sub-\(\mathrm{\AA{}}\) beam propagation in the sample [7.135, 7.136, 7.137]. This delocalization contribution enters the definition of resolution for energy-filtered elemental mapping (Sect. 7.6.2, Quantitative EELS Imaging). It is important to note that elastic scattering can affect the apparent resolution shown in inelastic images due to the modulation of the signal entering the spectrometer. If strong elastic contrast is present (for example lattice fringes, larger atomic number elements, or diffraction contrast), inelastic images might show features with apparent resolution that are not simply related to the inelastic scattering distribution but rather to the much more localized elastic scattering [7.139]. Lattice images obtained with inelastically scattered electrons can be obtained, therefore, at energy losses where the localization contribution is, in principle, larger than the interatomic spacing due to these effects. As discussed in the elemental mapping Sect. 7.6.2, approaches have been devised to circumvent these effects by selecting the collection aperture so that the phase contrast, giving rise to lattice images, disappears and by normalizing the inelastic images with the background prior to the edge threshold [7.140, 7.88].
Fig. 7.73

Localization of inelastic scattering as a function of energy loss for a collection angle \(\beta\) of \({\mathrm{10}}\,{\mathrm{mrad}}\). Curves are based on a Rayleigh approach to the evaluation of the localization. Brown symbols refer to FWHM (diamonds) and \(d_{80}\) (hexagons) to values from [7.16]. Open symbols refer to \(d_{80}\) (circles), \(d_{50}\) (squares), \(\Updelta x\) (triangles) and FWHM (diamonds) to values from [7.138]. Various experimental measurements based on the literature are presented (filled symbols). Please refer to [7.35] for more details

7.6 Elemental Mapping

7.6.1 Elemental Mapping with EDXS

With the control of scanning coils of the electron microscope (Sect. 7.2.2) and by recording x-ray counts within a particular energy window of interest under a characteristic peak or even the full spectrum at each pixel of a rastered area, it is possible to combine the intensity recorded for each element into a map in order to display the distribution of elements in the sample with a resolution appropriate to the experimental conditions and sample thickness. Because of the long recording time necessary to obtain a statistically significant number of counts at each pixel and still maintain a small probe size compatible with the spatial resolution of the technique, the acquisition of elemental maps with EDXS can take several minutes to a few hours, depending on the sample thickness, the detector solid angle, the size of the area, sampling etc. Examples of such maps in the case of semiconductor materials are shown in Figs. 7.69 and 7.74. Although simple unprocessed elemental maps provide a wealth of information on the distribution of elements, it is also possible to extract quantitative information on the concentration of elements based on the use of ratios of images and the use of \(k\)-factor analysis. Relative concentration maps for elements A and B can thus be obtained by processing the entire image rather than just integrated intensities under an energy window. The quantitative concentration map can be obtained as \(I_{C_{\mathrm{A}}/C_{\mathrm{B}}}(x,y)=k_{\text{AB}}[I_{A}(x,y)/I_{B}(x,y)]\) where \(k_{\text{AB}}\) is the Cliff–Lorimer factor discussed in Sect. 7.4.1, Quantification in EDXS and \(I_{A,B}(x,y)\) are the elemental maps for elements A and B, respectively. Each elemental map has a background image (obtained by recording the intensity within a window where no characteristic peak is visible) subtracted prior to the quantification. The division is carried out pixel by pixel with the constant \(k_{\text{AB}}\) factor.

Fig. 7.74

Color-coded elemental map of a device showing the distribution of elements in the rastered area: Red: Al-rich area, blue: Si-rich area, green: Ti-rich area. White: Tungsten (W). Interdiffusion of Si into the Al is noticed through the bottom barrier layer containing Ti

If absorption is not significant, relative concentration maps are insensitive to changes in thickness. Therefore, although raw elemental maps show changes in intensity related to variations in the projected density of atoms with changes in thickness, the ratio map does not (Fig. 7.75a-d). When absorption is significant, this simple approach fails and absorption correction is required. For this correction, the sample thickness must be determined. To deal with strong absorption cases without calculation of the absorption correction factor (Sec. 7.4.1, Absorption), quantification of maps through the \(\zeta\)-factor approach have been proposed [7.141, 7.142, 7.143].

Fig. 7.75a-d

Elemental maps of Ni-based alloy with intergranular product. (a) STEM image, (b) Ni map showing the variation of intensity due to thickness variations (from top-left corner to bottom-right corner), (c) Cr elemental map with similar variations in intensity due to thickness, (d) elemental ratio map for \(\text{Ni}/\text{Cr}\) demonstrating the correction of the thickness variations in the maps. The intergranular product contains a Cr-rich carbide phase surrounded by an oxidation product

The technique allows us to circumvent the problem of having to determine the sample thickness independently (with intrinsic large errors) by calculating the absorption correction composition and the sample thickness simultaneously. The \(\zeta\)-factor relates the intensity and composition of a standard sample to the mass-thickness for both elements A and B as
$$\rho t=\zeta_{A}\frac{I_{\mathrm{A}}}{C_{\mathrm{A}}}\text{ and }\rho t=\zeta_{\mathrm{B}}\frac{I_{\mathrm{B}}}{C_{\mathrm{B}}}\;,$$
(7.66)
where \(I_{A,B}\) are the intensities recorded for elements A and B, and \(C_{A,B}\) are the concentrations of the same elements. If the \(\zeta\)-factors are determined through measurements of standards of known composition and thickness via (7.66) the concentrations are deduced as [7.141]
$$\begin{aligned}\displaystyle C_{\mathrm{A}}&\displaystyle=\frac{I_{\mathrm{A}}\zeta_{\mathrm{A}}}{I_{\mathrm{A}}\zeta_{\mathrm{A}}+I_{\mathrm{B}}\zeta_{\mathrm{B}}}\;,\quad C_{\mathrm{B}}=\frac{I_{\mathrm{B}}\zeta_{\mathrm{B}}}{I_{\mathrm{A}}\zeta_{\mathrm{A}}+I_{\mathrm{B}}\zeta_{\mathrm{B}}}\\ \displaystyle\text{and}\quad\rho t&\displaystyle=I_{\mathrm{A}}\zeta_{\mathrm{A}}+I_{\mathrm{B}}\zeta_{\mathrm{B}}\;.\end{aligned}$$
(7.67)
Quantitative concentration and thickness maps accounting for this effect have been demonstrated (Fig. 7.76a-d).
Fig. 7.76a-d

Quantitative elemental maps of an NiAl multilayer obtained with the \(\zeta\)-factor approach. The gray scale represents the quantitative information on the sample composition including correction for absorption. Reprinted from [7.141], with permission from Elsevier. (a) BF STEM image, (b) Ni map, (c) Al map, (d) thickness map

A variant of (7.66) was subsequently proposed by Watanabe and Williams [7.103] which makes this \(\zeta\)-factor independent of the acquisition time, the beam current, the specimen composition, and mass-thickness so (7.66) is modified with a dose term \(D_{\mathrm{e}}\) as follows
$$\rho t=\zeta_{\mathrm{A}}\frac{I_{\mathrm{A}}}{C_{\mathrm{A}}D_{\mathrm{e}}}\text{ and }\rho t=\zeta_{\mathrm{A}}\frac{I_{\mathrm{B}}}{C_{\mathrm{B}}D_{\mathrm{e}}}$$
(7.68)
As shown in [7.103], this new definition accounts for the detector efficiency and absorption, thus the only factors that the \(\zeta\)-factor is dependent upon include the concerned x-ray energy and the acceleration voltage. Thus, by estimating the \(\zeta\)-factors for each element separately with standards of known composition, thickness and with known dose (7.68), and by measuring their characteristic x-ray intensities, the specimen composition and mass-thickness (\(\rho t\)) can be determined easily with (7.67) being rewritten as
$$\begin{aligned}\displaystyle&\displaystyle C_{\mathrm{A}}=\frac{\zeta_{\mathrm{A}}I_{\mathrm{A}}}{(\zeta_{\mathrm{A}}I_{\mathrm{A}}+\zeta_{\mathrm{B}}I_{\mathrm{B}})}\;,\\ \displaystyle&\displaystyle C_{\mathrm{B}}=\frac{\zeta_{\mathrm{B}}I_{\mathrm{B}}}{(\zeta_{\mathrm{A}}I_{\mathrm{A}}+\zeta_{\mathrm{B}}I_{\mathrm{B}})}\;,\\ \displaystyle&\displaystyle\varrho t=\frac{(\zeta_{\mathrm{A}}I_{\mathrm{A}}+\zeta_{\mathrm{B}}I_{\mathrm{B}})}{D_{\mathrm{e}}}\;.\end{aligned}$$
(7.69)
The estimation of the \(\zeta\)-factor requires the determination of:
  1. 1.

    Absolute specimen thickness (\(t\))

     
  2. 2.

    Specimen density (\(\rho\))

     
  3. 3.

    Total electron dose (\(D_{\mathrm{e}}\))

     
  4. 4.

    X-ray intensities (7.68) for a standard thin film.

     
The specimen density can be estimated from the crystallographic unit cell information, leading then to the estimation of specimen thickness. Estimation of the total electron dose (starting from the beam current and acquisition time), however, requires the measurement of beam current during the experiment (typically with a Faraday cup available in some of the sample holders or calibration with the EELS spectrometer or current readout on the microscope imaging screen).

In determining the \(\zeta\)-factor, the x-ray intensities should be measured from standard pure element thin-film samples with known compositions and thicknesses. The use of pure element thin films as standards makes the technique more practical to implement than alloys and \(k\)-factors discussed in Sect. 7.4.1, Quantification in EDXS. Depending on the fabrication method (in the case of physical evaporation), the specimen thickness can be independently determined and very thin samples (where absorption can be neglected) can be produced. One important factor to consider here is the critical sample thickness that accounts for a compromise between the amount of x-ray absorption in the sample and the x-ray intensities generated. Watanabe and Williams [7.103] compared the critical sample thickness for various elements and found that pure element thin films of about \({\mathrm{30}}\,{\mathrm{nm}}\) (at \({\mathrm{5}}\%\) x-ray absorption) are ideal.

Advanced statistical analysis techniques based on the full processing of the spectrum and multivariate analysis allow the occurrence of the various phases to be extracted without prior knowledge of the individual component phases within the samples. Examples of multivariate analysis at high spatial resolution have been demonstrated in AEM [7.144] following the initial developments in SEM [7.145] and earlier work on line profiles by [7.146, 7.147]. These techniques are particularly useful for the analysis of segregation at grain boundaries and interfacial phases. Further discussion on advanced statistical analysis methods can be found in Sect. 7.9.

7.6.2 EELS Mapping

Energy-Filtered TEM Mapping

With both postcolumn and in-column energy filters described in Sect. 7.2.4, Spectrometers, it is possible to retrieve images at specific energy losses by taking advantage of an energy-selecting slit. These images can show the distribution of elements following the extraction of the background from the images. Reviews of the technique and the applications have been given in [7.140, 7.148, 7.149] and we will summarize the approaches here. In the basic approach of energy-filtered elemental mapping, this extraction of element-specific images can be achieved through two different methods: the three-window technique and jump-ratio technique. In both cases, energy-selected images at selected energy losses must be taken before the ionization edge and at the edge of interest (Fig. 7.77).

Fig. 7.77

Various approaches to EFTEM imaging. Zero-loss filtered imaging (selecting only electrons that have lost no significant amounts of energy), plasmon imaging (selecting only electrons that have lost energy in the \(10{-}30\,{\mathrm{eV}}\) range), and core-loss imaging with the three-window technique (extrapolation of the background under the edge) and the jump-ratio technique

In the three-window method, two energy-filtered images (called pre-edge images) acquired prior to the edge onset are used to extrapolate the noncharacteristic background under the edge where the third energy-filtered image (called postedge image) has been acquired. The same power-law extrapolation model used for quantitative analysis is used (namely \(I_{\mathrm{b}}(E)=AE^{-\mathrm{r}}\)) using the two pre-edge windows. The elemental map is then obtained by subtraction of the background image from the postedge image. Since there are only two images for this extrapolation, the resulting error in the intensity of the postedge background image, and thus the subtraction, can be large. If the intensity in the recorded images is low and the background varies steeply at the energy loss of interest, spurious images with negative extracted intensity can often be obtained because the variance in the extrapolation can be larger than the intensity of the ionization edge. Since the signal in the ionization edge is related to the ionization cross section and the number of atoms (Sect. 7.4.2, Quantification Procedures) images represent the quantitative distribution of elements within the field of view.

The second approach to extract the distribution of elements is through jump-ratio images. In this case, the postedge image is divided by one of the pre-edge images (generally obtained just before the edge threshold). Although these images are not directly quantifiable, they have the advantage that the noise is lower (no extrapolation is involved), the elastic contrast (due to diffraction, high-atomic number, and phase contrast) is canceled as it affects both the pre-edge and postedge images in a similar way, and finally there is less drift involved as only two images are used (and thus shorter acquisition times) [7.149]. This latter point significantly affects the spatial resolution of images as discussed below. Both methods can be applied with energy filters (irrespective of whether they are in-column or postcolumn) and also in STEM instruments equipped with serial spectrometers allowing the acquisition of energy-filtered images through an energy-selecting slit. Early serial spectrometers on dedicated VG-STEM and the first serial spectrometers from Gatan both equipped with scintillators and photomultipliers served this purpose. Parallel spectrometers and imaging filters equipped with single channel photomultiplier detectors [7.150] or a fast array of photomultipliers (developed in the Ottensmeyer group in Toronto) were also developed in prototype systems but were superseded by the development of much faster detectors allowing the acquisition of full spectra at each pixel of the rastered area as discussed in the STEM imaging section (Sect. 7.6.2, STEM-EELS Mapping). More recently, very fast shutters have enabled the use of dual EELS imaging (allowing virtually simultaneous low-loss and core-loss acquisition) also discussed in Sect. 7.6.2, Quantitative EELS Imaging.

In addition to these basic EFTEM acquisition techniques, there have been variants of the three-window and jump-ratio methods for EELS imaging based on the developments of more advanced acquisition software controlling the spectrometer, the microscope and increased storage capabilities in desktop computers. One variant is the EFTEM-spectrum imaging method initially proposed by [7.151] and fully developed as very powerful tools for mapping and quantitative analysis by [7.152, 7.153, 7.154]. This technique is now implemented in commercial packages from energy-filter vendors. The method is based on the acquisition of an information data-cube consisting of a large number of energy-filtered images with narrow energy windows (from \(1{-}2\) to \(5{-}10\,{\mathrm{eV}}\)) and small step intervals allowing one to cover large energy ranges around the edges (in some cases even from \({\mathrm{0}}\,{\mathrm{eV}}\) to the characteristic core-losses [7.154]) so that the full information from the spatial distribution and energy loss can be retrieved with high spatial sampling (Fig. 7.78). If the spatial drift of the sample is well corrected in the image stack using automated procedures [7.155], these datasets have the advantage that the background extrapolation can be carried out with much higher precision than in the three-window technique. Energy windows as small as \({\mathrm{0.1}}\,{\mathrm{eV}}\) have been used on instruments equipped with monochromators and high-resolution filters corrected up to third-order aberrations [7.156]. Quantitative analysis to extract the absolute atomic density and deconvolution of the effects of multiple scattering are also possible since the full spectrum is retrievable at each pixel [7.154]. In fact, for each pixel of the image sequence the intensity can be measured and a spectrum of energy resolution equivalent to the width of the energy-selecting slit is deduced. On the basis of limited sequences of images with narrow energy windows (\(1{-}2\,{\mathrm{eV}}\)) and selection of spectra at the edge threshold, this approach has demonstrated that changes in the near-edge structure due to variations in the chemical bonding state of elements are visible [7.140, 7.157, 7.158, 7.159]. This capability suggests that, similar to x-ray absorption scanning transmission microspectroscopy [7.160], bonding changes can be visualized with EELS mapping albeit with a spatial resolution typical of EFTEM images (Sect. 7.6.2, Quantitative EELS Imaging).

