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Model-Based Electron Microscopy

  • Sandra Van AertEmail author
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

The growing interest in materials design and control of nanostructures explains the need for precise determination of the atomic arrangement of non-periodic structures. This includes, for example, locating atomic column positions with a precision in the picometer range, a precise determination of the chemical composition of materials, and counting the number of atoms with single atom sensitivity. In order to extract these quantitative measurements from atomic resolution (scanning) transmission electron microscopy ( ) images, statistical analysis methods are needed. For this purpose, statistical parameter estimation theory has been shown to provide reliable results. In this theory, observations are purely considered as data planes, from which structure parameters have to be determined using a parametric model describing the images. This chapter summarizes the underlying theory and highlights some of the recent applications of quantitative model-based (S)TEM.

model-based parameter estimation quantitative structure determination maximum likelihood estimation experiment design precision resolution probability of error atomic column position measurement composition analysis atom counting 

Aberration corrected transmission electron microscopy ( ) is an excellent technique to study nanostructures down to the atomic scale. As compared to x-rays, electrons interact very strongly with small volumes of matter providing local information on the material under study [12.1, 12.2, 12.3]. In this manner, TEM can be used to observe deviations from perfect crystallinity, which is of great importance when studying nanostructures. Because of the presence of defects, interfaces, and surfaces, the locations of atoms of nanostructures deviate from their equilibrium bulk positions. This results in strain playing a crucial role on the observed properties. For example, strain induced by the lattice mismatch between a substrate and a superconducting layer grown on top can change the interatomic distances by picometers and can in this manner turn an insulator into a conductor [12.4]. In order to unscramble the structure–properties relation, experimental characterization methods that can locally determine the unknown structure parameters with sufficient precision are required [12.2, 12.5, 12.6, 12.7]. A precision of the order of \(0.01{-}0.1\,{\mathrm{\AA{}}}\) is needed for the atomic positions [12.8, 12.9]. If we can determine the type and position of atoms with sufficient precision, the atomic structure can be linked to the physicochemical properties. A common approach to understanding materials' properties is to use theoretical ab-initio calculations that allow one to obtain equilibrium atomic positions for a given composition. Once this equilibrium structure has been obtained, properties can be computed, and predictions of how the material would behave under different environmental conditions, even beyond the capability of any laboratory, can be performed. In this manner, materials science is gradually evolving toward materials design, that is, from describing and understanding toward predicting materials with interesting properties [12.10, 12.11, 12.12, 12.13, 12.7]. The physicochemical properties of nanostructures are controlled by the shape, size, and atomic arrangement, as well as the electronic state and chemical composition. The aim of TEM is, therefore, to measure those structure parameters as accurately and precisely as possible from the experimental data.

In the last decade, remarkable high-technology developments in lens design have greatly improved the image resolution. Currently, a resolution of the order of \({\mathrm{50}}\,{\mathrm{pm}}\) can be achieved [12.14, 12.15, 12.16, 12.17, 12.18]. For most atomic types, this exceeds the point where the width of the electrostatic potential of the atoms is the limiting factor [12.19]. In addition, new data collection geometries that allow one to optimize the experimental settings are emerging [12.20, 12.21, 12.22, 12.23, 12.24]. Furthermore, detectors behave increasingly more as ideal quantum detectors [12.25]. In this manner, the microscope itself becomes less restricting, and the quality of the experimental images is set mainly by the unavoidable presence of electron counting noise. However, when aiming for precise structure parameters, signals need to be interpreted quantitatively. Therefore, the focus in TEM research has gradually moved from obtaining better resolution to improving precision. To reach this goal, the use of statistical parameter estimation theory is of great help [12.26, 12.27, 12.28].

In this chapter, a concise overview of the methods that can be applied for the solution of a general type of parameter estimation problem often met in materials characterization or applied science and engineering will be presented. In particular, the maximum likelihood estimator will be discussed, as well as the limits set to the precision that can be achieved. Next, an overview of applications of parameter estimation in the field of TEM will be given. In these applications, the goal is to determine unknown structure parameters, including atomic positions, chemical concentrations, and atomic numbers, as precisely as possible from experimentally recorded images. In this manner, it will be shown that statistical parameter estimation theory allows one to measure two-dimensional () atomic column positions with subpicometer precision, to measure compositional changes at interfaces, to count atoms with single atom sensitivity, and to reconstruct three-dimensional () atomic structures.

12.1 Model-Based Parameter Estimation

In general, the aim of statistical parameter estimation theory is to determine, or more correctly, to estimate, unknown physical quantities or parameters on the basis of observations that are acquired experimentally [12.29]. In many scientific disciplines, observations are usually not the quantities to be measured themselves but are related to the quantities of interest. Often, this relation is a known mathematical function derived from physical laws. The quantities to be determined are parameters of this function. Parameter estimation, then, is the computation of numerical values for the parameters from the available observations. In TEM, the observations of a specific object are, for example, the image pixel values recorded using a charge coupled device (CCD) camera. A parametric model describing these observations should then include all ingredients needed to perform a computer simulation of the images, i. e., the electron–object interaction, the transfer of the electrons through the microscope, and the image detection. If first principles-based models cannot be derived, or are too complex for their intended use, simplified empirical models may be used. Some of the model's parameters are the atomic positions and atomic types. The parameter estimation problem then becomes one of computing the atomic positions and atomic types from TEM images. Statistical parameter estimation theory provides an elegant solution for such problems. Indeed, based on the availability of a parametric model, the unknown structure parameters can be estimated by fitting the model to the experimental images in a refinement procedure, usually called estimation procedure or estimator. In general, different estimation procedures can be used to estimate the unknown parameters of this model, such as least squares (LS), least absolute deviations , or maximum likelihood (ML ) estimators [12.26, 12.29, 12.30]. In practice, the ML estimator is often used, since it is known to be the most precise. Section 12.1.1 discusses the derivation of a parametric statistical model of the observations. Subsequently, the ML estimator is reviewed in Sect. 12.1.2. For a detailed overview on statistical parameter estimation theory, the reader is referred to [12.26, 12.29, 12.30, 12.31, 12.32, 12.33].

12.1.1 Parametric Statistical Model of Observations

Generally, due to the inevitable presence of noise, sets of observations made under the same conditions differ from experiment to experiment. The usual way to describe the fluctuating behavior of 2-D images in the presence of noise is by modeling the noisy pixel values as stochastic variables. By definition, a set of observations, \(\boldsymbol{w}=(w_{11},\ldots,w_{\text{KL}})^{\mathrm{T}}\), is defined by its joint probability density function ( ) \(p_{\boldsymbol{w}}(\boldsymbol{\omega})\). The joint PDF defines the expectations, i. e., the mean value of each observation and the fluctuations about these observations. The expectation values \(E[w_{kl}]\equiv\lambda_{kl}\) are described by parametric models \(f_{kl}(\theta)\). The availability of such a model makes it possible to parameterize the PDF of the observations, which is of vital importance for quantitative structure determination , as will be shown in the remainder of this chapter. Obviously, it is important to test the validity of the expectation model before attaching confidence to the structure determination results obtained using the model. If the model is inadequate, it must be modified and the analysis continued until a satisfactory result is obtained. A review of statistical model assessment methods can be found in [12.26].

