Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope

CHAPTER THREE

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope Christian Dwyer Monash Centre for Electron Microscopy, Department of Materials Engineering, and Australian Research Council Centre of Excellence for Design in Light Metals, Monash University, Victoria 3800, Australia

Contents 1. Introduction 2. Practical Aspects 2.1. Experimental Setup 2.2. Factors Determining Spatial Resolution 2.2.1. 2.2.2. 2.2.3. 2.2.4.

Beam Size Beam Channeling Inelastic Delocalization Detector Geometry

2.3. Specimen Requirements 2.4. Data Processing 2.4.1. Conventional Background Fitting and Subtraction 2.4.2. More Advanced Data-Processing Techniques

3. Theoretical Aspects 3.1. The Role of Theory 3.2. General Theory of Inelastic High-Energy Electron Scattering 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6.

Preliminaries Yoshioka’s Equations Incoherent State Formalism Freedom in the Theory Relation to Density-Matrix Formalisms Paraxial Approximation to Yoshioka’s Equations

3.3. Atomic Inner-Shell Excitation by a Focused Electron Beam 3.3.1. 3.3.2. 3.3.3. 3.3.4. 3.3.5.

Transition Matrix Elements for Atomic Inner-Shell Excitation Projected Matrix Elements Chemical Imaging of Single Atoms Validity of the Dipole Approximation EDX Mapping and the Object-Function Approach

3.4. Combined Inner-Shell Excitation and Dynamical Elastic Scattering 3.4.1. Propagators in the Single Inelastic Scattering Approximation 3.4.2. Single Channeling Versus Double Channeling 3.4.3. Numerical Implementation Using Multislice Advances in Imaging and Electron Physics, Volume 175 ISSN 1076-5670, http://dx.doi.org/10.1016/B978-0-12-407670-9.00003-2

Ó 2013 Elsevier Inc. All rights reserved.

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4. Selected Applications 4.1. Chemical Mapping of Crystalline Compounds 4.2. Chemical Mapping at Solid-Solid Interfaces 4.3. ELNES Mapping 4.4. EDX Mapping 5. Concluding Remarks Acknowledgments References

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1. INTRODUCTION In the years since its feasibility was demonstrated independently by Batson (1993), Browning et al. (1993), and Muller et al. (1993), atomicresolution core-level spectroscopy in the scanning transmission electron microscope (STEM) has emerged as an exciting new tool for the advanced characterization of nanomaterials, materials defects, and interfaces (Okunishi et al., 2006; Bosman et al., 2007; Kimoto et al., 2007; Muller et al., 2008; Botton et al., 2010). The technique uses an atomicsized beam of high-energy electrons that is passed through the material before being dispersed by an energy-loss spectrometer. The positions of chemical elements in the material can be mapped, for example, by monitoring their core-level spectral “edges” while the electron beam is scanned across the material. Compared with conventional atomic-resolution imaging techniques in the (S)TEM, the power of this technique lies in its ability to not only locate the atoms in materials, but also identify them and provide information on their electronic environments. This capability has already proven extremely powerful for the analysis of a diverse range of materials problems, such as bulk properties (e.g., Varela et al., 2009; Botton et al., 2010; Lazar et al., 2010; Tan et al., 2011; Mundy et al., 2012; Turner et al., 2012), interfaces and layered compounds (e.g., Muller et al., 2008; Botton et al., 2010; Colliex et al., 2010; Fitting Kourkoutis et al., 2010; Garcia-Barriocanal et al., 2010; Zhu et al., 2011), point defects and clusters (e.g., Kaiser et al., 2002; Varela et al., 2004), nanoparticles (e.g., Turner et al., 2011; Xin et al., 2012b), and the range of applications appears set for continued growth for the foreseeable future. This chapter presents an account of recent advances, both practical and theoretical, that have enabled atomic-resolution core-level spectroscopy in

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the STEM to become a practical tool for characterizing materials at the atomic scale. We point out at the outset that the discussion here is mainly applicable to core-level spectroscopy using electron energy-loss spectroscopy (EELS), as is assumed unless otherwise stated. However, some consideration is also given to core-level spectroscopy using energy-dispersive X-ray (EDX) spectroscopy, which has recently been demonstrated to yield chemical maps at atomic resolution. Section 2 concerns practical aspects of the technique. An account of the significant advances in STEM instrumentation over the past decade or so is given. Some of these advances, such as the advent of aberration correctors, have literally transformed the field of atomic-resolution electron microscopy, though here we discuss their consequences only in the context of the technique in question. We also review other pertinent practical issues, e.g., data processing, that in many ways are equally important in enhancing the robustness of the technique. Section 3 concerns theoretical aspects and data interpretation. We present a theoretical formulation that enables a detailed understanding of the issues that can arise in the interpretation of chemical signals at the atomic scale. The theoretical development is presented in some detail, including an account of a general theory of inelastic electron scattering. We have striven to present a theoretical development accessible to readers with some understanding of quantum mechanics, but not necessarily an understanding of how quantum mechanics is applied in the field of (S)TEM. To this end, a serious attempt is made to convey, as often as possible, how the mathematical expressions and theoretical nomenclature relate to the experiments. During the development, particular attention is paid to pointing out the various approximations or assumptions that are often taken for granted in the literature. We also point out the connections between the theoretical formulation adopted here and those used by others. Section 4 discusses some selected applications of atomic-resolution corelevel spectroscopy in a variety of materials contexts. The examples are chosen to illustrate both the power of the technique for materials analysis and to draw attention to some of the issues arising in data interpretation and the understanding that can be gained by combining experiments with simulations. We also note the connection between this work and the several relevant reviews that have appeared in recent years (e.g., Egerton, 2009; Muller

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2009; Kociak et al., 2011; Varela et al., 2011; Botton 2012). Regarding practical aspects, the primary aim here is to provide a fairly concise, selfcontained account of the aspects that have directly enhanced the feasibility and robustness of the present technique. For more detailed discussions of STEM instrumentation, for example, readers are referred to the works cited and references therein. Regarding EELS, here we discuss only the aspects that are directly relevant to the present techniquednamely, atomic innershell excitation. Again, readers seeking an overview of EELS techniques are referred to the cited works.

2. PRACTICAL ASPECTS 2.1. Experimental Setup Experimentally, the main challenge for atomic-resolution core-level spectroscopy in the STEM arises from the weakness of the chemical signals, which typically consist of only 106 to 103 of the scattered electrons. In recent years, however, several advances in STEM instrumentation have allowed the technique to become a practical tool for analyzing materials. These advances include the advent of aberration correctors, improved spectrometer design, brighter electron sources, and better microscope stability, as detailed further below. The experimental setup uses a STEM equipped with a high-brightness (cold or Schottky-type field emission) source, an aberration corrector, and an electron energy-loss spectrometer (Figure 1). The instrument’s probeforming lenses and aberration corrector focus the high-energy (about 100 to 300 keV) electrons into an area of atomic dimensions at the specimen plane. The atomic-sized beam is raster-scanned across the specimen by means of deflector coils (not shown), and for each position of the beam the energy-loss spectrum is recorded, giving rise to a spectrum image ( Jeanguillaume and Colliex, 1989). A spectrum-image dataset can be twoor three-dimensional (3D), consisting of energy-loss spectra for beam positions that span either a line profile or an image, respectively. For chemical mapping, the energy-loss signal corresponding to the excitation of core electrons in a particular atomic species is extracted from each spectrum, resulting in an atomic-resolution map of that species. For chemical mapping via EDX, it is the flux of characteristic X-rays that is monitored as the beam is scanned. Typically, the instrument is also

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High-Brightness Source Beam-Defining Aperture Aberration Corrector Probe-Forming Lens(es) α

X-Ray Detector Specimen

t β

Post-Specimen Lens(es) Di raction Plane/ ADF Detector

Energy-Loss Spectrometer Figure 1 Schematic of the STEM setup used for atomic-resolution core-level spectroscopy, showing the major electron-optical elements (blue) and various detectors (orange). See the color plate.

equipped with a high-angle annular dark-field (ADF) detector, which enables an ADF image to be recorded simultaneously with a spectrum image (or EDX map). The often-direct relationship between the contrast in atomic-resolution ADF images and the specimen structure makes ADF images useful for determining the beam position, greatly facilitating the interpretation of the spectroscopic data.

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As mentioned above, in decades past the major factor limiting the practicality of atomic-resolution core-level spectroscopy in the STEM was the weakness of the chemical signals. Essentially, the small scattering cross sections involved meant that it was extremely challenging to record an adequate signal within a dwell time short enough to avoid beam and/or specimen drift at the atomic scale. In recent years, however, several advances in STEM instrumentation have allowed the technique to become a practical reality. The most notable advance is the advent of aberration correctors (Haider et al., 1998; Krivanek et al., 1999, 2003, 2008; M€ uller et al., 2006), which compensate for the adverse effects of third-order spherical and other inherent aberrations in the solenoidal lenses used to focus the electron beam onto the sample. This advance has not only allowed the formation of smaller electron beams but has also given rise to beams with 5 to 10 times more current at the specimen plane by virtue of the larger beam-defining apertures that can be used. Specimen damage aside, such improvements translate directly into stronger chemical signals (Okunishi et al., 2006; Bosman et al., 2007). Enhanced practicality has also resulted from significant advances in the design and operation of energy-loss spectrometers (Gubbens et al., 2010), enabling the use of larger collection angles while maintaining adequate energy resolution, faster and more efficient data acquisition, and greater operational flexibility. In the case of dedicated STEMs, important improvements have also occurred in column design (Krivanek et al., 2008), where post-specimen coupling lenses allow spectrometer collection angles approaching 100 mrad for a 100 kV instrument, ensuring that the amount of “wasted” chemical signal is minimized (Muller et al., 2008). In modern TEM/STEMs, recent post-specimen lens series have allowed similarly large (or even larger) collection angles to be achieved (Botton et al., 2010), greatly enhancing the practicality of the technique on such instruments. Recently, high-brightness Schottky-type field emission gun (FEG) sources have become available (Kisielowski et al., 2008). The brightness of these sources rivals that of current cold FEGs (which have a brightness of w2  109 A/cm2/sr at 100 kV), and hence significantly improves the practicality of atomic-resolution core-level spectroscopy with respect to conventional Schottky FEGs. Finally, improvements in instrumental (electronic and mechanical) and environmental (thermal, mechanical, and electromagnetic) stability have

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also played a significant role in increasing the usable dwell time of the beam while retaining atomic resolution.

