Model-Based605

PartA|12

12. Model-Based Electron Microscopy

Sandra Van Aert

The growing interest in materials design and con-trol of nanostructures explains the need for precisedetermination of the atomic arrangement of non-periodic structures. This includes, for example,locating atomic column positions with a precisionin the picometer range, a precise determinationof the chemical composition of materials, andcounting the number of atoms with single atomsensitivity. In order to extract these quantitativemeasurements from atomic resolution (scanning)transmission electron microscopy (STEM) images,statistical analysis methods are needed. For thispurpose, statistical parameter estimation theoryhas been shown to provide reliable results. Inthis theory, observations are purely consideredas data planes, from which structure parametershave to be determined using a parametric modeldescribing the images. This chapter summarizesthe underlying theory and highlights some of therecent applications of quantitative model-based(S)TEM.

12.1 Model-Based Parameter Estimation . .. 60612.1.1 Parametric Statistical Model

of Observations ................................. 60612.1.2 Maximum Likelihood Estimation ......... 607

12.2 Experiment Design ............................ 60812.2.1 Attainable Precision:

The Cramér–Rao Lower Bound ............ 60912.2.2 Probability of Error ............................ 610

12.3 Quantitative Atomic ColumnPosition Measurements ..................... 612

12.4 Quantitative Composition Analysis ..... 614

12.5 Atom Counting .................................. 616

12.6 Atomic Resolutionin Three Dimensions ......................... 619

12.7 Conclusions ...................................... 620

References ................................................... 621

Aberration corrected transmission electron microscopy(TEM) is an excellent technique to study nanostruc-tures down to the atomic scale. As compared to x-rays,electrons interact very strongly with small volumes ofmatter providing local information on the material un-der study [12.1–3]. In this manner, TEM can be usedto observe deviations from perfect crystallinity, whichis of great importance when studying nanostructures.Because of the presence of defects, interfaces, and sur-faces, the locations of atoms of nanostructures devi-ate from their equilibrium bulk positions. This resultsin strain playing a crucial role on the observed prop-erties. For example, strain induced by the lattice mis-match between a substrate and a superconducting layergrown on top can change the interatomic distances by pi-cometers and can in this manner turn an insulator intoa conductor [12.4]. In order to unscramble the structure–properties relation, experimental characterization meth-ods that can locally determine the unknown structure

parameters with sufficient precision are required [12.2,5–7]. A precision of the order of 0:01�0:1Å is neededfor the atomic positions [12.8, 9]. If we can determinethe type and position of atoms with sufficient precision,the atomic structure can be linked to the physicochem-ical properties. A common approach to understandingmaterials’ properties is to use theoretical ab-initio cal-culations that allow one to obtain equilibrium atomicpositions for a given composition. Once this equilib-rium structure has been obtained, properties can be com-puted, and predictions of how thematerial would behaveunder different environmental conditions, even beyondthe capability of any laboratory, can be performed. Inthis manner, materials science is gradually evolving to-ward materials design, that is, from describing and un-derstanding toward predictingmaterials with interestingproperties [12.7, 10–13]. The physicochemical proper-ties of nanostructures are controlled by the shape, size,and atomic arrangement, as well as the electronic state

© Springer Nature Switzerland AG 2019P.W. Hawkes, J.C.H. Spence (Eds.), Springer Handbook of Microscopy, Springer Handbooks,https://doi.org/10.1007/978-3-030-00069-1_12

PartA|12.1

606 Part A Electron and Ion Microscopy

and chemical composition. The aim of TEM is, there-fore, tomeasure those structure parameters as accuratelyand precisely as possible from the experimental data.

In the last decade, remarkable high-technology de-velopments in lens design have greatly improved theimage resolution. Currently, a resolution of the orderof 50 pm can be achieved [12.14–18]. For most atomictypes, this exceeds the point where the width of theelectrostatic potential of the atoms is the limiting fac-tor [12.19]. In addition, new data collection geometriesthat allow one to optimize the experimental settings areemerging [12.20–24]. Furthermore, detectors behaveincreasingly more as ideal quantum detectors [12.25].In this manner, the microscope itself becomes less re-stricting, and the quality of the experimental imagesis set mainly by the unavoidable presence of elec-tron counting noise. However, when aiming for precisestructure parameters, signals need to be interpretedquantitatively. Therefore, the focus in TEM researchhas gradually moved from obtaining better resolutionto improving precision. To reach this goal, the use

of statistical parameter estimation theory is of greathelp [12.26–28].

In this chapter, a concise overview of the methodsthat can be applied for the solution of a general typeof parameter estimation problem often met in materialscharacterization or applied science and engineering willbe presented. In particular, the maximum likelihood es-timator will be discussed, as well as the limits set tothe precision that can be achieved. Next, an overviewof applications of parameter estimation in the field ofTEM will be given. In these applications, the goal isto determine unknown structure parameters, includingatomic positions, chemical concentrations, and atomicnumbers, as precisely as possible from experimentallyrecorded images. In this manner, it will be shown thatstatistical parameter estimation theory allows one tomeasure two-dimensional (2-D) atomic column posi-tions with subpicometer precision, to measure composi-tional changes at interfaces, to count atoms with singleatom sensitivity, and to reconstruct three-dimensional(3-D) atomic structures.

12.1 Model-Based Parameter Estimation

In general, the aim of statistical parameter estimationtheory is to determine, or more correctly, to esti-mate, unknown physical quantities or parameters onthe basis of observations that are acquired experimen-tally [12.29]. In many scientific disciplines, observa-tions are usually not the quantities to be measuredthemselves but are related to the quantities of interest.Often, this relation is a known mathematical functionderived from physical laws. The quantities to be de-termined are parameters of this function. Parameterestimation, then, is the computation of numerical val-ues for the parameters from the available observations.In TEM, the observations of a specific object are, for ex-ample, the image pixel values recorded using a chargecoupled device (CCD) camera. A parametric modeldescribing these observations should then include allingredients needed to perform a computer simulationof the images, i. e., the electron–object interaction, thetransfer of the electrons through the microscope, andthe image detection. If first principles-based modelscannot be derived, or are too complex for their intendeduse, simplified empirical models may be used. Someof the model’s parameters are the atomic positions andatomic types. The parameter estimation problem thenbecomes one of computing the atomic positions andatomic types from TEM images. Statistical parameterestimation theory provides an elegant solution for suchproblems. Indeed, based on the availability of a para-

metric model, the unknown structure parameters canbe estimated by fitting the model to the experimentalimages in a refinement procedure, usually called es-timation procedure or estimator. In general, differentestimation procedures can be used to estimate the un-known parameters of this model, such as least squares(LS), least absolute deviations, or maximum likelihood(ML) estimators [12.26, 29, 30]. In practice, the MLestimator is often used, since it is known to be themost precise. Section 12.1.1 discusses the derivation ofa parametric statistical model of the observations. Sub-sequently, the ML estimator is reviewed in Sect. 12.1.2.For a detailed overview on statistical parameter estima-tion theory, the reader is referred to [12.26, 29–33].

12.1.1 Parametric Statistical Modelof Observations

Generally, due to the inevitable presence of noise, setsof observations made under the same conditions dif-fer from experiment to experiment. The usual wayto describe the fluctuating behavior of 2-D images inthe presence of noise is by modeling the noisy pixelvalues as stochastic variables. By definition, a set ofobservations, w D .w11; : : : ;wKL/

T, is defined by itsjoint probability density function (PDF) pw .!/. Thejoint PDF defines the expectations, i. e., the mean valueof each observation and the fluctuations about these

Model-Based Electron Microscopy 12.1 Model-Based Parameter Estimation 607Part

A|12.1

observations. The expectation values EŒwkl� � �kl aredescribed by parametric models fkl.�/. The availabil-ity of such a model makes it possible to parameterizethe PDF of the observations, which is of vital impor-tance for quantitative structure determination, as will beshown in the remainder of this chapter. Obviously, it isimportant to test the validity of the expectation modelbefore attaching confidence to the structure determina-tion results obtained using the model. If the model isinadequate, it must be modified and the analysis con-tinued until a satisfactory result is obtained. A reviewof statistical model assessment methods can be foundin [12.26].

As an example, a parametric model often used forthe description of high-resolution (scanning) (S)TEMimages will be discussed. For (S)TEM images, theintensity is sharply peaked at the atomic column posi-tions [12.34, 35]. Therefore, (S)TEM images are oftenmodeled as a superposition of Gaussian peaks. Theexpectation of the intensity of pixel .k; l/ at position.xk; yl/ can then be described by an expectation modelfkl.�/, with � the vector of unknown structure parame-ters, as

fkl.�/ D � CIX

iD1

MiXmi

�mi

� exp

��xk �ˇxmi

�2 C �yl �ˇymi

�22�2i

!;

(12.1)

with � a constant background, �i the column-dependentwidth of the Gaussian peak, �mi the column intensityof the mith Gaussian peak, and ˇxmi

and ˇymithe x and

y-coordinates of the mi-th atomic column, respectively.The index i refers to atomic columns of the same atomtype with I different types, and the indexmi refers to them-th column of type i with Mi the number of columnsof type i. The indices in the summation of (12.1) can besimplified in the case of a monotype crystalline nano-structure, since then only one column type is present.The unknown parameters of the parametric imagingmodel of (12.1) are given by the parameter vector �

� D .ˇx11; : : : ; ˇxMI

; ˇy11; : : : ; ˇyMI

;

�1; : : : ; �I; �11 ; : : : ; �MI ; �/T : (12.2)

When assuming that the observations are independentelectron counting results, which can be modeled asa Poisson distribution, the joint PDF is given by

pw .!I �/ DKY

kD1

LYlD1

�!klkl

!klŠexp.��kl/ : (12.3)

Appealing to the central limit theorem, the assumptionof normally distributed observations is often justifiedin practical cases, where also disturbances other thanpure counting statistics contribute. Furthermore, whenassuming equal variances �2, the joint probability den-sity function is given by

pw .!I �/ DKY

kD1

LYlD1

1p2 �

� exp

"�1

2

�!kl ��kl

�

�2#: (12.4)

Since the expectations �kl are described by the func-tional model fkl.�/, substitution of (12.1) in (12.3) orin (12.4) shows how the PDF depends on the unknownparameters to be measured.

