Optimal experimental design of STEM measurement of atom column positions

Ultramicroscopy 90 (2002) 273–289

Optimal experimental design of STEM measurementof atom column positions

S. Van Aerta,*, A.J. den Dekkera, D. Van Dyckb, A. van den Bosa

aDepartment of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The NetherlandsbDepartment of Physics, University of Antwerp (RUCA), Groenenborgerlaan 171, 2020 Antwerp, Belgium

Received 21 November 2000; received in revised form 6 September 2001

Abstract

A quantitative measure is proposed to evaluate and optimize the design of a high-resolution scanning transmission

electron microscopy (STEM) experiment. The proposed measure is related to the measurement of atom column

positions. Specifically, it is based on the statistical precision with which the positions of atom columns can be estimated.

The optimal design, that is, the combination of tunable microscope parameters for which the precision is highest, is

derived for different types of atom columns. The proposed measure is also used to find out if an annular detector is

preferable to an axial one and if a Cs-corrector pays off in quantitative STEM experiments. In addition, the optimal

settings of the STEM are compared to the Scherzer conditions for incoherent imaging and their dependence on the type

of object is investigated. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 07.05.FB; 61.16.Bg; 02.50.�r; 43.50.+y

Keywords: Scanning transmission electron microscopy (STEM); Electron microscope design and characterization; Data processing/

image processing

1. Introduction

For many years, it has been standard practice toevaluate the performance of STEM imagingmodes qualitatively, that is, in terms of directvisual interpretability. The performance criteriaused are resolution and contrast. For example,when axial bright-field coherent STEM is com-pared to annular dark-field incoherent STEM, thelatter imaging mode is preferred. The basic ideas

underlying this preference are the higher resolutionfor incoherent imaging than for coherent imaging[1] and the higher contrast in dark-field imagesthan in bright-field images [2]. In annular dark-field incoherent STEM, visual interpretation of theimages is optimal if the Scherzer conditions [3] forincoherent imaging are adapted [4]. As demon-strated in [5], the resolution can be improvedfurther if the main lobe of the probe is narrowed.However, visual interpretability is then reduced asa result of a considerable rise of the sidelobes ofthe probe.However, two important aspects are absent in

these widely used performance criteria. First, the

*Corresponding author. Tel.: +31-15-2781823; fax: +31-15-

2784263.

E-mail address: [email protected] (S. Van Aert).

0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 1 5 2 - 8

electron–object interaction is not taken into ac-count. Second, the dose efficiency, which is definedas the fraction of the primary electron dose that isdetected, is left out of consideration. Higherresolution and contrast are often obtained at theexpense of dose efficiency, which leads to adecrease in signal-to-noise ratio (SNR). Forexample, the incoherence in annular dark-fieldincoherent STEM is only attained by using anannular dark-field detector with a geometry muchlarger than the objective aperture [6]. Its corre-sponding higher resolution, by adapting theScherzer conditions for incoherent imaging, is thusobtained at the expense of dose efficiency. Anotherexample is the following. It is well known that inbright-field images, decreasing the size of an axialdetector leads to higher contrast, but also to adeterioration of the SNR. To compensate for sucha decrease in SNR, longer recording times arenecessary, which in turn increase the disturbinginfluence of specimen drift.In our opinion, the ultimate goal of STEM is

not providing optimal visual interpretability, butquantitative structure determination instead. Thereason for this is that the positions of atomcolumns should be known within sub-angstr .omprecision in order to understand the physical andchemical properties of materials such as super-conductors, nanoparticles, quantum transistors,etc. [7]. The images are then to be considered asdata planes from which the structural informationhas to be measured quantitatively. This can bedone as follows: one has a model for the object andfor the imaging process, including electron–objectinteraction, microscope transfer and image detec-tion. This model describes the expectations of theintensity observations [8] and it contains para-meters that have to be measured experimentally.The quantitative structure determination is doneby fitting the model to the experimental data byuse of criteria of goodness of fit, such as, leastsquares, least absolute values or maximum like-lihood. Thus, quantitative structure determinationbecomes a statistical parameter estimation pro-blem. Ultimately, the structural parameters, suchas the positions of the atom columns, have to bemeasured as precisely as possible. However, thisprecision will always be limited by the presence of