As an example, a parametric model often used for the description of high-resolution (scanning) (S)TEM images will be discussed. For (S)TEM images, the intensity is sharply peaked at the atomic column positions [12.34, 12.35]. Therefore, (S)TEM images are often modeled as a superposition of Gaussian peaks. The expectation of the intensity of pixel \((k,l)\) at position \((x_{k},y_{l})\) can then be described by an expectation model \(f_{kl}(\theta)\), with \(\theta\) the vector of unknown structure parameters, as
$$\begin{aligned}\displaystyle f_{kl}(\theta)&\displaystyle=\zeta+\sum_{i=1}^{I}\sum_{m_{i}}^{M_{i}}\eta_{m_{i}}\\ \displaystyle&\displaystyle\quad\,\times\exp\left(-\frac{\left(x_{k}-\beta_{x_{m_{i}}}\right)^{2}+\left(y_{l}-\beta_{y_{m_{i}}}\right)^{2}}{2\rho_{i}^{2}}\right),\end{aligned}$$
(12.1)
with \(\zeta\) a constant background, \(\rho_{i}\) the column-dependent width of the Gaussian peak, \(\eta_{m_{i}}\) the column intensity of the \(m_{i}\)th Gaussian peak, and \(\beta_{x_{m_{i}}}\) and \(\beta_{y_{m_{i}}}\) the \(x\) and \(y\)-coordinates of the \(m_{i}\)-th atomic column, respectively. The index \(i\) refers to atomic columns of the same atom type with \(I\) different types, and the index \(m_{i}\) refers to the \(m\)-th column of type \(i\) with \(M_{i}\) the number of columns of type \(i\). The indices in the summation of (12.1) can be simplified in the case of a monotype crystalline nanostructure, since then only one column type is present. The unknown parameters of the parametric imaging model of (12.1) are given by the parameter vector \(\theta\)
$$\begin{aligned}\displaystyle\theta&\displaystyle=(\beta_{x_{1_{1}}},\ldots,\beta_{x_{M_{I}}},\beta_{y_{1_{1}}},\ldots,\beta_{y_{M_{I}}},\\ \displaystyle&\displaystyle\qquad\rho_{1},\ldots,\rho_{I},\eta_{1_{1}},\ldots,\eta_{M_{I}},\zeta)^{\mathrm{T}}\;.\end{aligned}$$
(12.2)
When assuming that the observations are independent electron counting results, which can be modeled as a Poisson distribution, the joint PDF is given by
$$p_{w}(\boldsymbol{\omega};\theta)=\prod_{k=1}^{K}\prod_{l=1}^{L}\frac{\lambda{{}_{kl}^{\omega_{kl}}}}{\omega_{kl}!}\exp(-\lambda_{kl})\;.$$
(12.3)
Appealing to the central limit theorem, the assumption of normally distributed observations is often justified in practical cases, where also disturbances other than pure counting statistics contribute. Furthermore, when assuming equal variances \(\sigma^{2}\), the joint probability density function is given by
$$\begin{aligned}\displaystyle p_{w}(\boldsymbol{\omega};\theta)&\displaystyle=\prod_{k=1}^{K}\prod_{l=1}^{L}\frac{1}{\sqrt{2\uppi}\sigma}\\ \displaystyle&\displaystyle\quad\,\times\exp\left[-\frac{1}{2}\left(\frac{\omega_{kl}-\lambda_{kl}}{\sigma}\right)^{2}\right].\end{aligned}$$
(12.4)
Since the expectations \(\lambda_{kl}\) are described by the functional model \(f_{kl}(\theta)\), substitution of (12.1) in (12.3) or in (12.4) shows how the PDF depends on the unknown parameters to be measured.

12.1.2 Maximum Likelihood Estimation

The observations are considered as a data plane from which the unknown parameters have to be estimated in a statistical way. Different estimators can be used to estimate the same unknown parameters of the proposed parametric models . Each estimator will have a different precision, however, the variance of unbiased estimators will never be lower than the Cramér–Rao lower bound ( ), which is a theoretical lower bound on the variance and will be described in Sect. 12.2.1. The ML estimator achieves this theoretical lower bound asymptotically, i. e., for an increasing number of observations, and is, therefore, of practical importance. From the joint PDF of the observations, discussed in Sect. 12.1.1, the ML estimator may be derived. The ML estimates \(\hat{\theta}_{\text{ML}}\) of the parameters \(\theta\) are given by the values of \(\boldsymbol{t}\) that maximize the likelihood function \(p_{\boldsymbol{w}}(\boldsymbol{w};\boldsymbol{t})\) with \(\boldsymbol{t}\) independent variables replacing the true parameters and the observations replacing the stochastic variables in the joint PDF
$$\begin{aligned}\displaystyle\hat{\theta}_{\text{ML}}&\displaystyle=\text{arg }\max_{t}p_{\boldsymbol{w}}(\boldsymbol{w};\boldsymbol{t})\\ \displaystyle&\displaystyle=\text{arg }\max_{\boldsymbol{t}}\ln p_{\boldsymbol{w}}({\boldsymbol{w}};\boldsymbol{t})\;.\end{aligned}$$
(12.5)
The joint PDF \(p_{\boldsymbol{w}}(\boldsymbol{\omega};\hat{\theta}_{\text{ML}})\) with the ML estimates inserted generates the observations with higher probability than the joint probability distribution with another set of parameters \(\hat{\theta}\). The ML estimator equals the LS estimator for independent normally distributed observations. Therefore, for the joint PDF given by (12.4), the ML estimator simplifies to
$$\hat{\theta}_{\text{LS}}=\text{arg }\min_{\boldsymbol{t}}\sum_{k}\sum_{l}[w_{kl}-f_{kl}(\boldsymbol{t})]^{2}$$
(12.6)
with \(f_{kl}\) the parametric model.

The relative ease with which these estimators can be derived does not mean that the optimization is straightforward [12.26]. Finding the ML estimate corresponds to finding the global optimum in a parameter space of which the dimension corresponds to the number of parameters to be estimated. The search for this optimum is usually an iterative numerical procedure. In electron microscopy applications, the dimension of the parameter space is usually very high. Consequently, it is quite possible that the optimization procedure ends up at a local optimum, so that the wrong structure model is suggested, which introduces a systematic error (bias). To solve this dimensionality problem, that is, to find a pathway to the optimum in the parameter space, good starting values for the parameters are required. In other words, the structure has to be resolved. This corresponds to x-ray crystallography, where one first has to resolve the structure and afterwards one has to refine the structure. Resolving the structure is not trivial. It is known that details in TEM images do not necessarily correspond to features in the atomic structure. This is not only due to the unavoidable presence of noise, but also to the dynamic scattering of the electrons on their way through the object and the image formation in the electron microscope, which both have a blurring effect. As a consequence, the structure information of the object may be strongly delocalized, which makes it very difficult to find good starting values for the structure parameters. However, it has been shown that starting values for the parameters, and hence a good starting structure of the object, may be found by using so-called direct methods. Direct methods, in a sense, invert the imaging process and the dynamic scattering process using some prior knowledge, which is generally valid, irrespective of the (unknown) structure parameters of the object. The starting structure obtained with such a direct method can be obtained in different ways. Examples of such methods are high-voltage electron microscopy, aberration correction in the electron microscope, high-angle annular dark-field ( ) STEM, focal-series reconstruction, and off-axis holography. A common goal of these methods is to improve the interpretability of the experimental images in terms of the structure and may as such yield an approximate solution that can be used as a starting structure for ML estimation from the original images. Furthermore, a direct implementation of the ML estimator in which all parameters are estimated at the same time is computationally very intensive and is only feasible for images containing a limited number of projected atomic columns in the (S)TEM images, i. e., a limited field of view. To overcome this problem, a user-friendly software package, StatSTEM, which enables the quantitative analysis of large fields of view, has been developed [12.36]. The basic idea of the underlying algorithm is the segmentation of the image into smaller sections containing individual columns without ignoring overlap between neighboring columns.