Fig. 7.78

Schematic description of the energy-filtered TEM spectrum imaging ( ) technique. Each image obtained at a given energy loss is part of a three-dimensional data-cube containing information on the distribution of elements. The technique provides detailed sampling of the spatial information with little sampling of the energy-loss distribution (energy window widths can vary from less than 1 to \({\mathrm{10}}\,{\mathrm{eV}}\))

STEM-EELS Mapping

The other variant of the more advanced EELS imaging technique makes use of developments in fast detectors, large storage capacity, and fast computers. The approach is based on the use of STEM instruments equipped with parallel or dedicated 2-D fast detectors. The original idea of this technique was proposed by the Orsay group [7.161] and implemented in subsequent years by various groups [7.162, 7.163, 7.164]. As for EFTEM-spectrum imaging, the STEM EELS imaging is also commercially available from the spectrometer manufacturer Gatan. The technique involves the sequential acquisition of an energy-loss spectrum (acquired with the photodiode array or a 2-D detector) at each pixel of a rastered area. The filling of the data-cube (Fig. 7.79) is thus achieved by scanning the beam over each pixel of the area of interest (pixel by pixel in two dimensions or over a line across interfaces) with the third (and fourth) dimension in the data-cube being the energy-loss spectrum (energy and intensity).

Fig. 7.79

Schematic description of the STEM spectrum-imaging ( ) technique. For each pixel of the rastered area an energy-loss spectrum is acquired. Although the spatial sampling is typically lower than in the EFTEM-SI technique, the spectral sampling is higher with easy recording of the near-edge structure features at each pixel

The advantage of this approach is the availability of the full spectrum at each pixel making it easy to implement various data-mining approaches. Signals can be extracted with advanced methods, including multiple least-square techniques, and the detailed shapes of the near-edge structures can be fitted with reference standards from different phases. This technique can therefore be used to map changes in the chemical bonding environment of atoms in nanoscale structures as in EFTEM imaging but with much higher spectral sampling. Through the analysis of a single edge and with reference standards, it is thus possible to extract the distribution of the individual phases rather than just the chemical composition. This powerful technique can be implemented in one or two dimensions (Fig. 7.80) for core-loss near-edge structure features and also on low-loss features. Using the differences in low-loss spectra of various biological structures and ice [7.165] were able to map the distribution of the various functional components in cells (Fig. 7.81a-e). Similarly, for the analysis of polymer-based materials, the use of low-loss features related to the presence of \(\pi\) and \(\pi+\sigma\) plasmons has been used to distinguish polystyrene and polyethylene in composite blends (Fig. 7.82a-da) [7.166]. This approach has also been used to understand ceramic phase distributions by identification and matching of reference spectra [7.167]. In the area of Li battery materials, this approach has been used very successfully to map the Li content and the presence of reaction phases between the electrolyte and active Si anode material (Fig. 7.83a-e) [7.168]. This approach has also been used to map, using electron tomography approaches, the distribution and shape of Si nanocrystals in an \(\mathrm{SiO_{\mathit{x}}}\) matrix, by using reference spectra and the sharp differences between Si and \(\mathrm{SiO_{\mathit{x}}}\) [7.169].

Fig. 7.80

(a) Application of the STEM-SI technique to determine the distribution of elements across interfaces between a high dielectric constant material (Hf-O-N) and Si (STEM annular dark-field image, left). EDXS spectra and EELS spectra were recorded simultaneously to extract the elemental composition. By analysis of the near-edge structure shape of the Si \(\mathrm{L_{23}}\) edge and O K edge, it is possible to distinguish and map the contribution of pure Si, Si-O-N, and Hf-O-N. (Courtesy of M. Couillard, McMaster University). (b) Two-dimensional phase maps of B nanostructure with BN, \(\mathrm{B_{2}O_{3}}\), and metallic B separated according to the shape of the near-edge structure spectra of the B K edges using the STEM-Spectrum imaging technique and least-square fitting of spectra. The structure consists of a metallic B core with a thin BN shell and outer thick shell of \(\mathrm{B_{2}O_{3}}\). (Courtesy of O. Stephan and C. Colliex, U. Paris-Sud)

Fig. 7.81a-e

STEM images and phase-component maps in frozen hydrated liver tissue. (a) Low-dose dark-field STEM image sectioned sample showing no contrast, (b) bright-field image of the same region based on integration of all signals on the spectrum at each pixel of the STEM-SI (the intensity drop is due to the variation of beam current during the acquisition), (c) relative thickness map (\(t/\lambda\)), contrast is visible on the lipid droplets, (d) water maps and identification of the various biological components based on the multiple least-square fit of the low-loss spectra with water and protein reference compounds and the amount of water within the structure: \(L={}\)lipid droplet (zero water content), \(P={}\)plasma (\({\mathrm{91}}\%\) water), \(R={}\)erythrocyte (\({\mathrm{65}}\%\) water), \(M={}\)mitochrondria (\({\mathrm{57}}\%\) water). (e) Low energy-loss spectra of the various components differentiating the phases. Reprinted with permission from [7.165], John Wiley and Sons

Fig. 7.82a-d

Section of a polyefin-polycarbonate polymer composite. The low-loss spectrum of the polyefin (a) does not show the \(\pi\) plasmon resonance visible on the spectrum of the polycarbonate material (b). The zero-loss image (c) does not show strong contrast between the two phases while the \(\pi\) plasmon energy-filtered image (d) clearly allows one to distinguish the regions where the polycarbonate is present. Images courtesy of T. Oikawa, JEOL USA

Fig. 7.83a-e

Phase identification in an Si-Li anode battery material based on the shape of the low-loss spectrum. The color-coded regions are identified based on the shape of the single scattering distribution of pure phases taken as reference. Reprinted (adapted) with permission from [7.168]. Copyright 2016 American Chemical Society. (a) Reference EELS spectra of Li compounds. (b) EELS low-loss spectra acquired at different electron doses. (c) Reference EELS spectra of Si-Li alloys. (d) MLLS fitting of the experimental spectrum integrated over about 4 pixels in (e). (e) Color-coded maps of different compounds obtained by MLLS fitting of a parent spectrum image

Quantitative EELS Imaging

When one combines core-loss with low-loss spectra acquired from the same pixel, it is possible to retrieve more quantitative information on the sample. Hence, deconvolution techniques (Sect. 7.8.2) can be applied to retrieve the single scattering distribution for accurate quantification (Sect. 7.4.2), dielectric function measurements (Sect. 7.8.2), the thickness relative to the inelastic mean free path of the sample (Sect. 7.8.2), and also implement quantitative statistical analysis [7.170] to retrieve significant spectral components and analyze changes in bonding at interfaces in materials. Thickness maps, relative to the mean free path (Sect. 7.8.2), can be obtained with the EFTEM and STEM-imaging approaches by acquiring a zero-loss image \(I_{0}(x,y)\) and an unfiltered image \(I_{\mathrm{t}}(x,y)\). Following the approach discussed in Sect. 7.8.2 for the analysis of individual spectra to obtain relative thickness (\(t/\lambda\)) values, the ratio of the two images can be combined to give the relative thickness map \(I_{t/\lambda}(x,y)=\text{ln}[I_{t}(x,y)/I_{0}(x,y)]\). The variant of this EFTEM method is to acquire the low-loss spectrum at each pixel and process the individual spectra to deduce \(t/\lambda\) at each point of an image (Sect. 7.8.2). This information on the sample thickness is valuable for the determination of the volume of the sample under analysis (hence the volume fraction of particular phases or defects), or to determine whether the changes in thickness affect the apparent intensity of elemental maps. The thickness information is also useful to verify whether the thickness of the sample is beyond the critical thickness where accurate extraction and quantification can be carried out (Sect. 7.4.2, Limitations in Analysis and Quantification). Finally, the thickness information, if combined with EDXS maps, can lead to fully quantitative x-ray maps accounting for x-ray and absorption corrections.

As in the case of EDXS imaging, elemental maps can also be combined to retrieve fully quantitative concentration maps and deduce phase analysis histograms using experimental \(k\)-factors or cross sections [7.171, 7.172]. The advantage of concentration maps is, as in the case of EDXS mapping, the fact that within a range of relative thickness \(t/\lambda<{\mathrm{0.5}}\), images are independent of thickness as discussed in Sect. 7.4.2, Limitations in Analysis and Quantification. Diffraction effects due to elastic scattering of electrons outside the objective aperture can also lead to apparent variations in the intensity of elemental maps and can be canceled out using the jump-ratio imaging technique. Concentration maps based on the single scattering distribution of energy losses obtained after deconvolution of the full spectrum at each pixel show that reliable quantitative images can be obtained for thicknesses up to \(t/\lambda\cong 2\) [7.153].

Another useful technique demonstrating the removal of diffraction effects is the use of the rocking beam method during the acquisition of the energy-filtered images. In this approach, the incident electron beam is tilted over a cone of angles (of the order of the Bragg angle) so as to average out the local diffraction effects including deviations of the scattering due to dislocations. Energy-filtered images with virtually no diffraction contrast can thus be obtained even in bent and highly deformed samples [7.140, 7.173] (Fig. 7.84a-f). Removal of diffraction contrast for qualitative imaging and visualization of precipitates in highly deformed samples has also been demonstrated using ratios of plasmon images obtained at different energies [7.174].

Various aspects of optimization of signals for EFTEM and STEM EELS maps, including the position of energy windows, automatic detection of edges, illumination conditions, and magnification are discussed in the work of [7.172, 7.175, 7.176, 7.177, 7.178, 7.179]. Through image analysis of the quantitative maps, it is also possible to segment images based on the composition and relative fraction of elements [7.140, 7.171]. Algorithms to allow automatic detection of edges, for quantitative analysis of phase distributions, for the determination of thresholds for phase detection and problematic zones in the samples have also been developed with the use of full spectra recorded in STEM mode [7.172]. Corrections of drifts in EFTEM images for quantitative analysis have been discussed in detail in [7.155].

Fig. 7.84a-f

Imaging of precipitates in a steel sample. (a) TEM bright-field image of the \({\mathrm{10}}\%\) Cs steel with \(\mathrm{Cr_{23}C_{6}}\)-VN and Nb(C,N) precipitates. (b) EELS spectrum with the Fe-\(\mathrm{M_{23}}\) edge, (c) Fe-\(\mathrm{M_{23}}\) jump-ratio image recorded with the rocking beam illumination, (d) Cr-\(\mathrm{L_{23}}\) jump-ratio image, (e) V-\(\mathrm{L_{23}}\) jump-ratio image, (f) Nb-\(\mathrm{M_{45}}\) jump-ratio image. (From [7.140])

A particularly useful development is the ability to acquire both low-loss and core-loss virtually simultaneously for the same pixel position. This approach has been enabled by the development of ultrafast shutters implemented in the Quantum spectrometers of Gatan and subsequent models (e.g. Continuum). Typically, the dynamic range in an EELS spectrum is at least \(\mathrm{10^{6}}\) or more for an energy-loss range from \(\mathrm{0}\) to \({\mathrm{2000}}\,{\mathrm{eV}}\), and even the most sophisticated CCD detectors (dynamic range \(\approx{\mathrm{10^{4}}}\)) cannot record the whole spectrum in a single acquisition. Another challenge here is to have an adequate signal-to-noise ratio in the spectrum that is required for a detailed analysis of the sample, particularly when ELNES analysis is sought. The EELS dynamic range is basically the ratio of the saturation signal to the readout noise. Hence, if the signal generated is very low and on the order of the noise levels, the signal-to-noise ratio will be extremely poor. Thus, as a general rule, the minimum signal of \(\approx{\mathrm{1}}\%\) of saturation is usually required. Consequently, the useful spectral data can only be recorded over an intensity range of about \(\mathrm{10^{2}}\) in a single acquisition. In practice, this means that the low-loss (with intense zero-loss peak) and core-loss regions have to be recorded separately. Further, if a wide energy-loss range was sought, this would require more than a single acquisition.

The modern Quantum and Continuum spectrometers cope with the dynamic range problem described above via different approaches. Most common is the use of a CCD camera that is divided into four quadrants. One half is used for the recording of one spectrum (e. g., low-loss range) while the other half (in the nondispersive direction) is used for recording the second spectrum (e. g., core-loss). The spectral integration times for the two halves will have to be set independently, and by deflecting and shifting in energy the recorded spectrum from one half to the other and suitably changing the electron-optical conditions, the two spectra of different energy ranges (e. g., low-loss and core-loss) can be readily acquired. Commonly this process is referred to as Dual EELS in the Gatan Digital Micrograph acquisition environment.

Although the recording of the individual spectra is carried out separately in Dual EELS mode, a near-simultaneous acquisition can be possible with the use of electrostatic beam-shutters [7.180]. These shutters allow for switching between the two different electron-optical conditions, each suitable for the individual energy ranges (e. g., low-loss versus core-loss energy ranges) at fast spectral integration times. As opposed to the electromagnetic shutters used previously, the electrostatic shutters allow for the beam to be blanked and unblanked at much faster rates (\(\approx{\mathrm{1}}\,{\mathrm{\upmu{}s}}\) versus tens of ms). A significant advantage of the Dual EELS acquisition process is the more reliable comparison of energy shifts of features in the spectra as the mid-term and long-term instabilities in the energy scale (high-tension instabilities and power supplies) can be accounted for during the long acquisition times necessary for a map. In this case, the spectra can be locally recalibrated with the use of the low-loss part of the spectrum if the zero-loss peak is present and can be used as a reference feature. In this case, more quantitative (and reliable) maps with valence state determined based on the spectral feature positions and absolute atom density determination given the intensity of the low-loss spectrum and the (relative) thickness can be measured on the same spectrum [7.181].

Spatial Resolution in EFTEM Elemental Mapping

The resolution in EFTEM elemental mapping depends on several factors related to the operation parameters of the microscope and the energy loss of interest so that further discussion of this topic is required. When no angular limiting aperture is present (in imaging the objective aperture would be limiting the scattering angles) the dominant factor is related to the chromatic aberration term discussed in Sect. 7.5.2. For general conditions, however, we must present a summary of the contributions that need to be added in quadrature in order to retrieve the total broadening of an object point [7.182]:
  1. 1.
    Following the discussion in Sect. 7.5.2, the chromatic aberration broadening term when a limiting aperture is used can be described as
    $$d_{\mathrm{c}}=C_{\mathrm{c}}\frac{\Updelta E}{E_{0}}\beta\;,$$
    (7.70)
    where \(\beta\) is the collection angle (limited by the objective aperture in imaging mode), \(\Updelta E\) is the width of the energy window used to acquire the image, and \(E_{0}\) is the incident energy. This expression assumes that images are focused at the energy-loss \(E\) where the energy window is located (rather than at the elastic image) and that the aperture is filled with the electrons. This assumption is important since, as discussed in Sect. 7.5.2, the width of angular distribution of scattering can be smaller and the contribution to the chromatic aberration term would be modified. As experimental conditions often impose some convergence in the illumination and the chromatic aberration term is small compared to the subsequent broadening terms for low scattering angles (either limited by \(\beta\) or \(\theta_{\mathrm{E}}\)) this assumption is often assumed to be valid.
     
  2. 2.

    The delocalization of inelastic scattering term \(b\) (introduced in Sect. 7.5.2, (7.64)) contributes to the broadening with an energy- and angular-dependent term increasing at low energy losses and low scattering angles (large scattering angles imply small impact parameter).

     
  3. 3.
    The diffraction limit contribution arises from use of the objective aperture and dominates for small angles due to the denominator term
    $$d_{\mathrm{d}}=\frac{{\mathrm{0.6}}\lambda}{\beta}\;.$$
    (7.71)
     
  4. 4.

    The spherical aberration term \(d_{\mathrm{s}}=2C_{\mathrm{s}}\beta^{3}\) strongly varies with \(\beta\) and is considered to contribute to a uniform background in the image and a decrease of contrast when the conditions are optimized for minimal chromatic aberration contributions [7.182]. Egerton's work [7.134] showed that this term is smaller than the chromatic aberration term for typical energy windows used for EFTEM mapping but for small energy windows (few eV wide as used for example in EFTEM spectrum imaging), the term will dominate the resolution at high collection angles and will need to be included in the analysis. Experimental evidence suggests that the term should be neglected given the good spatial resolution of energy-filtered images with relatively large collection angles.

     

Additional terms affecting the resolution depend on the noise in the images (requiring averaging of signals, increase of acquisition time), radiation damage of the specimen due to the high doses required for imaging at core-losses, and instabilities of the sample and microscope leading to drift of the area under analysis during acquisition.