noise. Given the parametric model and knowledgeabout the statistics of the observations, use of theconcept of Fisher information [9] allows us toderive an expression for the highest attainableprecision with which the positions of the atomcolumns can be estimated. This expression, whichis called the Cram!er–Rao Lower Bound (CRLB),is a function of the object parameters, microscopeparameters, and dose efficiency. Therefore, it maybe used as an alternative performance measure inthe optimization of the design of a STEMexperiment for a given object. The optimal designis the set of microscope parameters resulting in thehighest overall attainable precision. In practice,the design is optimized by minimizing a scalarmeasure of the CRLB. An overview of the conceptof the CRLB and its use can be found in [9]. In[10], an example can be found in which the CRLBis computed in order to optimize the design of aquantitative HREM experiment.The microscope parameters considered are

related to the probe, the image recording and thedetector configuration. The probe is described bythe objective aperture radius, the defocus, thespherical aberration constant, the electron wave-length, the width of the source image, and thereduced brightness of the source. The parametersdescribing the image recording are the field of view(FOV), that is, the area scanned on the specimen,the probe sampling distances and the recordingtime. The detector configuration is described byeither the inner radius of an annular detector orthe outer radius of an axial detector.The paper is organized as follows. The para-

metric model for the expectations of the intensityobservations is described in Section 2. In Section 3,the joint probability density function of theobservations is introduced. From this, the CRLBwill be derived in Section 4. In Section 5, theCRLB is used to evaluate and optimize the designof a STEM experiment. It is assumed that theexperimental design is limited by specimen drift.

2. Parametric model for the intensity observations

A parametric model, describing the expectationsof the intensity observations recorded by the

S. Van Aert et al. / Ultramicroscopy 90 (2002) 273–289274

STEM, is needed in order to derive an expressionfor the CRLB. In this section, such a model will bederived by use of the simplified channeling theory[11–13] and of the fact that the scattering isdominated by the tightly bound 1s-type state ofthe atom columns. The source image will also betaken into account. The model contains bothmicroscope parameters, such as objective apertureradius, defocus, spherical aberration constant,detector radius, width of the source image, andobject parameters, such as the positions of theatom columns, the energy of the 1s-states and theobject thickness.

2.1. The exit wave

According to the simplified channeling theory,described in [11–13], an expression can be derivedfor the exit wave cðr; zÞ of an object consisting ofN atom columns. This is a complex wave functionin the plane at the exit face of the object, resultingfrom the interaction of the probe with the object:

cðr; zÞ ¼ pðr� r0Þ þXN

n¼1

cnðr0 � bnÞf1s;nðr� bnÞ

exp �ipE1s;n

E0kz

� �� 1

� �; ð1Þ

where r ¼ ðx; yÞT is a two-dimensional vector inthis plane. The symbol T denotes taking thetranspose. The specimen thickness is z; E0 is theincident electron energy, k is the inverse electronwavelength, and r0 ¼ ðxk; ylÞ

T is the position of theSTEM probe, described by the function pðr� r0Þ:The function f1s;nðr� bnÞ is the 1s-state of thecolumn at position bn ¼ ðbxn ; bynÞ

T: In accordancewith the 1s-state of an atom, it is the lowest energybound state with energy E1s;n: In Eq. (1), it isassumed that the dynamical motion of the electronin a column can be expressed primarily in terms ofthis tightly bound 1s-state. The excitation coeffi-cients cn can be found from

cnðr0 � bnÞ ¼Z

fn

1s;nðr� bnÞpðr� r0Þ dr; ð2Þ

where ‘*’ denotes taking the complex conjugate.Since the 1s-state is a real function and since theprobe is assumed to be radially symmetric, so that

pðrÞ ¼ pð�rÞ; Eq. (2) can be written as a convolu-tion product:

cnðr0 � bnÞ ¼ pðr0 � bnÞnf1s;nðr0 � bnÞ: ð3Þ

If the convolution theorem is used, Eq. (3) may bewritten as

cnðr0 � bnÞ

¼ I�1g-r0�bn

PðgÞF1s;nðgÞ

¼Z

PðgÞF1s;nðgÞ expð�i2pg � ðr0 � bnÞÞ dg; ð4Þ

where PðgÞ is the Fourier transform of the probepðrÞ; F1s;nðgÞ is the Fourier transform of the 1s-state f1s;nðrÞ; g is a two-dimensional spatialfrequency vector in reciprocal space, and ‘ � ’denotes the scalar product. In this paper, the two-dimensional Fourier transform F ðgÞ of an arbi-trary function f ðrÞ is defined as