12.1.2 Maximum Likelihood Estimation

The observations are considered as a data plane fromwhich the unknown parameters have to be estimated ina statistical way. Different estimators can be used to es-timate the same unknown parameters of the proposedparametric models. Each estimator will have a differentprecision, however, the variance of unbiased estimatorswill never be lower than the Cramér–Rao lower bound(CRLB), which is a theoretical lower bound on the vari-ance and will be described in Sect. 12.2.1. The MLestimator achieves this theoretical lower bound asymp-totically, i. e., for an increasing number of observations,and is, therefore, of practical importance. From the jointPDF of the observations, discussed in Sect. 12.1.1, theML estimator may be derived. The ML estimates O�ML

of the parameters � are given by the values of t thatmaximize the likelihood function pw .w I t/ with t in-dependent variables replacing the true parameters andthe observations replacing the stochastic variables in thejoint PDF

O�ML D arg maxt

pw .w I t/D arg max

tln pw .w I t/ : (12.5)

The joint PDF pw .!I O�ML/ with the ML estimatesinserted generates the observations with higher prob-ability than the joint probability distribution withanother set of parameters O� . The ML estimator equalsthe LS estimator for independent normally distributedobservations. Therefore, for the joint PDF given by(12.4), the ML estimator simplifies to

O�LS D arg mint

Xk

Xl

Œwkl � fkl.t/�2 (12.6)

with fkl the parametric model.

PartA|12.2

608 Part A Electron and Ion Microscopy

The relative ease with which these estimators can bederived does not mean that the optimization is straight-forward [12.26]. Finding the ML estimate correspondsto finding the global optimum in a parameter spaceof which the dimension corresponds to the number ofparameters to be estimated. The search for this opti-mum is usually an iterative numerical procedure. Inelectron microscopy applications, the dimension of theparameter space is usually very high. Consequently, itis quite possible that the optimization procedure endsup at a local optimum, so that the wrong structuremodel is suggested, which introduces a systematic er-ror (bias). To solve this dimensionality problem, thatis, to find a pathway to the optimum in the parameterspace, good starting values for the parameters are re-quired. In other words, the structure has to be resolved.This corresponds to x-ray crystallography, where onefirst has to resolve the structure and afterwards one hasto refine the structure. Resolving the structure is nottrivial. It is known that details in TEM images do notnecessarily correspond to features in the atomic struc-ture. This is not only due to the unavoidable presenceof noise, but also to the dynamic scattering of the elec-trons on their way through the object and the imageformation in the electron microscope, which both havea blurring effect. As a consequence, the structure in-formation of the object may be strongly delocalized,which makes it very difficult to find good starting val-ues for the structure parameters. However, it has been

shown that starting values for the parameters, and hencea good starting structure of the object, may be foundby using so-called direct methods. Direct methods, ina sense, invert the imaging process and the dynamicscattering process using some prior knowledge, whichis generally valid, irrespective of the (unknown) struc-ture parameters of the object. The starting structureobtained with such a direct method can be obtained indifferent ways. Examples of such methods are high-voltage electron microscopy, aberration correction inthe electron microscope, high-angle annular dark-field(HAADF) STEM, focal-series reconstruction, and off-axis holography. A common goal of these methods isto improve the interpretability of the experimental im-ages in terms of the structure and may as such yieldan approximate solution that can be used as a startingstructure for ML estimation from the original images.Furthermore, a direct implementation of the ML es-timator in which all parameters are estimated at thesame time is computationally very intensive and isonly feasible for images containing a limited numberof 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 en-ables the quantitative analysis of large fields of view,has been developed [12.36]. The basic idea of the un-derlying algorithm is the segmentation of the image intosmaller sections containing individual columns withoutignoring overlap between neighboring columns.

12.2 Experiment Design

In general, the purpose of experiment design in elec-tron microscopy is to set up experiments in such a waythat unknown structure parameters of the sample un-der study can be estimated as precisely as possible fromthe data obtained. Ultimately, the precision with whichthese parameters can be estimated is limited by noise.The goal of statistical experiment design is to answerthe question: which microscope settings are expectedto yield the highest precision with which structure pa-rameters, such as, atomic column positions, particlesize, and thickness, can be estimated? Previous workhas shown that the CRLB is a very efficient way toanswer this question [12.19, 26, 27, 37–45]. This lowerbound provides a theoretical lower bound on the vari-ance of unbiased estimators of these parameters and canbe computed from the parameterized PDF discussedin Sect. 12.1.1. By calculating the CRLB, the exper-imenter is able to compute the design so as to attainmaximum precision. So far, studies on the precisionof atomic scale measurements from (S)TEM images

considered the estimation of the position of atomsor atomic columns in projection [12.19, 26, 27, 37–43],the atomic column thickness [12.44], and nanoparti-cle’s sizes [12.45]. In these papers, it has been shownhow optimizing the design of quantitative electron mi-croscopy experiments may substantially enhance theprecision of the structure parameter estimators. Thecommon aspect in these studies is the continuous dif-ferentiability of the PDF with respect to the parameters.However, when estimating a so-called restricted (ordiscrete) parameter from a (S)TEM image, such asthe atomic number (Z), or the number of atoms ina projected atomic column, this condition is no longersatisfied and, hence, the CRLB is not defined. There-fore, an alternative approach has been developed forestimating discrete parameters using the principles ofdetection theory [12.22, 30, 46, 47]. This framework al-lows one to formulate a discrete parameter estimationproblem as a binary or multiple hypothesis test, whereeach hypothesis corresponds, for example, to the as-

Model-Based Electron Microscopy 12.2 Experiment Design 609Part

A|12.2

sumption of a specific Z value or a specific numberof atoms in the column. Furthermore, statistical detec-tion theory provides the tools to compute the probabilityto assign an incorrect hypothesis. This so-called prob-ability of error can be computed as a function of theexperimental settings and, hence, can be used insteadof the CRLB to optimize the experiment design for dis-crete parameter estimation problems.

12.2.1 Attainable Precision:The Cramér–Rao Lower Bound

Different estimators of the same parameters generallyhave a different precision. The question then arisesas to what precision may be ultimately achieved froma particular set of observations. For the class of unbi-ased estimators (bias equals zero), this answer is givenin the form of a lower bound on their variance, theCRLB [12.48–50]. Let pw .!I �/ be the joint PDF ofa set of observationsw D .w11; : : : ;wKL/

T. An exampleof this function is given in Sect. 12.1.1. The dependenceof pw .!I �/ on the R�1 parameter vector � can now beused to define the so-called Fisher information matrix

F D �E

@2 ln pw .w I �/

@� @�T

; (12.7)

which is an R�R matrix. The expression betweensquare brackets represents the Hessian matrix of thelogarithm of the joint PDF of which the .r; s/-thelement is defined by @2 ln pw .!I �/=@�r@�s. TheFisher information expresses the inability to knowa measured quantity [12.50]. Indeed, using the conceptof Fisher information allows one to determine the high-est precision, that is, the lowest variance, with whicha parameter can be estimated unbiasedly. Suppose thatb� is any unbiased estimator of � , that is,

Ehb�i

D � :

Then it can be shown that under general conditions thecovariance matrix cov.b�/ ofb� satisfies

cov.b�/ � F�1 ; (12.8)

so that cov.b�/�F�1 is positive semi-definite. A prop-erty of a positive semi-definite matrix is that its diagonalelements cannot be negative. This means that the diag-onal elements of cov.b�/, that is, the actual variances of

b�1; : : : ;b�R

are larger than or equal to the corresponding diagonalelements of F�1

var.b� r/ � ŒF�1�rr ; (12.9)

where r D 1; : : : ;R and ŒF�1�rr is the r-th diagonal el-ement of the inverse of the Fisher information matrix.In this sense, F�1 represents a lower bound for the vari-ances of all unbiased estimators b� . The matrix F�1 isthe CRLB on the variance ofb� .

Often, the question arises how to measure atomicpositions with picometer precision if the resolutionof the instrument is only 50 pm under optimal condi-tions. Resolution and precision are very different no-tions [12.31]. In (S)TEM, resolution expresses the abil-ity to visually distinguish neighboring atomic columnsin an image. Classical resolution criteria, such asRayleigh’s, are derived from the assumption that thehuman visual system needs a minimal contrast to dis-criminate two points in its composite intensity distribu-tion [12.51]. Therefore, they are expressed in terms ofthe width of the point spread function of the (S)TEMimaging system [12.52]. However, if the physics be-hind the image formation process is known, images nolonger need to be interpreted visually. Instead, atomiccolumn positions can be estimated by fitting this knownparametric model to an experimental image using theML estimator. In the absence of noise, this procedurewould result in infinitely precise atomic column loca-tions. However, since detected images are never noisefree, model fitting never results in a perfect reconstruc-tion, thus limiting the statistical precision with whichthe atom locations can be estimated. For continuousparameters, such as the atomic column positions, theattainable precision can be adequately quantified usingthe expression for the CRLB. Under certain assump-tions, it can then be shown that the attainable precision,expressed in terms of the standard deviation with whichthe position of a projected atomic column can be esti-mated, is approximately equal to [12.38, 40, 41]

�CR � �pN

; (12.10)

where � represents the width of the Gaussian peaks andN represents the number of detected electrons per atomor atomic column. The width of the peaks can be shownto be proportional to the Rayleigh resolution [12.19].This explains why the precision to estimate projectedatomic column positions can be down to 1 or a fewpicometers, although the resolution of modern instru-ments is 50�100 pm. In order to push the precisionfurther by a factor of 10, it is necessary to increase thedose by a factor of 100, which will require a very highincoming dose and/or a long exposure time.