12.2 Experiment Design

In general, the purpose of experiment design in electron microscopy is to set up experiments in such a way that unknown structure parameters of the sample under study can be estimated as precisely as possible from the data obtained. Ultimately, the precision with which these parameters can be estimated is limited by noise. The goal of statistical experiment design is to answer the question: which microscope settings are expected to yield the highest precision with which structure parameters, such as, atomic column positions, particle size, and thickness, can be estimated? Previous work has shown that the CRLB is a very efficient way to answer this question [12.19, 12.26, 12.27, 12.37, 12.38, 12.39, 12.40, 12.41, 12.42, 12.43, 12.44, 12.45]. This lower bound provides a theoretical lower bound on the variance of unbiased estimators of these parameters and can be computed from the parameterized PDF discussed in Sect. 12.1.1. By calculating the CRLB, the experimenter is able to compute the design so as to attain maximum precision. So far, studies on the precision of atomic scale measurements from (S)TEM images considered the estimation of the position of atoms or atomic columns in projection [12.19, 12.26, 12.27, 12.37, 12.38, 12.39, 12.40, 12.41, 12.42, 12.43], the atomic column thickness [12.44], and nanoparticle's sizes [12.45]. In these papers, it has been shown how optimizing the design of quantitative electron microscopy experiments may substantially enhance the precision of the structure parameter estimators. The common aspect in these studies is the continuous differentiability of the PDF with respect to the parameters. However, when estimating a so-called restricted (or discrete) parameter from a (S)TEM image, such as the atomic number (\(Z\)), or the number of atoms in a projected atomic column, this condition is no longer satisfied and, hence, the CRLB is not defined. Therefore, an alternative approach has been developed for estimating discrete parameters using the principles of detection theory [12.22, 12.30, 12.46, 12.47]. This framework allows one to formulate a discrete parameter estimation problem as a binary or multiple hypothesis test, where each hypothesis corresponds, for example, to the assumption of a specific \(Z\) value or a specific number of atoms in the column. Furthermore, statistical detection theory provides the tools to compute the probability to assign an incorrect hypothesis. This so-called probability of error can be computed as a function of the experimental settings and, hence, can be used instead of the CRLB to optimize the experiment design for discrete parameter estimation problems.

12.2.1 Attainable Precision: The Cramér–Rao Lower Bound

Different estimators of the same parameters generally have a different precision. The question then arises as to what precision may be ultimately achieved from a particular set of observations. For the class of unbiased estimators (bias equals zero), this answer is given in the form of a lower bound on their variance, the CRLB [12.48, 12.49, 12.50]. Let \(p_{\boldsymbol{w}}(\boldsymbol{\omega};\theta)\) be the joint PDF of a set of observations \(\boldsymbol{w}=(w_{11},\ldots,w_{KL})^{\mathrm{T}}\). An example of this function is given in Sect. 12.1.1. The dependence of \(p_{\boldsymbol{w}}(\boldsymbol{\omega};\theta)\) on the \(R\times 1\) parameter vector \(\theta\) can now be used to define the so-called Fisher information matrix
$$F=-E\left[\frac{\partial^{2}\ln p_{\boldsymbol{w}}(\boldsymbol{w};\theta)}{\partial\theta\,\partial\theta^{\mathrm{T}}}\right],$$
(12.7)
which is an \(R\times R\) matrix. The expression between square brackets represents the Hessian matrix of the logarithm of the joint PDF of which the \((r,s)\)-th element is defined by \(\partial^{2}\ln p_{\boldsymbol{w}}(\boldsymbol{\omega};\theta)/\partial\theta_{r}\partial\theta_{s}\). The Fisher information expresses the inability to know a measured quantity [12.50]. Indeed, using the concept of Fisher information allows one to determine the highest precision, that is, the lowest variance, with which a parameter can be estimated unbiasedly. Suppose that \(\widehat{\theta}\) is any unbiased estimator of \(\theta\), that is,
$$E\left[\widehat{\theta}\right]=\theta\;.$$
Then it can be shown that under general conditions the covariance matrix \(\text{cov}(\widehat{\theta})\) of \(\widehat{\theta}\) satisfies
$$\text{cov}(\widehat{\theta})\geq F^{-1}\;,$$
(12.8)
so that \(\text{cov}(\widehat{\theta})-F^{-1}\) is positive semi-definite. A property of a positive semi-definite matrix is that its diagonal elements cannot be negative. This means that the diagonal elements of \(\text{cov}(\widehat{\theta})\), that is, the actual variances of
$$\widehat{\theta}_{1},\ldots,\widehat{\theta}_{R}$$
are larger than or equal to the corresponding diagonal elements of \(F^{-1}\)
$$\text{var}(\widehat{\theta}_{r})\geq[F^{-1}]_{rr}\;,$$
(12.9)
where \(r=1,\ldots,R\) and \([F^{-1}]_{rr}\) is the \(r\)-th diagonal element of the inverse of the Fisher information matrix. In this sense, \(F^{-1}\) represents a lower bound for the variances of all unbiased estimators \(\widehat{\theta}\). The matrix \(F^{-1}\) is the CRLB on the variance of \(\widehat{\theta}\).
Often, the question arises how to measure atomic positions with picometer precision if the resolution of the instrument is only \({\mathrm{50}}\,{\mathrm{pm}}\) under optimal conditions. Resolution and precision are very different notions [12.31]. In (S)TEM, resolution expresses the ability to visually distinguish neighboring atomic columns in an image. Classical resolution criteria, such as Rayleigh's, are derived from the assumption that the human visual system needs a minimal contrast to discriminate two points in its composite intensity distribution [12.51]. Therefore, they are expressed in terms of the width of the point spread function of the (S)TEM imaging system [12.52]. However, if the physics behind the image formation process is known, images no longer need to be interpreted visually. Instead, atomic column positions can be estimated by fitting this known parametric model to an experimental image using the ML estimator. In the absence of noise, this procedure would result in infinitely precise atomic column locations. However, since detected images are never noise free, model fitting never results in a perfect reconstruction, thus limiting the statistical precision with which the atom locations can be estimated. For continuous parameters, such as the atomic column positions, the attainable precision can be adequately quantified using the expression for the CRLB. Under certain assumptions, it can then be shown that the attainable precision, expressed in terms of the standard deviation with which the position of a projected atomic column can be estimated, is approximately equal to [12.38, 12.40, 12.41]
$$\sigma_{\text{CR}}\approx\frac{\rho}{\sqrt{N}}\;,$$
(12.10)
where \(\rho\) represents the width of the Gaussian peaks and \(N\) represents the number of detected electrons per atom or atomic column. The width of the peaks can be shown to be proportional to the Rayleigh resolution [12.19]. This explains why the precision to estimate projected atomic column positions can be down to 1 or a few picometers, although the resolution of modern instruments is \(50{-}100\,{\mathrm{pm}}\). In order to push the precision further by a factor of 10, it is necessary to increase the dose by a factor of \(\mathrm{100}\), which will require a very high incoming dose and/or a long exposure time.