F ðgÞ ¼ Ir-g f ðrÞ ¼Z

f ðrÞ expði2pg � rÞ dr: ð5Þ

Consequently, the inverse Fourier transform isdefined as

f ðrÞ ¼ I�1g-rF ðgÞ ¼

ZF ðgÞ expð�i2pg � rÞ dg: ð6Þ

For radially symmetric 1s and probe functions,Eq. (4) can be written as

cnðr0 � bnÞ

¼ cnðjr0 � bnjÞ

¼ 2pZ

N

0

PðgÞF1s;nðgÞJ0ð2pgjr0 � bnjÞg dg: ð7Þ

This is an elementary result of the theory of Besselfunctions, where J0ð�Þ is the zeroth-order Besselfunction of the first kind and

jr0 � bnj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxk � bxn Þ

2 þ ðyl � bynÞ2

qð8Þ

is the distance from the probe to the atom columnposition.The illuminating STEM probe pðrÞ is the inverse

Fourier transform of the transfer function of the

S. Van Aert et al. / Ultramicroscopy 90 (2002) 273–289 275

objective lens PðgÞ:

pðrÞ ¼ I�1g-rPðgÞ: ð9Þ

The transfer function PðgÞ is radially symmetricand given by

PðgÞ ¼ PðgÞ ¼ AðgÞ expðiwðgÞÞ; ð10Þ

where g ¼ jgj is the spatial frequency. The circularaperture function AðgÞ is defined as

AðgÞ ¼1 if gpgap;

0 if g > gap;

(ð11Þ

with gap being the aperture radius. The phase shiftwðgÞ; resulting from the objective lens aberrations,is radially symmetric and given by

wðgÞ ¼ pelg2 þ 12pCsl

3g4; ð12Þ

with Cs being the spherical aberration constant, ethe defocus, and l the electron wavelength.According to [14], the 1s-state function is well

approximated by a two-dimensional quadraticallynormalized (radially symmetric) Gaussian func-tion:

f1sðrÞ ¼1

affiffiffiffiffiffi2p

p exp �r2

4a2

� �; ð13Þ

where r ¼ jrj and a represents the column-depen-dent width. This width is directly related to theenergy of the 1s-state. The Fourier transformF1sðgÞ of Eq. (13), that is, the Fourier spectrum ofthe object, is given by

F1sðgÞ ¼1

cffiffiffiffiffiffi2p

p exp �g2

4c2

� �ð14Þ

with

c ¼1

4pa; ð15Þ

the width of the Fourier transformed 1s-state.

2.2. The image intensity distribution

From the exit wave, which is given in Eq. (1),the total detected intensity in the Fourier detectorplane of a STEM can be derived [15] as

Ipsðr0Þ ¼Z

jCðg; zÞj2DðgÞ dg; ð16Þ

where Cðg; zÞ is the two-dimensional Fouriertransform of the exit wave cðr; zÞ; and DðgÞ is thedetector function, which is equal to one inthe detected field and equal to zero elsewhere.The two-dimensional Fourier transform of the exitwave can be derived by combining Eqs. (1) and (5):

Cðg; zÞ ¼PðgÞ expð2pig � r0Þ

þXN

n¼1

cnðr0 � bnÞF1s;nðgÞ expð2pig � bnÞ

� exp �ipE1s;n

E0kz

� �� 1

� �: ð17Þ

Thus far, it has been assumed that the source canbe modeled as a point. Therefore, the subscript ‘ps’in Eq. (16) refers to point source. Elaborating onthe ideas given in [16], it follows that the finite sizeof the source image can be taken into account by atwo-dimensional convolution of the intensitydistribution Ipsðr0Þ with the intensity distributionof the source image SðrÞ:

Iðk; lÞ ¼ Iðr0Þ ¼ Ipsðr0Þ*Sðr0Þ: ð18Þ

The effect of the source image is thus an additionalblurring. The indices ðk; lÞ correspond to the probeat position r0 ¼ ðxk; ylÞ

T: A realistic form for theintensity distribution of the source image isGaussian [16]. The function SðrÞ is thus a two-dimensional normalized Gaussian distribution:

SðrÞ ¼ SðrÞ ¼1

2ps2exp �

r2

2s2

� �ð19Þ

with s representing the width corresponding to theradius containing 39% of the total intensity of S:Up to now, no assumptions have been made

about the shape or size of the detector. From nowon, however, the detector is assumed to be radiallysymmetric. Mathematically this means that DðgÞ ¼DðgÞ:Eq. (18) can be split up into three terms:

Iðk; lÞ ¼ I0 þ I1ðk; lÞ þ I2ðk; lÞ: ð20Þ

The zeroth order term I0 corresponds to a non-interacting probe, the first order term I1ðk; lÞ tothe interference between the probe and the 1s-state and the second order term I2ðk; lÞ to the


interference between different 1s-states. The zerothorder term I0 is given by

I0 ¼Z

jPðgÞ expð2pig � r0Þj2DðgÞ dg� �

*Sðr0Þ: ð21Þ

It describes a constant background intensity,resulting from the non-interacting electrons col-lected by the detector. This equation can bewritten as

I0 ¼ 2pZ

A2ðgÞDðgÞg dg ð22Þ

by substitution of Eq. (10) and using the fact thatDðgÞ is radially symmetric. Due to the definition ofthe aperture function, given in Eq. (11), thefollowing equality may be used:

A2ðgÞ ¼ AðgÞ: ð23Þ

Therefore, Eq. (22) becomes

I0 ¼ 2pZ

AðgÞDðgÞg dg: ð24Þ

The first order term I1ðk; lÞ corresponds to theinterference between the incident probe pðr� r0Þand the 1s-state f1s;nðr� bnÞ:

I1ðk; lÞ ¼XN

n¼1

2 Re

"cnðjr0 � bnjÞ

� exp �ipE1s;n

E0kz

� �� 1

� �

� 2pZ

PnðgÞF1s;nðgÞJ0ð2pgjr0 � bnjÞ

� DðgÞg dg

#*Sðr0Þ: ð25Þ

This is a linear term in the sense that thecontributions of different atom columns areadded. The second order term I2ðk; lÞ describesthe mutual interference between different 1s-statesf1s;n and f1s;m:

I2ðk; lÞ ¼XN

n¼1

XN

m¼1

"cnðjr0 � bnjÞcnmðjr0 � bmjÞ

� exp �ipE1s;n

E0kz

� �� 1

� �

� exp þipE1s;m

E0kz

� �� 1

� �

� 2pZ

F1s;nðgÞF1s;mðgÞ

� J0ð2pg dAn�Am ÞgDðgÞ dg

#nSðr0Þ; ð26Þ

where

dAn�Am ¼ jbn � bmj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbxn � bxm Þ

2 þ ðbyn � bym Þ2

qð27Þ

is the distance between the atom columns atpositions bn and bm:It is only the last term I2ðk; lÞ of Eq. (20) that

remains if the detector is annular and has an innerradius greater than or equal to the aperture radius.Then,

PðgÞDðgÞ ¼ 0 ð28Þ

or, equivalently,

AðgÞDðgÞ ¼ 0: ð29Þ

Therefore, Eq. (26) describes the model for annu-lar dark-field STEM.

2.3. The image recording

In a STEM, the illuminating probe scans thespecimen in a raster fashion. The image is thusrecorded as a function of the probe position r0 ¼ðxk; ylÞ

T: Without loss of generality, the imagemagnification is ignored. Therefore, the probeposition r0 ¼ ðxk; ylÞ

T directly corresponds to animage pixel at the same position.The recording device is characterized as consist-

ing of K � L equidistant pixels of area Dx � Dy;where Dx and Dy are the probe sampling distancesin the x and y directions, respectively. Pixel ðk; lÞcorresponds to position ðxk; ylÞ

T ðx1 þ ðk �1ÞDx; y1 þ ðl � 1ÞDyÞT with k ¼ 1;y;K and l ¼1;y;L and ðx1; y1Þ

T represents the position of the


pixel in the bottom left corner of the field of viewFOV. The FOV is chosen to be centered aboutð0; 0ÞT:Assuming a recording time t for one pixel and a

probe current J; one calculates the number ofelectrons per probe position:

Jte

ð30Þ

with e ¼ 1:6� 10�19 C the electron charge. Therecording time t for one pixel is the ratio ofthe frame time t; that is, the recording time for thewhole FOV, to the total number of pixels KL:

t ¼t

KL: ð31Þ

The primary electron dose D is thus given by

D ¼ KLJte: ð32Þ

The probe current J is given by [17]