Note that the CRLB is not related to a particularestimation method and that the existence of a lowerbound on the parameter variance does not imply that anestimator can be found that reaches this lower bound.

PartA|12.2

610 Part A Electron and Ion Microscopy

However, it is known that there exists an estimatorthat achieves the CLRB at least asymptotically, thatis, for an increasing number of observations. This es-timator is the ML estimator, which has been discussedin Sect. 12.1.2. It is known to be asymptotically nor-mally distributed with a mean equal to the true valueof the parameter and a covariance matrix equal to theCRLB. In electron microscopy the number of obser-vations is usually sufficiently large for the asymptoticproperties of the ML estimator to apply. For this andother reasons, the use of the ML estimator in quan-titative electron microscopy is highly recommended.Moreover, approximate confidence regions and inter-vals for ML parameter estimates can be obtained basedon the asymptotic statistical properties of the ML es-timator. In this approach, the CRLB is approximatedby substituting the ML parameter estimates for the truevalues of the parameters [12.26].

12.2.2 Probability of Error

For discrete estimation problems, statistical detectiontheory provides the tools to optimize the experiment de-sign by using a statistical hypothesis test [12.46]. Thiscan be either a binary or a multiple hypothesis test, inwhich every hypothesis corresponds, for example, toa specific atomic number Z or a specific number ofatoms in a projected atomic column. The probabilityof deciding the wrong hypothesis, the so-called prob-ability of error, can be defined and decision rules aredetermined in such a way that the probability of error isminimized. In order to optimize the experiment designwhen measuring discrete parameters, such as, the pres-ence 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 optimalitycriterion, by computing it as a function of the experi-mental settings [12.22, 23, 30, 47, 53]. The optimal ex-periment design then corresponds to those experimentalsettings that result in the lowest probability of error.

When considering the problem of deciding betweentwo different atom types, detecting a light atom or de-ciding between the presence of n or nC 1 atoms ina projected atomic column, a binary hypothesis test canbe used. In these cases, the estimation problem can bedescribed as deciding between a so-called null hypoth-esis H0 and the alternative hypothesisH1

H0 W Z D Z0 ; H1 W Z D Z1 ; (12.11a)

H0 W Z D Z0 ; H1 W Z 2 ; ; (12.11b)

H0 W nH0 D n ; H1 W nH1 D nC 1 : (12.11c)

In the case of (12.11a), both hypotheses correspondto two different possible atomic numbers, Z0 and Z1,

in (12.11b); the hypotheses correspond to whether thelight atom is present or absent, and in (12.11c), thehypotheses correspond to two succeeding numbers ofatoms in a projected atomic column. In binary hypoth-esis testing problems, a priori knowledge is usuallyassumed assuring that only H0 or H1 is possible, sothat one of both hypotheses is always correct. In or-der to express a prior belief in the likelihood of thehypotheses, the prior probabilities P.H0/ and P.H1/associated with these hypotheses are assumed to beknown, such that P.H0/CP.H1/ D 1. If both hypothe-ses are equally likely, then it is reasonable to assignequal prior probabilities of 1=2. In a quantitative ap-proach, the goal is now to minimize the probability ofassigning the wrong hypothesis. In a so-called Bayesianapproach, this probability of error Pe is defined as

Pe D Prfdecide H0;H1 truegC Prfdecide H1;H0 trueg

D P.H0jH1/P.H1/CP.H1jH0/P.H0/ ;

(12.12)

with P.HijHj/ the conditional probability of decidingHi, while Hj is true. Using criterion (12.12), the twopossible errors are weighted appropriately to yield anoverall error measure. Decision rules are now definedsuch that the probability of error is minimized. It isshown in [12.46] that one, therefore, should decide H1

if

pw .w IH1/

pw .w IH0/>

P.H0/

P.H1/D � ; (12.13)

otherwiseH0 is decided. In this expression, pw .w IHi/is the conditional (joint) PDF pw .!IHi/ assuming Hi

to be true, evaluated at the available observations w .For equal prior probabilities of 1=2, it is clear that � in(12.13) corresponds to 1. Then, we decide H1 if

LR.w/ D pw .w IH1/

pw .w IH0/> 1 : (12.14)

The function LR.w/ is called the likelihood ratio, sinceit indicates for each set of observationsw the likelihoodof H1 versus the likelihood of H0. This test is, there-fore, also known as the likelihood ratio test. Similarly,decision rule (12.14) corresponds to deciding H1 if

ln LR.w/ D ln pw .w IH1/� ln pw .w IH0/ > 0 :

(12.15)

Otherwise H0 is decided. This decision rule corre-sponds to choosing the hypothesis for which the log-likelihood function is maximal. The left-hand side of

Model-Based Electron Microscopy 12.2 Experiment Design 611Part

A|12.2

0.43

0.42

0.41

0.40

0.39

0.38

0.37

0.36

0.170.160.150.140.130.120.110.100.090.08

100

80

60

40

20

0.0 0.000.00

0.05

0.4

0.3

0.2

0.1

0.5

0.0

0.10

0.15

0.20

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

–15 –10 –5 0 5 10 15–10 –5 0 5 10–2 –1 0 1 2

20 40 60 80 100

0.830.820.810.800.790.780.770.760.75

�0 �1

a) b) c)

d) e)

g)

f)

ln LR(ω) ln LR(ω) ln LR(ω)

Probability Probability Probability

Inner detector radius (mrad)

Outer detector radius (mrad)

Y H

�0 �1 �0 �1

Fig. 12.1a–g Simulated STEM images of a YH2 unit cell viewed from the [010] direction, for annular detector collectionranges of (a) 11�53mrad, (b) 22�53mrad D ADF, and (c) 11�17mrad D ABF. The corresponding log-likelihood ratiodistributions are shown in (d–f) for the presence (black) and absence (brown) of H in YH2. (g) The probability oferror as a function of the inner and outer detector angle at Scherzer conditions for an electron dose of 2000 e�=Å2 at300 kV and a probe semi-convergence angle of 21:8mrad for the detection of H in a 2:6 nm thick YH2 crystal. Reprintedfrom [12.22], with the permission of AIP Publishing

PartA|12.3

612 Part A Electron and Ion Microscopy

(12.15) is termed the log-likelihood ratio. When as-suming independent, Poisson distributed observations,for which the joint PDF is given by (12.3), the log-likelihood ratio can be rewritten as

ln LR.w /

DKX

kD1

LXlD1

�wkl ln

��H1;kl

�H0;kl

���H1;kl C�H0;kl

�;

(12.16)

where �Hi;kl corresponds to the expectation of the inten-sity of pixel .k; l/ under hypothesisHi. If an experimentis repeated under the same conditions, it can be shownthat the distribution of the log-likelihood ratio tends tofollow a normal distribution following the central limittheorem.

As an example, the problem of optimizing the annu-lar STEM detector in order to detect the lightest H atomin YH2 is considered. For this material, we consider theproblem of optimizing the annular STEM detector inorder to detect the lightest H atom. One has been ableto experimentally detect H in this material by usingan annular bright field (ABF) STEM detector [12.54].Here, our goal is to test whether the same optimal de-tector type is found using our quantitative approach andto determine the exact optimal detector angles [12.22].The expectation models are simulated using STEM-

sim [12.55] both for the crystal in the presence andabsence of hydrogen, corresponding to the hypothesesH0 W Z D 1 andH1 W Z 2 ;. The detailed simulation pa-rameters can be found in [12.22]. Simulated images areshown in Fig. 12.1a–c for detector collection anglesof 11�53, 22�53, and 11�17mrad. The correspond-ing distributions of the log-likelihood ratio in the caseof the presence (black) and absence (brown) of H inYH2 are shown in Fig. 12.1d–f. The black and browncolored areas correspond to the probability of decidingH0 while H1 is true and the probability of decidingH1 while H0 is true, respectively. The sum of bothareas represents the probability of error. These figuresillustrate that the probability of error depends on thechoice of the detector collection angles. By evaluatingthe probability of error as a function of the inner andouter detector angles, the experiment design can hencebe optimized. Results of the probability of error for thedetection of H in YH2 are shown in Fig. 12.1g. Asan optimal detector setting, the ABF STEM regime isfound with a detector ranging from 11 to 17mrad. Alsoa local optimum is observed in the low-angle annulardark field (LAADF) regime with both inner and outerdetector radius larger than the probe semi-convergenceangle. In general, it should be noticed that LAADFSTEM is often competitive and that the optimal de-tector settings largely depend on the sample type andthickness.