Note that the CRLB is not related to a particular estimation method and that the existence of a lower bound on the parameter variance does not imply that an estimator can be found that reaches this lower bound. However, it is known that there exists an estimator that achieves the CLRB at least asymptotically, that is, for an increasing number of observations. This estimator is the ML estimator, which has been discussed in Sect. 12.1.2. It is known to be asymptotically normally distributed with a mean equal to the true value of the parameter and a covariance matrix equal to the CRLB. In electron microscopy the number of observations is usually sufficiently large for the asymptotic properties of the ML estimator to apply. For this and other reasons, the use of the ML estimator in quantitative electron microscopy is highly recommended. Moreover, approximate confidence regions and intervals for ML parameter estimates can be obtained based on the asymptotic statistical properties of the ML estimator. In this approach, the CRLB is approximated by substituting the ML parameter estimates for the true values of the parameters [12.26].

12.2.2 Probability of Error

For discrete estimation problems, statistical detection theory provides the tools to optimize the experiment design by using a statistical hypothesis test [12.46]. This can be either a binary or a multiple hypothesis test, in which every hypothesis corresponds, for example, to a specific atomic number \(Z\) or a specific number of atoms in a projected atomic column. The probability of deciding the wrong hypothesis, the so-called probability of error, can be defined and decision rules are determined in such a way that the probability of error is minimized. In order to optimize the experiment design when measuring discrete parameters, such as, the presence or absence of a specific projected atomic column, or the number of atoms in a projected atomic column, this probability of error may be used as an optimality criterion, by computing it as a function of the experimental settings [12.22, 12.23, 12.30, 12.47, 12.53]. The optimal experiment design then corresponds to those experimental settings that result in the lowest probability of error.

When considering the problem of deciding between two different atom types, detecting a light atom or deciding between the presence of \(n\) or \(n+1\) atoms in a projected atomic column, a binary hypothesis test can be used. In these cases, the estimation problem can be described as deciding between a so-called null hypothesis \(\mathcal{H}_{0}\) and the alternative hypothesis \(\mathcal{H}_{1}\)
$$\mathcal{H}_{0}:Z =Z_{0}\;, \mathcal{H}_{1}:Z =Z_{1}\;,$$
(12.11a)
$$\mathcal{H}_{0}:Z =Z_{0}\;, \mathcal{H}_{1}:Z \in\emptyset\;,$$
(12.11b)
$$\mathcal{H}_{0}:n_{\mathcal{H}_{0}} =n\;, \mathcal{H}_{1}:n_{\mathcal{H}_{1}} =n+1\;.$$
(12.11c)
In the case of (12.11a), both hypotheses correspond to two different possible atomic numbers, \(Z_{0}\) and \(Z_{1}\), in (12.11b); the hypotheses correspond to whether the light atom is present or absent, and in (12.11c), the hypotheses correspond to two succeeding numbers of atoms in a projected atomic column. In binary hypothesis testing problems, a priori knowledge is usually assumed assuring that only \(\mathcal{H}_{0}\) or \(\mathcal{H}_{1}\) is possible, so that one of both hypotheses is always correct. In order to express a prior belief in the likelihood of the hypotheses, the prior probabilities \(P(\mathcal{H}_{0})\) and \(P(\mathcal{H}_{1})\) associated with these hypotheses are assumed to be known, such that \(P(\mathcal{H}_{0})+P(\mathcal{H}_{1})=1\). If both hypotheses are equally likely, then it is reasonable to assign equal prior probabilities of \(1/2\). In a quantitative approach, the goal is now to minimize the probability of assigning the wrong hypothesis. In a so-called Bayesian approach, this probability of error \(P_{\mathrm{e}}\) is defined as
$$\begin{aligned}\displaystyle P_{\mathrm{e}}&\displaystyle=\text{Pr}\{\text{decide }\mathcal{H}_{0},\mathcal{H}_{1}\text{ true}\}\\ \displaystyle&\displaystyle\quad\,+\text{Pr}\{\text{decide }\mathcal{H}_{1},\mathcal{H}_{0}\text{ true}\}\\ \displaystyle&\displaystyle=P(\mathcal{H}_{0}|\mathcal{H}_{1})P(\mathcal{H}_{1})+P(\mathcal{H}_{1}|\mathcal{H}_{0})P(\mathcal{H}_{0})\;,\end{aligned}$$
(12.12)
with \(P(\mathcal{H}_{i}|\mathcal{H}_{j})\) the conditional probability of deciding \(\mathcal{H}_{i}\), while \(\mathcal{H}_{j}\) is true. Using criterion (12.12), the two possible errors are weighted appropriately to yield an overall error measure. Decision rules are now defined such that the probability of error is minimized. It is shown in [12.46] that one, therefore, should decide \(\mathcal{H}_{1}\) if
$$\frac{p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{1})}{p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{0})}> \frac{P(\mathcal{H}_{0})}{P(\mathcal{H}_{1})}=\gamma\;,$$
(12.13)
otherwise \(\mathcal{H}_{0}\) is decided. In this expression, \(p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{i})\) is the conditional (joint) PDF \(p_{\boldsymbol{w}}(\boldsymbol{\omega};\mathcal{H}_{i})\) assuming \(\mathcal{H}_{i}\) to be true, evaluated at the available observations \(\boldsymbol{w}\). For equal prior probabilities of \(1/2\), it is clear that \(\gamma\) in (12.13) corresponds to 1. Then, we decide \(\mathcal{H}_{1}\) if
$$\mathrm{LR}(\boldsymbol{w})=\frac{p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{1})}{p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{0})}> 1\;.$$
(12.14)
The function \(\mathrm{LR}(\boldsymbol{w})\) is called the likelihood ratio, since it indicates for each set of observations \(\boldsymbol{w}\) the likelihood of \(\mathcal{H}_{1}\) versus the likelihood of \(\mathcal{H}_{0}\). This test is, therefore, also known as the likelihood ratio test. Similarly, decision rule (12.14) corresponds to deciding \(\mathcal{H}_{1}\) if
$$\ln\mathrm{LR}(\boldsymbol{w})=\ln p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{1})-\ln p_{\boldsymbol{w}}(\boldsymbol{w};\mathcal{H}_{0})> 0\;.$$
(12.15)
Otherwise \(\mathcal{H}_{0}\) is decided. This decision rule corresponds to choosing the hypothesis for which the log-likelihood function is maximal. The left-hand side of (12.15) is termed the log-likelihood ratio. When assuming independent, Poisson distributed observations, for which the joint PDF is given by (12.3), the log-likelihood ratio can be rewritten as
$$\begin{aligned}\displaystyle&\displaystyle\ln\mathrm{LR}(\boldsymbol{w})\\ \displaystyle&\displaystyle\quad=\sum_{k=1}^{K}\sum_{l=1}^{L}\left(w_{kl}\ln\left(\frac{\lambda_{\mathcal{H}_{1},kl}}{\lambda_{\mathcal{H}_{0},kl}}\right)-\lambda_{\mathcal{H}_{1},kl}+\lambda_{\mathcal{H}_{0},kl}\right),\end{aligned}$$
(12.16)
where \(\lambda_{\mathcal{H}_{i},kl}\) corresponds to the expectation of the intensity of pixel \((k,l)\) under hypothesis \(\mathcal{H}_{i}\). If an experiment is repeated under the same conditions, it can be shown that the distribution of the log-likelihood ratio tends to follow a normal distribution following the central limit theorem.