Considering the significant terms added in quadrature it is possible to determine the ultimate physical limits to EFTEM mapping resolution [7.182]
$$d^{2}_{\text{tot}}=d^{2}_{\mathrm{c}}+(2b)^{2}+d^{2}_{\mathrm{d}}\;.$$
(7.72)
The trends, as a function of collection aperture, suggest that there are optimal operating conditions for a given energy loss, energy window \(\Updelta E\), and microscope characteristics (Fig. 7.85). When energy losses of the order of few hundred eV are analyzed, for small collection angles the diffraction limit term dominates while, for large angles, the chromatic aberration term is the most important one. The width of the energy window has a significant effect on the resolution as it is linearly related to the chromatic contribution. Quite often, large energy windows are used to increase the counting statistics and reduce the noise in the images. When the energy window is large, the choice of the optimal collection aperture is more crucial as the chromatic aberration term rises steeply. For modern analytical TEMs operating at or above \({\mathrm{200}}\,{\mathrm{keV}}\), and for lenses with chromatic aberration parameters around \({\mathrm{1}}\,{\mathrm{mm}}\), the delocalization term (7.64) is not a limiting factor for core-losses above around \({\mathrm{500}}\,{\mathrm{eV}}\). For lower energy losses, in the order of \({\mathrm{100}}\,{\mathrm{eV}}\) or less, this term can dominate the resolution limit in theory. Experimental results in the literature, however, suggest that, for low losses, the delocalization contributions based on (7.64) are overestimated [7.132, 7.183]. For high core-losses, elemental maps show resolution limits consistent with the calculations of optimal values given above (e. g., below \({\mathrm{1}}\,{\mathrm{nm}}\) based on Fig. 7.84a-f) while for low energy losses (including plasmon losses around \(10{-}20\,{\mathrm{eV}}\)), images with a resolution around \({\mathrm{1}}\,{\mathrm{nm}}\) (significantly better than the prediction of \(4{-}5\,{\mathrm{nm}}\)) have been obtained [7.183].
Fig. 7.85

Calculations of the expected resolution in EFTEM elemental maps at the oxygen K edge (\({\mathrm{530}}\,{\mathrm{eV}}\)) as a function of the collection angle for two energy-filtering windows of width \(\Updelta E={\mathrm{5}}\,{\mathrm{eV}}\) (solid line) and \({\mathrm{20}}\,{\mathrm{eV}}\) (dashed line) and \(C_{\mathrm{s}}=C_{\mathrm{c}}={\mathrm{1}}\,{\mathrm{mm}}\) (at \({\mathrm{200}}\,{\mathrm{keV}}\))

On the basis of significant contributions of chromatic aberrations for EFTEM imaging, the optimal approach to achieve the ultimate spatial resolution limits in EELS mapping, as imposed by the unavoidable delocalization, is to use the STEM approach with small and high-intensity probes achievable today on modern analytical electron microscopes. With the progress of current technology making use of bright electron sources and aberration correctors it is possible to focus several hundred pA of current into a near \({\mathrm{2}}\,{\mathrm{\AA{}}}\) probe (Sects. 7.2.1 and 7.2.2 and Chap.  2).

Examples of chemical analysis with a noncorrected STEM demonstrate the capability to resolve individual atomic planes of Ca in the \(\mathrm{Bi_{2}Sr_{2}Ca_{1}Cu_{2}O_{8+\mathit{\delta}}}\) superconductor (Fig. 7.86a-c) and analyze half-unit cell defects where the Ca planes are not present suggesting the existence of a \(\mathrm{Bi_{2}Sr_{2}Ca_{0}Cu_{1}O_{8+\mathit{\delta}}}\) subunit cell intergrowth [7.184] while, with a probe-corrected microscope one can obtain atomic-resolved maps of interfaces in superconductors (\(\mathrm{YBa_{2}Cu_{3}O_{\mathit{x}}}\)) with oxide (\(\mathrm{La_{1-\mathit{x}}Ca_{\mathit{x}}CuO_{4}}\)) showing the termination of Ba-O planes in contact with La-O planes (Fig. 7.87a,b). Similar atomic maps and the limitations of the techniques are discussed in Sect. 7.9 of this chapter.

Fig. 7.86a-c

Chemical analysis in a \(\mathrm{Bi_{2}Sr_{2}Ca_{1}Cu_{2}O_{8-{\mathit{\delta}}}}\) superconductor with a noncorrected STEM probe. (a) High-resolution EELS profile of Ca in \(\mathrm{Bi_{2}Sr_{2}Ca_{1}Cu_{2}O_{8-{\mathit{\delta}}}}\) based on the Ca-\(\mathrm{L_{23}}\) edge showing the detection of a single plane of Ca in the half-unit cell of the structure. (b) High-angle annular dark field image of the sample. (c) Line profile of Ca-\(\mathrm{L_{23}}\) edge showing the presence of a half-unit cell defect of the \(\mathrm{Bi_{2}Sr_{2}Ca_{1}Cu_{2}O_{8-{\mathit{\delta}}}}\) phase (where there is a missing plane of Ca) in the structure. (Courtesy of Y. Zhu, McMaster University)

Fig. 7.87a,b

Atomic-resolved elemental maps of La, Ba, Mn, Cu, Ca, and O in a YBaCuO-LaCaMnO heterostructure. (a) High angle annular dark field image of the sample. (b) EELS Elemental maps over a selected region highlighted by green frame shown in (a). Data courtesy of Sorin Lazar, McMaster University and FEI-ThermoFisher Scientifics, sample from U. Kaiser and H. Keimer. The atomic-resolved maps allow the determination of the termination of each layer and site substitution of Ca in the La atomic columns

7.7 Detection Limits in Microanalysis

The very high spatial resolution of both EDXS and EELS results in relatively low detection limits for most elements as compared to bulk analysis methods. This effect is caused by a combination of factors including the small analyzed volumes, the low signals resulting from poor efficiency in signal collection, and/or low incident beam current, high background, short acquisition time, and instrumental contributions. Two quantities characterize the detection limits for EDXS and EELS depending on information of interest. The minimum detectable fraction ( ) refers to a dilute element uniformly distributed in the analysis area and represents the lowest concentration that can be detected. The minimum detectable number ( ) refers to the smallest number of atoms that can be detected. This applies to analyses where these atoms are clustered (such as biological molecules containing few atoms, atomic clusters, etc.) or when very small probes are used and only a few atoms of impurity are being analyzed in the interaction volume.

Predictions of the MDF are related to the process of signal generation (incident number of electrons, cross sections, fluorescence yield), collection efficiency of the detector system (solid angle, detector quantum efficiency, noise), sample characteristics (thickness, scattering outside apertures leading to loss of signal, background signal, and signal extraction variance), and potential sources of noncharacteristic signals arising from the instrument.

The overall principle for estimating the MDF is based on the statistical certainty of detecting a signal above a background containing some noise due to the variance in the background intensity. This detection is based on the Rose visibility criterion [7.185] stating that the signal should be three times the standard deviation of the background (the noise) for reliable identification of the signal as genuine in a \({\mathrm{98}}\%\) confidence level. This criterion assures that the signal can be clearly distinguishable from a simple statistical variation of the background with a high degree of confidence. This general criterion is applicable in both EELS and EDXS measurements (as in any other signal processing method).

7.7.1 Detection Limits for EDXS

We can describe the evaluation of the MDF for EDXS as follows. Assuming that the noise in the background signal follows a Poisson statistical distribution, the minimum detectable signal for element \(\mathrm{B}\,(I^{\text{min}}_{\mathrm{B}})\) in EDXS measurements is
$$I^{\text{min}}_{\mathrm{B}}\geq 3\,\sqrt{2\,I^{\mathrm{b}}_{\mathrm{B}}}$$
where the square root term represents the variance of the background signal under the peak of element \(\mathrm{B}\,(i^{\mathrm{b}}_{\mathrm{B}})\)—i. e., the noise (Fig. 7.88). Experimentally, one can easily determine the detection limit of an element B in a matrix A by using a standard of known composition \(C_{\mathrm{B}}\) and the detected signal \(I_{\mathrm{B}}\). The MDF is the concentration resulting in a minimum signal-to-noise ratio \(\text{SNR}=3\) hence
$$\text{SNR}^{C_{\mathrm{B}}}_{\text{std}}=\frac{I_{\mathrm{B}}-I^{\mathrm{b}}_{\mathrm{B}}}{\sqrt{2\,I^{\mathrm{b}}_{\mathrm{B}}}}\text{ and }\text{SNR}_{\text{MDF}}=3$$
(the signal-to-noise ratio corresponding to the Rose criterion and the MDF). This resulting concentration yielding an SNR\(=\)3 is the minimum detectable fraction
$$C^{\text{MMF}}_{\mathrm{B}}=\frac{C_{\mathrm{B}}\cdot 3\cdot\sqrt{2\,I^{\mathrm{b}}_{\mathrm{B}}}}{I_{\mathrm{B}}-I^{\mathrm{b}}_{\mathrm{B}}}\;.$$
(7.73)
If the standard containing element B is not available, we can still retrieve the detectable fraction based on a pure element A standard using \(k_{\text{AB}}\) factors (Sect. 7.4) and by assuming that the background under the peak of element B (if it was present) is not significantly altered when this element is in trace concentration in a sample. In this case, we obtain
$$C^{\text{MMF}}_{\mathrm{B}}=\frac{C_{\mathrm{A}}\cdot 3\cdot\sqrt{2\,I^{\mathrm{b}}_{\mathrm{B}}}}{k_{\text{AB}}\left(I_{\mathrm{A}}-I^{\mathrm{b}}_{\mathrm{B}}\right)}\;,$$
(7.74)
where \(I^{\mathrm{b}}_{\mathrm{B}}\) is the background under the peak of element A and \(C_{\mathrm{A}}\) is the concentration of element A (assumed to be 1 if the standard is pure).
Fig. 7.88

Definition of peak and background intensities in EDXS spectra. The peak intensity is defined with respect to intensities just beside the peak

From this simple empirical treatment, it can be seen that improvements in detection limits (i. e., lower values) can be achieved by increasing the peak intensity (through longer acquisition time and/or larger probe current) and by reducing the background. This latter contribution is affected by sample thickness, instrument operating conditions, and instrumental contributions (discussed in Sect. 7.7.2). The general treatment required to understand the trends in detection limits is due to Ziebold [7.186] who related the minimum mass fraction ( ) to microanalysis conditions [7.85]
$$\text{MMF}\propto\frac{1}{\sqrt{P\frac{P}{B}n\tau}}\;,$$
(7.75)
where \(P\) is the peak above the background count rate, \(P/B\) is the peak-to-background ratio (with integration of the background defined over the same energy window as the peak), \(n\) is the number of analyses, and \(\tau\) is the acquisition time for each analysis carried out. It is possible to increase the peak counts by increasing the electron-beam current, the thickness of the sample, the collection efficiency (larger solid angle of the detector) while one can increase the \(P/B\) ratio by increasing the accelerating voltage of the microscope and reducing the instrumental contribution leading to noncharacteristic signals. Although a large fraction of the background arises from bremsstrahlung radiation in the sample, significant contributions can come from spurious signals arising from electrons scattered at high angles (including backscattering) that, in turn, generate x-rays within the column. An additional contribution to the background arises from hard x-rays generated in the condenser lens system—when electrons hit apertures in the optic path—that fluoresce x-rays in the specimen area, including the sample holder. Such spurious signals are known as instrumental contributions.

As discussed above, increased count rates for a given spot size can be achieved through brighter electron sources (Sect. 7.2.1) and thicker samples although the latter will also lead to increased undesired contributions (increase in background) and loss of spatial resolution (Sect. 7.5). Increased total analysis time \(n\tau\) can only be achieved if the sample is stable under the electron beam (due to electron-beam damage, contamination, etc.) and sample drift is minimal or can be corrected via alignment algorithms. A clean vacuum system and clean samples are of utmost importance. Dry pumping systems, bakeable columns, clean sample holders always kept in vacuum, and plasma cleaning of the samples prior to the TEM sessions are key components of improved analytical performance of the microscope and are ultimately as important as the quality of the EDXS detector and the microscope. With the use of analytical electron microscopes equipped with small electron beams, large current (with the aberration correctors) and, thin samples, it has been possible to achieve fractions of \({\mathrm{1}}\%\) detection with sub-nm spatial resolution (Fig. 7.89) [7.126].

Fig. 7.89

Detection limits reached in modern AEM and various examples of performance reached with different instruments. WDS is data obtained with a dedicated microprobe (\({\mathrm{30}}\,{\mathrm{keV}}\)), AEM (1) \({\mathrm{120}}\,{\mathrm{keV}}\), AEM (2) \({\mathrm{100}}\,{\mathrm{keV}}\), LaB6 instruments, \({\mathrm{100}}\,{\mathrm{keV}}\) FEG, \({\mathrm{300}}\,{\mathrm{KeV}}\) [7.143] and aberration-corrected FEG. (Adapted after [7.126, 7.2])

7.7.2 Instrumental Contributions in EDXS

These spurious signals can be minimized by improvements in the column, detector and sample holder designs. For example, a significant reduction in hard x-rays generated in the upper part of the column can be achieved by using thick top-hat-shaped Pt apertures. These are now available in most modern analytical microscopes as part of the selection of \(\mathrm{C_{2}}\) apertures (some might just be Mo or lighter elements allowing transmission of hard x-rays). Good collimation of the detector is also important to minimize the line of sight between the specimen and the active area of the detector so that x-rays generated elsewhere in the specimen chamber area by scattered electrons reaching apertures and the pole-piece of the objective lens do not reach the detector (Figs. 7.23a-c and 7.90). Using small samples (rather than the standard \(\mathrm{3}\)-\(\mathrm{mm}\) disks with bulk edges) also minimizes the contributions due to fluorescence and electron scattering (electrons returning back to the sample after hitting parts of the column). Finally, the sample holder must be designed for analytical purposes (these are available commercially), namely built with light element materials, ideally Be as one of the main components in the specimen-supporting area, with a Be ring to tighten the sample and a design geometry allowing the acquisition of spectra without significant tilt of the sample (Fig. 7.91a,b). This configuration leads to minimal instrumental contributions so that x-rays generated in the sample have a direct line of sight to the detector without being absorbed (and causing fluorescence) in the sample holder. Sample grids made of Cu (or other metals such as Ni, Mo, etc.) to support continuous, lacey, or holey C films where the sample is distributed (in the form of dispersed particles, replicas, or lift-out focused ion beam samples) generate significant spurious signals as shown by the presence of peaks of the grid material even if the electrons are not directly illuminating the grid. Recently, diamond grids have been developed to minimize these contributions when these problems affect the analysis. In many cases, the selection of the grid material can be made judiciously to avoid quantification problems given the commercial availability of several grid materials.

Fig. 7.90

Instrumental contributions to the characteristic signals detected from the sample. Spurious x-rays are produced by the scattered electrons interacting with the microscope components (labeled 1, 2, 4). Spurious signals are also produced by hard-x-rays generated on the top part of the column hitting the sample holder, grid, and other parts of the specimen chamber (labels 5, 6, and 3). Nonoptimal collimation allows x-rays generated elsewhere than the sample area to reach the detector

Fig. 7.91a,b

Sample holders for AEM: (a) JEOL 2010F microscope holder and (b) Philips/FEI CM-Tecnai series holder with machined grove for minimal tilt of sample

On the basis of these instrumental considerations, the overall quality of the microscope analytical performance can be evaluated with measurements of the hole-count signals. By measuring the signal generated when the electron beam is directed into the hole of the sample, contributions from the grid, the microscope chamber, and holder can potentially be observed (and identified from the element present) and should be, in a good analytical TEM, less than \({\mathrm{1}}\%\) of the signal generated on the sample. Quantitative evaluation of the performance can be carried out using standard samples of Cr (\({\mathrm{100}}\,{\mathrm{nm}}\) thick) and NiO and well-defined tests based on \(P/B\) ratios [7.187, 7.2]. Overall, low instrumental contributions lead to lower peak-to-background values and improved detection limits. Well-designed analytical microscopes (Sect. 7.2.3, Geometry of the EDXS Detector in the AEM) and the use of analytical conditions (correct analytical apertures, sample geometry, tilt) lead to significantly improved detection limits.

7.7.3 EELS Detection Limits

The basic statistical principle for detection of signals used in EDXS (the Rose criterion) is also applicable in the case of EELS. The empirical approach to estimate the detection limit based on experiments and known standards is the same as in EDXS but with additional complications due to the determination of the noise component arising from the extrapolation of the background rather than the simple interpolation. The noise, and thus the signal-to-noise, cannot be simply determined based on the simple variance of the number of counts at a given energy loss. The noise must be estimated directly from a detailed statistical analysis of the spectra and the errors related to the determination of the extrapolation parameters [7.188] or simpler approximations of the extrapolation error based on the width of the fitting and extrapolation windows [7.35].