J ¼BrE0p2d2I50a

2

4eð33Þ

with Br being the reduced brightness of the source,E0 the beam energy, dI50 the diameter of the sourceimage containing 50% of the current and a; whichis equal to gapl; the aperture angle. From Eq. (19),it follows that

dI50 ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln 0:5

ps: ð34Þ

As a consequence of the detector shape and size inSTEM, only the electrons within a selected part ofthe diffraction pattern are used to produce theimage. Mathematically, this is expressed inEq. (16). The expected number of detected elec-trons per pixel position ðk; lÞ equals [18]

fkl

Jte; ð35Þ

where fkl denotes the fraction of electrons recordedby the detector ðfklo1Þ: This fraction can beexpressed as

fkl ¼Iðk; lÞID¼1

; ð36Þ

with ID¼1 being the constant intensity obtained ifthe detector function DðgÞ is uniform:

ID¼1 ¼ 2pZ

AðgÞg dg; ð37Þ

following from straightforward calculations, usingEqs. (20)–(26). The total number of detectedelectrons to form the image is now equal to

Ddet ¼XK

k¼1

XL

l¼1

fkl

Jte: ð38Þ

Then, the dose efficiency DE, which is defined asthe fraction of the primary electron dose that isdetected, becomes

DE ¼Ddet

D¼PK

k¼1

PLl¼1 fkl

KL: ð39Þ

The result of Eq. (35) defines the expectationvalues of the intensity observations recorded bythe detector, which is needed to derive the jointprobability density function in the next section.

3. The joint probability density function of the

observations

In any STEM experiment, the observations will‘‘contain errors’’. These errors have to be specified,which is the subject of this section.Generally, sets of observations made under the

same conditions nevertheless differ from experi-ment to experiment. The usual way to describe thisbehavior is to model the observations as stochasticvariables. The reason is that there is no viablealternative and that it has been found to work.Stochastic variables are defined by probabilitydensity functions [19]. In a STEM experiment theobservations are counting results, for which theprobability density function can be modeled as aPoisson distribution. In what follows, the under-lined characters represent stochastic variables.Consider a set of KL stochastic observations

f%wkl ; k ¼ 1;y;K; l ¼ 1;y;Lg; where the index

kl corresponds to the pixel at the position ðxk; ylÞT:

In what follows, the vector w will represent theKL � 1 column vector of these observations. Theobservations are assumed to be statistically in-dependent and have a Poisson distribution. Theprobability that the observation

%wkl is equal to okl

is given by [20]

loklkl

okl !expð�lklÞ; ð40Þ


where the parameter lkl is equal to the expectationE½

%wkl : The expectation values E½

%wkl are given in

Eq. (35):

E½%wkl ¼ fkl

Jte: ð41Þ

The probability fwðx; bÞ that a set of observationsis equal to fokl ; k ¼ 1;y;K ; l ¼ 1;y;Lg is theproduct of all probabilities described by Eq. (40)since the observations are assumed to be statisti-cally independent:

fwðx; bÞ ¼YKk¼1

YLl¼1

loklkl

okl !expð�lklÞ; ð42Þ

where the elements of x correspond with those ofw and the column vector b contains the unknownparameters. In our case b ¼ ðbx1 ; by1 ;y; bxN

; byNÞT

contains the coordinates in the x- and y-directionof the N atom column positions. The functiondescribed in Eq. (42) is called the joint probabilitydensity function of the observations.The joint probability density function, which

has been derived here, will now be used for thecomputation of the CRLB.

4. The Cram!er–Rao Lower Bound

In this section, the CRLB is discussed. TheCRLB is a lower bound on the variance of anyunbiased estimator. What does this mean? Sup-pose that an experimentalist wants to measure theposition parameters b ¼ ðbx1 ; by1 ;y; bxN

; byNÞT of

a set of N atom columns quantitatively. For thispurpose, one can use many estimators (estimationmethods) such as least squares, least absolutevalues or maximum likelihood estimators. Theprecision of an estimator is represented by thevariance or by its square root, the standarddeviation. Generally, different estimators will havedifferent precisions. It can be shown, however,that the variance of unbiased estimators will neverbe lower than the CRLB. Fortunately, there existsa class of estimators (including the maximumlikelihood estimator) that achieves this bound atleast asymptotically, that is, for the number ofobservations going to infinity. For the details ofthis lower bound we refer to [9].