2.2. Factors Determining Spatial Resolution Spatial resolution is one of the most important practical aspects of atomicresolution chemical mapping. Issues of specimen damage and signal-to-noise aside, the spatial resolution is determined by four factors: (1) the size and nature of the focused electron beam, (2) the channeling of the beam in the sample caused by dynamical scattering, (3) the inherent delocalization of the atomic-excitation events, and (4) the detector geometry. Of these, only factors (1) and (4) are controllable by instrumentation, but (2) and (3) are also discussed here for continuity. 2.2.1. Beam Size Achieving atomic resolution in chemical mapping requires an atomic-sized beam, which is greatly assisted by aberration correction. When the aberrations of the probe-forming lenses can be made sufficiently small (e.g., to satisfy the requirements of so-called aberration-free imaging [Zach, 2009]), then the two major factors that govern the size of the electron beam at focus are its convergence angle and the size and shape of the effective source distribution. An atomic-sized beam is produced by allowing a highly coherent electron wave field to be brought accurately to focus by the probe-forming lenses. The convergence semi-angle (a) of the beam at focus is directly proportional to the size of the beam-defining aperture (see Figure 1). According to the diffraction limit (e.g., Erni, 2010), achieving a 0.1 nm beam in a 100 kV instrument, for example, requires a convergence semiangle of 22 mrad, which is fairly large by STEM standards. The wave fronts of the converging electron wave field must be close to spherical across the range of angles admitted by the aperture. These wave fronts are generally distorted by the aberrations of the probe-forming lenses, hence the need for aberration correction. The other major factor affecting the beam size at focus is the so-called effective source distribution. The importance of the effective source for atomic-resolution ADF imaging has been emphasized in several recent works (LeBeau et al., 2008; Dwyer et al., 2008a, 2010, 2012; Maunders et al., 2011), with many of the considerations extending directly to atomic-resolution chemical mapping (Dwyer et al., 2010; Xin et al., 2011). The concept of an effective source (Hopkins, 1951) encapsulates any factors deleterious to

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the spatial coherence of the wave field, which includes the finite size of the (virtual) electron source and any instabilities that induce an uncertainty in the beam’s position at the specimen (i.e., beam and/or specimen instabilities). Such factors lead directly to a degradation of the spatial resolution with respect to that expected for an ideal (point-like) effective source. 2.2.2. Beam Channeling Even with an intense atomic-sized electron beam, atomic-resolution is not guaranteed, and it is usually greatly assisted if the specimen is a zone-axisaligned crystal. This occurs for two reasons: First, zone-axis-aligned crystals often consist of columns of atoms that remain well separated in projection (unlike the atoms in an amorphous material, for example). Second, when the beam is focused onto an atomic column, dynamical scattering causes the beam intensity to become confined i.e., channeled, along the column and, in the case of specimens with low or moderate atomic numbers, retain an atomic size laterally for a greater distance along the column than it would in free space (Hillyard et al., 1993). Loosely speaking, such channeling is a refocusing of the electron beam caused by its electrostatic attraction to the nuclei in the atomic column (Van Dyck and Op de Beeck, 1996); in this sense, the specimen could be considered as forming a supplementary part of the probe-forming system (somewhat analogous to the situation in atom probe microscopy, for example). In the case of atomic columns composed of high atomic numbers, the dynamical scattering of the beam is so strong that the confinement of intensity persists only over a short segment of the column before the beam becomes dechannelled, implying that the chemical signal can be dominated by that part of the specimen in the vicinity of the focused beam. If the specimen is not aligned close to a zone axis, or if it is non-crystalline, then the beam will retain atomic dimensions for only a limited distance along the optic axis before spreading out again (comparable to its behavior in free space). In this case, the atoms of interest must constitute a very thin object to achieve atomic resolution. 2.2.3. Inelastic Delocalization Delocalization in STEM core-level spectroscopy refers to fact that a beam electron can induce an atomic excitation even when it passes the atom at a distance that lies beyond the spatial extent of the atomic electron’s initial orbital. The maximum distance at which excitations remain probable decreases with increasing energy loss. Hence, chemical maps extracted at higher energy losses tend to exhibit better spatial resolution and contrast

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 153

(with the caveat of a generally weaker signal). This effect has been demonstrated in the works of Kimoto et al. (2007), Botton et al. (2010), and Fitting Kourkoutis et al. (2010), for example, where the same atomic species was mapped using two edges at different energy losses. On the other hand, delocalization means that is generally not possible to obtain atomic-resolution chemical maps from the low-loss region of the spectrum, and any atomic-scale contrast from that region is likely due to the preservation of elastic contrast, though an intriguing counterexample has been reported by Lazar et al. (2010). For typical beam energies and interatomic spacings, the threshold energy loss at which atomic-resolution chemical mapping becomes possible is of the order 100 eV. In fact, the relationship between spatial resolution and energy loss is a general phenomenon of inelastic electron scattering. This behavior has been rationalized using a variety of (related) arguments, including energytime uncertainty (Howie, 1979), momentum transfer (Ritchie and Howie, 1988), classical theory (Muller and Silcox, 1995), and optical arguments (Egerton, 2009). Here we provide a relatively simple explanation based on quantum electrodynamics (QED). In QED, charged particles interact by exchanging so-called virtual photons. In the case of atomic excitation in the (S)TEM, a virtual photon transfers energy (and momentum) from the beam electron to an atomic electron. The mathematical function describing the propagation of the virtual photon over a distance jxj is DðE; xÞ ¼

eiEjxj=Zc ; 4pjxj

(1)

where E is the energy transfer and we have legitimately regarded the photon as a so-called scalar particle. For a probable excitation, we require the phase in Eq. (1) to be of the order unity or less1 (i.e., jxj(Zc=E). Hence if the energy loss is increased, the distance with which the excitation can take place decreases, potentially giving rise to a chemical map with better spatial resolution. However, it is important to bear in mind that, while providing useful estimates, arguments such as the one given here or those cited above should not be expected to provide quantitative predictions of the resolution (for example, the argument given here turns out to supply an upper limit for the delocalization). The reason is that the resolution ultimately involves specifics of the scattering process that are not incorporated by such arguments. This point is demonstrated in Section 3.3.2. 1 The contributions of phases greater than order unity tend to cancel out.

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2.2.4. Detector Geometry The detector geometry refers to the size and shape of the spectrometer entrance aperture and its position with respect to the optic axis. Although some novel detector geometries have been explored as a means of obtaining better spatial resolution (Rafferty and Pennycook, 1999; Ruben et al., 2011), here we discuss only the conventional case of a circular aperture centered on the optic axis. For chemical mapping, it is generally accepted that using a collection angle significantly larger than the probe convergence angle has several advantages. First, larger collection angles increase the detection efficiency. Second, they often minimize unwanted artifacts that can be observed with smaller collection angles. Common artifacts2 are intensity minima located at the atomic sites (Kohl and Rose, 1985; Allen et al., 2003a; Dwyer, 2005b; Bosman et al., 2007; Oxley et al., 2007; D’Alfonso et al., 2008; Lazar et al., 2010), often called “volcanos” (D’Alfonso et al., 2008), which confound the interpretation of chemical maps. The reduction of such artefacts with increasing aperture size has been demonstrated experimentally by Lazar et al., 2010. Moreover, the reduction of artefacts is often accompanied by an increase in the spatial resolution (Cosgriff et al., 2005; Dwyer, 2005a). The latter point, which is demonstrated in Figure 2, can be understood by appreciating that, in the limit of an infinitely large collection angle, atomic excitation can be described within an incoherent (Ritchie and Howie, 1988; Muller and Silcox, 1995; Dwyer, 2005a) or ‘local’ (Allen and Josefsson, 1995, 1996) model, whereby the interaction volume for atomic excitation takes on a minimum size (this point is discussed further in Section 3.3.5). In line with the points above, larger collection angles also tend to reduce the fraction of chemical signal coming from atomic columns adjacent to the nominal probe position, again resulting in a more localized signal (Dwyer, 2005a,b).

2.3. Specimen Requirements The specimen requirements for obtaining good-quality atomic-resolution chemical maps are typically more stringent than for other STEM techniques. First, we have the specimen requirements to fulfill the criteria of conventional core-level EELSdnamely, that the specimen is thin enough to obtain an adequate jump ratio and minimize the complications of multiple inelastic 2 Artifacts is perhaps not the right word since the effects are genuine physical effects arising

from the scattering geometry. However, its use is intended to imply that the signals do not relate directly to the atomic structure of the sample.

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 155

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0.8 0.6 0.4 0.2 0.0 45 40 35 30 25 20

0.8 0.6 0.4 0.2 0.0 45 40

35 30 25 20

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Figure 2 “Volcano” effects in atomic-resolution chemical mapping. (a) Calculated chemical signal from the L1 edge of a single titanium atom as a function of the beam position and the collection semi-angle. (b) Analogous data for the L2;3 edge. The calculations assume a 100 keV aberration-free electron beam with a 37.5 mrad convergence semi-angle, and energy windows 40 eV wide centered 20 eV beyond threshold. The titanium atom is located at the position origin. Note how both the degree of the volcano effect and the apparent size of the atom decrease with increasing collection semi-angle. (From D’Alfonso et al., 2008).

scattering (Egerton, 1996). These requirements are generally met for thicknesses (l=4, where l is the inelastic mean-free path (typically w100 nm for conventional beam energies). Notwithstanding this, good-quality maps have been demonstrated even for thicknesses approaching l (Xin et al., 2011). Two other pertinent considerations for atomic-scale core-level spectroscopy are beam-induced charging and damage of the specimen. The latter effect, in particular, can be greatly exacerbated by the longer dwell times generally required for core-level spectroscopy than for other STEM techniques, and it often imposes the ultimate practical limitation on what can be achieved in terms of the effective spatial resolution and/or sensitivity of the technique. Beam-induced damage often can be significantly reduced by changing to a different beam energy, usually a lower beam energy to avoid so-called knock-on damage (e.g., Botton et al., 2010; Krivanek et al., 2010), which is a feat made practical by the range of beam energies (< 80–300 keV) available in modern TEM/STEMs.

2.4. Data Processing The chemical edges in an electron energy-loss spectrum invariably reside on a background signal that arises from alternative inelastic scattering events, such as plasmon-loss events, single-electron excitations, lower-lying chemical edges, etc. Hence the chemical signal must be isolated by means of data processing.