From first principles, one can estimate the detection limits accounting for these statistical effects and physical principles as described in detail by [7.35, 7.78]. The variance of the signal must take into account the error due to the extrapolation of the background under the edge and the introduction of noise related to the detector quantum efficiency ( ) of the spectrometer. The DQE is defined [7.35, 7.38] as \(\text{SNR}^{2}_{\text{outut}}/\text{SNR}^{2}_{\text{input}}\) (where the indices input and output are based on the incident counting statistic and measured statistic, respectively) and varies with the number of incident electrons but, for the purpose of our estimations, is assumed constant and not equal to one as generally assumed in the case of EDXS spectra. The determination of the EELS detection limits must also include the effect of elastic scattering reducing the efficiency of collection of the signals due to scattering outside the aperture. Accounting for the detector response, the signal-to-noise ratio is [7.35],
$$\text{SNR}\approx\sqrt{\text{DQE}}\frac{I_{\mathrm{k}}}{\sqrt{h\cdot I^{\mathrm{b}}}}\;,$$
(7.76)
where \(I^{\mathrm{b}}\) is the background under the edge, \(h\) is an error parameter due to the extrapolation, and \(I_{\mathrm{k}}\) is the integrated edge intensity. Large fitting intervals close to the edge threshold and small integration windows lead to small \(h\) parameters (around \(5{-}10\)), while small fitting intervals further from the edge and large integration windows results in large \(h\) values (as large as \(20{-}30\)) [7.35]. The number of electrons causing a signal is related to the dose \(D\) (\(\mathrm{C}/(\text{unit area}\))) and the area illuminated by a probe with diameter \(d\) as
$$I(\beta,\Delta)\approx\left(\frac{\uppi}{4}\right)d^{2}\left(\frac{D}{e}\right)\exp\left(\frac{-t}{\lambda_{\mathrm{e}}}\right),$$
(7.77)
where the exponential term represents the loss of electrons following elastic scattering outside the collector aperture \(\beta\) in a specimen of thickness \(t\) relative to the elastic mean free path \(\lambda_{\mathrm{e}}\) and \(e\) is the electron charge. The background signal (\(I^{\mathrm{b}}\)) related to the number of atoms of the matrix \(N_{\mathrm{t}}\) and the number of atoms of the trace element \(N\) giving rise to edge signals \(I_{\mathrm{k}}\) are related to the respective cross sections and the number of incident electrons and number of atoms in the volume as
$$\begin{aligned}\displaystyle&\displaystyle I^{\mathrm{b}}\approx N_{\mathrm{t}}I(\beta,\Delta)\sigma_{\mathrm{b}}(\beta,\Delta)\quad\text{and}\\ \displaystyle&\displaystyle I_{\mathrm{k}}=NI(\beta,\Delta)\sigma_{\mathrm{k}}(\beta,\Delta)\;.\end{aligned}$$
(7.78)
Since the fraction of trace element \(f=N/N_{\mathrm{t}}\) we obtain
$$f=\frac{\text{SNR}}{\sigma_{\mathrm{k}}(\beta,\Delta)}\left[\frac{h\sigma_{\mathrm{b}}(\beta,\Delta)}{N_{\mathrm{t}}I(\beta,\Delta)}\right]^{\frac{1}{2}}(\text{DQE})^{-\frac{1}{2}}\;.$$
(7.79)
The MDF is the fraction \(f_{\text{min}}\) corresponding to an SNR\(=\)3 (as in the case of EDXS following the Rose criterion). This yields [7.35]
$$\begin{aligned}\displaystyle\text{MDF}&\displaystyle=f_{\text{min}}\approx\frac{3}{\sigma_{\mathrm{k}}(\beta,\Delta)}\left(\frac{1.1}{d}\right)\\ \displaystyle&\displaystyle\quad\,\times\left[\dfrac{h\sigma_{\mathrm{k}}(\beta,\Delta)}{(\text{DQE})\left(\frac{D}{e}\right)N_{\mathrm{t}}}\right]^{\frac{1}{2}}\\ \displaystyle&\displaystyle\quad\,\times\exp\left(\frac{t}{2\lambda_{\mathrm{e}}}\right).\end{aligned}$$
(7.80)
The minimum detectable number quantity is related to the number of atoms within the analyzed area corresponding to the minimum detectable fraction determined above [7.35]
$$\begin{aligned}\displaystyle\text{MDN}&\displaystyle=\frac{\uppi}{4}d^{2}f_{\text{min}}N_{\mathrm{t}}=\frac{{\mathrm{2.7}}d}{\sigma_{\mathrm{k}}(\beta,\Delta)}\\ \displaystyle&\displaystyle\quad\,\times\left[\dfrac{N_{\mathrm{t}}h\sigma_{\mathrm{k}}(\beta,\Delta)}{(\text{DQE})\left(\frac{D}{e}\right)N_{\mathrm{t}}}\right]^{\frac{1}{2}}\exp\left(\frac{t}{2\lambda_{\mathrm{e}}}\right).\end{aligned}$$
(7.81)

By calculating the various cross sections (Sect. 7.3), one can estimate detection limits. Assuming a \(\text{DQE}\approx{\mathrm{0.5}}\) typical of parallel spectrometers, \(h=9\), \(\lambda_{\mathrm{e}}={\mathrm{200}}\,{\mathrm{nm}}\) and a carbon matrix of \({\mathrm{30}}\,{\mathrm{nm}}\) thickness, Egerton [7.35] calculated, using the expression given above, the detection limit of Ca atoms for a range of doses for \(\mathrm{100}\)-\(\mathrm{keV}\) electrons and with hydrogenic cross sections. The detection of a few atoms within a \(\mathrm{1}\)-\(\mathrm{nm}\) probe is possible assuming the sample can withstand doses of the order of \({\mathrm{10^{6}}}\,{\mathrm{C/cm^{2}}}\). For elements with large cross sections (such as L and M lines) detection of a few ppm has been demonstrated [7.105] whilst higher detection limits are possible for K edges and delayed edges of elements such as Ag, Au, S etc. A significant improvement in the detection limits is achieved if the spectra are processed using multiple least-square analysis of data acquired in spectral difference mode (Sect. 7.4.2, Quantification Procedures) where reference data is fitted to the experimental spectrum. This approach results in much lower signal extraction error with \(h\) values approaching 1. Detection of single atoms of Thorium was demonstrated by Krivanek et al [7.189] and individual Gd atoms were detected by Suenaga et al [7.190]. Systematic work on Ca- and Fe-containing molecules demonstrated elemental maps with the detection of just a few (\(7{-}8\)) base pairs of a DNA molecule under the electron beam (within the pixel size) containing \(14{-}16\,{\mathrm{P}}\) atoms and \(\mathrm{4}\) atoms of Fe in one single hemoglobin molecule [7.191] and single Ca atom detection [7.192]. Experiments demonstrating detection of a few ppm in standard reference materials of known composition has been reported by [7.105, 7.193, 7.194] (Fig. 7.92a,b). The work assumes detection of the edges based on the spectral difference technique (Sect. 7.4.2, Quantification Procedures), parallel detection spectrometers, and very long acquisition times. The advent of direct electron detectors (discussed in Sect. 7.2.4, Direct Electron Detectors for EELS) is expected to significantly improve the detection limits. These detection limits also assume very thin samples, typically \(t/\lambda<0.3{-}0.5\) (Sect. 7.4.2, Limitations in Analysis and Quantification). If thicker samples are used, the detection limits degrade significantly up to the point that even pure elements would not be detected in samples due to the increase in the background due to multiple scattering (Sect. 7.8.2) which masks the edges. A summary of the detection limits based on the work of Leapman and Newbury [7.105, 7.193, 7.194] is presented in Table 7.6.

Fig. 7.92a,b

Extraction of signals using the second-difference technique for the detection of trace concentration of reference materials in a standard sample (SRM 610 glass from NIST). (a) In the raw spectrum nearly all edges of trace constituents are not visible. (b) In the second difference spectrum the edges of trace elements are well resolved. Elements Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, and Yn are present in the \(\mathrm{60}\) to \({\mathrm{80}}\,{\mathrm{ppm}}\) range. Sc, Ti, V, Cr, Mn, Fe, and Co are present in the \(150{-}240\,{\mathrm{ppm}}\) range. Adapted with permission from [7.105]. Copyright 1993 American Chemical Society

Table 7.6

Edge shapes and labels for elements of the periodic table as detectable in EELS experiments

Detection limit

Element and edges

Edge type (following the Gatan Chart Convention)

\(10{-}100\,{\mathrm{ppm}}\)

3d-Transition metals and preceding alkaline earth elements: L edges (Ca, Sc, …, Ni)

Open image in new window

 

4f-Lanthanides and preceding alkaline earth elements: \(\mathrm{M_{45}}\) edges (Ba, La, …, Yb)

Open image in new window

\(100{-}1000\,{\mathrm{ppm}}\) (\({\mathrm{0.1}}\%\))

Light elements: K edges (Li, Be, B, …, P) \(\mathrm{L_{23}}\) edges (Mg, Al, Si, P, S, Cl)

Open image in new window

 

4d transition-metal elements and preceding alkaline earth elements: \(\mathrm{L_{23}}\) edges (Sr, Y, …, Rh)

Open image in new window Open image in new window

\(> {\mathrm{1000}}\,{\mathrm{ppm}}\) (\({\mathrm{0.1}}\%\))

Ga, Ge, As, Se, Ag, Cd, In, Sn, Sb, Te, I, W, …

Open image in new window

A detailed simulation package predicting the detection limits including the effects of increased background, multiple scattering, and angular collection has been developed commercially (available from Gatan as the EELS advisor in Digital Micrograph) based on the initial work of [7.195, 7.196]. Applications of these simulations are extremely useful to predict whether elements present at low concentration levels in any matrix can be detected and to suggest experimental conditions to detect said elements.

7.7.4 Comparison of EELS and EDXS Detection Limits

An expression for the detected x-ray signal based on the incident current \(I\) and acquisition time \(T\) can be deduced so that the relative merits of EDXS and EELS can be compared [7.35].

Following the description given for x-ray signal quantification (Sect. 7.4.1) we describe the intensity
$$I_{\mathrm{x}}=N\left(\frac{I}{e}\right)T\omega_{k}\sigma_{k}\eta_{\mathrm{x}}\;,$$
(7.82)
where \(\omega_{k}\) is the fluorescence yield for the particular line, \(\sigma_{k}\) is the total cross section (integrated over all angles; Sect. 7.3.4, Partial and Total Ionization Cross Sections), and \(\eta_{x}\) is the collection efficiency including all detector components and any absorption in the sample.
The relative sensitivity of EELS and EDXS signals can be calculated by considering the ratio of EELS and EDXS intensities
$$\frac{I_{k}}{I_{\mathrm{x}}}=\frac{\sigma_{k}(\beta,\Delta)}{\omega_{k}\sigma_{k}}\frac{1}{\eta_{\mathrm{x}}}\exp\left(\frac{-t}{\lambda_{\mathrm{e}}}\right).$$
(7.83)

The ratio of the partial and total cross-section term is typically in the order of \(\mathrm{0.1}\) [7.35] although large scattering angles and energy windows (\(\Delta> {\mathrm{50}}\,{\mathrm{eV}}\)) would tend to include most of the inelastic distribution given the asymptotic behavior of the scattering distribution and the cross sections (Sect. 7.3.4). The fluorescence term is the major contributor to the greater effectiveness of EELS for the detection of a large number of elements in ideal samples. For K edges of heavy elements with \(Z> {\mathrm{40}}\) (difficult to reach in TEM-EELS experiments given their high energy loss), \(\omega\) approaches one (Fig. 7.2) but, for light elements or L edges, it drops very sharply (for K edges it is around \(\mathrm{0.1}\) for Al, \(\mathrm{0.02}\) for Na, down to \(\mathrm{0.001}\) for B while for L edges it is less than \(\mathrm{0.001}\) for \(Z<{\mathrm{20}}\)). The detector efficiency term also has a significant effect on the relative merit of EELS. The small solid angle for a single Si(Li) EDXS detector implies that only \({\mathrm{1}}\%\) of the emitted x-rays are collected by the detector. This can be improved by a factor of \(\mathrm{10}\) with multiple detectors and good collimation. Furthermore, low-energy x-rays can be absorbed in the thin window and the dead layer of the detector (Sect. 7.2.3, Detector Windows). Because of the combined terms of fluorescence and detector efficiency, the EELS signals are \(3{-}4\) orders of magnitude stronger than EDXS signals for light elements while they are slightly stronger for most heavy elements. The situation improves in EDXS for multiple large-area detectors consistent with the increase in the collection efficiency. The relative merits can therefore be summarized in Fig. 7.93. The drawback with EELS, however, is not fully accounted for in (7.83) because of the strong increase of background with thickness arising from multiple scattering. For thick samples, even of pure elements, EELS would not show any edge irrespective of the atomic number while EDXS spectra would still show peaks (even for light elements, albeit small)! Absence of EELS edges in a spectrum from a thick area of a sample does not obviously imply absence of any elements.

Fig. 7.93

Comparison of the relative sensitivity of EELS/EDXS detection limits. (After [7.78]). Note that the calculations assume samples are very thin

7.8 Energy-Loss Fine Structures

As discussed in the introduction section of this chapter, there are fine modulations in the structure of spectra that yield useful information on the chemical environment of the atoms and the dielectric properties of the material. These fine structures can be subdivided into three parts. The energy-loss near-edge structures ( ) are modulations appearing in the first \(10{-}20\,{\mathrm{eV}}\) from the ionization edge threshold (Fig. 7.94). These are nowadays used almost routinely to characterize the chemical environment of atoms, including the type of phases and valence state.

Fig. 7.94

Regions and energy ranges for the energy-loss near-edge structures (ELNES) and extended energy-loss fine structures (EXELFS) of core edges

At higher energy losses (from about \(30{-}50\,{\mathrm{eV}}\) of the threshold up to several hundred eV), the extended energy-loss fine structures ( ) provide information on the radial distribution function of the material (similar to x-ray absorption fine structures ) (Fig. 7.94). These modulations arise from the backscattering of the ejected electron in the solid and the creation of interference between the ejected and backscattered wavefunctions and are particularly useful to provide the bond distances in amorphous solids at the nanometer scale. Since current applications of the technique in the AEM literature are limited we will refer the reader to a good overview of this technique in [7.35] and to XAFS literature describing the principles of the analysis method.

Finally, the fine modulations in the low-loss part of the EEL spectra (from \({\mathrm{0}}\,{\mathrm{eV}}\) to \(50{-}100\,{\mathrm{eV}}\)) also provide a wealth of information on the dielectric properties of materials. Quantitative analysis makes it possible to compare optical spectroscopy measurements to low-loss energy-loss data and measure electron density using some simple approximations for metals. Qualitative analysis allows one to use the differences in spectra for various materials to map the distribution of phases. Given the impact of the ELNES and low-loss spectra in AEM, details of these two techniques are presented below.

7.8.1 Energy-Loss Near-Edge Structure

As discussed in the introductory section of this chapter, ELNES provide information on the electronic structure and bonding environment of the atoms probed by the incident fast electrons. An example of the information is demonstrated in Fig. 7.95a,b showing the relationship between the spectrum and the energy states along with examples of near-edge structures for different compounds (Fig. 7.96a,b). The features visible in the near-edge structure represent the unoccupied energy states as modified with respect to a free atom by effects such as hybridization, coordination changes, and solid-state effects. The technique provides, therefore, data equivalent to the well-established x-ray absorption near-edge structure ( ) spectroscopy carried out in synchrotrons. As such, reference data and literature from XANES can often be used to identify compounds, understand trends and electronic structure effects in ELNES. Examples of changes in the ELNES derived from changes in the electronic structure demonstrate the sensitivity to the structural environment and the chemical state (Fig. 7.96a,b). In addition to the changes in the shape of near-edge structures, the energy position of the edges can vary with the oxidation state in a similar way to x-ray photoelectron spectroscopy. Changes in the core energy level and the position of unoccupied states can result from charge-transfer effects due to oxidation, bonding, and coordination changes. Systematic trends are therefore observed for several metals with oxidation state both for x-ray absorption spectroscopy [7.198] and for EELS [7.199]. Detailed reviews on applications of ELNES can be found in [7.197, 7.200, 7.201, 7.202]. Examples demonstrating the application of ELNES at high spatial resolution are shown in [7.203, 7.204, 7.205, 7.206, 7.207, 7.208]. Examples of surface reconstructions based on numerical analysis of a series of measurements can be found in [7.209, 7.210]. Similarly, examples of mapping of orbitals at atomic resolution can be found in [7.211].