12.3 Quantitative Atomic Column Position Measurements

When the goal is to measure shifts of the atomicpositions, aberration-corrected TEM, exit wave recon-struction methods, or combinations of both are oftenused in practice. Whereas aberration correction has animmediate impact on the resolution of the experimen-tal images, the exit plane of the sample under study isreconstructed using exit wave reconstruction. Usually,a series of images taken at different defocus values,an electron holographic image, or a series of imagesrecorded with different illuminating beam tilts is usedas an input to reconstruct the exit wave [12.56–60].Ideally, the exit wave is free from any imaging arti-facts, thus enhancing the visual interpretability of theatomic structure. Exit wave reconstruction has becomea powerful tool in high-resolution TEM because ofits potential to visualize light atomic columns, suchas oxygen or nitrogen, with atomic resolution [12.61,62]. In particular, its combination with quantitativemethods nowadays demonstrates its potential to pre-cisely measure atomic column positions [12.63–65].As an example, the quantification of localized dis-

placements at a f110g twin boundary in orthorhombicCaTiO3 will be discussed [12.66]. Theoretical studiesshow that such domain boundaries are mainly ferri-electric with maximum dipole moments at the wall.To investigate these boundaries experimentally, the exitwave has been reconstructed using aberration-correctedTEM. Figure 12.2a shows the reconstructed phase whenimaging the sample along the [001]-direction with a res-olution of 80 pm. This phase is directly proportionalto the projected electrostatic potential of the struc-ture. Next, statistical parameter estimation is used toobtain quantitative numbers for the atomic column po-sitions [12.26, 27, 31]. In this manner, atomic columnscan be located with a precision of a few picometerswithout being restricted by the information limit of themicroscope. To reach this result, the phase of the recon-structed exit wave is considered as a data plane fromwhich the atomic column positions are estimated us-ing the LS estimator, defined by (12.6). The estimatedcolumn positions thus correspond to the numbers forwhich the LS sum is minimal with wkl corresponding to

Model-Based Electron Microscopy 12.3 Quantitative Atomic Column Position Measurements 613Part

A|12.3

Twin wall

Parallel mean displacement (pm)a) Perpendicular mean displacement (pm)b) c)6420–2–4–6–8–10 8 106420–2–4–6–8–10 8 10

TiCa

1 nm

3 pm6 pm6 pm3 pm

Fig. 12.2 (a) Experimental phase image of a (110) twin boundary in orthorhombic CaTiO3. Mean displacements of the Tiatomic columns from the center of the four neighboring Ca atomic columns are indicated by arrows. (b,c) Displacementsof Ti atomic columns perpendicular and parallel to the twin wall averaged along and in mirror operation with respect tothe twin wall together with their 90% confidence intervals. From [12.66]

the pixel values of the reconstructed phase and fkl theparametric model, which is given by (12.1). The esti-mated numbers could be used to measure shifts in theatomic column positions. It has been found that pos-sible shifts in the Ca atomic positions are too smallto be identified, whereas shifts in the Ti atomic posi-tions in the vicinity of the twin wall are statisticallysignificant [12.66]. Therefore, this analysis is focusedon the off-centering of the Ti atomic positions with re-spect to the center of the neighboring four Ca atomicpositions. First, we average all displacements in planesparallel to the twin wall. Next, we average the resultsin the planes above with the corresponding planes be-low the twin wall. This second operation identifies theoverall symmetry of the sample with the twin wall rep-resenting a mirror plane. The resulting displacementsperpendicular to and along the twin wall are shownin Fig. 12.2b and c, respectively, together with their90% confidence intervals. In the direction perpendic-ular to the wall, systematic deviations for Ti of 3:1 pmin the second closest layers pointing toward the twinwall are found. A larger displacement is measured inthe direction parallel to the wall in the layers adja-cent to the twin wall. The averaged displacement in

these layers is 6:1 pm. In layers further away fromthe twin wall, no systematic deviations are observed.These experimental results confirm the theoretical pre-dictions [12.67].

Another efficient technique to measure shifts ofthe atomic positions is so-called negative-spherical-aberration imaging in which the spherical aberrationconstant Cs is tuned to negative values by employ-ing an aberration corrector [12.68, 69]. As comparedto traditional positive Cs imaging, this imaging modeyields a negative phase contrast of the atomic structure,with atomic columns appearing bright against a darkerbackground. For thin objects, this leads to a substan-tially higher contrast compared to the dark atom imagesformed under positive Cs imaging. This enhanced con-trast has the effect of improving the measurement preci-sion of the atomic positions and explains the use of thistechnique to measure atomic shifts of the order of a fewpicometers. Examples show measurements of the widthof ferroelectric-domain walls in PbZr0:2Ti0:8O3 [12.16],measurements of the coupling of elastic strain fieldsto polarization in PbZr0:2Ti0:8O3=SrTiO3 epitaxial sys-tems [12.70], and of oxygen-octahedron tilt and polar-ization in LaAlO3=SrTiO3 interfaces [12.17].

PartA|12.4

614 Part A Electron and Ion Microscopy

12.4 Quantitative Composition Analysis

HAADF STEM imaging, in which an annular detectoris used with a collection range outside of the illumi-nation cone, is a very convenient method for struc-ture characterization at the atomic level. Because ofthe high-angle scattering, the signal is dominated byRutherford and thermal diffuse scattering and, hence,approximately scales with the square of the atomicnumber Z. One of the advantages of this Z-contrast isthe possibility to visually distinguish between chemi-cally different atomic column types. Furthermore, theresolution observed in an HAADF STEM image ismainly set by the intensity distribution of the illumi-nating probe. Nowadays, a probe size of the orderof 50 pm can be attained when using aberration cor-rected probe-forming optics [12.18]. This high spatialresolution combined with the high chemical sensitiv-ity means that HAADF STEM images are to a certainextent directly interpretable. Despite this advantage,HAADF STEM is also of great benefit when analyzingthe resulting images using statistical parameter estima-tion theory [12.71]. Especially when the difference inatomic number of distinct atomic column types is smallor when the signal-to-noise ratio is low, visual interpre-tation becomes insufficient.

Cross-sections are well known and often used inparticle scattering experiments as a measure of theprobability of scattering. In STEM, a scattering cross-section approach was first proposed by Retsky in1974 [12.72]. However, because HAADF STEM im-ages are to some extent interpretable directly, scatteringcross-sections were little used until their importancewas recently realized to interpret images quantitativelyin terms of structure and composition. In HAADFSTEM imaging, scattering cross-sections correspondto the total scattered intensity for each atomic col-umn and can be measured using statistical parameterestimation theory [12.71] or by integrating intensitiesover the probe positions in the vicinity of a singlecolumn of atoms [12.73]. The advantage of using scat-tering cross-sections over other metrics, such as peakintensities, is their robustness to magnification, defo-cus, source size, astigmatism, and small sample mis-tilt [12.73–75]. Moreover, this measure is very sensitiveto changes in composition and/or thickness [12.23].The estimated scattering cross-sections allow us to de-tect differences in averaged atomic number of only 3.This is illustrated in the following example, in whicha La0:7Sr0:3MnO3-SrTiO3 multilayer structure is inves-tigated. Figure 12.3a shows an enlarged area from anexperimental image using an FEI Titan 50-80 oper-ated at 300 kV. No visual conclusions could be drawnconcerning the sequence of the atomic planes at the in-

terfaces. To overcome this problem, the parameters ofthe parametric model, which is given by (12.1), havebeen 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 experimentaldata. Figure 12.3c shows the experimental observationstogether with an overlay indicating the estimated po-sitions of the columns and their atomic column types.Cross-sections can be derived from the estimated pa-rameters,

Vmi D 2 �mi�2i : (12.17)

The composition of the columns away from the inter-faces is assumed to be in agreement with the compo-sition in the bulk compounds. Histograms of the esti-mated scattering cross-sections of these known columnsare presented in Fig. 12.3d and show the random na-ture of the result. The colored vertical bands correspondto 90% tolerance intervals. It is important to notethat these tolerance intervals do not overlap, whichmeans that columns, for which the difference in av-eraged atomic number is only 3 (TiO and MnO) inthis example, can clearly be distinguished. Based onthis histogram, the composition of the unknown (pur-ple colored) columns in the planes close to the interfacecould be identified as shown on the right-hand sideof Fig. 12.3c. Single-color dots are used to indicatecolumns whose estimated scattering cross-section fallsinside a tolerance interval, whereas pie charts, indicat-ing the presence of intermixing or diffusion, are usedotherwise.

In the previous example, the chemical compositionwas quantified in a relative manner by comparing scat-tering cross-sections of unknown columns with those ofknown columns. When aiming for an absolute quantifi-cation, intensity measurements relative to the intensityof the incoming electron beam are required [12.76, 77].In this manner, experimental scattering cross-sectionscan be directly compared with simulated scatteringcross-sections [12.77–79]. Reference cross-section val-ues are then simulated by carefully matching experi-mental imaging conditions for a range of sample condi-tions, including thickness and composition. To illustratethis, Fig. 12.4a shows part of a normalized imageof a Pb1:2Sr0:8Fe2O5 compound where the intensitiesare normalized with respect to the incoming electronbeam [12.79]. By comparing experimental scatteringcross-sections for each atomic column with simulatedvalues, the thickness values for the PbO columns andthe composition for the SrPbO columns was determinedas shown in Fig. 12.4b. Figure 12.4c compares the av-

Model-Based Electron Microscopy 12.4 Quantitative Composition Analysis 615Part

A|12.4

La

Sr

MnO

TiO

La Sr MnOTiO

Estimated peak volume (arb. u.)