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Conventionally, this is done by first subtracting the background and then, in the case of chemical mapping, integrating the resulting spectrum over an energyloss window of the order 10 eV wide. Often, the most difficult processing step is to obtain a reliable subtraction of the background, and since the qualitative and quantitative content of a map can be sensitive to the extrapolation procedure and position of the integration window (Dudeck et al., 2012), the potential misinterpretation of data caused by an inaccurate background subtraction is a challenging aspect of core-level spectroscopy at atomic resolution. Many of the difficulties are well described in the recent works by Cueva et al., (2012) and Dudeck et al., (2012). Here we provide a brief summary. 2.4.1. Conventional Background Fitting and Subtraction The conventional approach to background subtraction uses a power-law fit to the pre-edge spectrum; that is, the pre-edge spectrum intensity is modeled as r for some exponent r, and this functional form is used to extrapolate IfEloss and then subtract the background from the chemical edge (Egerton, 1996). This procedure is performed for each pixel in the map. The advantage of this conventional approach is its simplicity, and the results can be adequate as long as the background estimation is not hampered by either noise or the presence of pre-edge features that can cause a significant departure from the assumed functional form. In particular, the noise requirement can be very difficult to satisfy in atomic-resolution mapping, motivating the development of more advanced and reliable processing techniques. 2.4.2. More Advanced Data-Processing Techniques More advanced techniques for extracting atomic-resolution chemical signals include statistical methods, such as principal component analysis (PCA) (Bosman et al., 2006, 2007; Varela et al., 2009; Watanabe et al., 2009; Dudeck et al., 2012), and a priori techniques, such as spectrum modeling (Verbeeck and Van Aert, 2004), least-squares fitting of reference spectra (Browning et al., 1993; Muller et al., 1999), linear combination of fixed power laws (Cueva et al., 2012), local averaging (Cueva et al., 2012), and periodic averaging (Varela et al., 2009; Botton et al., 2010). PCA, for example, extracts those components of a spectrum image that exhibit maximum correlation, which effectively improves the signal-tonoise ratio by eliminating uncorrelated noise (Bosman et al., 2006; Dudeck et al., 2012). Often, such components are of direct significance for chemical mapping, though this is not guaranteed (Mundy et al., 2012), particularly in the presence of interfaces or defects (Cueva et al., 2012). Perhaps the major

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 157

Figure 3 Comparison of background-fitting methods in atomic-resolution chemical mapping of a YBa2Cu3O7x/manganite interface (manganite at left of each image). “Power Law” designates the conventional power-law approach, “LCPL” designates a linear combination of (fixed) power laws, and “LBA” designates local averaging of the background. (a) Individual chemical maps. (b) Color-coded composite maps showing copper (red), manganese (blue), and oxygen (red). (From Cueva et al., 2012). See the color plate

advantage of statistical techniques such as PCA is that they yield results that are essentially unbiased. On the other hand, a priori methods can yield very reliable results if care is taken. Local averaging methods, for example, adopt the premise that the spatial variation of the background is often relatively slow compared with the pixel size. Hence the signal-to-noise ratio of the background, and thus the background estimation and subtraction, can be improved by local-spatial averaging (Cueva et al., 2012) (Figure 3). Alternatively, if the specimen exhibits one- or two-dimensional periodicity lateral to the beam direction, periodic averaging offers a conceptually simple and robust approach for improving the signal-tonoise ratio of the entire spectrum image before any attempt is made to extract the signals (Varela et al., 2009; Botton et al., 2010).

3. THEORETICAL ASPECTS 3.1. The Role of Theory Since transmission electron microscopy is usually viewed as an experimental field, it is worthwhile to emphasize the important role of theory in the

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interpretation of chemical signals at atomic resolution. Essentially, the necessity of theory can be attributed to the strong interaction of electrons with matter, which, on the one hand, provides the sensitivity required to analyze nanoscale volumes of material and, on the other hand, causes significant multiple scattering of the beam electrons that can give rise to a nontrivial relationship between the specimen structure and the measured signals. Other factors, such as inelastic delocalization and coherent effects arising from the detector geometry, can also contribute to this nontrivial relationship. Therefore, requisite to a solid interpretation of atomic-resolution core-level spectroscopy experiments are reliable and tractable theories with which to conduct simulations. Furthermore, if there is sufficient a priori information about the specimen, then accurate theories make it possible to extract quantitative information about the specimen that would otherwise remain inaccessible. The theoretical development below is presented in some detail. As mentioned in the introduction, an attempt is made to outline the various approximations that are often taken for granted in the recent literature, and to point out the connections between the different theoretical formulations that exist. We begin by considering a general theory of inelastic electron scattering before restricting ourselves once again to the inelastic scattering processes relevant to core-level spectroscopy. To aid readers, the theory of atomic-resolution core-level spectroscopy is presented in two stages, first in relation to chemical imaging of single atoms where elastic scattering can be neglected (Section 3.3) and then in relation to chemical mapping of materials where elastic scattering must be included (Section 3.4). Figure 4 serves as a guide to the various stages of approximation.

3.2. General Theory of Inelastic High-Energy Electron Scattering 3.2.1. Preliminaries To consider inelastic electron scattering, it is necessary to describe the electron as a charged particle, moving not merely in some fixed external potential (as in theories of elastic scattering) but under the influence of a target3, which is itself a quantum object capable of quantum transitions. Hence we consider a physical system composed of a beam electron and 3 While the target is usually called the specimen in the field of electron microscopy, we adopt

here the terminology of scattering theory.

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 159

(a)

Von Neumann Equation (beam e− + target in mixed state)

Schrodinger ¨ Equation (Sec. 3.2.1) (beam e− + target in pure state, no thermal scattering) Yoshioka’s Equations (Sec. 3.2.2) (beam e− only, multiple inelastic channels) Paraxial Approx. to Yoshioka’s Eqns. (Sec. 3.2.6) (fast beam e− , multiple inelastic channels) Single Inelastic Scattering Approx. (Sec. 3.4.1) (atomic inner-shell excitations, multiple inelastic channels)

Single Channeling Approx. (Sec. 3.4.3) (no elastic scattering in inelastic channels)

Object-Function Approach (Sec. 3.3.5) (EDX, effectively no outgoing channels)

(b) Solid-State Theory (many e− + nuclei, ELNES)

Atomic Structure Theory (Sec. 3.3.1) (many e− , multiplet effects, no ELNES from solid-state effects)

Dipole Approx. (Sec. 3.3.4) (small scattering angles)

Single Electron Theory (Secs. 3.3.1 and 3.3.2) (single e− only, no multiplet effects)

Figure 4 Hierarchy of approximations used in the theory of atomic-resolution chemical mapping. (a) Approximations in inelastic electron scattering theory. (b) Approximations in atomic inner-shell excitation theory. Note that neither the Von Neumann equation nor the solid-state theory are discussed in the present work.

a target, and the system as a whole is subject to the laws of quantum mechanics. The beam electron-target system is assumed to be closeddthat is, isolated from the rest of the world. Hence the system has a constant energy E, and, in

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the Schr€ odinger picture of quantum mechanics (Messiah, 1961), its state vector contains a trivial oscillatory time dependence that can be factored out. This gives rise to a time-independent formalism, where the system’s state vector jJi satisfies the time-independent Schr€ odinger equation
2  p (2) þ HT þ V jJi ¼ EjJi: 2m In this equation, p2 =2m represents the kinetic energy of the beam electron, HT is the Hamiltonian for the target that depends only on the positions x1 ; .; xN and momenta p1 ; .; pN of the N constituent particles (electrons and nuclei) of the target, and V represents the Coulomb interaction between the beam electron and each charged particle in the target. Relativistic effects are included here to sufficient accuracy by using the relativistic values for the beam electron’s mass and wavelength. For simplicity, magnetic interactions between the beam electron and target particles have not been included, although it is certainly possible to do so within the current formalism. On the other hand, we should mention that, strictly speaking, the assumption of a closed beam electron-target system precludes any description of thermaldiffuse scattering at, for example, room temperature, because that would require the target to be described by a so-called mixed state (see Figure 4). In practice, however, the effects of this often-important form of scattering can be incorporated at a later stage. The ðN þ 1Þ-particle Hilbert space H to which the system state jJi belongs can be considered as a tensor product He 5HT of the singleparticle Hilbert space He of the beam electron and the N-particle Hilbert space HT of the target. This approach allows the identity operator on H to be written as a direct product of the identity operators on He and HT : XX b 1T ¼ ðjme i5jaiÞðhaj5hme jÞ; 1 ¼ b 1 e 5b (3) me

a

where jme i and jai are complete orthonormal bases in He and HT , respectively, so that the states jme i5jai span H. In particular, the states jai are taken to be those of the target in the absence of the beam electrond these states satisfy HT jai ¼ Ea jai, where the eigenvalue Ea is an allowed energy level of the target. Using the form of the identity operator b 1 given above, we see that the system state jJi can be expanded in the form XX ðjme i5jaiÞðhaj5hme jÞjJi: (4) jJi ¼ me

a

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 161

Figure 5 Graphical representation of the ingoing and outgoing channels. The diagram shows one ingoing channel corresponding to the target in its ground state (elastic channel) and three outgoing channels (one elastic and two inelastic).

Now, by defining the new beam electron states on He X jja i ¼ jme iðhaj5hme jÞjJi;

(5)

me

the system state can be written in the form (Taylor, 1972) X jJi ¼ jja i5jai:

(6)

a

As shown below, the new beam electron states jja i have a special significance in that they are each coupled to a unique target state jai. It should be noted that the beam electron states jja i are normalized in the sense that X (7) hja jja i ¼ 1; a

such that for a specific value of a we have hja jja i  1. This normalization corresponds to the interpretation that the beam electron must be “somewhere” among the various states, or channels, jja i. This normalization is consistent with recent quantitative experiments in the STEM that measure the image intensity as a fraction of the incident beam intensity (LeBeau and Stemmer, 2008; LeBeau et al., 2008, 2009, 2010; Rosenauer et al., 2009; Dwyer et al., 2011, 2012; Kim et al., 2011; Xin et al., 2012a, c). Figure 5 shows a schematic representation of the ingoing and outgoing channels described by the theory. 3.2.2. Yoshioka’s Equations Substituting the state vector for the system into the Schr€ odinger equation, multiplying on the left by haj, and making use of the orthonormality of target states, we obtain the celebrated Yoshioka’s equations for inelastic electron scattering (Yoshioka, 1957):
2  X   p (8) þ Vaa  ðE  Ea Þ jja i ¼  Vab jb ; 2m bðsaÞ

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where Vab is a matrix element of the Coulomb interaction Vab ¼ hajV jbi:

(9)

The physical interpretation of Yoshioka’s equations is as follows. The homogeneous equation, which is obtained by setting the terms on the right-hand side equal to zero, describes the elastic scattering of a beam electron with energy E  Ea . Elastic scattering results from the electrostatic interaction P Vaa . Inelastic scattering is represented by the inhomogeneous terms bðsaÞ Vab jjb i, which describe inelastic scattering processes of the form jjb i/jja i (accompanied by a target transition jbi/jai). The inhomogeneous terms can be regarded as sources for the state jja i. Since the number of target states is, in general, infinite, Yoshioka’s equations form an infinite set of coupled differential equations. However, for practical purposes, it is sufficient to consider only a finite number n þ 1 of target states. While this, strictly speaking, precludes target states belonging to a continuous range of energy eigenvalues, such as the continua relevant to core-level spectroscopy, in practice a set of continuum states can be approximated to any desired accuracy by a finite number of discrete states. Thus, a finite subset of target states, labeled in order of increasing energy j0i; j1i; .; jni, where j0i denotes the target’s ground state, will be assumed below. By defining the following matrices, which are specified by subscript “M”, 1 1 0 0 0 1 k0 / 0 V00 / V0n jj0 i C C B B B C C B B C 1 « C jjM i ¼ B A; @ « A; kM ¼ @ « 1 « A; VM ¼ @ « 0 / kn Vn0 / Vnn jjn i (10) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ka ¼ 2mðE  Ea Þ=Z, Yoshioka’s equations can be expressed succinctly as the single matrix equation (Wang, 1989, 1990):
2  p Z2 k2M þ VM jjM i ¼ jjM i: (11) 2m 2m This equation has the appearance of a time-independent Schr€ odinger equation, and in this sense, Yoshioka’s equations may be regarded as a multi-dimensional extension of the time-independent Schr€ odinger