Fig. 7.95a,b

Relationship between the near-edge structure observed on the EELS edges and the unoccupied electronic states. (a) Transitions are observed from core levels to unoccupied electronic states above the Fermi level. (b) Example of near-edge structure (experimental spectrum) for the C K edge in graphite with the relationship between the \(\pi\) and \(\sigma\) orbitals (and the antibonding orbitals \(\pi^{\ast}\) and \(\sigma^{\ast}\)) in the hybridized atoms and the related bands in the solid

Fig. 7.96a,b

Examples of near-edge structures in (a) various carbon-based compounds showing the sensitivity to the structural environment and hybridization, (b) Fe based compounds. Reprinted from [7.197]

At the more quantitative level, the sensitivity of the ELNES to bonding can be explained by the general oscillator strength and the form factor terms of the partial cross sections (Sect. 7.3.3). These two terms are dependent on the initial and final state wavefunctions of the interacting electrons and thus contain information on the chemical state and electronic structure of the probed atoms as modified in the solid. In the dielectric formulation of the cross sections (Sect. 7.3.3, Outer-Shell Excitations), it is also possible to understand how the energy-loss spectrum relates to other spectroscopy measurements
$$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}\propto\left(\frac{1}{\theta^{2}+\theta^{2}_{\mathrm{E}}}\right)\Im\left\{\frac{-1}{\varepsilon(q,E)}\right\},$$
(7.84)
where \(\varepsilon\) is the dielectric function of the material (Sect. 7.3.3, Outer-Shell Excitations) which can be expressed by its real part \(\varepsilon_{1}\) (related to the screening process of the electrons) and its imaginary part \(\varepsilon_{2}\) (related to the absorption process and thus to optical and x-ray absorption measurements). The first term in (7.84) is general and is related to the kinematics of scattering. It imposes a simple Lorentzian angular distribution to the scattering and the rapid drop in intensity with increasing scattering angle. The second term is the loss function that is related to the dielectric response of the solid to an electromagnetic radiation and is therefore ultimately linked to the intrinsic properties of the solid. At high energy losses (from about \(50{-}100\,{\mathrm{eV}}\) and above) where the screening of the incident electron charge by collective effects is not important, \(\varepsilon_{1}\rightarrow 1\) and \(\varepsilon_{2}\) is small (\(\varepsilon_{2}\ll\varepsilon_{1}\)) so that the loss function is reduced to
$$\Im\left\{\frac{-1}{\varepsilon}\right\}=\frac{\varepsilon_{2}}{\left(\varepsilon^{2}_{1}+\varepsilon^{2}_{2}\right)}=\varepsilon_{2}$$
and is thus directly related to the absorption part of the dielectric function and therefore to absorption measurements. In this particular condition and in the one-electron approximation, the cross section can be calculated based on Fermi's golden rule describing the transition rate of an electron from an initial energy level \(i\) described by an initial wavefunction \(\psi_{i}\) of energy \(E_{i}\) to a final level \(f\) described by a wavefunction \(\psi_{f}\) of energy \(E_{f}\)
$$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}\propto\sum_{i,f}\left|\left\langle\psi_{f}|\exp(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r})|\psi_{i}\right\rangle\right|^{2}\updelta(E-E_{f}+E_{i})\;,$$
(7.85)
where we have omitted the first (kinematic) term from (7.84) because it is independent of the solid-state effects. As defined in Sect. 7.3.3, the scattering vector \(\boldsymbol{q}=\boldsymbol{k}_{0}-\boldsymbol{k}_{1}\) where \(\boldsymbol{k}_{0}\) and \(\boldsymbol{k}_{1}\) are the incident and final wavevectors of the incident electron. The exponential operator is derived from the Hamiltonian describing the interaction between the incident electron, the nucleus and the atomic electrons in the solid ( [7.212] for a detailed derivation of this term). The sum is carried out over all possible final energy states limited by the \(\updelta\) function to ensure the conservation of energy in the scattering event so that the energy-loss \(E\) corresponds to the difference in the energies of the final and initial states. The exponential operator can be expanded as a Taylor series
$$\exp(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r})=1+\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r}+\frac{1}{2}(\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{r})^{2}\ldots$$
(7.86)
The first term of this expansion is zero due to the orthogonality of the initial and final wavefunctions (i. e., \(\langle\psi_{f}|\psi_{i}\rangle=0\)). For most conditions used in EELS experiments (i. e., small spectrometer collection angles \(\beta\) and edges above around \({\mathrm{100}}\,{\mathrm{eV}}\)), only the term related to dipole excitation \(\boldsymbol{q}\cdot\boldsymbol{r}\) is retained because the scattering angles are small and the core wavefunctions are very localized in typical EELS experiments. Under these conditions, \(\boldsymbol{q}\cdot\boldsymbol{r}\ll 1\) leads to what is known as the dipole approximation and the second term in the Taylor expansion can be neglected (the treatment is thus similar to x-ray absorption spectroscopy). For larger scattering angles and/or for very low core-losses (the core state wavefunctions are less and less localized as the core energy decreases, i. e., the mean radius of the core wavefunction increases), this approximation is, strictly speaking, no longer valid but it gives a good representation of the spectra (nonetheless, full calculations with nondipole terms are possible with current methodologies) [7.213]. In the dipole approximation, we obtain,
$$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}\propto\sum_{i,f}\left|\left\langle\psi_{f}|\boldsymbol{q}\cdot\boldsymbol{r}|\psi_{i}\right\rangle\right|^{2}\delta(E-E_{f}+E_{i})\;.$$
(7.87)
The squared term is the dipole matrix element representing the transition rate from a core state to a final state. In this condition, the sum in (7.87) is simplified as
$$\begin{aligned}\displaystyle\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E}&\displaystyle\propto\left|M_{\ell,\ell-1}(E)\right|^{2}\rho_{\ell-1}(E)\\ \displaystyle&\displaystyle\quad\,+\left|M_{\ell,\ell+1}(E)\right|^{2}\rho_{\ell+1}(E)\;,\end{aligned}$$
(7.88)
where \(\rho(E)\) is the density of states ( ) resolved in angular momentum components \(\ell\) (s, p, d, f). In the development of the matrix elements of (7.87), the decomposition of the wavefunctions in radial and angular terms imposes two very interesting effects summarized by (7.88). From the radial part of the matrix elements one can deduce that, for a transition to be observed (i. e., \(M\neq 0\)), there must be an overlap between the core states and the final state wavefunctions. This implies that the energy-loss spectra probe the site-dependent electronic structure projected on the excited atom. Practically speaking, since the core states are very localized, the information is also very local and not just limited to the electron-beam illuminated area but it is specific to the atom type excited by the electron beam: different atoms in the same area illuminated by the electron beam will have a different local electronic structure environment. Second, from the development of the angular parts of (7.87), the matrix element of a transition from a state of angular momentum component \(\ell\) is zero unless the final state is of angular momentum \(\ell\pm 1\) as suggested by the subscripts in (7.88). Hence, transitions will therefore occur for states with a change in angular momentum component \(\ell=\pm 1\). This implies that if the initial state is of s character (\(\ell=0\)), the transition will occur only to states of p character (\(\ell=1\)). This is observed for K edges where the core state is 1s. For an initial state of p character (\(\ell=1\)), transitions will be observed to states of s (\(\ell=0\)) and d character (\(\ell=2\)). This applies to \(\mathrm{L_{23}}\) edges arising from the 2p\({}_{3/2}\) and 2p\({}_{1/2}\) initial states.

The matrix elements therefore allow us to probe the local DOS of each element separately by selecting the edge corresponding to the atomic number of the element of interest and the edge type (K, L, etc.) according to different principal and angular momentum quantum numbers. This latter sensitivity is very illuminating since EELS experiments allow one to probe not only the unoccupied states but also the atom-site specific and symmetry projected density of unoccupied states.

The calculations of near-edge structures using (7.88) have been demonstrated to be equivalent for x-ray absorption spectroscopy near-edge structures (XANES) and ELNES in most experimental collection conditions respecting the dipole approximation. The only difference in the formulation between XANES and ELNES derives from the fact that the scattering vector \(\boldsymbol{q}\) is replaced by the electric field \(E\) in (7.86) and (7.87). The ELNES is therefore an extremely powerful probe of the electronic structure of solids with high spatial resolution as compared to other valence or conduction band spectroscopies where the entire bands (valence or conduction) are probed irrespective of the angular momentum character.

The calculations based on (7.88) with density of states determined by first principle methods assume an infinite lifetime of the excited state. To account for more realistic conditions where the decay occurs via de-excitation processes, a finite lifetime must be considered. This can be achieved artificially by broadening the \(\delta\) function with a Lorentzian distribution to account for the finite life time of the excited state, the life time of the core state as well as the instrumental function accounting for the energy spread of the incident electrons and the resolution of the spectrometer.

The derivation of (7.85)–(7.88) have been described here very briefly and we refer the interested reader to the work of Fink [7.214], Vvedensky [7.215], Saldin [7.216] and Schattschneider and Jouffrey [7.217] and a review by Paxton [7.212] for further details. It is important to mention, however, some of the limitations of this description in order to give an idea of what can be expected from first principle calculations. First of all, the one electron derivation of the transition probability is based on the single particle approach. This simplification assumes that the excited state (where there is a hole in the core state and an ejected electron) can be represented by ground state wavefunctions (no excitation effects are accounted for). This is the most important approximation which can be improved on, in principle, by considering the final state rule proposed by von Barth and Grossman [7.218] which considers the electronic structure of the system in the potential probed by the ejected electron i. e., with a core-hole in the initial level. In spite of this approximation, the single particle approach is a first useful step to understand general features in the spectra and to assess the need for more refined models accounting for the more realistic final state. For many systems including metallic materials and even some insulators where the screening of the core-hole is effective, this description is successful while in others, the interactions between the core-hole and the ejected electron significantly modify the ground state electron wavefunctions of the solid.

A second limitation is related to the specific approaches used to describe the electronic structure of the solid. The predictions of the DOS are based on the use of density functional theory and the different implementations to calculate the electron wavefunctions (for a review, see [7.219]). For the most part, electronic structure calculations for solids have focused on the description of occupied states and low-lying unoccupied states. This presents a limitation for the calculations of energy-loss spectra that probe unoccupied states \(10{-}30\,{\mathrm{eV}}\) above the threshold (and thus above the Fermi energy). To simplify the computation, linear band structure methods are often used (such as linear muffin-tin orbital methods, linear augmented plane wave etc.), but these are limited in the energy range that will be reliably reproduced (\(5{-}20\,{\mathrm{eV}}\) from the edge threshold). Alternatives to these techniques are the multiple scattering-based techniques such as the Korringa–Kohn–Rostoker method (used for transition-metal edges [7.220]) and the real-space multiple scattering technique [7.221], and the pseudopotential technique based on the use of plane waves that has given impressive results at about \(40{-}50\,{\mathrm{eV}}\) above the threshold in diamond [7.222].

Besides these calculations of ELNES based on band structure techniques or real-space methods, other approaches must be used to deal with systems demonstrating strong electron–electron interactions due to electron localization. For these systems the band structure and real-space methods completely fail to describe the spectra and a completely different scheme must be used. For transition-metal L edges and rare-earth M edges, a description based on atomic multiplet theory is much more effective to describe the spectra. These techniques include solid-state effects by including crystal field and charge-transfer effects to describe the structural and bonding environment. A full description of the methods is given in the excellent reviews by de Groot [7.223, 7.224]. Examples of calculations with this technique and a range of other methods are shown in Fig. 7.97a-c. A description of the hierarchy of these methods starting from the molecular orbital approach and multiple scattering methods can be found in [7.225] and in the review by Radtke and Botton [7.201].

Fig. 7.97a-c

Examples of calculations of ELNES with (a) the multiple scattering method for the Al K edge in AlN; (b) band structure technique: O K edge in Rutile experiments (dots) and calculations (full line) with ab initio code Wien2K; and (c) atomic multiplet calculations of the Mn \(\mathrm{L_{23}}\) edge in \(\mathrm{MnTiO_{3}}\)

In addition, the flexibility of EELS experiments in the TEM make it possible to tune very effectively and elegantly the scattering vector of the incident electron using momentum-resolved energy-loss experiments to study anisotropic materials. Experiments demonstrating this effect in low-loss spectra [7.214, 7.226], core-losses [7.227, 7.228, 7.229], and to record inelastic scattering distributions [7.230, 7.231] can be found in the literature.

Various tools are therefore available to materials scientists and microscopists to understand and model near-edge structures should the need arise. In many cases of AEM applications, however, a detailed comparison of various edges with reference materials is sufficient to understand the trends and associate the spectral features to bonding bands and materials properties. Databases are currently in development allowing users to access libraries of spectra from various compounds [7.232, 7.233].

7.8.2 Low-Loss Spectroscopy

Fundamentals

At low energy losses (\(\mathrm{0}\) to \(50{-}100\,{\mathrm{eV}}\)), the most intense part of the spectrum is directly related to the collective response of the electrons in the solid to the fast incident electron. The incident charges with their associated electric field polarize the medium and set up oscillations of the weakly bound electrons of the solid at particular eigenfrequencies related to the mass of the electrons and their density in the solid. This is analogous to the resonance frequency of an object of mass \(m\) attached to a spring with spring constant \(k\) of classical mechanics. The collective behavior of these electrons in a solid is best characterized by treating the collective response as an effective particle called a plasmon oscillating at a frequency \(\omega_{\mathrm{p}}\) that results in an energy loss of the primary electron equivalent to the energy \(E_{\mathrm{p}}=\hbar\omega_{\mathrm{p}}\).

The link between the spectrum, the general oscillator strength and loss function presented in Sect. 7.3.3, Outer-Shell Excitations can be discussed in the context of low energy losses. Whereas at high energies (Sect. 7.8.1) the loss function is essentially determined by \(\varepsilon_{2}\), at low energies the screening of the electrons is very effective and \(|\varepsilon_{1}|\) becomes much greater (in simple metals) than (or is about the same order of) \(\varepsilon_{2}\). The loss-function
$$\Im\left\{\frac{-1}{\varepsilon}\right\}=\frac{\varepsilon_{2}}{\left(\varepsilon^{2}_{1}+\varepsilon^{2}_{2}\right)}$$
must then consider both the polarizability and the absorption terms essentially describing the dielectric response of the solid to electromagnetic radiation.

Detailed derivations of the models describing the formulation of the dielectric response have been reviewed by [7.217, 7.234] and only a brief summary is given here to understand the most significant features in the spectra and the relation to the dielectric response of the medium.

The Drude model is the simplest case that deals with free-electron metals and the dielectric function of the material. This model considers electrons in the solid as free particles interacting with the medium via a simple damping term \(\tau\) describing the relaxation time due to friction in the electron gas. The dielectric function becomes then
$$\varepsilon(\omega)=1-\frac{n_{\mathrm{a}}\mathrm{e}^{2}}{m\varepsilon_{0}}\,\dfrac{1}{\omega^{2}+\frac{\mathrm{i}\omega}{\tau}}\;,$$
(7.89)
where we define the term
$$\omega_{\mathrm{p}}=\sqrt{\frac{n_{\mathrm{a}}\mathrm{e}^{2}}{m\varepsilon_{0}}}$$
(7.90)
as the eigenfrequency of the electron plasma oscillation with
$$\begin{aligned}\displaystyle&\displaystyle\varepsilon_{1}=\Re\{\varepsilon\}=1-\dfrac{\omega^{2}_{\mathrm{p}}}{\omega^{2}+\frac{1}{\tau^{2}}}\quad\text{and }\\ \displaystyle&\displaystyle\varepsilon_{2}=\Im\{\varepsilon\}=\frac{1}{\omega\tau}-\dfrac{\omega^{2}_{\mathrm{p}}}{\omega^{2}+\frac{1}{\tau^{2}}}\;.\end{aligned}$$
(7.91)
\(\tau\) is related to the FWHM of the plasmon peaks \(\Updelta E_{\mathrm{p}}=\hbar/\tau\), \(m\) is the effective mass of the electrons, \(\varepsilon_{0}\) is the vacuum dielectric constant, and \(n_{\mathrm{a}}\) is the free-electron density and the plasmon energy
$$E_{\mathrm{p}}=\hbar\omega_{\mathrm{p}}=\hbar\sqrt{\frac{n_{\mathrm{a}}\mathrm{e}^{2}}{m\varepsilon_{0}}}\;.$$
(7.92)
The free-electron model, although very applicable for many sp metals is not really realistic for all materials or even all metals. To treat more realistically the full range of transition metals, insulators, and semiconductors there must be provision for transitions from various occupied to unoccupied bands in addition to contributions from the free electrons. This is done in the Drude–Lorentz model, where discrete transitions are added by considering independent oscillators having eigenfrequencies equal to the transition energies (for example to account for band-to-band transitions related to, for example, \(\mathrm{d}\) electrons, interband transitions from the valence to conduction band etc.). The expression contains the free oscillators (such as the Drude oscillators) and the bound oscillators. The dielectric function is therefore the sum over all these oscillators \(j\)
$$\varepsilon(\omega)=1+\frac{\mathrm{e}^{2}}{m\varepsilon_{0}}\sum_{j}\dfrac{n_{j}}{\omega^{2}_{j}-\omega^{2}-\mathrm{i}\frac{\omega}{\tau}}\;,$$
(7.93)
where \(n_{j}\) is the number of electrons able to oscillate at the eigenfrequency \(\omega_{j}\).