As an example, the problem of optimizing the annular STEM detector in order to detect the lightest H atom in \(\mathrm{YH_{2}}\) is considered. For this material, we consider the problem of optimizing the annular STEM detector in order to detect the lightest \(\mathrm{H}\) atom. One has been able to experimentally detect \(\mathrm{H}\) in this material by using an annular bright field ( ) STEM detector [12.54]. Here, our goal is to test whether the same optimal detector type is found using our quantitative approach and to determine the exact optimal detector angles [12.22]. The expectation models are simulated using STEMsim [12.55] both for the crystal in the presence and absence of hydrogen, corresponding to the hypotheses \({\mathcal{H}_{0}}:Z=1\) and \(\mathcal{H}_{1}:Z\in\emptyset\). The detailed simulation parameters can be found in [12.22]. Simulated images are shown in Fig. 12.1a-ga–c for detector collection angles of \(11{-}53\), \(22{-}53\), and \(11{-}17\,{\mathrm{mrad}}\). The corresponding distributions of the log-likelihood ratio in the case of the presence (black) and absence (brown) of H in \(\mathrm{YH_{2}}\) are shown in Fig. 12.1a-gd–f. The black and brown colored areas correspond to the probability of deciding \(\mathcal{H}_{0}\) while \(\mathcal{H}_{1}\) is true and the probability of deciding \(\mathcal{H}_{1}\) while \(\mathcal{H}_{0}\) is true, respectively. The sum of both areas represents the probability of error. These figures illustrate that the probability of error depends on the choice of the detector collection angles. By evaluating the probability of error as a function of the inner and outer detector angles, the experiment design can hence be optimized. Results of the probability of error for the detection of H in \(\mathrm{YH_{2}}\) are shown in Fig. 12.1a-gg. As an optimal detector setting, the ABF STEM regime is found with a detector ranging from \(\mathrm{11}\) to \({\mathrm{17}}\,{\mathrm{mrad}}\). Also a local optimum is observed in the low-angle annular dark field ( ) regime with both inner and outer detector radius larger than the probe semi-convergence angle. In general, it should be noticed that LAADF STEM is often competitive and that the optimal detector settings largely depend on the sample type and thickness.

Fig. 12.1a-g

Simulated STEM images of a \(\mathrm{YH_{2}}\) unit cell viewed from the [010] direction, for annular detector collection ranges of (a\(11{-}53\,{\mathrm{mrad}}\), (b\(22{-}53\,{\mathrm{mrad}}=\text{ADF}\), and (c\(11{-}17\,{\mathrm{mrad}}=\text{ABF}\). The corresponding log-likelihood ratio distributions are shown in (df) for the presence (black) and absence (brown) of H in \(\mathrm{YH_{2}}\). (g) The probability of error as a function of the inner and outer detector angle at Scherzer conditions for an electron dose of \({\mathrm{2000}}\,{\mathrm{e^{-}/\AA{}^{2}}}\) at \({\mathrm{300}}\,{\mathrm{kV}}\) and a probe semi-convergence angle of \({\mathrm{21.8}}\,{\mathrm{mrad}}\) for the detection of \(\mathrm{H}\) in a \({\mathrm{2.6}}\,{\mathrm{nm}}\) thick \(\mathrm{YH_{2}}\) crystal. Reprinted from [12.22], with the permission of AIP Publishing

12.3 Quantitative Atomic Column Position Measurements

When the goal is to measure shifts of the atomic positions, aberration-corrected TEM, exit wave reconstruction methods, or combinations of both are often used in practice. Whereas aberration correction has an immediate impact on the resolution of the experimental images, the exit plane of the sample under study is reconstructed using exit wave reconstruction. Usually, a series of images taken at different defocus values, an electron holographic image, or a series of images recorded with different illuminating beam tilts is used as an input to reconstruct the exit wave [12.56, 12.57, 12.58, 12.59, 12.60]. Ideally, the exit wave is free from any imaging artifacts, thus enhancing the visual interpretability of the atomic structure. Exit wave reconstruction has become a powerful tool in high-resolution TEM because of its potential to visualize light atomic columns, such as oxygen or nitrogen, with atomic resolution [12.61, 12.62]. In particular, its combination with quantitative methods nowadays demonstrates its potential to precisely measure atomic column positions [12.63, 12.64, 12.65]. As an example, the quantification of localized displacements at a {110} twin boundary in orthorhombic \(\mathrm{CaTiO_{3}}\) will be discussed [12.66]. Theoretical studies show that such domain boundaries are mainly ferrielectric with maximum dipole moments at the wall. To investigate these boundaries experimentally, the exit wave has been reconstructed using aberration-corrected TEM. Figure 12.2a shows the reconstructed phase when imaging the sample along the [001]-direction with a resolution of \({\mathrm{80}}\,{\mathrm{pm}}\). This phase is directly proportional to the projected electrostatic potential of the structure. Next, statistical parameter estimation is used to obtain quantitative numbers for the atomic column positions [12.26, 12.27, 12.31]. In this manner, atomic columns can be located with a precision of a few picometers without being restricted by the information limit of the microscope. To reach this result, the phase of the reconstructed exit wave is considered as a data plane from which the atomic column positions are estimated using the LS estimator, defined by (12.6). The estimated column positions thus correspond to the numbers for which the LS sum is minimal with \(w_{kl}\) corresponding to the pixel values of the reconstructed phase and \(f_{kl}\) the parametric model, which is given by (12.1). The estimated numbers could be used to measure shifts in the atomic column positions. It has been found that possible shifts in the Ca atomic positions are too small to be identified, whereas shifts in the Ti atomic positions in the vicinity of the twin wall are statistically significant [12.66]. Therefore, this analysis is focused on the off-centering of the Ti atomic positions with respect to the center of the neighboring four Ca atomic positions. First, we average all displacements in planes parallel to the twin wall. Next, we average the results in the planes above with the corresponding planes below the twin wall. This second operation identifies the overall symmetry of the sample with the twin wall representing a mirror plane. The resulting displacements perpendicular to and along the twin wall are shown in Fig. 12.2b and c, respectively, together with their \({\mathrm{90}}\%\) confidence intervals. In the direction perpendicular to the wall, systematic deviations for Ti of \({\mathrm{3.1}}\,{\mathrm{pm}}\) in the second closest layers pointing toward the twin wall are found. A larger displacement is measured in the direction parallel to the wall in the layers adjacent to the twin wall. The averaged displacement in these layers is \({\mathrm{6.1}}\,{\mathrm{pm}}\). In layers further away from the twin wall, no systematic deviations are observed. These experimental results confirm the theoretical predictions [12.67].