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 163

equation governing the elastic scattering of a beam electron by a fixed electrostatic field. 3.2.3. Incoherent State Formalism When the system’s state vector jJi is considered as a tensor, the summation over a indicates that it is not a simple tensor. In other words, as emphasized by Verbeeck et al., (2009), jJi represents an entangled state, and it is this property of jJi that is associated with inelastic scattering processes. To explore this further, we first note that, by virtue of the Hermiticity of the target Hamiltonian HT , target states belonging to different energy eigenvalues are orthogonal. While it can and does happen that some target states are degenerate, the degeneracy almost always arises from a symmetry of the target4, in which case any set of symmetrically-degenerate target states can be chosen to be orthogonal. Now, considering the expectation value for observing the beam electron irrespective of any target particle, where the observable is represented by the b e on He , we obtain Hermitian operator O X    X      b e 5b b e jb hajbi ¼ b e ja ; (12) ja  O ja  O 1 T jJi ¼ hJj O a

a;b

where the orthonormality of the target states has been used. As an example, b e might represent the energy-/momentum-selecting process the operator O involved in EELS. Also of particular interest for experiments in the (S)TEM is the probability density, or intensity, of observing the beam electron at b e ¼ jxihxj: a position x, in which case O X X IðxÞ ¼ (13) hja jxihxjja i ¼ jja ðxÞj2 ; a

a

where ja ðxÞ is the wave function for the state jja i. The expectation value Eq. (13), or, more generally, Eq. (13), applies to the majority of experiments conducted in a (S)TEMdthat is, the target particles are not observed. In such cases, the different beam electron states jja i are regarded as incoherent. This point of view will be referred to as the incoherent state formalism. Contrary to popular belief, however, this incoherence is not a consequence of the beam electron states having different energies (which is simply 4 Degeneracy can also be accidental (i.e., not the result of symmetry), but such cases are rare

enough to be neglected.

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not true for beam electron states coupled to degenerate target states). Rather, the incoherence arises because of (1) the orthogonality of the target states, and (2) failure to observe the target particles in the experiment. 3.2.4. Freedom in the Theory It was mentioned above that degeneracy of the target states is assumed to arise from symmetry. In fact, the situation can be stated precisely (see, for example, Tinkham, 1964): If the Hamiltonian of a given system is invariant under the operations of a symmetry group G, then there exists an la -fold degenerate eigenvalue Ea of the Hamiltonian, where la is the dimensionality of the ath irreducible representation of G. The set of degenerate states ja1 i; ja2 i; .; jala i belonging to the eigenvalue Ea forms a basis for the ath irreducible representation of G. These states can be chosen to be orthogonal, in which case the irreducible representation is unitary. For our purposes, the key point is that, even if we demand that the states 1 ja i; ja2 i; .; jala i are orthogonal, they are not determined uniquely. Specifically, given a choice of orthogonal states, any unitary transformation applied to them will result in a new orthogonal set (forming a basis for a new unitary representation of G). This freedom can be applied to simulations of inelastic electron scattering, and particularly for simulations of atomicresolution core-level spectroscopy, to choose degenerate target states that are as efficient as possible (this point is discussed further in Section 3.3.1). 3.2.5. Relation to Density-Matrix Formalisms We divert briefly to demonstrate the formal equivalence of the above formulation of inelastic scattering with formulations based on the density matrix, the latter pioneered in the field of (S)TEM by Dudarev et al., (1993), and extended and used extensively by Schattschneider et al., (1999, 2000, 2009), Schattschneider and Jouffrey (2003), and others. The power of density-matrix formulations lies in their ability to elegantly describe the partial coherence of the electron wave field, which can arise from either inelastic scattering or (classical) uncertainty regarding the initial state of the beam electron and/or the target, within one mathematical quantitydthe density matrix. On the other hand, the price to be paid for such elegance is that the density matrix is a function of two coordinates, which can make both the interpretation and the numerical implementation of the formalism more challenging than the incoherent-state formalism (where the inelastic waves are a function of one coordinate).

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 165

In this work, we neglect any (classical) uncertainty in the initial state of the beam electron and/or the target. Because of this assumption, the state of the beam electron-target system is pure, and its density operator is given by X ðjja i5jaiÞðha0 j5hja0 jÞ: r ¼ jJihJj ¼ (14) a;a0

On the other hand, the state of the beam electron, irrespective of the target, can be represented by the reduced density operator re ¼ trT r   P P   ¼ jb 5jbi hb0 j5 jb0  jai haj a

b;b0

P ¼ jja ihja j;

(15)

a

where trT indicates a trace over the target states, and the summation over a in the last line indicates a mixed state. If the experiment consists of observing the beam electron irrespective of the target particles, then the expected outcome of the experiment is given by X     b e ¼ tre re O be ¼ b e ja ; O ja  O (16) a

in agreement with the incoherent state formalism. 3.2.6. Paraxial Approximation to Yoshioka’s Equations Since the scattering of beam electrons in a (S)TEM is predominantly paraxialdtheir direction of motion lies within a narrow cone about the optic axisdit is desirable, for practical reasons, to obtain the paraxial approximation to Yoshioka’s equations. For this purpose, it is convenient to work in the position representation of the beam electron, so that jM ðxÞ ¼ hxjjM i now denotes a column matrix of beam electron wave functions, and the matrix VM ðxÞ is now a function of the beam electron position x. Taking the optic axis to be the z-axis, the paraxial approximation to Yoshioka’s equations is derived by making the replacement jM ðxÞ/jM ðxÞeikM z , where eikM z is a diagonal matrix of rapidly varying plane-wave components, and the new column matrix jM ðxÞ varies slowly with z. The assumption that jM varies slowly with z is valid provided that (1) the direction of the beam electron’s motion nearly coincides with the þz-axis (i.e., forwardparaxial motion), and (2) the beam electron’s kinetic energy is significantly larger than the magnitude of the interaction energy. Both assumptions are

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valid for the majority of electron scattering in a (S)TEM. Making the above replacement in Yoshioka’s equations, and neglecting terms containing v2z jM on account of jM varying slowly with z, we can obtain (bold symbols denote 2D vectors transverse to the optic axis), 
1 2 ikM z ikM z ðx; jM ðx; zÞ; (17) zÞe ivz jM ðx; zÞ ¼  k1 V þ s e V M M 2 M 2 1 where sM ¼ mk1 M =Z is a diagonal matrix of interaction constants, and kM is the inverse of the matrix kM . The matrix equation (17) is the paraxial approximation to Yoshioka’s equations. The main advantage of the paraxial equation (17) is that it takes the form of a time-dependent Schr€ odinger equation (with z acting as time), and hence we can use what is essentially standard nonrelativistic propagator theory (see, for example, Bjorken and Drell, 1964) to obtain a formal solution for the beam electron wave functions at the target’s exit surface ðz ¼ tÞ in terms of those at the target’s entrance surface ðz ¼ 0Þ Z jM ðx; tÞ ¼ d2 x0 iGM ðx; t; x0 ; 0ÞjM ðx0 ; 0Þ; (18)

where the propagator matrix GM encompasses all possible scattering processes that can occur between the entrance and exit surfaces, with the only restriction being that the motion is always forward-paraxial. As is typical, we impose the boundary condition that the beam electron initially has a well-defined kinetic energy E and the target is initially in its ground state, which is defined to have energy E0 ¼ 0, so that: 0 B B B jM ðx; 0Þ ¼ B B @

j0 ðx; 0Þ 0 «

1 C C C C: C A

(19)

0 To link with the graphical representation of ingoing and outgoing channels in Figure 5, the boundary condition (19) applies on the left of that figure and the z parameter increases from left to right. Of course, the expression (18) does not represent an explicit solution because we do not yet know how to compute GM . However, as demonstrated in Sections 3.3.2 and 3.4.1, considerable simplification is achieved by

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 167

introducing the so-called single inelastic scattering approximation, which is valid for atomic-resolution chemical mapping.

3.3. Atomic Inner-Shell Excitation by a Focused Electron Beam Having outlined a general theory of inelastic electron scattering, we now focus on the inelastic scattering processes relevant to core-level spectroscopy: atomic inner-shell exicitations. As a stepping stone to a theory of atomicresolution core-level spectroscopy of materials, we first consider the simpler case where the target is a single atom. This allows us to introduce key quantities in the theorydnamely, the projected matrix elements for atomic inner-shell excitationdin a simpler context. 3.3.1. Transition Matrix Elements for Atomic Inner-Shell Excitation We adopt an isolated atom model of atomic inner-shell excitation, which neglects energy-loss near-edge structure (ELNES) effects that arise from electronic bonding (Leapman et al., 1980; Rez, 1989). For an excellent discussion on ELNES, including bonding effects, readers are referred to Radtke and Botton (2011). We assume that, (1) the isolated atom is initially in its ground (discrete energy) state j0i, and (2) the interaction with the beam electron induces the atom to make a transition to an excited (discrete or continuum energy) state jai containing a hole in an inner shell. We also assume that this interaction is sufficiently weak that we need to calculate the probability for such a transition only to the first order in the interaction potential. Hence we compute the matrix element for scattering as Va0 ðxÞ ¼

N X e2 1   j0i; haj   4pε0 j ¼ 1 x  xj

(20)

where x is the position of the beam electron and xj is the position of one of the N atomic electrons. Notwithstanding the fact that the initial and final states of the atom are antisymmetric under the permutation of any two atomic electrons (Slater, 1960), these electrons are indistinguishable, so that the matrix element reduces to Va0 ðxÞ ¼

Ne2 1 j0i; haj 4pε0 jx  x1 j

(21)

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which involves only the coordinate x1 of one of the atomic electrons. Note that expression (21) contains the electrostatic potential generated by the atomic electrons as a special case, which is obtained by setting jai ¼ j0i. The atomic states j0i and jai are assumed to be constructed according to the methods of self-consistent field theory in the single-configuration approximation (see Cowan, 1981). In this method, N-electron states jbi are first constructed from antisymmetrized products of one-electron spin orbitals. There are M such states jb1/M i, corresponding to the number of ways the N electrons can fill the spin orbitals consistent with the configuration. Then, a unitary transformation5 applied to the states jb1/M i produces the eigenstates ja1/M i of the atomic Hamiltonian HT . The energy eigenvalues associated with the configuration typically exhibit splitting representative of the so-called multiplet structure of the atom. For the present purposes, however, we make the simplifying assumption that the multiplet splitting is not resolved by the experiment. In atomicresolution core-level spectroscopy, this assumption is valid provided the chemical signal is extracted using an energy window that is wider than the multiplet splitting. For chemical mapping, 10 eV windows are common, so that this assumption is often satisfied in practice.6 We then need to consider only transitions between the simpler atomic states jb0 i and jbi. We obtain hbj

1 1 jb0 i ¼ hff1 f2 /gj jff01 f02 /gi; jx  x1 j jx  x1 j

(22)

where the f’s are one-electron spin-orbitals and the curly braces indicate an antisymmetrized product. Our final simplifying assumption concerning the atomic states is the socalled frozen-core approximation, whereby we assume that all spin-obitals not directly involved in the transition are identical in jb0 i and jbi. In this case, we finally obtain for the matrix element Vf;f0 ðxÞ ¼

e2 1 hfj jf i; 4pε0 jx  x1 j 0

(23)

5 Or, in practice, two unitary transformations, the first producing states in a pure coupling

scheme, such as LS coupling, and the second producing from these the actual eigenstates, which are in so-called intermediate coupling. 6 The assumption is not valid for the transition-element L and rare-earth M edges, for 2;3 4;5 example, where considerable splitting is observed. However, in such cases the splitting can usually be put in by “hand” at a later stage (see, for example, Rez, 1989).