The presence of oscillators due to interband transitions or low-energy edges cause the peaks in the absorption part of the dielectric function and generate shifts in the position of the plasmon peaks. Similarly, the effect of the plasmon peaks is to shift the apparent energy of the interband transitions in the energy-loss spectra with respect to the energy in the absorption part of the dielectric function. Such shifts therefore imply that caution should be taken when interpreting peaks in spectra: the position of a peak at a given energy-loss \(E\) does not imply that there is an interband transition with the same exact energy \(E\).

Applications in Low-Loss Spectroscopy

The effects discussed in the previous section, even if not always fully quantified in analytical electron microscopy work, can be exploited in energy-filtered plasmon images to identify the presence of phases with different electron densities (hence plasmon position) or dielectric function. Systematic variations of the plasmon energy as a function of the atomic number have been demonstrated in the early work of Colliex [7.235] (Fig. 7.98). Tabulations of plasmon energies in alloys, metallic hydrides as a function of alloying element concentration and hydrogen content have been tabulated from various sources in [7.35]. This technique of plasmon measurement has also been applied to study the ratio of \(\text{sp}^{2}/\text{sp}^{3}\) hybridization bonding in C films and the water content in biological structures (Fig. 7.81a-e) [7.165].

Fig. 7.98

Variation of the plasmon energy for a series of pure elements. Triangles are experimental values and full circles are theoretical values calculated with the free electron model of (7.92). (After [7.235])

Although the detailed analysis of the low-loss spectra is not one of the routine AEM techniques yet, a detailed study of the spectra can be extremely useful to understand some of the functional properties of materials. For example, the relationship between the spectrum and the dielectric properties of solids can help elucidate some local variations of optical properties of materials [7.236, 7.237, 7.238]. Such studies make use of some key properties of the dielectric function and in particular the Kramers–Kronig relations [7.239] that link the real and imaginary parts of the dielectric function with the help of the Kramers–Kronig analysis ( ): if the loss function \(\Im\{-1/\varepsilon\}\) is known from the measurement of a spectrum, then one can determine \(\Re\{1/\varepsilon\}\) by KKA (in order to retrieve the loss function from a spectrum, the single scattering distribution of scattering must be retrieved as demonstrated below). On the basis of this relationship it is possible to retrieve both \(\varepsilon_{1}\) and \(\varepsilon_{2}\) and gain full knowledge of the dielectric function of a material (Fig. 7.99a,b). Once the dielectric function is known, the results can be compared to optical measurements at low energies and be used to associate peaks to features in electronic transitions. One important advantage of the dielectric function measurement with EELS (as compared to optical techniques) is in the energy range of the measurements. With EELS, one can analyze energies up to a few hundred eV covering, with the use of monochromators energies as low as the infrared region of the spectrum (down to about \({\mathrm{0.2}}\,{\mathrm{eV}}\)), the visible, and the UV up to the soft x-ray region (a few \({\mathrm{100}}\,{\mathrm{eV}}\)). Examples of KKA with EELS to study the dielectric function of a material is given in [7.240, 7.241]. A systematic work on the modeling of low-loss spectra using first principles and a discussion of the limitation of this approach has been presented by Keast [7.242].

Fig. 7.99a,b

Real and imaginary part of the dielectric function based on Kramers–Kronig analysis of energy-loss spectra for (a) TiC and (b) VC. (From [7.217])

Another very useful application of low energy losses is related to the strong dependence of the spectra on the thickness of the samples (Fig. 7.100). As the sample thickness increases, the probability of multiple inelastic losses increases. For example, the incident electron can excite one or more plasmon losses when traveling in the sample. If the mean distance between inelastic losses is \(\lambda\), the probability of multiple loss events \(n\) is given by
$$P_{\mathrm{n}}=\frac{1}{n!}\left(\frac{t}{\lambda}\right)^{n}\exp\left(-\frac{t}{\lambda}\right).$$
(7.94)
If we consider the intensity of the zero-loss peak \(I_{0}\) (corresponding to zero plasmon loss events, i. e., \(n=0\)) with respect to the total intensity in the spectrum \(I_{\mathrm{t}}\), we can deduce the probability \(P_{0}\) for \(n=0\) and
$$\frac{I_{0}}{I_{\mathrm{t}}}=P_{0}=\exp\left(-\frac{t}{\lambda}\right)$$
(7.95)
where this expression is typically used as
$$\frac{t}{\lambda}=\text{ln}\left(\frac{I_{\mathrm{t}}}{I_{0}}\right)$$
(7.96)
with the mean free path \(\lambda\) values based either on those tabulated in the literature (for a standard collection condition), on measurements with standards of known thickness, or on local measurements using complementary techniques (such as convergent beam or contamination spot measurements) [7.2, 7.94]. Measurements of the intensities for \(I_{\mathrm{t}}\) and \(I_{0}\) are based on the integration of the spectra over windows as demonstrated in Fig. 7.101. The measurements of thickness have been demonstrated to be reliable for thicknesses up to \(t/\lambda=4\) [7.243]. Values of the inelastic mean free path are dependent on the energy of the primary electrons, the material, and the angular collection conditions. Estimates of \(\lambda\) can be obtained based on the work of [7.244].
Fig. 7.100

Example of the variation of the low-loss spectrum in Al for two different relative thicknesses. The increase in thickness (from \(t/\lambda={\mathrm{0.2}}\) to \({\mathrm{2.2}}\)) causes multiple plasmon losses visible at energies values \(E\) corresponding to multiple integers \(n\) of single losses (\(E=n\,E_{\mathrm{p}}\)). For broad plasmon peaks, multiple peaks are not well resolved

Fig. 7.101

Features used for the determination of the relative thickness \(t/\lambda\) according to (7.96)

The use of a finite collection angle \(\beta\) results in scattering outside the collection aperture and affects the thickness measurements. For multiple inelastic losses, the total angular distribution emerging from the sample is broader as compared to the single scattering distribution due to multiple scattering events and the convolution of the inelastic scattering distributions. A finite collection angle results, therefore, in loss in collection efficiency of electrons having lost higher energies by multiple events and leads to changes in the apparent mean free path. Approaches to convert mean free paths from one aperture collection condition to another have been presented by Botton et al [7.243] while iterative methods and computer programs have been developed by Egerton [7.35]. Mean free paths for \(\mathrm{100}\)-\(\mathrm{keV}\) electrons and \(\beta={\mathrm{10}}\,{\mathrm{mrad}}\) are in the range of \({\mathrm{115}}\,{\mathrm{nm}}\) for C, \({\mathrm{100}}\,{\mathrm{nm}}\) for Al, \(\mathrm{110}\) for Si, \({\mathrm{75}}\,{\mathrm{nm}}\) for Fe and have been tabulated in [7.244, 7.35]. One can estimate the mean free path with some basic assumptions on the sample composition if \(\beta\ll(E/E_{0})\) where \(E\) is the energy loss and \(E_{0}\) is the incident energy (so as to satisfy the dipole condition). If these conditions are respected, values of mean free paths can be calculated using [7.244]
$$\lambda(\beta)\approx\dfrac{106F\left(\frac{E_{0}}{E_{\mathrm{m}}}\right)}{\text{ln}\left(\frac{2\beta E_{0}}{E_{\mathrm{m}}}\right)}$$
(7.97)
with \(E_{0}\) values given in keV, \(\lambda\) in nm, \(\beta\) in mrad. \(F\) is a relativistic factor that can be calculated as
$$F=\dfrac{1+\frac{E_{0}}{1022}}{\left(1+\frac{E_{0}}{511}\right)^{2}}$$
(7.98)
and \(E_{\mathrm{m}}\) is the mean energy loss in eV. \(E_{\mathrm{m}}\) can be approximated as \(E_{0}\approx{\mathrm{7.5}}Z^{0.36}\) without taking into account any contributions from changes in density or the crystallographic structure. For compounds containing elements of known atomic fraction \(f_{i}\), a mean effective atomic number can be calculated by summing over each element \(i\) as
$$Z_{\text{eff}}=\frac{\sum_{i}f_{i}Z_{i}^{1.3}}{\sum_{i}f_{i}Z_{i}^{0.3}}$$
(7.99)
so that the effective atomic number \(Z_{\mathrm{eff}}\) can be used to determine the mean energy-loss \(E_{\mathrm{m}}\). Using the Kramers–Kronig analysis of the scattering distribution, the absolute sample thickness can also be calculated [7.245] without prior knowledge of the sample.

From the recorded spectra containing multiple inelastic losses (Fig. 7.98), one can retrieve the single scattering distribution using deconvolution methods. These techniques are founded on the principle that the recorded spectrum is based on the sum of the single scattering distribution, the double scattering distribution, …, the \(i\) scattering distribution, up to the \(n\)-th scattering distribution. Each \(i\) distribution is obtained by the convolution of the single scattering distribution with itself \(i\) times. This sum of convolutions can be analyzed with Fourier techniques to retrieve, from the experimental spectrum containing multiple inelastic losses, the original single scattering distribution. These methods are available in commercial EELS analysis packages as standard functions. In this context, the Fourier-log technique is used to process the full spectrum [7.246] from \({\mathrm{0}}\,{\mathrm{eV}}\) to a few hundred eV. The analysis of the spectrum to retrieve the distribution representative of single scattering is necessary for all quantitative work related to the KKA and the measurement of absolute thickness discussed in [7.247, 7.35]. The additional application of the deconvolution method is to remove multiple scattering effects from core-loss spectra, such as plasmon peaks occurring through double scattering, following an inner-shell ionization edge. The probability of multiple scattering including one core-loss and plasmon losses increases in thick samples and the effect is visible as an increased intensity further away from the edge threshold: the spectra appear as a convolution of the low-loss spectra with the expected edge profile (Fig. 7.102). These multiple scattering effects must be removed by deconvolution methods (in this context one uses the Fourier ratio method) in order to correctly interpret modulations in the spectra at high energy from the threshold. Structures within a few eV of the edge threshold, however, are less sensitive to these multiple scattering effects.

Fig. 7.102

Effect of multiple inelastic losses on the shape of the ELNES

With the development of monochromators, it has also been possible to explore the measurement of bandgaps in semiconductors with localized electron beams. For example, Gu et al [7.248] used a monochromated TEM to measure local bandgaps while Horak and Stöger-Pollach [7.249] have explored the conditions that enable one to obtain reliable measurements in Si, GaAs, and GaP. They showed that it is important to avoid Cerenkov losses (because of the higher speed of electrons than light in the solid) interfering with the bandgap measurement [7.250] particularly using low-voltage TEM [7.251, 7.252].

In confined metallic nanoscale structures, surface plasmons have quantized energies and have been shown to exhibit resonances that are strongly dependent on the shape and size of the structures. These resonances are known as surface plasmon resonances first detected with EELS by Nelayah et al [7.253] and Bosman et al [7.254]. With the advent of efficient monochromators, high-order resonances were observed first by Rossouw et al [7.255] extending down to the mid-infrared part of the electromagnetic spectrum [7.256] (Fig. 7.103). The fundamental theory of this work, detailed in [7.257, 7.258] relates the EELS measurements to the photonic density of states above the sample. Such resonances in increasingly complex nanostructures have been investigated for squares [7.259], dimers [7.260], and fractals [7.261] and demonstrate the unique value of low-loss EELS for studies requiring high-spatial resolution below the diffraction limit of light-based methods.

Fig. 7.103

(a) Surface plasmon resonances in a single Ag nanowire rod (\(\approx{\mathrm{500}}\,{\mathrm{nm}}\) length) excited by a scanning electron beam. Top panel STEM-dark-field image, lower panels energy distribution of surface plasmon resonance losses at the indicated energy ranges in the panels from \(0.48{-}0.6\,{\mathrm{eV}}\) to higher energies up to the surface plasmon loss in the \(3.36{-}3.6\,{\mathrm{eV}}\) energy range. (b) Surface plasmon resonances for longer Ag nanorods (in the range \(1{-}2\,{\mathrm{\upmu{}m}}\)) exhibiting kinks and bends. The surface plasmon resonance losses are at significantly lower energies, in the mid-infrared part of the electromagnetic spectrum down to \({\mathrm{0.17}}\,{\mathrm{eV}}\) energy loss

7.8.3 Phonon and Vibrational Spectroscopy

Vibrational spectroscopy based on infrared radiation, Raman scattering, neutrons, and low-energy electrons (in HR-EELS on a surface analysis instrument) makes it possible to explore the different phonon modes (longitudinal optical ( ), shear (SH) and the transverse optical ( )) in the specimen. Typically, the spatial resolution provided by these spectroscopic techniques, however, is about one micrometer at best. Thus, given the high spatial resolution and much enhanced energy-resolution capabilities demonstrated recently, the modern electron microscopes can also enter this field of spectroscopy. Note that although some tip-based spectroscopy techniques (using sharp metallic tips) provide high resolution even as high as a few angstroms, these techniques typically sample signals from a surface layer, whereas the geometry in the TEM can provide information from thin sections of the specimen just like regular EELS measurements. Until recently the poor energy resolution in the microscopes was a major limitation for this prospect. But, the recent improvements in monochromator design (discussed earlier), the use of ultrabright cold field-emission guns, and the latest aberration-corrected electron optics have all allowed significant improvements demonstrating the potential for carrying out vibrational spectroscopy inside the microscopes.

Figure 7.104a-ga illustrates the \(\mathrm{18}\)-\(\mathrm{meV}\)-wide LO phonon identified in a typical hexagonal-BN sample, at the peak energy of about \({\mathrm{173}}\,{\mathrm{meV}}\) [7.262]. On the left, the \(\mathrm{14}\)-\(\mathrm{meV}\)-wide zero-loss peak (FWHM) acquired is also shown. Both spectra were obtained using a monochromated electron probe without which, even with a cold field-emission source with \(\mathrm{250}\)-\(\mathrm{meV}\)-wide FWHM of the ZLP shown in Fig. 7.104a-gb, would not be possible. Similarly, Fig. 7.104a-gc illustrates vibrational excitations recorded in a number of materials: h-BN, \(\mathrm{SiO_{2}}\), SiC, \(\mathrm{TiH_{2}}\), and epoxy resin. Further work on a biological molecule (guanine) has shown the evident similarities between the EELS spectrum and the FTIR data [7.54]. This pioneering work has potentially opened the prospect of molecular spectroscopy in the electron microscope. For the \(\mathrm{TiH_{2}}\) specimen, in particular, the nanoscale electron probe was placed about \({\mathrm{5}}\,{\mathrm{nm}}\) away from the sample—a technique popularly called the aloof mode [7.68], which we discuss below. Using the same instrumentation, energy-loss spectra have also been obtained in MgO cubes showing surface phonon polariton excitations as low as \({\mathrm{70}}\,{\mathrm{meV}}\) and their spatial distribution (Fig. 7.104a-gd) fully consistent with theoretical calculations of phonon modes [7.263]. More recently, other instruments with improved spectrometers have also demonstrated the detection of similar signals (Fig. 7.105a-c) and the detection of edge and corner modes in MgO cubes.