Frequency

a) b)

d)

c)

Fig. 12.3 (a) Area from an experimental HAADF STEM image of a La0:7Sr0:3MnO3-SrTiO3 multilayer structure. (b) Re-fined parametric model. (c) Overlay indicating the estimated positions of the columns together with their atomic columntypes. (d) Histograms of the estimated scattering cross-sections of the known columns. Reprinted from [12.71], withpermission from Elsevier

eraged experimental intensity profile along the verticaldirection of the unit cell indicated in Fig. 12.4b witha frozen lattice simulation, where the estimated thick-ness and composition values were used as an input. Theoverall match between the simulated and experimentalimage intensities further confirms the results that have

been obtained when using the scattering cross-sectionsapproach. However, it should be noted that small de-viations between simulated and experimental imageintensities cannot be avoided because of, for example,remaining uncertainties in the microscope settings suchas defocus, source size, or astigmatism.

PartA|12.5

616 Part A Electron and Ion Microscopy

0.060

0.055

0.050

0.045

0.040

0.035

0.030

0.025

a)

b)

c)

0.0 0.5 1.0 1.5 2.0c (nm)

Averaged intensity in b direction

Experiment

FP simulation

Amount of Pb in Sr column (%)Thickness (nm)

50

40

30

20

10

12

11

10

9

8 0

[100] PbO SrPbO Fe OFig. 12.4 (a) Area from an exper-imental HAADF STEM image ofa Pb1:2Sr0:8Fe2O5 compound wherethe intensities are normalized withrespect to the incoming electronbeam. (b) Quantification results show-ing the estimated thickness valuesat the PbO site and the estimatedcomposition for the SrPbO atomiccolumns. (c) Comparison of the av-eraged experimental intensity profilealong the vertical direction of the unitcell indicated in (b) together witha frozen lattice simulation assumingthe thickness and composition valuesshown in (b). Reprinted from [12.79],with permission from Elsevier

12.5 Atom Counting

Scattering cross-sections are not only sensitive for thecomposition but also for the number of atoms in anatomic column. Figure 12.5 illustrates how this advan-tage can be used to count the number of Au atomsfrom an experimental image of an Au nanorod. Theintensities of the HAADF STEM image shown inFig. 12.5a were normalized with respect to the inci-dent beam [12.77, 80]. Next, in a similar manner tothe analysis presented in Sect. 12.4, scattering cross-sections of all Au columns were estimated. Figure 12.5bshows the refined parametric model. The histogram ofall scattering cross-sections is shown in Fig. 12.5d. Ow-ing to a combination of experimental detection noise

and residual instabilities, broadened rather than discretecomponents are observed in such a histogram. There-fore, these results cannot directly be interpreted in termsof the number of atoms. By evaluation of the so-calledintegration classification likelihood criterion (ICL) incombination with Gaussian mixture model estimation,the presence of 47 components and their respectivelocations were found for the scattering cross-sectionsof the Au columns [12.81–84]. This is illustrated inFig. 12.5d. From the estimated locations of the compo-nents, the number of Au atoms can be quantified, lead-ing to the result shown in Fig. 12.5c. It is important tonote that this statistics-based method to count the num-

Model-Based Electron Microscopy 12.5 Atom Counting 617Part

A|12.5

2 nm 2 nm 2 nm

Number of atoms

Scattering cross-section (Å2)

Scattering cross-section (Å2)

Number of columns

0 20 40 60 80 1000 0.1 0.2 0.3 0.4 0.5

SimulationExperiment

0.8

0.6

0.4

0.2

0

45

40

35

30

25

20

15

10

5

0

100

50

0

a)

d) e)

c)b)

ICL–3000–4000–5000–6000–7000–8000

0 20 40Number of Gaussian components

60 80 100

Fig. 12.5 (a) Experimental HAADF STEM image of an Au nanorod. (b) Refined parametric model. (c) Number of Auatoms per column. (d) Histogram of scattering cross-sections of the Au columns together with the estimated mixturemodel and its individual components (colored curves, which correspond to the colors for the number of atoms in (c)); theinset shows the order selection criterion by ICL indicating the presence of 47 components. (e) Comparison of experimen-tal and simulated scattering cross-sections. Reprinted with permission from [12.84]. Copyright 2013 by the AmericanPhysical Society

ber of atoms does not require the use of simulations.This approach is robust against systematic errors whentwo conditions are met; the number of experimentalscattering cross-sections per unique thickness should belarge enough and the spread of scattering cross-sectionsshould be small enough as compared to the differencebetween those of differing thicknesses [12.83].

An alternative method to count atoms is throughcomparison with image simulations [12.78]. However,a main drawback is that systematic errors are difficultto detect, since the assignment of numbers of atoms willalways find a match by comparing experimental scatter-ing cross-section values or peak intensities with simu-lated values. The reliability of the atom counting resultsthen purely depends on the accuracy with which, for ex-ample, the detector inner and outer angles have beendetermined and the accuracy with which the simula-tions have been carried out. The use of the simulations-based and independent statistics-based methods allowsone to validate the accuracy of the obtained atom

counts [12.84, 85]. This is illustrated in Fig. 12.5e,which shows the experimental mean scattering cross-sections—corresponding to the component locations inFig. 12.5d—together with the scattering cross-sectionsestimated from the frozen phonon calculations usingthe STEMsim program under the same experimentalconditions [12.55]. The excellent match of the exper-imental and simulated scattering cross-sections withinthe expected 5�10% error range validates the accuracyof the obtained atom counts [12.78, 86]. Furthermore,this step also validates the choice of the local mini-mum present in the ICL criterion shown in the inset ofFig. 12.5d. The precision of the atom counts is limitedby the unavoidable presence of noise in the experi-mental images, resulting in an overlap of the Gaussiancomponents, as shown in Fig. 12.5d. When the overlapincreases, the probability to assign an incorrect numberof atoms increases. In this example, the probability tohave an error of 1 atom is only 20%, whereas the num-ber of atoms of 80% of all columns can be determined

PartA|12.5

618 Part A Electron and Ion Microscopy

1 nm

a) b) c)

12

10

8

6

4

2

1

0

–1

Fig. 12.6 (a)Hypothetical ADF STEM image corresponding to a dose of 1000 e�=Å2. (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

4035302520151050

70

60

50

40

30

20

10

0

a) b) c)Number of Au atomsNumber of Ag atoms

2 nm 2 nm

Fig. 12.7 (a,b) Number of Ag and Au atoms counted from an experimental HAADF STEM image of an Ag-coated Aunanorod and (c) a 3-D reconstructed atomic model. Reprinted with permission from [12.88]. Copyright 2016 by theAmerican Physical Society

without error. The combination of a simulations-basedand statistics-based method thus allows for reliableatom-counting with single atom sensitivity.

Ultimately, the simulations-based method and thestatistics-based method are combined into a hybrid ap-proach, which overcomes the limitations of both meth-ods. To reach this goal, prior knowledge resulting fromimage simulations can be incorporated in the statisti-cal framework while taking possible inaccuracies in theexperimental parameters into account [12.87]. The useof this hybrid method has clear benefits when analyzinglow-dose images of small nanoparticles. This is demon-strated in Fig. 12.6, where the number of Pt atoms wasaccurately determined from a simulated ADF STEMimage when assuming an input electron dose of only1000 e�=Å2

.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, sincesmall changes in atom ordering in the column havean effect on the scattering cross-sections. Therefore,the amount of required simulations increases exponen-tially with the thickness and the number of elementspresent in the sample. For example, already more than2�106 column configurations exist for a 20-atom thickbinary alloy. When all possible outcomes need to beconsidered, this becomes an impossible task in terms ofcomputing time. To help solve this problem, the avail-ability of a model to predict scattering cross-sectionsas a function of composition, configuration, and thick-ness is desirable [12.88]. In the simplest model, theassumption of longitudinal incoherence is considered,where the scattered intensity of an atomic column iswritten as the sum of the scattered intensities of the in-dividual atoms constituting this column. However, this

Model-Based Electron Microscopy 12.6 Atomic Resolution in Three Dimensions 619Part

A|12.6

methodwill not only lead to large deviations, it will alsoneglect the information about the configuration of thecolumn. A more accurate prediction is obtained whendescribing each atom as a lens focussing the electronson the next atom [12.89, 90]. As shown in [12.88], thelensing factors of the individual atoms in monotypeatomic columns can be calculated in order to predictthe scattering cross-sections of mixed columns. Thisnew approach leads to an accurate prediction of scatter-ing cross-sections, which is not restricted to the number

of atom types or detector angles. This atomic lensingmodel can be used to unravel the 3-D composition at theatomic scale. This is demonstrated in Fig. 12.7, wherethe number of both Ag and Au atoms were countedfrom an experimental HAADF STEM image of an Ag-coated Au nanorod. In combination with atom countingresults obtained from an additional viewing directionand prior knowledge concerning the shape of the nano-rod, a 3-D atomic model could be reconstructed.