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 169

where jf0 i and jfi are the spin orbitals directly involved in the transition. The matrix element given by (23) comprises a single-electron picture of atomic excitation (Manson, 1972; Saldin and Rez, 1987). In practice, the spin orbitals are extracted from self-consistent field calculations of atomic structures, such as the Hartree–Slater (e.g., Herman and Skillman, 1963) or Hartree–Fock schemes (e.g., Cowan, 1981). Furthermore, owing to the relatively small scattering angles experienced by the beam electron, electron spin is effectively a “spectator degree of freedom”, so that the spin components of the spin orbitals jf0 i and jfi have very little influence and can be omitted. We conclude this subsection by mentioning that the literature contains essentially two different (but ultimately equivalent) schools for computing the matrix elements in Eq. (23), corresponding to two different representations of the final atomic states jfi. These are the angular-momentum representation (Manson, 1972; Leapman et al., 1980; Saldin and Rez, 1987; Dwyer, 2005a,b) and the asymptotic-linear-momentum representation (Maslen and Rossouw, 1983, 1984; Rossouw and Maslen, 1984; Allen and Josefsson, 1995; Oxley and Allen, 2001). In the case of inner-shell excitations in solids (which are considered in Section 3.4), if certain elastic scattering events are neglected it is often feasible to perform the summation over final states in advance by use of a mixed-dynamic form factor (Kohl and Rose, 1985; Schattschneider et al., 2000) or closely-related quantity, in which case the choice of representation is not important. If, on the other hand, such an approximation is not made, as in the work of Dwyer et al., (2008b), then it is often impractical to perform this summation in advance, in which case each of the two representations has advantages and disadvantages depending on the specific situation. For example, in the case of energy losses near threshold, which is highly relevant to the present work, the angular-momentum representation offers greater efficiency, particularly for K- or L-shell excitations where it provides accurate results using just a small number of degenerate states per atom (Dwyer, 2005a). 3.3.2. Projected Matrix Elements Because it is confined to a cone of a small solid angle about the optic axis, the direction of motion of beam electrons allows us to transform the above matrix elements, which are a function of the 3D position of the beam electron, into functions of the beam electron’s two-dimensional position in a plane perpendicular to the optic axis. The resulting projected matrix elements are, in fact, exactly analogous to the projected atomic potentials

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used in theories of dynamical elastic scattering. The only essential difference is that the projected matrix elements are complex (as opposed to real) quantities that describe inelastic (as opposed to elastic) scattering. To obtain an expression for the projected matrix elements, we use the so-called small-angle approximation, whereby all terms containing secondor higher-order powers of the scattering angle are neglected. In terms of the paraxial Yoshioka’s equations (17), this amounts to neglecting the secondorder transverse derivative, in which case we obtain ivz jM ðx; zÞ ¼ sM eikM z VM ðx; zÞeikM z jM ðx; zÞ; the formal solution being 0 jM ðx; zÞ ¼ exp@  isM

Zz

(24)

1 dz0 eikM z VM ðx; z0 ÞeikM z AjM ðx; z0 Þ: (25) 0

0

z0

The exponential factor in Eq. (25) represents elastic and inelastic scattering to all orders (within the small-angle approximation). We assume that there is, at most, one inelastic event leading to an atomic inner-shell excitation, a valid assumption because the mean-free path for such events typically is several microns (Brydson, 2001). Moreover, in the present section, where we have assumed that the target is a single atom, we neglect any effects arising from elastic scattering. Hence, we can replace the exponential factor in (25) with its first-order Taylor expansion. Then, using the boundary condition (19), we can obtain for the ath component of the column matrix jM ja ðx; zÞ ¼ isa Va0 ðxÞj0 ðx; z0 Þ;

(26)

where we have defined the projected matrix element Zz 0 Va0 ðxÞ ¼ dz0 Va0 ðx; z0 Þeiqz z ;

(27)

z0

where qz is the change in the z-component of the beam electron’s momentum in the small-angle approximation. By taking z0 and z to define the positions of planes before and after the atom, respectively, Eqs. (26) and (27) enable us to calculate inelastic scattering arising from a given incident wave function j0 , such as the wave function of an incident beam electron in a STEM.

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 171

For atomic inner-shell excitation, the 3D matrix elements Vf;f0 ðxÞ given by Eq. (23) are sufficiently localized around the atom that we can legitimately extend the limits of integration to N. Hence the projected matrix elements for atomic inner-shell excitation are defined by the expression (Ritchie and Howie, 1988; Holbrook and Bird, 1995; Dwyer, 2005a,b) ZN Vf;f0 ðxÞ ¼

0

dz0 Vf;f0 ðx; z0 Þeiqz z :

(28)

N

As an example, Figure 6 shows projected matrix elements for the excitation of the Si-K shell, where the angular-momentum representation is used for the final atomic states. In this case, the energy loss is close to threshold and there are four significant projected matrix elements, three of which are shown in Figure 6 (the omitted case l0 ¼ 1; m0 ¼ 1 is similar to l0 ¼ 1; m0 ¼ þ1). The qualitative forms of the projected matrix elements exhibit a strong dependence on the final atomic state. For example, those for which m0 ¼ 0 are peaked at the atomic nucleus and exhibit rotational symmetry, whereas those for m0 s0 vanish at the nucleus. In the general case, which includes the excitation of L and higher-order shells (not shown), the qualitative form of the projected matrix elements is determined by the change in angular momentum experienced by the atomic electron. It is also noteworthy that in Figure 6 the extent of the initial atomic orbital is essentially the same as that of the projected matrix element for l 0 ¼ 0; m0 ¼ 0. Hence, the considerably greater spatial extent of the projected matrix elements for l 0 ¼ 1 is a manifestation of the inelastic delocalization (see Section 2.2.3). Reversing this argument, we see that the projected matrix element for l0 ¼ 0; m0 ¼ 0 does not exhibit delocalization (which turns out to be a consequence of the fact that the orthogonality of the initial and final states is upheld by the radial parts of the atomic wave functions in this case). This represents an example where order-of-magnitude estimates regarding inelastic delocalization (see Section 2.2.3) are not applicable, although there is no contradiction with such arguments since they provide an upper limit of the delocalization. 3.3.3. Chemical Imaging of Single Atoms To complete the theory as it applies to single atoms, all that remains is to provide an expression for the elastic wave j0 and demonstrate how the inelastic waves ja relate to the chemical signal measured in an experiment.

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Figure 6 Projected matrix elements for excitation of the Si-K shell. In each case l 0 ; m0 denotes the angular momentum of the final atomic state. A beam energy of 100 keV and energy loss of 10 eV beyond threshold are assumed. (From Dwyer, 2005a).

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 173

The wave function of an electron in an incident STEM beam is given by the expression (Spence and Cowley, 1978) Z d2 kAðkÞeigðkÞ eik,ðXX0 Þ ; j0 ðx  x0 ; z0 Þ ¼ (29) where x0 is the beam position in the plane defined by z0 , and g is the aberration function of the probe forming lenses, given by
 1 2 gðkÞ ¼ 2p lC1 jkj þhigher-order-aberration terms : (30) 2 Here C1 is the defocus, which we have defined to be positive when the beam cross over is upstream of the plane z0 . In chemical imaging experiments, the chemical signal (inelastic intensity) is measured by a detector situated in the far field of the target. For the purposes of calculating the chemical signal from a single atom, we can consider the planes defined by z and z0 to coincide at the center of the atom [but still retain the infinite limits of integration in the projected matrix element, as in Eq. (28)]. Then, neglecting unimportant phase factors, the inelastic wave function at a point k in the far field is related to the inelastic wave function in plane z (or z0 ) by a Fourier transformation: Z ~ f;f ðkÞ ¼ j d2 xjf;f0 ðx; zÞeik,x ; (31) 0 ~ f;f denotes the Fourier where jf;f0 is given by (26) and the tilde on j 0 transform. The chemical signal is given by integrating the inelastic intensity over the detector  2 XZ ~  2 Iðx0 Þ ¼ d k DðkÞjf;f0 ðkÞ ; (32) f;f0

where the detector function DðkÞ is defined to be unity (zero) for points k on (off) the detector. In Eq. (32), the summation over f and f0 incorporates any atomic transitions that contribute to the chemical signal, the latter being obtained experimentally by integrating the background-subtracted spectrum over an energy window. Hence, the summation over f0 includes the degenerate initial spin orbitals with different angular momenta, while the summation over 4 includes final spin orbitals with different energies consistent with the energy-loss window, and for each such energy,

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degenerate final spin orbitals with different angular momenta. Finally, the chemical signal in Eq. (32) is written as a function of the beam position x0 , so that it can be considered as a 2D image or chemical map of the atomic species. Examples of computed chemical signals were presented in Section 2.2.4, and further examples are provided in the next section. 3.3.4. Validity of the Dipole Approximation Since the dipole approximation is often used in theories of core-level EELS, it is worthwhile to address the question of its validity in core-level spectroscopy at atomic resolution. From a computational perspective, the dipole approximation enables considerable simplification of the matrix elements for atomic excitation. This attraction has resulted in its application at times in theories of chemical mapping of single atoms (e.g., Kohl and Rose, 1985) and materials (e.g., Tan et al., 2011). However, as will be demonstrated, the accuracy of the dipole approximation is highly questionable in the case of the scattering geometries used in state-of-the-art instruments, leading most workers to refrain from applying it to simulations of chemical signals obtained at atomic resolution. The approximation in question is easily obtained by considering the matrix element for atomic excitation when the initial and final states of the beam electron are plane waves: ~ f;f0 ðqÞ ¼ ðhfj5hkjÞV ðjk0 i5jf0 iÞ V e2 ¼ 4pε0 ¼

e2 ε0 q2

Z

3

d x Z

Z

d3 x1 fðx1 Þ

eiq,x f ðx1 Þ jx  x1 j 0

(33)

d3 x1 fðx1 Þeiq,x1 f0 ðx1 Þ;

where k0 and k are the initial and final wave vectors of the beam electron, q ¼ k  k0 is the scattering vector, and the last equality uses the Fourier transform of the Coulomb potential. The dipole approximation is obtained by using the first-order Taylor expansion eiq,x1 z1  iq$x1 , which holds provided that jqj and/or jx1 j is sufficiently small. The contributing values of jx1 j are determined by the forms of the initial and final atomic states and therefore cannot be controlled. Conversely, the contributing values of jqj are determined by the experimental geometry, so that the validity of the dipole approximation generally corresponds to small convergence and collection angles. Using the first-order Taylor expansion, and noticing that the unity