4.1. Fisher information

The CRLB follows from the concept of Fisher

information. The Fisher information matrix Fb forestimation of the position parameters of a set of N

atom columns b ¼ ðbx1 ; by1 ;y; bxN; byN

ÞT ¼ðb1;y; b2N Þ

T is defined as

Fb ¼ �E@2ln fwðw; bÞ

@b@bT

� �; ð43Þ

where fwðw; bÞ is the joint probability densityfunction of the observations described in Eq. (42)and

@2ln fwðw; bÞ

@b@bTð44Þ

is the 2N � 2N Hessian matrix of ln f%wð%w; bÞ

defined by its ðp; qÞth element

@2ln f%wð%w; bÞ

@bp@bq

; ð45Þ

where bp and bq correspond to the p and qthelement of the vector b; respectively.

4.2. Cram !er–Rao inequality

Suppose that #b ¼ ð #bx1; #by1

;y; #bxN; #byN

ÞT ¼ð #b1;y; #b2N Þ

T is an unbiased estimator of b: TheCram!er–Rao inequality then states that [21]

Covð#b; #bÞXF�1b ; ð46Þ

where Covð#b; #bÞ is the 2N � 2N variance–covar-iance matrix of the estimator #b; defined by itsðp; qÞth element covð#bp; #bqÞ: Its diagonal elementsare thus the variances of the elements of #b: Thematrix F�1

b is called the Cram!er–Rao lower boundon the variance of #b: The Cram!er–Rao inequality(46) expresses that the difference between the left-hand and right-hand member is positive semi-definite. A property of a positive semi-definitematrix is that its diagonal elements cannot benegative. This means that the diagonal elements ofCovð#b; #bÞ will always be larger than or equal to thecorresponding diagonal elements of the inverse ofthe Fisher information matrix. Therefore, thediagonal elements of F�1

b define lower bounds onthe variances of the elements of #b

Varð #bpÞXF�1b ðp; pÞ; ð47Þ


where p ¼ 1;y; 2N and F�1b ðp; pÞ is the ðp; pÞth

element of the inverse of the Fisher informationmatrix. The elements Fbðp; qÞ may be calculatedexplicitly using Eqs. (41)–(45):

Fbðp; qÞ ¼ �EXK

k¼1

XL

l¼1

@2

@bp@bq %wkl ln fkl

Jte

� ��"

� fkl

Jte

!#

¼ �XK

k¼1

XL

l¼1

E½%wkl

@

@bp

1

fkl

@fkl

@bq

� ��

@2fkl

@bp@bq

Jte

� �

¼Jte

XK

k¼1

XL

l¼1

1

fkl

@fkl

@bp

@fkl

@bq

: ð48Þ

The derivative of fklðbÞ; with respect to b may becalculated from the parametric model of theintensity observations described in Section 2.In this section, it has been shown how from the

parametric model of the intensity observationsdescribed in Section 2 and the joint probabilitydensity function described in Section 3, theelements of the Fisher information matrix Fb ofEq. (48) may be calculated explicitly. From thelatter, the CRLB may be computed. The diagonalelements of the CRLB give a lower bound on thevariance of any unbiased estimator of the x- and y-coordinates of a set of N atom columns. TheCRLB is a function of microscope and objectparameters. In the following section, this lowerbound will be used to study the dependence of theprecision on the microscope parameters fordifferent objects.

5. Experimental design

The CRLB, which is discussed in Section 4, willbe used to evaluate and optimize the experimentaldesign of a quantitative STEM experiment. First,it will be stated which optimality criterion ischosen given the purpose of this paper. Theoptimal experimental design will be determinedby minimizing a scalar measure s2CR of the CRLBas a function of the microscope parameters.Second, an overview of the microscope parameters

will be given. Some of them are tunable, whileothers are fixed. Third, the results of the numericalevaluation of the dependence of s2CR on theseparameters will be discussed. Fourth, an inter-pretation of these results will be given. Fifth, theresulting optimal design will be compared with thedesign that is assumed to be optimal in theconventional approach using performance criteriathat are based on resolution and contrast.