12.6 Atomic Resolution in Three Dimensions

As described in the previous sections of this chap-ter, new developments within the field of TEM enablethe investigation of nanostructures at the atomic scale.Structural as well as chemical information can be ex-tracted in a quantitative manner. However, such imagesare mostly 2-D projections of a 3-D object. To over-come this limitation, 3-D imaging by TEM or electrontomography can be used. Atomic resolution in 3-D hasbeen the ultimate goal in the field of electron tomogra-phy during the past few years. The underlying theoryfor atomic resolution tomography has been well under-stood [12.91, 92], but it was nevertheless challenging toobtain the first experimental results. A first approachis based on the acquisition of a limited number ofHAADF STEM images that are acquired along dif-

98765432

10

1987654321

[100]

[101]-

2.04 Å

c

b

c

a

– b

c

a Fig. 12.8 Parts of experimentalHAADF STEM images of a nanosizedAg cluster embedded in an Almatrix in [10N1] and [100] zone-axisorientation together with the numberof Ag atoms per column and thecomputed 3-D reconstruction of thecluster viewed along three differentdirections. From [12.82]

ferent zone axes [12.82]. As illustrated in Sect. 12.5,advanced quantification methods enable one to countthe number of atoms in an atomic column from a 2-D(HA)ADF STEM image. In a next step, such atomcounting results can be used as an input for discretetomography. The discreteness that is exploited hereis the fact that crystals can be thought of as discreteassemblies of atoms [12.92]. In this manner, a verylimited number of 2-D images is sufficient to obtaina 3-D reconstruction with atomic resolution. This ap-proach was applied to Ag clusters embedded in anAl matrix as illustrated in Fig. 12.8 [12.82]. A 3-Dreconstruction was obtained using only two HAADFSTEM images. An excellent match was found whencomparing the 3-D reconstruction with additional pro-

PartA|12.7

620 Part A Electron and Ion Microscopy

a)

b)

Fig. 12.9 (a) Sta-tistical countingresults for threedifferent config-urations of anultra-small Gecluster. Green,red, and blue dotscorrespond to 1,2, and 3 atoms,respectively. Theresults of the ab-initio calculationsare shown in (b).From [12.94]

jection images that were acquired along different zoneaxes. In a similar manner, the core of a free-standingPbSe-CdSe core-shell nanorod could be reconstructedin 3-D [12.93].

Tomography typically requires several images de-manding a substantial electron dose, which hampersthe study of beam-sensitive materials. To overcomethis problem, atom counting results obtained from justa single ADF STEM image can be used as an inputto retrieve the 3-D atomic structure. In combinationwith prior knowledge about a material’s crystal struc-ture, an initial 3-D configuration is generated. Next,an energy minimization using ab-initio calculations ora Monte Carlo approach is performed to relax a na-noparticle’s 3-D structure. This technique was appliedto investigate the dynamical behavior of ultra-smallGe clusters consisting of less than 25 atoms [12.94].Ultra-small nanoparticles or clusters, with sizes below1 nm, form a challenging subject of investigation. Oneof the main bottlenecks is that these clusters may ro-tate or show structural changes during investigation byTEM [12.95]. Obviously, conventional electron tomog-raphy methods, even those that are based on a limitednumber of projections, can no longer be applied. On theother hand, the intrinsic energy transfer from the elec-tron beam to the cluster can be considered as a unique

possibility to investigate the transformation between en-ergetically excited configurations of the same cluster.Image series were collected using aberration correctedHAADF STEM. From a set of selected frames, thenumber of atoms at each position could be deter-mined, as illustrated in Fig. 12.9. In order to extract3-D structural information from these images, ab initiocalculations were carried out. Several starting config-urations were constructed, which are all in agreementwith the experimental 2-D projection images. Althoughall of the cluster configurations stay relatively close totheir starting structure after full relaxation, only thoseconfigurations in which a planar base structure was as-sumed, were found to still be compatible with the 2-Dexperimental images. In this manner, reliable 3-D struc-tural models were obtained for these small clusters,and also the transformation of a predominantly 2-Dconfiguration into a compact 3-D configuration couldbe characterized. In a related manner, the 3-D atomicstructure of catalytic Pt nanoparticles [12.96], the inter-face between individual PbSe building blocks in a 2-Dsuperlattice formed by oriented attachment [12.97], anda model-like Au nanoparticle [12.98] have been stud-ied. All of these studies could not have been realizedwithout these new developments circumventing con-ventional electron tomography.

12.7 Conclusions

In this chapter, it was shown how quantitative TEMgreatly benefits from statistical parameter estimationtheory to estimate unknown structure parameters. Ul-timately, these parameters need to be estimated as

accurately and precisely as possible from the observa-tions available. In Sects. 12.1 and 12.2 of this chapter,a framework was outlined to reach this goal. Theparameterized joint PDF of the observations was in-

Model-Based Electron Microscopy References 621Part

A|12

troduced as a model to describe statistical fluctuationsin the observations. This description requires a (usuallyphysics-based) model describing the expectation valuesof the observations. Furthermore, it requires detailedknowledge about the disturbances that are acting on theobservations. A thus parameterized PDF can then beused for two purposes. First, it can be used to derive theML estimator. The use of this estimator is motivated bythe fact that it has favorable statistical properties. Sec-ond, from the PDF of the observations, the CRLB canbe derived. This is a theoretical lower bound on the vari-ance of any unbiased estimator of the parameters andcan be used for the optimization of the experiment de-sign so as to attain the highest precision. However, fordiscrete parameters, such as the atomic number Z, thislower bound is no longer applicable. Therefore, alter-native solutions using the principles of detection theorywere proposed.

In Sects. 12.3–12.6, statistical parameter estimationtheory was applied to various kinds of observationsacquired by means of TEM. This is becoming increas-ingly important, since it allows one to quantitativelydetermine unknown structures at a local scale. Applica-tions in the field of high-resolution (S)TEM show howstatistical parameter estimation techniques can be usedto overcome the traditional limits set by modern elec-

tron microscopy. The precision that can be achievedin this quantitative manner far exceeds the resolutionperformance of the instrument. The characterizationlimits are, therefore, no longer imposed by the qual-ity of the lenses but are determined by the underlyingphysical principles. Structural, but also chemical, elec-tronic, and magnetic information can be obtained at theatomic scale. As demonstrated in this chapter, quantita-tive structure determination can not only be carried outin 2-D, but also 3-D analyses are currently becomingstandard.

Acknowledgments. The author would like to ac-knowledge all colleagues who contributed to this workover 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, andJ. Verbeeck. The author also expresses many thanks forall fruitful and enlightening theoretical discussions, aswell as all the shared experimental expertise and knowl-edge. Sincere thanks are due to A. van den Bos, whounfortunately passed away too soon, for his enthusias-tic and expert guidance in the author’s understanding ofstatistical parameter estimation theory.

References

12.1 D. Zanchet, B.D. Hall, D. Ugarte: X-ray character-ization of nanoparticles. In: Characterization ofNanophase Materials, ed. by Z.L. Wang (Wiley,Weinheim 2001) pp. 13–36

12.2 J.C.H. Spence: The future of atomic resolution elec-tron microscopy for materials science, Mater. Sci.Eng. 26(1/2), 1–49 (1999)

12.3 R. Henderson: The potential and limitations ofneutrons, electrons and x-rays for atomic resolu-tion microscopy of unstained biological molecules,Q. Rev. Biophys. 28, 171–193 (1995)

12.4 J.P. Locquet, J. Perret, J. Fompeyrine, E. Machler,J.W. Seo, G. Van Tendeloo: Doubling the criticaltemperature of La1.9Sr0.1CuO4, Nature 394, 453–456(1998)

12.5 D.A. Muller, M.J. Mills: Electronmicroscopy: Probingthe atomic structure and chemistry of grain bound-aries, interfaces and defects, Mater. Sci. Eng. A 260,12–28 (1999)

12.6 M. Springborg:Methods of Electronic-Structure Cal-culations: FromMolecules to Solids (Wiley, Hoboken2000)

12.7 G.B. Olson: Designing a new material world, Sci-ence 288, 993–998 (2000)

12.8 D.A. Muller: Why changes in bond lengths andcohesion lead to core-level shifts in metals, and

consequences for the spatial difference method,Ultramicroscopy 78, 163–174 (1999)

12.9 C. Kisielowski, E. Principe, B. Freitag, D. Hubert:Benefits ofmicroscopywith super resolution, Phys-ica B 308–310, 1090–1096 (2001)

12.10 Y. Wada: Atom electronics: A proposal of nano-scale device based on atom/molecule switching,Microelectron. Eng. 30, 375–382 (1996)

12.11 G.B. Olson: Computational design of hierarchicallystructured materials, Science 277, 1237–1242 (1997)

12.12 M.A. Reed, J.M. Tour: Computing with molecules,Sci. Am. 282, 68–75 (2000)

12.13 N.D. Browning, I. Arslan, P. Moeck, T. Topuria:Atomic resolution scanning transmission electronmicroscopy, Phys. Status Solidi (b) 227, 229–245(2001)

12.14 M. Haider, S. Uhlemann, E. Schwan, H. Rose,B. Kabius, K. Urban: Electronmicroscopy image en-hanced, Nature 392, 768–769 (1998)

12.15 K. Urban: Studying atomic structures by aberra-tion-corrected transmission electron microscopy,Science 321, 506–510 (2008)

12.16 C.L. Jia, S.B. Mi, K. Urban, I. Vrejoiu, M. Alexe,D. Hesse: Atomic-scale study of electric dipolesnear charged and uncharged domain walls in fer-roelectric films, Nat. Mater. 7, 57–61 (2008)

PartA|12

622 Part A Electron and Ion Microscopy

12.17 C.L. Jia, S.B. Mi, M. Faley, U. Poppe, J. Schu-bert, K. Urban: Oxygen octahedron reconstruc-tion in the SrTiO3/LaAlO3 heterointerfaces inves-tigated using aberration-corrected ultrahigh-res-olution transmission electron microscopy, Phys.Rev. B 79, 081405 (2009)

12.18 R. Erni, M.D. Rossell, C. Kisielowski, U. Dahmen:Atomic resolution imaging with a sub-50 pm elec-tron probe, Phys. Rev. Lett. 102, 096101 (2009)

12.19 D. Van Dyck, S. Van Aert, A.J. den Dekker, A. vanden Bos: Is atomic resolution transmission electronmicroscopy able to resolve and refine amorphousstructures?, Ultramicroscopy 98(1), 27–42 (2003)