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 175

term does not contribute because of the orthogonality of the initial and final atomic states, we obtain 2 ~ f;f0 ðqÞz  ie q$xf;f0 ; V ε0 q2

where xf;f0 is the dipole matrix element Z xf;f0 ¼ d3 x1 fðx1 Þx1 f0 ðx1 Þ:

(34)

(35)

To obtain the corresponding projected matrix element in real space (Section ~ f;f ðqÞ a 2D inverse Fourier transform: 3.3.2), we can apply to V 0 Z ~ f;f0 ðq; qz Þeiq$x ; Vf;f0 ðxÞ ¼ d2 qV (36) where qz was introduced above. Figure 7 compares simulated chemical signals from a single scandium atom obtained using the dipole approximation and the full theory. Calculations are presented for a broad beam (5 mrad convergence) and small collection angle (2 mrad), and an atomic-sized beam (32 mrad convergence) and large collection angle (80 mrad). The latter geometry is relevant to state-of-the-art experiments at atomic resolution. The dipole approximation and the full theory compare well if the convergence and collection semi-angles are small, as expected. For the atomic-sized beam and large collection angle, the discrepancy is large when the beam is close to the atom, that is, when the scattering is strongest, while positions away from the atom show good agreement because the scattering is weak and the scattering angles experienced by the beam are small in that case. Similar discrepancies between the dipole approximation and the more accurate theory have been found by D’Alfonso et al., (2008), who additionally reported that the apparent shape of the atom can be incorrect if the dipole approximation is used. To summarize, we have seen that the dipole approximation can be inaccurate for beam positions close to the atom (i.e., when the chemical signal is strongest), so its application in the context of the present technique warrants considerable caution. 3.3.5. EDX Mapping and the Object-Function Approach For EDX mapping, whereby chemical maps are obtained by monitoring the flux of characteristic X-rays as a function of the beam position, the motion of the beam electron after the atomic excitation event is completely immaterial.

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(a)

(b)

Figure 7 Validity of the dipole approximation in atomic-resolution chemical mapping. The graphs show calculated chemical signals from the L2;3 “white lines” ð2p/3dÞ of a single scandium atom as a function of the position of a 100 keV aberration-free electron beam, where the dipole approximation (lines) is compared with the full theory (points). (a) Convergence and collection semi-angles of 5 mrad and 2 mrad, respectively. (b) Convergence and collection semi-angles of 32 mrad and 80 mrad, respectively.

For simplicity, we also assume that the excitation of a particular atomic innershell always results in a characteristic X-ray (i.e., we neglect Auger processes) and that the characteristic X-rays are detected with 100% efficiency.7 With these assumptions, an expression for EDX mapping can be obtained by using Eq. (32) and taking the limit of an infinitely large detector (i.e., DðkÞ ¼ 1):  2 XZ ~  d2 kj ðkÞ (37) IEDX ðx0 Þ ¼  ; f;f0 f;f0

7 The quantitative corrections required as a result of these assumptions can be non-trivial,

especially in the case of solids (see, for example, Rossouw et al. 1997).

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 177

where the summation now includes all possible final states of the atomic electron (D’Alfonso et al., 2010). Using Parseval’s theorem, the above expression can be written in terms of integrals over coordinate space:  2 XZ   d2 xjf;f0 ðxÞ IEDX ðx0 Þ ¼ f;f0

¼

X f;f0

s2f;f0

Z

 2 d xVf;f0 ðxÞ jj0 ðxÞj2 : 2

(38)

Now, unlike the EELS case, the summation over initial and final atomic states can be easily applied before the integration is carried out. In this way, it is possible to define an object function for EDX mapping that acts on the intensity distribution jj0 ðxÞj2 in the ingoing elastic channel: Z IEDX ðx0 Þ ¼ d2 xWEDX ðxÞjj0 ðxÞj2 ; (39) where the object function is given by X 2  s2f;f0 Vf;f0 ðxÞ : WEDX ðxÞ ¼

(40)

f;f0

The expression (39) for chemical mapping based on EDX is considerably simpler than that based on EELS, since it only requires knowledge of the elastic intensity distribution in the plane of the atom (as opposed to the elastic wave function). Hence the EDX signal can be viewed as incoherent, i.e., it does not involve the beam electron’s phase. Moreover, the fact that the summation over initial and final atomic states is contained in the object function leads to considerably faster computation times in numerical work. The simplicity and numerical efficiency of the object-function approach embodied in expression (39) has led some workers to use a similar expression in the calculation of chemical maps based on EELS, where it is sometimes referred to as a local approximation (Allen and Josefsson, 1995; Oxley and Allen, 1998). In this case, some of the effects arising from the finite detector size can be retained by defining an object function of the form (e.g., Dwyer, 2005a) Z X 2 ~  x0 ÞVf;f0 ðx0 Þ; WEELS ðxÞ ¼ sf;f0 d2 x0 V f;f0 ðxÞDðx (41) f;f0

~ where DðxÞ is the inverse Fourier transform of the detector function DðkÞ, and the summation over final atomic states is restricted to those

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Figure 8 EELS object function for excitation of the Si-K shell . Also shown are the contributions of individual transitions to final atomic states labeled by their angular momemtum l0 ; m0 . A beam energy of 100 keV and an energy loss of 10 eV beyond threshold are assumed. See also Figure 6.

commensurate with the energy-loss window. It follows from Eq. (41) that the accuracy of an object-function approach for chemical mapping based on EELS is dependent on the detector geometry, with the results more accurate for larger detectors (Dwyer, 2005a). Accurate results can also be obtained if the beam electron’s wave function varies slowly on the scale of the projected matrix elements (Dwyer, 2005a), though that situation is of less relevance here since it is not associated with atomic spatial resolution. Figure 8 shows an EELS object function for excitation of the Si-K shell where an infinitely large detector is assumed. The assumption of an infinitely large detector implies a correspondingly large uncertainty in the final transverse momentum of the scattered beam electron. In accordance with Heisenberg’s uncertainty principle, the transverse extent of the volume in which inelastic scattering took place is therefore minimized. Hence the width of the object function reflects the best possible spatial resolution that can be achieved for a given excitation and beam energy (see also Section 2.2.4). These statements are even more pertinent to the case of EDX, where the assumption of an infinite detector needs no justification because it is equivalent to the statement that the beam electron’s final transverse momentum is immaterial. We conclude this section by attempting to clarify some of the confusing nomenclature that often accompanies analyses of coherent versus incoherent

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conditions in the literature on atomic excitations by fast electrons. As mentioned previously, the term local approximation is sometimes used to describe the object-function approach, because the signal (i.e., what is measured) is written mathematically as a sum of components, each of which depends only on the beam electron’s intensity at a single position (see Eq. (39)). Also, local potential is sometimes used to describe the object function itself, even though the object function does not have the correct units for a potential. In addition, the term nonlocal is sometimes used to describe those situations where the object-function approach is not used/ applicable, i.e., when the signal is (at least partially) coherent and the more fundamental and, in the case of EELS, more accurate, theory must be retained. The reason for use of the term ‘nonlocal’ is that the mathematical expression for the signal depends on the beam electron’s wavefield at two positions (see, for example, Oxley and Allen, 1998). However, we emphasize that such nonlocality is always present in quantum mechanics, e.g., the quintessential double-slit experiment. Furthermore, we strongly emphasize that, contrary to many discussions in the literature, the beam electron-sample interactions leading to inelastic scattering are fundamentally the same as those leading to elastic scattering, and these interactions are described by local quantities, such as the matrix element Eq. (23). Finally, delocalization and nonlocality should, strictly speaking, be distinguished, as the former persists even in the absence of the latter (e.g., Figure 8, where the object function extends beyond the bounds of the initial atomic wave function).

3.4. Combined Inner-Shell Excitation and Dynamical Elastic Scattering As is well known, the strength of the electrostatic interaction of electrons with matter causes the high-energy electrons in (S)TEM to scatter elastically multiple times within the specimen. For atomic-resolution core-level spectroscopy, an important consequence of such dynamical scattering is that it gives rise to so-called channeling effects, whereby the high-energy electrons tend to propagate along the atomic columns in the specimen (see Section 2.2.2). This effect can prove crucial to the interpretation of experimental data on both the qualitative and quantitative levels, and hence a reliable theory of core-level spectroscopy at atomic resolution must combine the theories of atomic inner-shell excitation and dynamical elastic scattering. In addition, thermal-diffuse scattering is also important, but we delay its discussion until Section 3.4.3.

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While many (ultimately equivalent) theories of dynamical electron scattering have been formulated over the years, today only two such formulations are commonly used for numerical calculations: the Bloch-state approach, pioneered by Bethe (1928) and further developed by Blackman (1939) and others (see Humphreys, 1979 for a concise presentation), and the multislice approach, originally developed by Cowley and Moodie (1957), and further developed by Goodman and Moodie (1974); Ishizuka and Uyeda (1977), and others (see Kirkland, 2010 for a concise presentation). Extensions of the Bloch-state approach to include atomic inner-shell excitations have been presented by various workers, notably Maslen and Rossouw (1984), Rossouw and Maslen (1984), Saldin and Rez (1987), Weickenmeier and Kohl (1989), Allen and Josefsson (1995, 1996), and Oxley and Allen (1998). Since they are based on the eigenstates of a periodic Hamiltonian, Bloch-state approaches are well-suited for the description of dynamical scattering in crystals. For specimens with lower symmetry, such as interfaces or defects, a multislice approach usually proves more tractable. Notable works describing extensions of the multislice approach to include atomic inner-shell excitations have been presented by Spence (1980), Wang (1989, 1990), Allen et al., (2003b), and Dwyer (2005a,b). The work of Verbeeck et al., (2009) is also of interest in expounding connections to a density-matrix approach. Here, combined atomic excitation and dynamical scattering is formulated using propagator theory, which has the benefit of including both the Bloch-state and multislice approaches as special cases. A numerical implementation of the theory using the multislice approach is presented in Section 3.4.3. 3.4.1. Propagators in the Single Inelastic Scattering Approximation As mentioned in Section 3.3.2, the lengths of the mean-free paths for atomic inner-shell excitation by high-energy electrons in solids mean that each beam electron can be assumed to experience only one such event. This is a fortunate circumstance as it leads to considerable simplification. We pick up the theory from Section 3.2.6. By application of the boundary condition Eq. (19) to Eq. (18), we immediately obtain Z ja ðx; tÞ ¼ d2 x0 iGa0 ðx; t; x0 ; 0Þj0 ðx0 ; 0Þ; (42) where the propagator Ga0 describes all possible scattering processes starting with the target in its ground state j0i and finishing in an excited state jai.