Fig. 12.2

(a) Experimental phase image of a (110) twin boundary in orthorhombic \(\mathrm{CaTiO_{3}}\). Mean displacements of the Ti atomic columns from the center of the four neighboring Ca atomic columns are indicated by arrows. (b,c) Displacements of Ti atomic columns perpendicular and parallel to the twin wall averaged along and in mirror operation with respect to the twin wall together with their \({\mathrm{90}}\%\) confidence intervals. From [12.66]

Another efficient technique to measure shifts of the atomic positions is so-called negative-spherical-aberration imaging in which the spherical aberration constant \(C_{\mathrm{s}}\) is tuned to negative values by employing an aberration corrector [12.68, 12.69]. As compared to traditional positive \(C_{\mathrm{s}}\) imaging, this imaging mode yields a negative phase contrast of the atomic structure, with atomic columns appearing bright against a darker background. For thin objects, this leads to a substantially higher contrast compared to the dark atom images formed under positive \(C_{\mathrm{s}}\) imaging. This enhanced contrast has the effect of improving the measurement precision of the atomic positions and explains the use of this technique to measure atomic shifts of the order of a few picometers. Examples show measurements of the width of ferroelectric-domain walls in \(\mathrm{PbZr_{0.2}Ti_{0.8}O_{3}}\) [12.16], measurements of the coupling of elastic strain fields to polarization in \(\mathrm{PbZr_{0.2}Ti_{0.8}O_{3}/SrTiO_{3}}\) epitaxial systems [12.70], and of oxygen-octahedron tilt and polarization in \(\mathrm{LaAlO_{3}/SrTiO_{3}}\) interfaces [12.17].

12.4 Quantitative Composition Analysis

HAADF STEM imaging, in which an annular detector is used with a collection range outside of the illumination cone, is a very convenient method for structure characterization at the atomic level. Because of the high-angle scattering, the signal is dominated by Rutherford and thermal diffuse scattering and, hence, approximately scales with the square of the atomic number \(Z\). One of the advantages of this \(Z\)-contrast is the possibility to visually distinguish between chemically different atomic column types. Furthermore, the resolution observed in an HAADF STEM image is mainly set by the intensity distribution of the illuminating probe. Nowadays, a probe size of the order of \({\mathrm{50}}\,{\mathrm{pm}}\) can be attained when using aberration corrected probe-forming optics [12.18]. This high spatial resolution combined with the high chemical sensitivity means that HAADF STEM images are to a certain extent directly interpretable. Despite this advantage, HAADF STEM is also of great benefit when analyzing the resulting images using statistical parameter estimation theory [12.71]. Especially when the difference in atomic number of distinct atomic column types is small or when the signal-to-noise ratio is low, visual interpretation becomes insufficient.

Cross-sections are well known and often used in particle scattering experiments as a measure of the probability of scattering. In STEM, a scattering cross-section approach was first proposed by Retsky in 1974 [12.72]. However, because HAADF STEM images are to some extent interpretable directly, scattering cross-sections were little used until their importance was recently realized to interpret images quantitatively in terms of structure and composition. In HAADF STEM imaging, scattering cross-sections correspond to the total scattered intensity for each atomic column and can be measured using statistical parameter estimation theory [12.71] or by integrating intensities over the probe positions in the vicinity of a single column of atoms [12.73]. The advantage of using scattering cross-sections over other metrics, such as peak intensities, is their robustness to magnification, defocus, source size, astigmatism, and small sample mistilt [12.73, 12.74, 12.75]. Moreover, this measure is very sensitive to changes in composition and/or thickness [12.23]. The estimated scattering cross-sections allow us to detect differences in averaged atomic number of only 3. This is illustrated in the following example, in which a \(\mathrm{La_{0.7}Sr_{0.3}MnO_{3}}\)-\(\mathrm{SrTiO_{3}}\) multilayer structure is investigated. Figure 12.3a shows an enlarged area from an experimental image using an FEI Titan 50-80 operated at \({\mathrm{300}}\,{\mathrm{kV}}\). No visual conclusions could be drawn concerning the sequence of the atomic planes at the interfaces. To overcome this problem, the parameters of the parametric model, which is given by (12.1), have been estimated using the LS estimator given by (12.6). The refined parametric model is shown in Fig. 12.3b, which illustrates a close match with the experimental data. Figure 12.3c shows the experimental observations together with an overlay indicating the estimated positions of the columns and their atomic column types. Cross-sections can be derived from the estimated parameters,
$$V_{m_{i}}=2\uppi\eta_{m_{i}}\rho_{i}^{2}\;.$$
(12.17)
The composition of the columns away from the interfaces is assumed to be in agreement with the composition in the bulk compounds. Histograms of the estimated scattering cross-sections of these known columns are presented in Fig. 12.3d and show the random nature of the result. The colored vertical bands correspond to \({\mathrm{90}}\%\) tolerance intervals. It is important to note that these tolerance intervals do not overlap, which means that columns, for which the difference in averaged atomic number is only 3 (TiO and MnO) in this example, can clearly be distinguished. Based on this histogram, the composition of the unknown (purple colored) columns in the planes close to the interface could be identified as shown on the right-hand side of Fig. 12.3c. Single-color dots are used to indicate columns whose estimated scattering cross-section falls inside a tolerance interval, whereas pie charts, indicating the presence of intermixing or diffusion, are used otherwise.
Fig. 12.3

(a) Area from an experimental HAADF STEM image of a \(\mathrm{La_{0.7}Sr_{0.3}MnO_{3}}\)-\(\mathrm{SrTiO_{3}}\) multilayer structure. (b) Refined parametric model. (c) Overlay indicating the estimated positions of the columns together with their atomic column types. (d) Histograms of the estimated scattering cross-sections of the known columns. Reprinted from [12.71], with permission from Elsevier

In the previous example, the chemical composition was quantified in a relative manner by comparing scattering cross-sections of unknown columns with those of known columns. When aiming for an absolute quantification, intensity measurements relative to the intensity of the incoming electron beam are required [12.76, 12.77]. In this manner, experimental scattering cross-sections can be directly compared with simulated scattering cross-sections [12.77, 12.78, 12.79]. Reference cross-section values are then simulated by carefully matching experimental imaging conditions for a range of sample conditions, including thickness and composition. To illustrate this, Fig. 12.4a shows part of a normalized image of a \(\mathrm{Pb_{1.2}Sr_{0.8}Fe_{2}O_{5}}\) compound where the intensities are normalized with respect to the incoming electron beam [12.79]. By comparing experimental scattering cross-sections for each atomic column with simulated values, the thickness values for the PbO columns and the composition for the SrPbO columns was determined as shown in Fig. 12.4b. Figure 12.4c compares the averaged experimental intensity profile along the vertical direction of the unit cell indicated in Fig. 12.4b with a frozen lattice simulation, where the estimated thickness and composition values were used as an input. The overall match between the simulated and experimental image intensities further confirms the results that have been obtained when using the scattering cross-sections approach. However, it should be noted that small deviations between simulated and experimental image intensities cannot be avoided because of, for example, remaining uncertainties in the microscope settings such as defocus, source size, or astigmatism.