Fig. 7.104a-g

Improvements in the monochromator design for vibrational spectroscopy. (a\(\mathrm{18}\)-\(\mathrm{meV}\)-wide LO phonon identified in a typical hexagonal-BN sample using a monochromated electron probe. (b) Comparison of FWHM of ZLP recorded using monochromated and unmonochromated electron probes with the NION design. (c) Vibrational excitations recorded in a number of materials: h-BN, \(\mathrm{SiO_{2}}\), SiC, \(\mathrm{TiH_{2}}\), and epoxy resin. After [7.262]. (dg) Surface phonon polariton maps of MgO nanocubes at losses corresponding to corner and edge modes (\(\mathrm{70}\) and \({\mathrm{77}}\,{\mathrm{meV}}\), respectively). From [7.263]

Fig. 7.105a-c

Monochromated and deconvoluted spectra on an MgO cube showing the \(\mathrm{7}\)-\(\mathrm{meV}\) shift between the corner and edge modes. (a) Bright field image of the cube with the location of the spectra. (b) Energy filtered map corresponding to the peak position at the corners (red intensity distribution), at the edges (white intensity distribution). (c) Energy loss spectra obtained after numerical deconvolution of the data showing the energy shift of \({\mathrm{7}}\,{\mathrm{meV}}\). Data obtained on a Themis microscope courtesy of S. Lazar and P. Tiemeijer, ThermoFisher Scientifics, data deconvoluted by I.C. Bicket, McMaster University

The microscope operation in the aloof mode can be extremely useful for vibrational spectroscopy since the signal delocalization for the energy-loss processes can be in the order of a few tens of nm and the excitation can occur when the beam is placed away from the sample, i. e., when the impact parameter \(b\) (the distance between the specimen and the beam) is nonzero. In this type of beam configuration, the radiation damage due to unnecessary placement of the beam on the specimen can be avoided [7.68]. Calculations performed by Egerton [7.68] indicate that an aloof beam with a small distance \(b\) of about \({\mathrm{20}}\,{\mathrm{nm}}\) away from the sample can reduce damage by a factor of \(\mathrm{1000}\) (relative to electrons of the same energy transmitted through the specimen) for the same signal strength and spatial resolution. This is attributed to the fact that about \({\mathrm{50}}\%\) of the vibrational-loss signal comes from the material lying within a distance \(b\) of the edge of the specimen and extending over a length \(\mathrm{2.5}\)\(b\) parallel to the edge. Additionally, since the knock-on displacement damage arises from large-angle elastic collisions and the corresponding delocalization distance has subatomic dimensions, the knock-on damage is expected to be completely absent in the aloof mode [7.68]. Conversely, it is important to note that the elastic scattering should have a negligible effect on such an aloof recorded spectrum, since the angular distribution would be broader, and simultaneously, the delocalization distance can be much smaller (compared to the impact parameter \(b\)).

One major disadvantage of the aloof mode is that the spatial resolution is reduced because of the extent of the delocalization. As such, energy-filtered images cannot be generated under the aloof mode. However, Egerton [7.68] suggests that it should be possible to take advantage of the delocalization of inelastic scattering in transmission mode by forming an energy-selected STEM image with a digital raster having an interpixel spacing much larger than the probe diameter. Under such undersampling conditions, the radiation damage is expected to be produced only within a few nanometers of the probe position, and the vibrational signals would arise mainly from the intervening undamaged material, meanwhile a large fraction of the scanned area remains undamaged.

7.9 Atomic-Scale Spectroscopy

7.9.1 Atomic-Spectroscopy with EELS

As discussed in Sect. 7.5.2, the delocalization of inelastic scattering is an important issue for obtaining the ultimate spatial resolution in the EELS technique. In simple terms, this means that even if we are capable of placing a sub-angstrom-resolution electron probe over a single atomic column, it still does not guarantee that the EELS signal is generated from that column. A similar cross-talk phenomenon can also be found for the \(Z\)-contrast imaging where from the virtue of the dynamical nature of the probe propagation in the sample there could be an appreciable contribution to the signal from the adjacent atomic columns as discussed in Sect. 7.3.2.

Allen et al have modeled the effects of probe parameters on delocalization [7.264]. For example, Allen et al considered three probe sizes characterized by the aberration conditions and the width of the probe (\(x\)) such that they are of increasing sizes (\(x_{1}<x_{2}<x_{3}\)). Figure 7.106a,ba,b shows the signals for excitation of the Ti L shell and O K shell for different probes rastered over single Ti or O atoms. It can be seen that, irrespective of the ionization cross sections the FWHM of the profile becomes progressively broader as we move from the finest probe (i. e., probe-1) to the broader probe (i. e., probe-3). More significant is the evidence that, for the smallest probe sizes, the intensity profile is not a simple intuitive function with the maxima at the nominal atom position. The simulations also highlight that the profile is strongly dependent on the angular collection conditions. However, it is important to note that these two cases are simulated for a single isolated atom scenario. Allen et al have also considered the case of a real crystalline specimen, in which case, a direct interpretation of the EELS signal without a priori information on the delocalization effects, probe propagation, and thermal diffuse scattering can be difficult. For the same probes (1, 2, and 3) Allen et al [7.264] show simulated and experimental Ti L shell EELS STEM images from a \(\mathrm{SrTiO_{3}}\) specimen as in Fig. 7.107. The calculations show that, when the probe is on the heavier Sr columns, there is an apparent Ti L local maximum signal even if there is no Ti on that particular column. Also, using the intermediate probes, there is a dip in the intensity at the center of the Ti columns. Thus, although finer features in EELS maps can be obtained with the finest probes, a detailed interpretation of the contribution from specific columns (even those that do not contain the atomic species where the signal is detected in the energy-loss spectrum) would require atomistic simulations modeling the delocalization effects. The experimental demonstration of such fine variations in the atomic contrast, particularly as a function of the collection angle for the spectra has been shown in [7.62, 7.63] (Figs. 7.108a-j and 7.109a,b). Here the calculations show that the intensity decay at an atomically sharp interface can be attributed to the delocalization: for an atomically sharp interface of \(\mathrm{BaTiO_{3}}\)/\(\mathrm{SrTiO_{3}}\), the Ba M edge signal drops to about \({\mathrm{50}}\%\) of the maximum intensity recorded on a Ba-containing plane on the next atomic plane away from the interface (Fig. 7.108a-j). Also, the effect of the collection angle on the intensity distribution was shown for two different edges of La, the M edge at \({\mathrm{849}}\,{\mathrm{eV}}\) and the N edge at \({\mathrm{99}}\,{\mathrm{eV}}\) in a single crystal of \(\mathrm{LaB_{6}}\). This work shows that, when the angle collection conditions are not well controlled, a simple inspection of the intensity distribution or core-loss intensities in a crystal does not necessarily reflect the atomic positions (Fig. 7.109a,b). These effects are particularly important for any quantification procedures aiming to determine either concentrations of elements in particular atomic columns, the number of dopant atoms in an atomic column, or the determination of hole carriers in a specific unit cell site based on fine structures. For example, Bugnet et al [7.207] studied the localization of holes in an SrCaCuO cuprate superconductor based on the fine structure of the O K edge and the presence of a prepeak in the O K edge corresponding to the presence of hole carriers (Fig. 7.110). With quantification and detailed calculations of the beam propagation, the measurements revealed that twice as many holes are present in specific sites compared to others. Similarly, Loeffler et al [7.211] explored the prospects of mapping hybridization orbitals in between the Ti 3d states and O 2p states and demonstrated this effect with both experiments and detailed calculations.

Fig. 7.106a,b

Excitation of a Ti L shell and O K shell over a single Ti (a) or O (b) atom with different probes, such that the widths of the probe (\(x\)) are of increasing size (\(x_{1}<x_{2}<x_{3}\)) for probes 1, 2, and 3, respectively. After [7.264]

Fig. 7.107

Simulated (lines) and experimental (dots) data plots showing the Ti L shell EELS results from a \(\mathrm{SrTiO_{3}}\) specimen using probes 1, 2, and 3 described in the Fig. 7.104a-g. After [7.264]

Fig. 7.108a-j

Atomic-resolved chemical analysis of an interface between \(\mathrm{BaTiO_{3}}\) and \(\mathrm{SrTiO_{3}}\) ultrathin films. (a) Low-magnification HAADF STEM image. (b) Region in (a) over which the mapping is carried out. (df) Atomic-resolved maps for Ba (blue), Sr (red) and Ti (green). (c) Color-coded combined map of Ba, Sr and Ti signals. (gi) Line profiles of the Ti signals from EELS (g), along with calculations with dynamical beam propagation for Ti (h) and Sr and Ba (experiments (i) and theory (j)). Reprinted from [7.62], with permission from Elsevier

Fig. 7.109a,b

Effects of collection angles on atomic resolved maps for B and La in a crystalline \(\mathrm{LaB_{6}}\) crystal. (a) Large collection angle data set (\({\mathrm{110}}\,{\mathrm{mrad}}\) spectrometer entrance collection angle): Annular dark-field images at low and high magnifications subpanels respectively La \(\mathrm{N_{45}}\) map, B K edge map \(188{-}220\,{\mathrm{eV}}\) energy window, and \(228{-}260\,{\mathrm{eV}}\) energy window, La \(\mathrm{M_{45}}\) map and single spectrum. (b) Small entrance collection angle of \({\mathrm{27}}\,{\mathrm{mrad}}\) with low magnification and high magnification images subpanels, La \(\mathrm{N_{45}}\) map, B K edge map within the \(188{-}220\,{\mathrm{eV}}\) range, La \(\mathrm{M_{45}}\) and line scan for the La \(\mathrm{N_{45}}\) intensities and single pixel spectrum. From [7.63] reproduced with permission

Fig. 7.110

Evolution of the O K edge fine structure for the \(\mathrm{Sr_{3}Ca_{11}Cu_{24}O_{41}}\) chain ladder compound in two different crystal orientations. Color plots show the spectra in a temperature scale normalized to 1 obtained in two different orientations. The channel scale represents the integrated spectra horizontally in the related HAADF images (two center panels). The peaks identified as H and U correspond to the presence of holes as carriers in the energy-loss fine structure and the Hubbard energy U. From the increased intensity of the H peak, it is possible to conclude that the holes are preferentially located in the chains. From [7.206]

While not fully ignoring the delocalization effects stated above, a number of recent works have demonstrated the successful implementation of single-atom EELS for chemical and bonding analysis [7.265, 7.266, 7.267, 7.268, 7.269, 7.270, 7.271, 7.61], in oxide heterostructures and in low-dimensional materials (since the effects of channeling and beam propagation are not significant) following on from the pioneering work of Suenaga et al [7.190] before the aberration correctors and several later works (Suenaga et al [7.266], Suenaga and Koshino [7.272]). One notable application is in identifying the nature of bonding of Si dopant atoms present in a graphene sheet, and recent work performed by Ramasse et al [7.273] is discussed below. Figure 7.111a-d shows high-angle annular dark field ( ) images of Si atoms (identified by the yellow box) substituted within a pristine (Fig. 7.111a-da) and a defected area (Fig. 7.111a-db) of the graphene sheet. Figure 7.111a-dc,d show the corresponding EELS spectra for these two individual cases that were acquired by a subscanning procedure Ramasse et al [7.273] described as follows. Once a suitable Si atom is identified in both cases from the HAADF signal, a small subscan window is defined around the selected site (in this case an area of \({\mathrm{2.5}}\,{\mathrm{\AA{}}}\times{\mathrm{2.5}}\,{\mathrm{\AA{}}}\) was used). The window is then scanned repeatedly at a high frame rate (as high as \({\mathrm{50}}\,{\mathrm{ms}}\) per frame over a \(\mathrm{200}\)-\(\mathrm{s}\) total acquisition time per spectrum). To account for any sample drift in the process, the position of the window needs to be suitably readjusted. The final Si-L\({}_{2,3}\) edge spectrum for the two individual cases generated by accumulating \(\mathrm{200}\) consecutive \({\mathrm{1}}\,{\mathrm{s}}\) exposures is shown in Fig. 7.111a-dc,d, respectively. The insets shown illustrate a \(\mathrm{50}\)-frame average of the subscanning procedure described above. On the basis of ab initio calculations supporting the coordination configuration, clear evidence of 3-fold planar and 4-fold tetrahedral coordination in the two individual cases (Fig. 7.111a-da,c and 7.111a-db,d, respectively) can be noted. The two spectra show clear differences in terms of their fine structures. For example, the 3-fold planar Si atom has an intense first maximum (at \({\mathrm{102.5}}\,{\mathrm{eV}}\)), which, by contrast, is shifted to \(\approx{\mathrm{100}}\,{\mathrm{eV}}\) for the tetravalent Si, and is also much weaker. Similarly, the secondary doublet at \(\mathrm{105}\) and \({\mathrm{108}}\,{\mathrm{eV}}\) for the tetravalent Si atom is not well developed in the case of the 3-fold planar configuration for the Si atom. Ramasse et al supported these conclusions based on ab initio calculations and an SiC reference spectrum. According to these calculations the trivalent Si substitution causes a slight off-plane distortion of the sp\({}^{2}\) structure, whereas the tetravalent Si atom acts like a true impurity, thus generating a noticeable distortion in the local electron density.

Fig. 7.111a-d

HAADF images of Si atoms (identified by a yellow box) substituted within a pristine (a) and a defected area (b) of the graphene sheet. (c,d) Shows the corresponding EELS spectra for these two individual cases that were acquired by a subscanning procedure. Adapted with permission from [7.273]. Copyright 2013 American Chemical Society

Thanks to the long acquisition times, the RMS noise in both spectra is low (\(<{\mathrm{1}}\%\)), thus providing the excellent statistics required for a detailed interpretation of the fine structure. The results of quantitative analysis of the spectra and beam current (\({\mathrm{31}}\,{\mathrm{pA}}\)) show that, in the particular collection conditions used, about \({\mathrm{90}}\%\) of the Si-L\({}_{2,3}\) inelastically scattered electrons were detected. Additional work was carried out to explore the localization of valence and holes in superconductors and related compounds [7.269, 7.270].

7.9.2 Atomic Spectroscopy with EDXS

The recent advancements in regards to the formation of sub-angstrom resolution probes using aberration correctors, the concomitant increase in the probe current, the use of silicon-drift detectors (Sect. 7.2.3, The EDXS Detector), and the larger solid angle acquisitions (Sect. 7.2.3, Geometry of the EDXS Detector in the AEM) have all revolutionized the capabilities of the EDXS technique. As a result, the modern EDS systems now allow elemental mapping on the level of an individual atomic column [7.138, 7.274, 7.275, 7.276, 7.277, 7.66]. Figure 7.112a,ba illustrates an example of the Si atom embedded (or absorbed) in/on a graphene sheet performed by Lovejoy et al [7.274]. As can be seen, the Si atoms appear brighter in the HAADF image compared to neighboring carbon atoms. Figure 7.112a,bb shows the corresponding EDXS spectrum obtained over an area covering the Si atom as shown in the inset. To obtain the data a subscanning procedure (as that described earlier; Sect. 7.9.1) was carried out. In this case the subscanning procedure involved setting a window (\({\mathrm{6}}\,{\mathrm{\AA{}}}\times{\mathrm{6}}\,{\mathrm{\AA{}}}\) here) over which the EDXS signals were acquired at high frame rates (\(\approx{\mathrm{60}}\,{\mathrm{ms}}\) per frame). As shown in Fig. 7.112a,bb, the final stacked spectrum reveals signatures characteristic of Si and C atoms: Si-K (\({\mathrm{1.74}}\,{\mathrm{keV}}\)) and Carbon-K (\({\mathrm{0.277}}\,{\mathrm{keV}}\)). Over the total acquisition time of about \({\mathrm{224}}\,{\mathrm{s}}\), the probe (\({\mathrm{1.4}}\,{\mathrm{\AA{}}}\)) had spent about \({\mathrm{10}}\,{\mathrm{s}}\) directly over the Si atom and the rest over the graphene carbon sheet. A total of \(\mathrm{51}\) Si K line counts was recorded in the process, whereas about \(\mathrm{115}\) C K line counts were recorded for carbon. A similar dataset acquired for a Pt substitutional atom under identical conditions led to much higher Pt M line counts. This is most likely due to increased scattering from the Pt atom compared to Si [7.274]. Interestingly, the evidence of the Pt atom was not obvious from the EELS analyses carried out under similar conditions (not shown here). This demonstrates the complementarity of the EDXS technique for material characterization. However, performing simultaneous EELS analyses did contribute to information, which is otherwise inaccessible to EDXS. For example, the EELS spectra showed no evidence of Cu, Fe and Co signatures, thereby confirming that the evidence of peaks corresponding to these elements in the EDXS spectrum must have been due to the system background [7.274] (discussed in Sect. 7.7.2 on instrumental contributions in this chapter). These examples suggest that being able to perform both EDS and EELS in the same instrument leads to the most reliable outcome.