5.1. Optimality criterion

The purpose of this paper is to optimize thedesign of a STEM so as to measure the atomcolumn positions as precisely as possible. Theprecision with which the atom column coordinatescan be measured is represented by the diagonalelements of the CRLB, that is, the right-hand sidemembers of inequalities (47). However, simulta-neous minimization of the diagonal elements of theCRLB as a function of the microscope parametersis in most cases impossible. A decrease of aparticular diagonal element has often the unfavor-able side effect of an increase of others. Therefore,the optimal experimental design has to be deter-mined by minimizing a scalar measure of theCRLB. Several measures can be found in literature[22,23]. Among these optimality criteria theobvious choice given the purpose of this paper isthe A-optimality criterion. The A-optimality cri-terion is defined by the scalar measure s2CR:

s2CR ¼ trace F�1b ; ð49Þ

that is, the trace of F�1b ; or equivalently, the sum of

the variances of the estimators of the atom columncoordinates. Minimization of s2CR as a function ofthe microscope parameters results in the optimalexperimental design.

5.2. Microscope parameters

An overview of the microscope parameters,which enter the parametric model for the STEMintensity observations, described in Section 2, isgiven here. For simplicity, some of these para-meters will be kept constant in the evaluation andoptimization of the experimental design.


The parameters describing the detector config-uration are related to the detector function DðgÞ:In principle, the detector can have any shape orsize. However, in the present work we will confineourselves to the more common ones, which are,annular and axial detectors. The inner or outerradius gdet of the annular or axial detector,respectively, is tunable.The parameters describing the probe are: the

defocus e; the spherical aberration constant Cs; theobjective aperture radius gap; the electron wave-length l; the width of the source image s; and thereduced brightness Br: In the evaluation, l and Brhave been kept constant.The parameters describing the image recording

are the FOV ; the probe sampling distances or,equivalently, the pixel sizes, Dx and Dy; thenumber of pixels K and L in the x- and y-direction, respectively, and the recording time t:The FOV is chosen fixed, but large enough so as toguarantee that the tails of the probe are collected.The pixel sizes Dx and Dy are kept constant. It canbe shown that the precision will generally improvewith smaller pixel sizes, with all other parameterskept constant. However, below a certain pixel size,no more improvement is gained. This has to dowith the fact that the pixel SNR decreases with adecreasing pixel size. Therefore, the pixel sizes arechosen in the region where no more improvementmay be gained. This is similar to what is describedin [10,24]. It is directly clear from Eqs. (33), (48)and (49) that the precision will increase propor-tionally to the recording time t and the reducedbrightness Br: In what follows, the recording time tis kept constant, presuming that specimen driftsets practical limits on the exposure time. Asalready mentioned Br has been kept constant too.

5.3. Numerical results

In this subsection, the results of the numericalevaluation of the dependence of s2CR on themicroscope parameters will be discussed. Thecases for which the parametric model consists ofan isolated atom column and of two atom columnsare considered in the first and second part,respectively. It will be discussed how the optimal

experimental design of an isolated atom column isinfluenced by a neighboring atom column.

5.3.1. Isolated atom column

In this part, the experimental design is evaluatedand optimized for the special case of a parametricmodel consisting of an isolated atom column.Therefore, s2CR which is defined in Eq. (49), isequal to the sum of the lowest variance with whichthe x- and y-coordinate of the atom columnposition can be estimated. The FOV is chosencentered about the atom column, because in thatcase it may be shown that the variances on the x-and y-coordinate are equal.To begin with, it is assumed that the width of

the source image is determined by the apertureangle, following the relation

dI50 ¼0:54la

; ð50Þ

where 0:54l=a is equal to the diameter of thediffraction-error disc containing 50% of the totalintensity. In this way, the contribution of thesource image to the total probe size is small. Then,meeting Eq. (50), it follows from Eq. (33) that theprobe current is constant and equal to JB ¼10�18Br [17].Next, the dependence of the precision on the

aperture radius is studied. From the evaluation ofs2CR; it is found that the optimal aperture radius ismainly determined by the object and that it is thesame for annular or axial detectors. The optimalaperture radius is proportional to the width of thecolumn’s 1s-state F1sðgÞ described by Eq. (14).Fig. 1 compares the optimal aperture radius (forCs ¼ 0:5 mm and at optimal defocus) with thewidth c ¼ 1=ð4paÞ of the 1s-state F1sðgÞ: Theoptimal aperture radii are plotted as a functionof ðd2=Z þ 0:276BÞ�1=2; since this is proportionalto the width c; as shown in [25]. For a given atomcolumn, d represents the interatomic distance, Z

the atomic number and B the Debye–Wallerfactor. The widths a of the corresponding 1s-statesf1sðrÞ in real space are given in Table 1. It is clearfrom Fig. 1 that the influence of the object on theoptimal aperture radius is substantial. In contrastto what one may expect, the resulting probe in theoptimal design is not as small as possible. It is even


larger than the 1s-function f1sðrÞ; which is shownin Fig. 2 for a silicon and a gold column in ½1 0 0 -direction. Furthermore, an increase of the sphe-rical aberration constant results in a decrease ofthe optimal aperture radius and vice versa. Thiseffect is especially important for heavy atomcolumns, such as a gold column in ½1 0 0 -direction,where the optimal aperture radius for Cs ¼ 0 mmis equal to 0:75 (A