12.20 N. Shibata, Y. Kohno, S.D. Findlay, H. Sawada,Y. Kondo, Y. Ikuhara: New area detector for atomic-resolution scanning transmission electron mi-croscopy, J. Electron Microsc. 59(6), 473–479 (2010)

12.21 R. Hovden, D.A. Muller: Efficient elastic imaging ofsingle atoms on ultrathin supports in a scanningtransmission electron microscope, Ultramicroscopy109, 59–65 (2012)

12.22 J. Gonnissen, A. De Backer, A.J. den Dekker,G.T. Martinez, A. Rosenauer, J. Sijbers, S. Van Aert:Optimal experimental design for the detection oflight atoms from high-resolution scanning trans-mission electron microscopy images, Appl. Phys.Lett. 105, 063116 (2014)

12.23 A. De Backer, A. De wael, J. Gonnissen, S. VanAert: Optimal experimental design for nano-parti-cle atom-counting from high-resolution STEM im-ages, Ultramicroscopy 151, 46–55 (2015)

12.24 H. Yang, T.J. Pennycook, P.D. Nellist: Efficient phasecontrast imaging in STEM using a pixelated de-tector. Part II: Optimisation of imaging conditions,Ultramicroscopy 151, 232–239 (2015)

12.25 R. Ishikawa, A.R. Lupini, S.D. Findlay, S.J. Pen-nycook: Quantitative annular dark field electronmicroscopy using single electron signals, Microsc.Microanal. 20, 99–110 (2014)

12.26 A.J. den Dekker, S. Van Aert, D. Van Dyck, A. vanden Bos: Maximum likelihood estimation of struc-ture parameters from high resolution electron mi-croscopy image. Part I: A theoretical framework,Ultramicroscopy 104(2), 83–106 (2005)

12.27 S. Van Aert, A.J. den Dekker, A. van den Bos, D. VanDyck, J.H. Chen: Maximum likelihood estimation ofstructure parameters from high resolution electronmicroscopy images. Part II: A practical example, Ul-tramicroscopy 104(2), 107–125 (2005)

12.28 S. Van Aert, A. De Backer, G.T. Martinez, A.J. denDekker, D. Van Dyck, S. Bals, G. Van Tendeloo: Ad-vanced electron crystallography through model-based imaging, IUCrJ 3, 71–83 (2016)

12.29 A. van den Bos: Parameter Estimation for Scientistsand Engineers (Wiley, Hoboken 2007)

12.30 A.J. den Dekker, J. Gonnissen, A. De Backer, J. Sij-bers, S. Van Aert: Estimation of unknown structureparameters from high-resolution (S)TEM images:What are the limits?, Ultramicroscopy 134, 34–43(2013)

12.31 A. van den Bos, A.J. den Dekker: Resolution recon-sidered—Conventional approaches and an alterna-tive, Adv. Imaging Electron Phys. 117, 241–360 (2001)

12.32 G.A.F. Seber, C.J. Wild: Nonlinear Regression (Wiley,Hoboken 1989)

12.33 H. Cramér: Mathematical Methods of Statistics(Princeton Univ. Press, Princeton 1946)

12.34 P.D. Nellist, S.J. Pennycook: The principles andinterpretation of annular dark-field Z-contrastimaging, Adv. Imaging Electron Phys. 113, 147–203(2000)

12.35 D. Van Dyck: High-resolution electron microscopy,Adv. Imaging Electron Phys. 123, 105–171 (2002)

12.36 A. De Backer, K.H.W. van den Bos, W. Van denBroek, J. Sijbers, S. Van Aert: StatSTEM: An effi-cient approach for accurate and precise model-based quantification of atomic resolution electronmicroscopy images, Ultramicroscopy 171, 104–116(2016)

12.37 A.J. den Dekker, J. Sijbers, D. Van Dyck: How to opti-mize the design of a quantitative HREM experimentso as to attain the highest precision?, J. Microsc.194, 95–104 (1999)

12.38 E. Bettens, D. Van Dyck, A.J. den Dekker, J. Sijbers,A. van den Bos: Model-based two-object resolu-tion from observations having counting statistics,Ultramicroscopy 77, 37–48 (1999)

12.39 A.J. den Dekker, S. Van Aert, D. Van Dyck, A. van denBos, P. Geuens: Does amonochromator improve theprecision in quantitative HRTEM?, Ultramicroscopy89, 275–290 (2001)

12.40 S. Van Aert, A.J. den Dekker, D. Van Dyck, A. vanden Bos: High-resolution electron microscopy andelectron tomography: Resolution versus precision,J. Struct. Biol. 138, 21–33 (2002)

12.41 S. Van Aert, A.J. den Dekker, A. van den Bos, D. VanDyck: High resolution electron microscopy: Fromimaging toward measuring, IEEE Trans. Instrum.Meas. 51(4), 611–615 (2002)

12.42 S. Van Aert, A.J. den Dekker, D. Van Dyck, A. vanden Bos: Optimal experimental design of STEMmeasurement of atom column positions, Ultrami-croscopy 90(4), 273–289 (2002)

12.43 S. Van Aert, D. Van Dyck, A.J. den Dekker: Resolu-tion of coherent and incoherent imaging systemsreconsidered—Classical criteria and a statistical al-ternative, Opt. Express 14, 3830–3839 (2006)

12.44 A. Wang, S. Van Aert, P. Goos, D. Van Dyck: Preci-sion of 3-D atomic scale measurements for HRTEMimages: What are the limits?, Ultramicroscopy 114,20–30 (2012)

12.45 W. Van den Broek, S. Van Aert, P. Goos, D. Van Dyck:Throughput maximization of particle radius mea-surements through balancing size versus currentof the electron probe, Ultramicroscopy 111, 940–947(2011)

12.46 S.M. Kay: Detection Theory, Fundamentals of Statis-tical Signal Processing, Vol. 2 (Prentice-Hall, UpperSaddle River 2009)

12.47 J. Gonnissen, A. De Backer, A.J. den Dekker, J. Sij-bers, S. Van Aert: Atom-counting in high resolution

Model-Based Electron Microscopy References 623Part

A|12

electron microscopy: TEM or STEM – That’s thequestion, Ultramicroscopy 174, 112–120 (2017)

12.48 R.I. Jennrich: An Introduction to ComputationalStatistics—Regression Analysis (Prentice-Hall, Up-per Saddle River 1995)

12.49 A. Stuart, K. Ord: Kendall’s Advanced Theory ofStatistics (Arnold, London 1994)

12.50 B.R. Frieden: Physics from Fisher Informa-tion—A Unification (Cambridge Univ. Press,Cambridge 1998)

12.51 Lord Rayleigh: Wave theory of light. In: ScientificPapers, Vol. 3, ed. by J.W. Strutt (Cambridge Univ.Press, Cambridge 1902) pp. 47–189

12.52 M.A. O’Keefe: “Resolution” in high-resolutionelectron microscopy, Ultramicroscopy 47, 282–297(1992)

12.53 J. Gonnissen, A. De Backer, A.J. den Dekker, J. Sij-bers, S. Van Aert: Detecting and locating light atomsfrom high-resolution STEM images: The quest fora single optimal design, Ultramicroscopy 170, 128–138 (2016)

12.54 R. Ishikawa, E. Okunishi, H. Sawada, Y. Kondo,F. Hosokawa, E. Abe: Direct imaging of hydrogen-atom columns in a crystal by annular bright fieldelectron microscopy, Nat. Mater. 10, 278–281 (2011)

12.55 A. Rosenauer,M. Schowalter: STEM SIM—A new soft-ware tool for simulation of STEM HAADF Z-contrastimaging. In: Microscopy of Semiconducting Materi-als, Springer Proceedings in Physics, Vol. 120, ed. byA.G. Cullis, P.A. Midgley (Springer, Dorchester 2008)pp. 170–172

12.56 D. Van Dyck, W. Coene: A new procedure for wavefunction restoration in high resolution electronmi-croscopy, Optik 77, 125–128 (1987)

12.57 H. Lichte: Electron holography approaching atomicresolution, Ultramicroscopy 20, 293–304 (1986)

12.58 D. Van Dyck, M. Op de Beeck, W. Coene: A new ap-proach to object wave function reconstruction inelectron microscopy, Optik 93, 103–107 (1993)

12.59 A.I. Kirkland, W.O. Saxton, K.L. Chau, K. Tsuno,M. Kawasaki: Super-resolution by aperture syn-thesis: Tilt series reconstruction in CTEM, Ultrami-croscopy 57, 355–374 (1995)

12.60 S. Haigh, H. Sawada, A.I. Kirkland: Optimal tiltmagnitude determination for aberration-correctedsuper resolution exit wave reconstruction, Philos.Trans. R. Soc. A 367, 3755–3771 (2009)

12.61 W. Coene, G. Janssen: Phase retrieval through fo-cus variation for ultra-resolution in field-emissiontransmission electron microscopy, Phys. Rev. Lett.69(26), 3743–3746 (1992)

12.62 C. Kisielowski, C.J.D. Hetherington, Y.C. Wang, R. Ki-laas, M.A. O’Keefe, A. Thust: Imaging columns oflight elements carbon, nitrogen and oxygen withsub Ångstom resolution, Ultramicroscopy 89, 243–263 (2001)

12.63 C.L. Jia, A. Thust: Investigation of atomic dis-placements at a ˙3 f111g twin boundary in BaTiO3by means of phase-retrieval electron microscopy,Phys. Rev. Lett. 82(25), 5052–5055 (1999)

12.64 J. Ayache, C. Kisielowski, R. Kilaas, G. Passerieux,S. Lartigue-Korinek: Determination of the atomic

structure of a ˙ 13 SrTiO3 grain boundary, J. Mater.Sci. 40, 3091–3100 (2005)