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Analogous to conventional propagator theory (Bjorken and Drell, 1964), we have that Gg0 satisfies the following Dyson-type equation: Gg0 ðx; t; x0 ; 0Þ ¼ Gg0 ðx; t; x0 ; 0Þdg0 Z þ

2 00

Zt

d x

dzGgb ðx; t; x00 ; zÞsb Vba ðx00 ; zÞGa0 ðx00 ; z; x0 ; 0Þ;

(43)

0

or, symbolically, Gg0 ¼ Gg0 dg0 þ Ggb Vba Ga0 :

(44)

Recalling that inelastic scattering is associated with the matrix elements Vba for which bsa, the single inelastic scattering approximation is given by, symbolically, Ga0 zG0 da0 þ Ga Va0 G0 ;

(45)

where the propagators G0 and Ga (containing only one subscript) include elastic scattering only. Hence, from Eq. (42), for as0, we obtain (written explicitly) Zt Z 2 00 dziGa ðx; t; x00 ; zÞðisa ÞVa0 ðx00 ; zÞ ja ðx; tÞz d x (46) 0 Z  d2 x0 iG0 ðx00 ; z; x0 ; 0Þj0 ðx0 ; 0Þ: This expression makes it clear that we can picture the scattering processes, quite rigorously, in terms of dynamical elastic scattering, followed by a single inelastic scattering event, followed by more dynamical elastic scattering. As in Section 3.3.3, the chemical signal for a given beam position x0 is obtained by integrating the inelastic intensity over the detector. However, here we must also include an additional summation over the different atoms (of a given species) in the specimendthat is, the label a ¼ f; f0 ; n, where n labels the different atoms. Hence the chemical signal is written in the form  2 X Z ~  d2 kDðkÞj Iðx0 Þ ¼ ðkÞ (47)  ; f;f0 ;n f;f0 ;n

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~ f;f ;n is the Fourier transform of jf;f n , and the latter is given by Eq. where j 0 0 (46). Note that this assumes that the chemical signals from different atoms add incoherently (for a discussion on this point, see Maslen, 1987). 3.4.2. Single Channeling Versus Double Channeling Given that the number of atoms of a given species typically sampled by the electron beam is large, and that each atom gives rise to several inelastic channels, the total number of inelastic channels included in the summation in Eq. (47) can be extremely large ( 105 is not uncommon). While this does not pose any conceptual difficulty, it leads to serious challenges for numerical work because the mutual incoherence of the inelastic waves effectively means that a separate elastic scattering calculation is necessary for each inelastic channel. Hence approximations are often sought to make the numerical calculations more tractable, and one common such approximation is the so-called single-channeling approximation, which includes elastic scattering in the elastic channel but neglects any elastic scattering in the inelastic channels. The term double channeling then refers to cases where elastic scattering is included in the inelastic channels as well. Essentially, the validity of the single-channeling approximation is dependent on (1) the strength of elastic and thermal-diffuse scattering and (2) the size of the detector. The first point is easily understood. To appreciate the second point, we need only understand that neglecting elastic scattering in the inelastic channels has the effect of redistributing the inelastic intensity in the diffraction plane, and that if the redistribution occurs on a (reciprocal) length scale that is smaller than the detector, then the total inelastic intensity impinging on the detector will be largely unchanged. Thus the single-channeling approximation generally improves as the detector size increases. This point is illustrated in Figure 9, which compares the single- and double- channeling predictions of the O-K signal from SrTiO3 for various collection angles. As anticipated, the single- and double-channeling prediction are in close agreement for the largest collection angle (60 mrad). On the other hand, the two models differ substantially for the smallest collection angle (4 mrad), highlighting the importance of double channeling in such cases. From Figure 9 we also see that the validity of the single-channeling approximation goes hand-in-hand with a chemical signal that is intuitively interpretable in terms of the positions of relevant atomic species. While this is an important observation and is in line with much of the discussion in Sections 2.2.4 and 3.3.5, this behavior is not guaranteed. For example, in the

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(A)

(B)

(C)

Figure 9 Comparison of single- (dashed lines) and double-channeling (solid lines) calculations of the O-K signal from a line trace across ½100 SrTiO3. Results are presented for collection semi-angles of (A) 4, (B) 25, and (C) 60 mrad. The beam path is indicated by the dashed line across the SrTiO3 unit cell at bottom (C), with Sr (green), Ti (blue), and O (red). Results of an approximate double-channeling model (dashed-dotted line) are also presented in (A). The calculations assume a 100 keV aberration-free beam with convergence semi-angle 25 mrad, an energy loss of 10 eV above threshold, and a specimen thickness of w100 Å. (Adapted from Dwyer et al., 2008b). See the color plate.

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case of very strongly scattering elements, the elastic scattering in the ingoing channel can be so strong that it leads to a nonintuitive chemical map (Xin et al., 2011); nonetheless the single- and double-channeling theories yield qualitatively similar results. 3.4.3. Numerical Implementation Using Multislice In order to conduct detailed simulations, one must ultimately choose a means for calculating numerically the effect of the elastic propagators in Eq. (46). Here we outline how this can be achieved using the multislice approach. As mentioned previously, the multislice approach is well suited to calculating elastic scattering in specimens of arbitrary atomic structure, and hence it offers the flexibility required to handle a wide range of materials systems. We consider a full double-channeling calculation (the procedure for the single-channeling and other approximations should then be obvious). In multislice theory, the specimen is regarded as composed of slices perpendicular to the optic axis, and the effect of dynamical elastic scattering is treated on a sequential, or slice-by-slice, basis. Let us consider the chemical signal arising from a particular excitation of a particular atom, say atom n, at a depth z in the material, which, in terms of the multislice approach, lies in the mth slice. As mentioned after expression (46), we can rigorously picture the scattering processes in terms of dynamical elastic scattering ‘up to’ the mth slice, followed by an inelastic scattering event in the mth slice, followed by more dynamical elastic scattering up to the specimen exit surface. In terms of the multislice approach: 1. The elastic wave impinging on the mth slice is computed using the form

j0 ðxm Þ ¼ iG0 ðxm1 ; DzÞ  Qðxm1 Þ /

  (48)

 / iG0 ðx2 ; DzÞ  Uðx2 Þ iG0 ðx1 ; DzÞ  Uðx1 Þj0 ðx1 Þ / ; where j0 ðx1 Þ is the elastic wave impinging on the first slice (i.e. the ingoing elastic wave), G0 is the free-space propagator, Dz is the slice thickness, Qðxk Þ ¼ eis0 V00 ðxk Þ is the so-called phase grating for the kth slice, and the convolutions are performed in order: the convolution with respect to x1 producing a function of x2 , and so on. In practice, the convolutions are

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handled numerically using fast Fourier transforms (see, for example, Kirkland, 2010). 2. The inelastic wave originating at the mth slice is then computed using a projected matrix element, as in expression (26), which, in the current notation, reads jf;f0 ;n ðxm Þ ¼ isf;f0 Vf;f0 ;n ðxm Þj0 ðxm Þ:

(49)

3. The inelastic wave at the exit surface is calculated using an expression analogous to that given in the first listed item, except that it accounts for elastic scattering from the mth slice to the final slice (inclusive). 4. The contribution to the chemical signal is computed by taking the Fourier transform of the inelastic wave at the exit surface and then integrating its intensity across the detector, exactly as done for each term in the summation of expression (47). As mentioned earlier, the simulation must respect the incoherence of the various inelastic waves. Hence, separate multislice calculations are required for the other inelastic waves originating from the atom, as well as the inelastic waves originating from all other atoms of the same species. For the simulation of a chemical map, we also require that the entire calculation be repeated for the different beam positions in the map. Needless to say, the amount of computation involved in this approach can be quite large. Recently, Dwyer (2010) has demonstrated how multislice calculations of ADF-STEM images can be sped up by factors of 10 or more by applying efficient sampling in conjunction with the high floating-point performance of general-purpose graphics processing units. The same ideas are applicable to chemical map and often provide a means of simulating chemical maps, including double channeling, in a feasible time frame. In addition to the single-channeling approximation and the objectfunction approach, we briefly mention one other approximation that can be used to speed up multislice computations incorporating atomic innershell excitations. The approach recognizes that the form of the elastic wave often varies relatively slowly from one slice to the next, so that the inelastic waves originating from adjacent slices give rise to very similar contributions. In this case, we can compute inelastic waves from only a subset of slices and make a correction for the omitted slices. Such an approach has been used by, for example, Dwyer and Barnard (2006) in the context of core-loss diffraction and Verbeeck et al., (2009) in the context of energyfiltered TEM imaging.

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Finally, the inclusion of thermal-diffuse scattering (TDS) is often important for accurate results regarding the channeling of the beam and hence for predicting the chemical signal at atomic resolution, particularly for specimens containing elements of high atomic numbers. Currently, various approaches for incorporating TDS exist. Here we mention only the so-called frozen phonon approach (Loane et al., 1991), which is generally accepted as very accurate. This approach models TDS as the ensemble average of elastic scattering from displaced atoms, and it is fairly straightforward to incorporate into multislice calculations (see Kirkland, 2010). For a detailed discussion on the effects of TDS on chemical signals at atomic resolution, readers are referred to the work of Findlay et al., (2005).

4. SELECTED APPLICATIONS In this section, we present selected applications of core-level spectroscopy at atomic resolution. The examples included here by no means form a complete or systematic representation of the published literature to date, but rather illustrate the power of the techniques for analyzing different materials systems and highlight the benefits of simulations in gaining a sound interpretation of experimental results.

4.1. Chemical Mapping of Crystalline Compounds The application to crystalline compounds is a rather straightforward, though nonetheless powerful, example of atomic-resolution chemical mapping. Numerous examples exist in the literature, including works reporting demonstrations and developments of the technique as well as applications to industrially relevant materials (e.g., Okunishi et al., 2006; Bosman et al., 2007; Kimoto et al., 2007; Gunawan et al., 2009; Varela et al., 2009; Botton et al., 2010; Lazar et al., 2010; Mundy et al., 2012). Figure 10 shows the results of Gunawan et al., (2009), who analyzed the compound Bi3.25La0.75Ti3O12 (BLT), a potential candidate for nonvolatile ferroelectric RAM applications. The atomic structure of this compound is related to that of Bi4Ti3O12, the latter being an Aurivillius compound composed of alternating fluorite-like and perovskite-like layers (see Figure 10a). Previous works were inconclusive as to whether La substitutes Bi in the fluorite-like and/or perovskite-like layers in the BLT unit cell. To address this question, Gunawan et al., (2009) performed simultaneous

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Figure 10 Atomic-resolution chemical mapping of [110] Bi3.25La0.75Ti3O12 (BLT). (A) Atomic structure of the parent compound Bi4Ti3O12 showing Bi sites (green) and TiO6 octahedra in the perovskite-like layers (red). (B) ADF image of the BLT specimen with the green box indicating the area for chemical mapping. (C) ADF image acquired simultaneously with the spectrum image. (D) Colour-coded composite map showing Ti (red) and La (green). (E) La-N4;5 map exhibiting stronger signal but greater delocalization. (Adapted from Gunawan et al. 2009). See the color plate.

atomic-resolution ADF imaging and chemical mapping. A relatively low beam energy of 80 keV was used to minimize specimen damage. While the ADF images in Figures 10b and 10c clearly show the Bi-containing and Ti atomic columns, the ADF images provide little information regarding the substitution of La. In contrast, the La M4;5 and N4;5 chemical maps in Figures 10d and 10e clearly indicate a preference for La to occupy Bi sites in the second atomic layer (with respect to the growth direction) in the fluorite-like layers. Such a result vividly demonstrates the power of atomicresolution chemical mapping in locating and unambiguously identifying atomic species in materials.