Fig. 7.112a,b

Illustration of EDXS elemental mapping on the level of an individual atom in the case of an Si atom embedded (or absorbed) in a graphene sheet. (a) HAADF-STEM image. (b) EDXS spectrum obtained over an area covering the Si atom (shown in (a)) as illustrated in the inset. Reprinted from [7.274], with the permission of AIP Publishing

The development of improved detectors has also led to dramatic improvements in the apparent spatial resolution of EDS measurements. For example, Fig. 7.113a-f illustrates another example of atomic-level elemental mapping carried out on a \(\langle 001\rangle\) \(\mathrm{SrTiO_{3}}\) thin sample by D'Alfonso et al [7.275]. Figure 7.113a-fe shows the atomic-resolution HAADF image of the sample, and Fig. 7.113a-fa,c show the elemental distribution of Ti and Sr, respectively, with apparent evidence of atomic-resolved signals. In each case, the experimental images are overlayed with simulations (based on a frozen phonon model outlined in [7.276]). Figure 7.113a-fb,d,f illustrates the same elemental information in the form of one-dimensional ( ) line scans. It can be seen that both HAADF and Sr-EDXS profiles are in good agreement in regard to the Sr position (Fig. 7.113a-fb,d). In the HAADF profile (Fig. 7.113a-f), this is clear from the expected stronger signals for the heavier Sr atomic columns compared to Ti columns. Similarly, a good agreement between the HAADF and Ti profiles in regard to Ti position is also obvious (Fig. 7.113a-fd). The alternating Sr-Ti-Sr structure is even clearer from Fig. 7.113a-fd. While this data shows apparent modulations at the atomic column levels, the complex probe propagation, the signal generation, and the characteristics of the emission of signals must be accounted for, particularly in consideration that these signals are obtained from relatively thick samples in which channeling to adjacent atomic columns is expected to be present. Hence detailed calculations and experiments on atomically sharp interfaces must be considered to assess the actual resolution in what appears to be atomic column spectroscopy. The most comprehensive discussion of such effects measured with a large-area SDD detector with detailed quantification of signals and electron-beam propagation is shown in the work of Kothleitner et al [7.67]. Here, by accounting for the full propagation of the signal, the collection efficiency, absorption and delocalization, a quantitative comparison was made of the signals detected in the zone axis experiments for both EDXS and EELS with expected values obtained from nonchanneling conditions. The full detailed calculations demonstrated that it is possible to retrieve quantitative information on the atomic areal density using an inversion procedure whereby, by assuming a structure, it is possible to deduce the scattered intensity within a given probe position.

Fig. 7.113a-f

Improved spatial resolution in the EDXS mapping of elements—example of an atomic-level mapping carried out on a \(\langle 001\rangle\) \(\mathrm{SrTiO_{3}}\) thin sample (a,c). Elemental maps of Ti and Sr, respectively. (e) HAADF-STEM image. Line profiles of HAADF intensity, (b) Sr and (d) Ti signals. (f) EDS sum spectrum of the sample. Reprinted with permission from [7.275]. Copyright 2010 by the American Physical Society

7.10 Signal Processing with Advanced Statistical Methods

7.10.1 Multivariate Statistical Analysis of Spectra

Multivariate statistical analyses ( ) refers to a suite of statistical methods that are useful in analyzing large datasets common to spectrum imaging. Unlike other numerical approaches where the EELS spectrum needs to be fitted to a parametric model based on the physics of interaction, the MSA is based on three nonphysical assumptions. These include assumptions that:
  1. 1.

    The problem is linear

     
  2. 2.

    The signal in the original data set has a higher variance then the noise

     
  3. 3.

    The so-called principal components (discussed below) are orthogonal.

     
Principal component analysis ( ) is one popular MSA method, which is basically a reduction in the dimensionality of an original dataset by finding a minimum number of variables that best describes the original dataset without losing much information [7.278, 7.279, 7.280]. This simply means that the systematic deviations from the average of the data set can be highlighted from PCA.
Mathematically, the 3-D spectrum image \(\mathbf{D}(x,y,E)\) acquired during the experiment is decomposed by applying PCA (readily available in the Digital Micrograph platform) into matrices, \(\mathbf{S}\) (\(=\) score) and \(\mathbf{L}\) (\(=\) loading) (see (7.100) below). The decomposition is carried out either by eigen-analyses or singular-value decomposition. The outcomes of this decomposition are: (a) the component \(n_{\mathrm{x}}\), which is calculated as the individual product of each row of the loading and the score matrices, (b) the eigenvalues of the data matrix \(\mathbf{D}\), whose relative magnitude indicates the amount of variance that the corresponding component contributes to the data set [7.279, 7.280].
$$\mathbf{D}=\mathbf{S}\times\mathbf{L}^{\textsf{T}}$$
(7.100)
Each row in the loading matrix \(\mathbf{L}\) contains a spectral feature uncorrelated to other rows, and each row in the score matrix \(\mathbf{S}\) represents the spatial amplitude of the corresponding \(\mathbf{L}\). The dominant features stored in the loading and score matrices represent the principal component of the matrix, which can be evaluated as the magnitudes of respective eigenvalues. Typically, the total number of principal components measured this way is far less than the rank of the data matrix \(\mathbf{D}\). Commonly a logarithmic scree plot is plotted to visualize this trend, wherein the eigenvalues of corresponding components is plotted against the index of the components. A typical scree plot is illustrated in Fig. 7.114. A high eigenvalue in the scree plot indicates that the corresponding component is statistically significant, in other words, occurs frequently in the data matrix. Conversely, smaller eigenvalues usually represent random noise resulting from the data that is not repeated very frequently in the data matrix. As a general rule, the series of components appearing in the form of a plateau represents noise. However, care is essential before discarding these components, particularly at the onset of the plateau. One common protocol here is to visualize the corresponding 2-D image for the individual components and identify if any features characteristic of the sample (e. g., grain boundary, interface, defect) is visible.
Fig. 7.114

An example of a typical scree plot used in PCA. After [7.143]

One of the earliest examples of implementation of PCA at high spatial resolution is shown in the context of atomic-resolved images. Figure 7.115a-d illustrates an application of PCA to analyze the STEM-EELS spectrum image of an \(\mathrm{Si_{3}N_{4}}\) sample. Figure 7.115a-da–c are the score images and the loading matrices, respectively. Figure 7.115a-da, being the most significant PC (i. e., component 1) represents the average of the entire data set. One can identify the Si-L, N-K, and C-K edges in Fig. 7.115a-db, which represents the average composition of the sample. Component #2 shown in Fig. 7.115a-dd corresponds to a deviation from the average represented by component #1. Although the positive and negative regions of the spectrum visible in Fig. 7.115a-dd are not physically meaningful, the corresponding score image shown in Fig. 7.113a-fb reveals brighter intensities, which correspond to Si atom locations along the [001] zone axis. Interestingly, these intensities only appear at signals beyond the Si-L\({}_{1}\) edge, and not at the Si-L\({}_{2,3}\) edge (see loading spectrum in Fig. 7.115a-dd). The authors suggested that the energy-loss process driving these excitations is more localized around the s electron state than the other electron states. Such unexpected features would have been impossible to identify without the use of statistical methods of analysis such as PCA. However, using improved signals, such an effect is already visible in raw data with simple extraction of the signals from spectrum images [7.63] and does not need to invoke any consideration of s states.

Fig. 7.115a-d

Illustrations of the score (a,c) and loading matrices (b,d) in the PCA of the STEM-EELS spectrum image of a \(\mathrm{Si_{3}N_{4}}\) sample. From [7.279], by permission of Oxford University Press

For further applications of PCA in mapping and noise reduction the reader is referred to the articles by [7.279, 7.280]. Two main disadvantages with PCA need to be mentioned here. First, depending on the problem it is generally not easy to establish a physical interpretation of all the principal components. Second, the PCA assumes that the principal components are orthogonal. Although this simplifies what would otherwise be a degenerate problem, the orthogonality may not necessarily always be true. For these reasons, PCA is regularly used as a technique to reduce noise in spectral analyses and as a first step in further data processing.

7.10.2 Independent Component Analysis

Another statistical method that has attracted great attention in analyzing large spectral data sets is independent component analysis ( ). As with PCA, the large data set is decomposed into smaller sets of components. However, whereas the PCA tries to identify the variability in the data by computing eigenvalues, ICA separates the original data set into statistically independent subelements. The method works because it has been shown empirically that the first derivative of the EELS spectra (of \(n\) compounds in the sample) conforms to the mutual independence hypothesis. A detailed procedure for the implementation of ICA can be found elsewhere [7.281] and only a summary is provided here.

The ICA algorithm changes the coordinate space by minimizing the Gaussianity of the data projected on the axes. This property is a result of the so-called central limit theorem, according to which any linear mixture of two independent random variables is more Gaussian than the original variables. Finally, the projected joint density distributions on the original axis are called the individual components, which are now mutually independent of one another.

The above principle holds well for treating data sets of even higher dimensions, and importantly, unlike PCA, the axes do not have to remain orthogonal. However, it is still advisable to perform PCA before ICA, because (a) current ICA algorithms still suffer from a condition called over-learning in cases where the noise is higher (i. e., the number of components is much smaller than the dimensions of the spectrum image), (b) ICA is more computationally intensive, therefore a reduction in the dimensionality with PCA can be very useful. There are many algorithms to implement ICA, but the fast-ICA (fixed-point ICA) method is the most popular [7.281].

In the recent past ICA has been a popular routine in the phase analysis of materials [7.282, 7.283, 7.284, 7.285, 7.286, 7.287]. For example, a combined PCA-ICA analysis of a Nickel-based super alloy performed by Rossouw et al [7.282, 7.283] is illustrated in Fig. 7.116a-c. Here, by retaining only the first five principal components, the noise in the original data is significantly reduced. The five ICA spectral components and their spatial distributions can be understood side-by-side. Component number 1 reveals strong Cr, Fe, and Co peaks that are concentrated particularly in the matrix of the alloy. Component number 2 contains only the Cu K and L peaks that appear to be uniformly distributed over the material. This is interpreted as a spurious signal due to the support of the thin sample. Component number 3 reveals strong O and Ni peaks, respectively. The negative peaks visible on component-1 may not carry a physical meaning, a possible explanation for the origin of such negative peaks could be that the matrix spectra are composed of a linear combination of components \(\mathrm{-0}\) and \(\mathrm{-1}\). A very powerful example of the use of ICA is in chemical tomography (also known as four-dimensional ( ) tomography) where it has been used to map the chemical state of Fe oxide in nanoparticles and extract the thickness, as well as the difference in valence on a surface layer [7.286].

Fig. 7.116a-c

ICA as a popular routine in the phase analysis of materials—example of a combined PCA-ICA analysis of a Nickel-based super alloy. (a) Scree plot of principal components from PCA, (b) independent spectral components resulting from FastICA, and (c) spatial distributions of the spectral components. The corresponding HAADF image is also shown in (c). From [7.282] published under CC-BY 4.0 license

7.11 Advanced STEM Detectors

Until recently, TEMs have heavily relied on charge coupling devices (CCDs) for electron imaging. These devices exhibited a number of advantages compared to their predecessors, namely emulsion films and imaging plates. For instance, although emulsion films provided good spatial resolution, a larger field of view and good contrast, the dynamic range was much lower, and the detector response was nonlinear [7.288]. These issues were improved upon in imaging plates that came later along with the added advantage of a digital readout. However, the signal-to-noise ratio provided by the imaging plates was still relatively poor. Furthermore, both emulsion films and imaging plates were inherently slow in nature, thus any advanced imaging experiments requiring automation or dynamic processing were limited. The CCDs on the other hand (discussed in detail earlier) provide improved performance in areas noted above particularly with the advent of direct electron detection. However, there are limitations in acquisition speed and dynamic range. These limitations are overcome by the recently developed pixel area detectors ( s) [7.289]. The PADs are compact and have single electron sensitivity, offering high-speed electron imaging, where a single frame can be read out in less than \({\mathrm{1}}\,{\mathrm{ms}}\). Importantly, the PADs provide high dynamic ranges on the order of \(\mathrm{10^{6}}\), in other words the capability to detect between \(\mathrm{1}\) and \(\mathrm{1000000}\) electrons for every pixel. These characteristics enable them to essentially record an image of all the transmitted electrons (corresponding to ZLP to higher order laue zones ( ) lines) for every probe position on the sample.

One particular implementation of the PAD arrangement developed by Tate et al [7.289] and Muller et al [7.290] is shown in Fig. 7.117a-da. The detector consists of Si-based pixelated sensors (\(\approx{\mathrm{500}}\,{\mathrm{\upmu{}m}}\) thick) that are bump-bonded pixel-by-pixel to an underlying integrated circuit, which processes the charge generated in each sensor pixel. A pixel array of \({\mathrm{128}}\times{\mathrm{128}}\) is used in sensing, by further dividing it into eight banks of \({\mathrm{128}}\times{\mathrm{16}}\) pixels each. As shown in Fig. 7.117a-db, the entire detector is housed separately, sliding in and out pneumatically as and when required. Figure 7.117a-dc illustrates the practical advantage of using a PAD detector, showing the diffraction pattern recorded in \({\mathrm{1}}\,{\mathrm{ms}}\) for a \(\mathrm{BaFiO_{3}}\) sample (along the [010] zone axis). Even at such small readout time, the recorded pattern clearly shows the central beam and the details of the Kikuchi bands. Figure 7.117a-dd shows the accumulation of this data over \(\mathrm{100}\) frames, and as can be seen, the Kikuchi band and the HOLZ lines and the central beam are much clearer. More details on the construction and the advantages of these PAD detectors can be found elsewhere [7.289, 7.290].

Fig. 7.117a-d

Pixel area detectors (PADs) for electron imaging. Illustration of a typical PAD as implemented by Tate et al [7.289] and Muller et al [7.290] (a,b). (c) Diffraction pattern recorded in \({\mathrm{1}}\,{\mathrm{ms}}\) for a \(\mathrm{BaFiO_{3}}\) sample (along the [010] zone-axis). (d) Accumulation of data similar to (c) over \(\mathrm{100}\) frames. Scale bar represents \({\mathrm{20}}\,{\mathrm{mrad}}\) for the diffraction patterns in (c,d). Reprinted with permission from [7.289, 7.290]

The MEDIPIX-type detector adopted in the field of TEM by the University of Glasgow [7.291, 7.292, 7.293] is an alternative implementation of the PAD architecture discussed above. One class of such PAD detectors employ a pulse-counting method which yields a very high signal-to-noise ratio for single electron hits, but limits the maximum rate of incoming electrons per pixel to about \({\mathrm{10^{6}}}\,{\mathrm{Hz}}\). Thus, these are more suitable for low-dose work and saturate quickly upon exposure to high fluxes (common in diffraction pattern detection). Another variation includes measuring the integrated charge, thus avoiding the need to distinguish between the individual pulses with the drawback of lower sensitivity. The third architecture is a combination of the high-gain charge integration with the in-pixel logic that resets the pixel during integration (as in the electron microscope pixel area detector (EMPAD ) architecture discussed above). The advantage here is the increase of the dynamic range per frame while the high-gain amplifier maintains single electron sensitivity [7.289, 7.292, 7.293].

Notes

Acknowledgements

Gianluigi Botton is grateful for the patience and understanding of Peter Hawkes and John Spence who enabled him to do this work, particularly the first edition of this chapter despite on-going academic commitments, conference organization, and setting up of a new facility. GB is indebted to past members of his group for providing some figures and for feedback, in particular to N. Braidy, G. Radtke, M. Couillard, C. Maunders, Y. Zhu, G. Zhu, M. Bugnet, and S. Lazar. GB wants to thank several collaborators and friends who have provided, over the years, interesting samples, motivating discussions, and moralsupport.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dept. of Materials Science & EngineeringMcMaster UniversityHamiltonCanada

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