Fig. 12.4

(a) Area from an experimental HAADF STEM image of a \(\mathrm{Pb_{1.2}Sr_{0.8}Fe_{2}O_{5}}\) compound where the intensities are normalized with respect to the incoming electron beam. (b) Quantification results showing the estimated thickness values at the PbO site and the estimated composition for the SrPbO atomic columns. (c) Comparison of the averaged experimental intensity profile along the vertical direction of the unit cell indicated in (b) together with a frozen lattice simulation assuming the thickness and composition values shown in (b). Reprinted from [12.79], with permission from Elsevier

12.5 Atom Counting

Scattering cross-sections are not only sensitive for the composition but also for the number of atoms in an atomic column. Figure 12.5 illustrates how this advantage can be used to count the number of Au atoms from an experimental image of an Au nanorod. The intensities of the HAADF STEM image shown in Fig. 12.5a were normalized with respect to the incident beam [12.77, 12.80]. Next, in a similar manner to the analysis presented in Sect. 12.4, scattering cross-sections of all Au columns were estimated. Figure 12.5b shows the refined parametric model. The histogram of all scattering cross-sections is shown in Fig. 12.5d. Owing to a combination of experimental detection noise and residual instabilities, broadened rather than discrete components are observed in such a histogram. Therefore, these results cannot directly be interpreted in terms of the number of atoms. By evaluation of the so-called integration classification likelihood criterion ( ) in combination with Gaussian mixture model estimation, the presence of 47 components and their respective locations were found for the scattering cross-sections of the Au columns [12.81, 12.82, 12.83, 12.84]. This is illustrated in Fig. 12.5d. From the estimated locations of the components, the number of Au atoms can be quantified, leading to the result shown in Fig. 12.5c. It is important to note that this statistics-based method to count the number of atoms does not require the use of simulations. This approach is robust against systematic errors when two conditions are met; the number of experimental scattering cross-sections per unique thickness should be large enough and the spread of scattering cross-sections should be small enough as compared to the difference between those of differing thicknesses [12.83].

Fig. 12.5

(a) Experimental HAADF STEM image of an Au nanorod. (b) Refined parametric model. (c) Number of Au atoms per column. (d) Histogram of scattering cross-sections of the Au columns together with the estimated mixture model and its individual components (colored curves, which correspond to the colors for the number of atoms in (c)); the inset shows the order selection criterion by ICL indicating the presence of 47 components. (e) Comparison of experimental and simulated scattering cross-sections. Reprinted with permission from [12.84]. Copyright 2013 by the American Physical Society

An alternative method to count atoms is through comparison with image simulations [12.78]. However, a main drawback is that systematic errors are difficult to detect, since the assignment of numbers of atoms will always find a match by comparing experimental scattering cross-section values or peak intensities with simulated values. The reliability of the atom counting results then purely depends on the accuracy with which, for example, the detector inner and outer angles have been determined and the accuracy with which the simulations have been carried out. The use of the simulations-based and independent statistics-based methods allows one to validate the accuracy of the obtained atom counts [12.84, 12.85]. This is illustrated in Fig. 12.5e, which shows the experimental mean scattering cross-sections—corresponding to the component locations in Fig. 12.5d—together with the scattering cross-sections estimated from the frozen phonon calculations using the STEMsim program under the same experimental conditions [12.55]. The excellent match of the experimental and simulated scattering cross-sections within the expected \(5{-}10\%\) error range validates the accuracy of the obtained atom counts [12.78, 12.86]. Furthermore, this step also validates the choice of the local minimum present in the ICL criterion shown in the inset of Fig. 12.5d. The precision of the atom counts is limited by the unavoidable presence of noise in the experimental images, resulting in an overlap of the Gaussian components, as shown in Fig. 12.5d. When the overlap increases, the probability to assign an incorrect number of atoms increases. In this example, the probability to have an error of 1 atom is only \({\mathrm{20}}\%\), whereas the number of atoms of \({\mathrm{80}}\%\) of all columns can be determined without error. The combination of a simulations-based and statistics-based method thus allows for reliable atom-counting with single atom sensitivity .

Ultimately, the simulations-based method and the statistics-based method are combined into a hybrid approach, which overcomes the limitations of both methods. To reach this goal, prior knowledge resulting from image simulations can be incorporated in the statistical framework while taking possible inaccuracies in the experimental parameters into account [12.87]. The use of this hybrid method has clear benefits when analyzing low-dose images of small nanoparticles. This is demonstrated in Fig. 12.6, where the number of Pt atoms was accurately determined from a simulated ADF STEM image when assuming an input electron dose of only \({\mathrm{1000}}\,{\mathrm{e^{-}/{\AA{}}^{2}}}\).

Fig. 12.6

(a) Hypothetical ADF STEM image corresponding to a dose of \({\mathrm{1000}}\,{\mathrm{e^{-}/{\AA{}}^{2}}}\). (b) Atom counts obtained with the hybrid atom counting method. (c) Difference between the input and the estimated atom counts. Reprinted from [12.87], with permission from Elsevier

The previous examples demonstrate how the use of scattering cross-sections has enabled atom counting for monotype crystalline structures. Applications to hetero-nanostructures are significantly more complex, since small changes in atom ordering in the column have an effect on the scattering cross-sections. Therefore, the amount of required simulations increases exponentially with the thickness and the number of elements present in the sample. For example, already more than \(\mathrm{2\times 10^{6}}\) column configurations exist for a 20-atom thick binary alloy. When all possible outcomes need to be considered, this becomes an impossible task in terms of computing time. To help solve this problem, the availability of a model to predict scattering cross-sections as a function of composition, configuration, and thickness is desirable [12.88]. In the simplest model, the assumption of longitudinal incoherence is considered, where the scattered intensity of an atomic column is written as the sum of the scattered intensities of the individual atoms constituting this column. However, this method will not only lead to large deviations, it will also neglect the information about the configuration of the column. A more accurate prediction is obtained when describing each atom as a lens focussing the electrons on the next atom [12.89, 12.90]. As shown in [12.88], the lensing factors of the individual atoms in monotype atomic columns can be calculated in order to predict the scattering cross-sections of mixed columns. This new approach leads to an accurate prediction of scattering cross-sections, which is not restricted to the number of atom types or detector angles. This atomic lensing model can be used to unravel the 3-D composition at the atomic scale. This is demonstrated in Fig. 12.7, where the number of both Ag and Au atoms were counted from an experimental HAADF STEM image of an Ag-coated Au nanorod. In combination with atom counting results obtained from an additional viewing direction and prior knowledge concerning the shape of the nanorod, a 3-D atomic model could be reconstructed.