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4.2. Chemical Mapping at Solid-Solid Interfaces Solid-solid interfaces constitute a pertinent application of atomic-resolution chemical mapping and have been the subject of study in several recent works (e.g., Muller et al., 2008; Botton et al., 2010; Colliex et al., 2010; Fitting Kourkoutis et al., 2010; Garcia-Barriocanal et al., 2010). Questions regarding interfacial roughness, interdiffusion, stoichiometry, and atomic ordering are all amenable to analysis. Moreover, information regarding electronic bonding at interfaces can also be accessed by analyzing near-edge structure (see Section 4.3). Figure 11 shows results from Botton et al., (2010), who obtained atomicresolution maps from a BaTiO3/SrTiO3 interface in a multilayer material. The experiments used a beam energy of 80 keV to avoid specimen damage and a collection semi-angle of 110 mrad. Here the large collection angle ensures that the chemical signals are largely incoherent (see Section 3.3.5). The positions of Ba, Sr, and Ti are clearly revealed by the chemical map in Figure 11c, with a qualitative interpretation of the results indicating a confinement of Ba and Sr to their respective layers, while Ti appears continuous across the interface, as expected for a sharp interface. On the other hand, the plots of the experimental signals across the interface in Figure 11d suggest some interdiffusion of Ba and Sr. However, the simulated results in Figure 11d, which assume an atomically sharp interface, reveal significant delocalization of the Ba and Sr chemical signals. In this case, the simulations highlight the dangers of a literal interpretation of the chemical signals in the presence of beam channeling and inelastic delocalization (even when the chemical signals are largely incoherent).

4.3. ELNES Mapping Mapping based on ELNES goes beyond extracting chemical signals and extracts changes in the near-edge structure as a function of the beam position. Hence, if performed at atomic resolution, ELNES mapping accesses not only the positions of sites occupied by a given atomic species, but also the local electronic structure(s) specific to those sites. Such an ability is, of course, extremely powerful for understanding electronic effects in materials. On the other hand, this ability comes at a price in terms of much more stringent signal-to-noise requirements to extract spectral changes with energy losses of the order 1 eV or less (as opposed to the order 10 eV needed for chemical mapping). ELNES mapping at near-atomic spatial resolution dates back to the work of Batson (1993), who extracted information on the valence states of Si

Atomic-Resolution Core-Level Spectroscopy in the Scanning Transmission Electron Microscope 189

Figure 11 Atomic-resolution chemical mapping at a BaTiO3/SrTiO3 interface. (A) ADF image of the specimen with the green box indicating the area for chemical mapping. (B) ADF image acquired simultaneously with the spectrum image. (C) Colour-coded composite map showing Ba (blue), Sr (red) and Ti (green). (D) Plots of the experimental (solid lines) and simulated (dashed lines) Ba and Sr signals across the interface. The simulation assumes an atomically sharp interface. (Adapted from Botton et al. 2010). See the color plate.

across an Si/SiO2 interface by monitoring changes in the Si-L2;3 near-edge structure. Recently, atomic-resolution ELNES has been demonstrated in a number of works in conjunction with 2D chemical mapping on nextgeneration instruments (Muller et al., 2008; Varela et al., 2009; Lazar et al., 2010; Tan et al., 2011; Mundy et al., 2012; Turner et al., 2012). Figure 12 shows the results of Tan et al., (2011), who demonstrated 2D mapping of the manganese valence states in Mn3O4. The experiments were performed on an aberration-corrected TEM/STEM operating at 120 kV and fitted with a monochromator excited to produce an energy resolution of 0.4 eV. The mapping involved tracking changes in the Mn fine structure.

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Figure 12 Atomic-resolution mapping of Mn valence states in ½100 Mn3O4. A, As-extracted spectral weights for Mn2þ and Mn3þ. B, Low-pass filtered data. C, Simulated data. The bottom row shows colour-coded composite maps with Mn2þ (green) and Mn3þ (red). (From Tan et al. (2011)). See the color plate.

In this case, the experiment must resolve the 1.7-eV splitting of the Mn L2 and L3 white lines, justifying the use of a monochromator. Changes in the Mn L2,3 fine structure were extracted using a multiplelinearleast squares fit of reference spectra, where the reference spectra were obtained by periodic averaging. The atomic-scale changes in the Mn L2,3 fine structure are attributed to the different Mn valence states in this compound, so that the results in Figure 12 reveal the Mn2þ and Mn3þ sites within the projected unit cell. Also shown in Figure 12 are simulated valence maps, which exhibit good qualitative agreement with the experimental data. Here the simulations were useful in providing an understanding of the mixing of the reference spectra, that is, the reference spectra do not exhibit purity with respect to either Mn2þ and Mn3þ but contain some mixing due to beam channeling and delocalization (see Sections 2.2.2 and 2.2.3).

4.4. EDX Mapping In our final example, we consider chemical mapping based on EDX spectroscopy. The chemical signals from EDX provide an alternative to those from EELS and offer advantages in terms of accessing deeper core levels in the heavier elements. Another attraction of EDX over EELS is that the maps can

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be simpler to interpret in terms of the atomic structure, which results from the validity of an object-function description of the underlying excitations (see Section 3.3.5). Essentially, the validity of an object-function description implies that certain artefacts that can be manifest in EELS-based mapping, such as “volcanoes” (see Figure 2), are less likely in EDX-based mapping. On the other hand, some disadvantages of EDX include the impossibility of analyzing electron bonding due to the absence of fine structure and the relatively low detection efficiency. Regarding the latter, in EELS it is often the case that virtually the entire chemical signal can be collected by using a collection semi-angle of the order 100 mrad. In contrast, characteristic Xrays are emitted from the specimen in all directions, so that achieving collection efficiency in EDX requires a detector covering a significant

Figure 13 Atomic-resolution chemical mapping of ½110 In0.53Ga0.47As using energydispersive X-ray spectroscopy. (A) Colour-coded chemical maps with overlaid white circles indicating the ‘dumbbell’ atomic structure. (B) Integrated EDX spectrum showing the chemical signals used for mapping and the corresponding ADF image (inset). (From Chu et al. 2010). See the color plate.

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proportion of 4p sr. Next-generation EDX detectors make considerable progress toward this goal, with detection solid angles approaching 1 sr. Atomic-resolution EDX mapping has been recently demonstrated independently by Chu et al., (2010), D’Alfonso et al., (2010), Watanabe et al., (2010), and others. Figure 13 shows the results of Chu et al., (2010), who demonstrated atomic-resolution EDX mapping of ½110 In0.53Ga0.47As using an aberration-corrected TEM/STEM operating at 200 kV and equipped with an EDX spectrometer subtending a solid angle of x0:13 sr. Their results clearly reveal the atomic structure in this crystallographic orientation, with both the dumbbell arrangement of the atomic columns and the mixed composition of the In-Ga atomic columns clearly shown.

5. CONCLUDING REMARKS Before concluding, we briefly mention some current limitations as well as future prospects for core-level spectroscopy at atomic resolution. Most workers seem to agree that, with the instruments now available, it is usually beam-induced specimen damage rather than the instrument itself that imposes ultimate limitations in terms of the spatial resolution and sensitivity that can be achieved. To this end, reduced knock-on damage associated with the lower beam energies (< 100 keV) available in the latest aberration-corrected instruments are yielding promising results, and this appears set to drive the trends in instrumentation over the coming years. As always, however, the advantages of low beam energies are likely to come with their own price tag, such as the additional challenges in data interpretation arising from increased multiple inelastic scattering, and also the more complex electron channeling that goes hand in hand with the reduced depth of field if atomic lateral resolution is to be maintained (for a diffraction-limited beam, the depth-of-field:lateral-resolution ratio is proportional to a1 and is independent of l). Some of these issues have been examined in the recent work of Lugg et al., (2011). Demonstrations of ELNES mapping at atomic resolution are still relatively few (especially 2D maps), and this is certainly one area where the rich potential of the technique is yet to be fully realized. For ELNES mapping, however, the greater signal-to-noise ratio required makes the issue of specimen damage even more pertinent. Apart from the lower beam energies mentioned above, further advances and refinements in data processing methods, some of which were mentioned here, may help in providing optimized means for the extraction of noisy signals.

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Regarding future prospects for theory, it is the author’s view that (1) considerable progress can and should be made on the efficiency of the mathematical formulation(s) and (2) we cannot simply rely on advances in computing power to make advances in data interpretation. Significant progress would also be made by refining and simplifying the formalism and nomenclature to make it more accessible to the wider electron microscopy community. While the present work has made some attempt toward this goal, the author is aware that the level of the theoretical presentation makes it difficult for those with little background in quantum mechanics to grasp the essential details. Finaly, we mention the prospect of quantitative core-level spectroscopy at atomic resolution. This approach combines accurate simulations with experiments in which the chemical signals are acquired in terms of the fractional intensity of the incident beam. In the case of chemical mapping, such an approach offers the prospect of element-specific atom counting at atomic resolution, with significant implications for materials characterization. A first attempt at such experiments has been reported by Xin et al., (2011, 2012a), who demonstrated very good quantitative agreement for the absolute cross sections and contrasts of experimental and simulated EELS-based chemical maps of a crystalline material extracted from chemical edges of a simple nature. In the case of more complex chemical edges, such as those featuring a pronounced core-hole effect, their work indicates the need for more accurate matrix elements than those provided by the single-electron description. In summary, this chapter has presented an account of the recent experimental and theoretical advances that have made atomic-resolution core-level spectroscopy an exciting new tool for analyzing the atomic and electronic structure of materials. In addition, it is hoped that the selected applications presented here make some progress in demonstrating the power of the technique in different contexts. Interested readers are strongly urged to consult the many excellent recent works cited herein, most especially those that were not discussed in detail.

ACKNOWLEDGMENTS The author acknowledges the many relevant and fruitful collaborations and discussions with various colleagues and other workers over the years, including L.J. Allen, J.S. Barnard, C.B. Boothroyd, G.A. Botton, S.L.Y. Chang, A.J. D’Alfonso, S. Dudarev, R.E. Dunin-Borkowski, J. Etheridge, S.D. Findlay, N. Gauquelin, C.J. Humphreys, V.J. Keast, S. Lazar, C. Maunders, A.F. Moodie, D.A. Muller, E. Okunishi, T.C. Petersen, G. Radtke, P. Rez, C.J. Rossouw, H. Sawada, P. Schattschneider, M. Weyland, H.L. Xin, and Y. Zhu. He

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particularly thanks S.L.Y. Chang and C.J. Rossouw for their patient proof reading of the draft and suggestions for the manuscript. Financial support by the Australian Research Council (DP110104734) is also acknowledged.

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