12.65 S. Bals, S. Van Aert, G. Van Tendeloo, D. Ávila-Brande: Statistical estimation of atomic positionsfrom exit wave reconstruction with a precision inthe picometer range, Phys. Rev. Lett. 96, 096106(2006)

12.66 S. Van Aert, S. Turner, R. Delville, D. Schryvers,G. Van Tendeloo, E.K.H. Salje: Direct observation offerrielectricity at ferroelastic domain boundaries inCaTiO3 by electron microscopy, Adv. Mater. 24, 523–527 (2012)

12.67 L. Goncalves-Ferreira, S.A.T. Redfern, E. Artacho,E.K.H. Salje: Ferrielectric twin walls in CaTiO3, Phys.Rev. Lett. 101, 097602 (2008)

12.68 C.L. Jia, M. Lentzen, K. Urban: Atomic-resolutionimaging of oxygen in perovskite ceramics, Science299, 870–873 (2003)

12.69 C.L. Jia, L. Houben, A. Thust, J. Barthel: On the ben-efit of the negative-spherical-aberration imagingtechnique for quantitative HRTEM, Ultramicroscopy110, 500–505 (2010)

12.70 C.L. Jia, S.B. Mi, K. Urban, I. Vrejoiu, M. Alexe,D. Hesse: Effect of a single dislocation in a het-erostructure layer on the local polarization of a fer-roelectric layer, Phys. Rev. Lett. 102, 117601 (2009)

12.71 S. Van Aert, J. Verbeeck, R. Erni, S. Bals,M. Luysberg,D. Van Dyck, G. Van Tendeloo: Quantitative atomicresolutionmapping using high-angle annular darkfield scanning transmission electron microscopy,Ultramicroscopy 109, 1236–1244 (2009)

12.72 M. Retsky: Observed single atom elastic cross sec-tions in a scanning electron microscope, Optik 41,127 (1974)

12.73 K.E. MacArthur, T.J. Pennycook, E. Okunishi,A.J. D’Alfonso, N.R. Lugg, L.J. Allen, P.D. Nellist:Probe integrated scattering cross sections in theanalysis of atomic resolution HAADF STEM images,Ultramicroscopy 133, 109–119 (2013)

12.74 G.T. Martinez, A. De Backer, A. Rosenauer, J. Ver-beeck, S. Van Aert: The effect of probe inaccura-cies on the quantitative model-based analysis ofhigh angle annular dark field scanning transmis-sion electron microscopy images, Micron 63, 57–63(2014)

12.75 K.E. MacArthur, A.J. D’Alfonso, D. Ozkaya, L.J. Allen,P.D. Nellist: Optimal ADF STEM imaging parame-ters for tilt-robust image quantification, Ultrami-croscopy 156, 1–8 (2015)

12.76 J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer:Quantitative atomic resolution scanning transmis-sion electron microscopy, Phys. Rev. Lett. 100,206101 (2008)

12.77 A. Rosenauer, K. Gries, K. Müller, A. Pretorius,M. Schowalter, A. Avramescu, K. Engl, S. Lutgen:Measurement of specimen thickness and compo-sition in AlxGa1-xN/GaN using high-angle annulardark field images, Ultramicroscopy 109, 1171–1182(2009)

12.78 J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer:Standardless atom counting in scanning transmis-

PartA|12

624 Part A Electron and Ion Microscopy

sion electronmicroscopy, Nano Lett. 10, 4405–4408(2010)

12.79 G.T. Martinez, A. Rosenauer, A. De Backer, J. Ver-beeck, S. Van Aert: Quantitative composition de-termination at the atomic level using model-basedhigh-angle annular dark field scanning transmis-sion electron microscopy, Ultramicroscopy 137, 12–19 (2014)

12.80 T. Grieb, K. Müller, R. Fritz, M. Schowalter,N. Neugebohrn, N. Knaub, K. Volz, A. Rosenauer:Determination of the chemical composition ofGaNAs using STEM HAADF imaging and STEM strainstate analysis, Ultramicroscopy 117, 15–23 (2012)

12.81 G. McLachlan, D. Peel: Finite Mixture Models, Prob-ability and Statistics (Wiley, Hoboken 2000)

12.82 S. Van Aert, K.J. Batenburg, M.D. Rossell, R. Erni,G. Van Tendeloo: Three dimensional atomic imag-ing of crystalline nanoparticles, Nature 470, 374–377 (2011)

12.83 A. De Backer, G.T. Martinez, A. Rosenauer, S. VanAert: Atom counting in HAADF STEM using a statis-tical model-based approach: Methodology, possi-bilities, and inherent limitations, Ultramicroscopy134, 23–33 (2013)

12.84 S. Van Aert, A. De Backer, G.T. Martinez, B. Goris,S. Bals, G. Van Tendeloo, A. Rosenauer: Procedureto count atoms with trustworthy single-atom sen-sitivity, Phys. Rev. B 87, 064107 (2013)

12.85 A. De Backer, G.T. Martinez, K.E. MacArthur,L. Jones, A. Béché, P.D. Nellist, S. Van Aert: Doselimited reliability of quantitative annular darkfield scanning transmission electronmicroscopy fornano-particle atom-counting, Ultramicroscopy 151,56–61 (2015)

12.86 A. Rosenauer, T. Mehrtens, K. Müller, K. Gries,M. Schowalter, P.V. Satyam, S. Bley, C. Tessarek,D. Hommel, K. Sebald, M. Seyfried, J. Gutowski,A. Avramescu, K. Engl, S. Lutgen: Compositionmap-ping in InGaN by scanning transmission electronmicroscopy, Ultramicroscopy 111, 1316–1327 (2011)

12.87 A. De wael, A. De Backer, L. Jones, P.D. Nellist, S. VanAert: Hybrid statistics-simulations based methodfor atom-counting from ADF STEM images, Ultrami-croscopy 177, 69–77 (2017)

12.88 K.H.W. van den Bos, A. De Backer, G.T. Martinez,N. Winckelmans, S. Bals, P.D. Nellist, S. Van Aert:Unscrambling mixed elements using high angle

annular dark field scanning transmission electronmicroscopy, Phys. Rev. Lett. 116, 246101 (2016)

12.89 D. Van Dyck, M. Op de Beeck: A simple intuitivetheory for electron diffraction, Ultramicroscopy 64,99–107 (1996)

12.90 J.M. Cowley, J.C.H. Spence, V.V. Smirnov: The en-hancement of electron microscope resolution byuse of atomic focusers, Ultramicroscopy68, 135–148(1997)

12.91 Z. Saghi, X. Xu, G. Möbus: Model based atomicresolution tomography, J. Appl. Phys. 106, 024304(2009)

12.92 J.R. Jinscheck, K.J. Batenburg, H.A. Calderon, R. Ki-laas, V. Radmilovic, C. Kisielowski: 3-D recon-struction of the atomic positions in a simulatedgold nanocrystal based on discrete tomography:Prospects of atomic resolution electron tomogra-phy, Ultramicroscopy 108, 589–604 (2008)

12.93 S. Bals, M. Casavola, M.A. van Huis, S. Van Aert,K.J. Batenburg, G. Van Tendeloo, D. Vanmaekel-bergh: 3-D atomic imaging of colloidal core-shellnanocrystals, Nano Lett. 11(8), 3420–3424 (2011)

12.94 S. Bals, S. Van Aert, C.P. Romero, K. Lauwaet,M.J. Van Bael, B. Schoeters, B. Partoens, E. Yücelen,P. Lievens, G. Van Tendeloo: Atomic scale dynamicsof ultrasmall germanium clusters, Nat. Commun. 3,897 (2012)

12.95 Z.Y. Li, N.P. Young, M. Di Vece, S. Palomba,R.E. Palmer, A.L. Bleloch, B.C. Curley, R.L. John-ston, J. Jiang, J. Yuan: 3-D atomic-scale structure ofsize-selected gold nanoclusters, Nature 451, 46–48(2008)

12.96 L. Jones, K.E. MacArthur, V.T. Fauske, A.T.J. vanHelvoort, P.D. Nellist: Rapid estimation of catalystnanoparticle morphology and atomic-coordina-tion by high-resolution Z-contrast electron mi-croscopy, Nano Lett. 14, 6336–6341 (2014)

12.97 J.J. Geuchies, C. van Overbeek, W.H. Evers, B. Goris,A. de Backer, A.P. Gantapara, F.T. Rabouw, J. Hil-horst, J.L. Peters, O. Konovalov, A.V. Petukhov,M. Dijkstra, L.D.A. Siebbeles, S. Van Aert, S. Bals,D. Vanmaekelbergh: In situ study of the formationmechanism of two-dimensional superlattices fromPbSe nanocrystals, Nat. Mater. 15, 1248–1254 (2016)

12.98 M. Yu, A.B. Yankovich, A. Kaczmarowski, D. Morgan,P.M. Voyles: Integrated computational and experi-mental structure refinement for nanoparticles, ACSNano 10, 4031–4038 (2016)

Sandra Van AertElectron Microscopy for Materials Research(EMAT)University of AntwerpAntwerp, Belgiumsandra.vanaert@uantwerpen.be

Sandra Van Aert received her PhD from Delft University of Technology in 2003.Thereafter, she joined the Electron Microscopy for Materials Research (EMAT)group of the University of Antwerp, where she became a Senior Lecturer in 2009and Professor in 2016. Her research focuses on new developments in the fieldof model-based electron microscopy aiming at precise measurements of structureparameters.