Fig. 12.7

(a,b) Number of Ag and Au atoms counted from an experimental HAADF STEM image of an Ag-coated Au nanorod and (c) a 3-D reconstructed atomic model. Reprinted with permission from [12.88]. Copyright 2016 by the American Physical Society

12.6 Atomic Resolution in Three Dimensions

As described in the previous sections of this chapter, new developments within the field of TEM enable the investigation of nanostructures at the atomic scale. Structural as well as chemical information can be extracted in a quantitative manner. However, such images are mostly 2-D projections of a 3-D object. To overcome this limitation, 3-D imaging by TEM or electron tomography can be used. Atomic resolution in 3-D has been the ultimate goal in the field of electron tomography during the past few years. The underlying theory for atomic resolution tomography has been well understood [12.91, 12.92], but it was nevertheless challenging to obtain the first experimental results. A first approach is based on the acquisition of a limited number of HAADF STEM images that are acquired along different zone axes [12.82]. As illustrated in Sect. 12.5, advanced quantification methods enable one to count the number of atoms in an atomic column from a 2-D (HA)ADF STEM image. In a next step, such atom counting results can be used as an input for discrete tomography. The discreteness that is exploited here is the fact that crystals can be thought of as discrete assemblies of atoms [12.92]. In this manner, a very limited number of 2-D images is sufficient to obtain a 3-D reconstruction with atomic resolution. This approach was applied to Ag clusters embedded in an Al matrix as illustrated in Fig. 12.8 [12.82]. A 3-D reconstruction was obtained using only two HAADF STEM images. An excellent match was found when comparing the 3-D reconstruction with additional projection images that were acquired along different zone axes. In a similar manner, the core of a free-standing PbSe-CdSe core-shell nanorod could be reconstructed in 3-D [12.93].

Fig. 12.8

Parts of experimental HAADF STEM images of a nanosized Ag cluster embedded in an Al matrix in [\(10\bar{1}\)] and [100] zone-axis orientation together with the number of Ag atoms per column and the computed 3-D reconstruction of the cluster viewed along three different directions. From [12.82]

Tomography typically requires several images demanding a substantial electron dose, which hampers the study of beam-sensitive materials. To overcome this problem, atom counting results obtained from just a single ADF STEM image can be used as an input to retrieve the 3-D atomic structure. In combination with prior knowledge about a material's crystal structure, an initial 3-D configuration is generated. Next, an energy minimization using ab-initio calculations or a Monte Carlo approach is performed to relax a nanoparticle's 3-D structure. This technique was applied to investigate the dynamical behavior of ultra-small Ge clusters consisting of less than 25 atoms [12.94]. Ultra-small nanoparticles or clusters, with sizes below \({\mathrm{1}}\,{\mathrm{nm}}\), form a challenging subject of investigation. One of the main bottlenecks is that these clusters may rotate or show structural changes during investigation by TEM [12.95]. Obviously, conventional electron tomography methods, even those that are based on a limited number of projections, can no longer be applied. On the other hand, the intrinsic energy transfer from the electron beam to the cluster can be considered as a unique possibility to investigate the transformation between energetically excited configurations of the same cluster. Image series were collected using aberration corrected HAADF STEM. From a set of selected frames, the number of atoms at each position could be determined, as illustrated in Fig. 12.9. In order to extract 3-D structural information from these images, ab initio calculations were carried out. Several starting configurations were constructed, which are all in agreement with the experimental 2-D projection images. Although all of the cluster configurations stay relatively close to their starting structure after full relaxation, only those configurations in which a planar base structure was assumed, were found to still be compatible with the 2-D experimental images. In this manner, reliable 3-D structural models were obtained for these small clusters, and also the transformation of a predominantly 2-D configuration into a compact 3-D configuration could be characterized. In a related manner, the 3-D atomic structure of catalytic Pt nanoparticles [12.96], the interface between individual PbSe building blocks in a 2-D superlattice formed by oriented attachment [12.97], and a model-like Au nanoparticle [12.98] have been studied. All of these studies could not have been realized without these new developments circumventing conventional electron tomography.

Fig. 12.9

(a) Statistical counting results for three different configurations of an ultra-small Ge cluster. Green, red, and blue dots correspond to 1, 2, and 3 atoms, respectively. The results of the ab-initio calculations are shown in (b). From [12.94]

12.7 Conclusions

In this chapter, it was shown how quantitative TEM greatly benefits from statistical parameter estimation theory to estimate unknown structure parameters. Ultimately, these parameters need to be estimated as accurately and precisely as possible from the observations available. In Sects. 12.1 and 12.2 of this chapter, a framework was outlined to reach this goal. The parameterized joint PDF of the observations was introduced as a model to describe statistical fluctuations in the observations. This description requires a (usually physics-based) model describing the expectation values of the observations. Furthermore, it requires detailed knowledge about the disturbances that are acting on the observations. A thus parameterized PDF can then be used for two purposes. First, it can be used to derive the ML estimator. The use of this estimator is motivated by the fact that it has favorable statistical properties. Second, from the PDF of the observations, the CRLB can be derived. This is a theoretical lower bound on the variance of any unbiased estimator of the parameters and can be used for the optimization of the experiment design so as to attain the highest precision. However, for discrete parameters, such as the atomic number \(Z\), this lower bound is no longer applicable. Therefore, alternative solutions using the principles of detection theory were proposed.

In Sects. 12.312.6, statistical parameter estimation theory was applied to various kinds of observations acquired by means of TEM. This is becoming increasingly important, since it allows one to quantitatively determine unknown structures at a local scale. Applications in the field of high-resolution (S)TEM show how statistical parameter estimation techniques can be used to overcome the traditional limits set by modern electron microscopy. The precision that can be achieved in this quantitative manner far exceeds the resolution performance of the instrument. The characterization limits are, therefore, no longer imposed by the quality of the lenses but are determined by the underlying physical principles. Structural, but also chemical, electronic, and magnetic information can be obtained at the atomic scale. As demonstrated in this chapter, quantitative structure determination can not only be carried out in 2-D, but also 3-D analyses are currently becoming standard.

Notes

Acknowledgements

The author would like to acknowledge all colleagues who contributed to this work over the years, in particular S. Bals, K.J. Batenburg, A. De Backer, A. De wael, R. Erni, A.J. den Dekker, J. Gonnissen, L. Jones, G.T. Martinez, P.D. Nellist, A. Rosenauer, M.D. Rossell, D. Schryvers, J. Sijbers, K. van den Bos, D. Van Dyck, G. Van Tendeloo, and J. Verbeeck. The author also expresses many thanks for all fruitful and enlightening theoretical discussions, as well as all the shared experimental expertise and knowledge. Sincere thanks are due to A. van den Bos, who unfortunately passed away too soon, for his enthusiastic and expert guidance in the author's understanding of statistical parameter estimation theory.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Electron Microscopy for Materials Research (EMAT)University of AntwerpAntwerpBelgium

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