�1and for Cs ¼ 0:5 mm it is

equal to 0:5 (A�1: For a lighter atom column, suchas silicon in ½1 0 0 -direction, the optimal apertureradius for Cs ¼ 0 and 0:5 mm are the same andequal to 0:28 (A

�1:

Then, a comparison between an annular and anaxial detector is presented. In Fig. 3, s2CR isevaluated for a Si [1 0 0] column as a function ofthe detector-to-aperture radius, for an annular aswell as an axial detector. The aperture radius and

0.8 1.2 1.60.2

0.4

0.6

Width of 1s-stateOptimal aperture radius

Å-1

(d2/Z+0.276B)-1/2 (Å-1)

Si[100]

Si[110]Sr[100]

Sn[100]Cu [100]

Au [100]

Fig. 1. Comparison of the optimal aperture radius (for

Cs ¼ 0:5 mm) with the width of the 1s-state F1sðgÞ; which isproportional to ðd2=Z þ 0:276BÞ�1=2: The Debye–Waller factorB is set to 0:6 (A

2: The optimal aperture radius increases with

the width of the 1s-state. Therefore, it varies strongly for

different atom columns.

Table 1

Width of the 1s-state, and its energy (Debye–Waller factor ¼ 0:6 (A2), interatomic distance, atomic number, and intercolumn distance

for different atom columns

Si [1 0 0] Si [1 1 0] Sr [1 0 0] Sn [1 0 0] Cu [1 0 0] Au [1 0 0]

a ð (AÞ 0.34 0.27 0.22 0.20 0.18 0.13

E1s ðeVÞ �20.2 �37.4 �57.3 �69.8 �78.3 �210.8d ð (AÞ 5.43 3.84 6.08 6.49 3.62 4.08

Z 14 14 38 50 29 79

dA�A ð (AÞ 1.92 1.36 4.03 2.29 2.56 2.88

-10 -5 50 100.0

0.4

0.8

1.2

ProbeSi [100]

Am

plitu

de

Spatial coordinate (Å) Spatial coordinate (Å)

-10 -5 0 5 100.0

0.8

1.6

2.4

3.2ProbeAu [100]

Fig. 2. The left and right figure represent the 1s-state f1sðrÞ fora Si ½1 0 0 column and a Au ½1 0 0 column, respectively. Theamplitude jpðrÞj of their associated optimal probes (forCs ¼ 0:5 mm) are also shown.

0.4 0.8 1.2 1.60.0

0.1

0.2 Annular detectorAxial detector

σ2 CR

(Å2 )

gdet /gap

Fig. 3. The criterion s2CR; which is defined in Eq. (49),computed as a function of the detector-to-aperture radius for

a Si ½1 0 0 column, using an annular and an axial detector. Theaperture radius and the defocus are set to their optimal values

0:28 (A�1and �8 nm; respectively. The detector radius is

variable. All other parameters are fixed (see Tables 2 and 3).


the defocus are set to their optimal values 0:28 (A�1

and �8 nm; respectively. All other parameters areheld fixed (Tables 2 and 3). Three relations can bederived from this figure. First, for an annulardetector, the optimal design is obtained when thedetector radius equals the optimal aperture radius.Second, for an axial detector, the optimal detectorradius is slightly smaller than the optimal apertureradius. Third, an annular detector may result inhigher precisions than an axial detector, whenoperating at the optimal conditions.Subsequently, the dependence of the precision

on the defocus is evaluated. In Fig. 4, s2CR isplotted for a Si ½1 0 0 column as a function of thedefocus for an annular detector, whereas in Fig. 5this is done for an axial detector. These figuresshow that the optimal defocus is the same for anannular and an axial detector, for a given sphericalaberration constant. Furthermore, the optimaldefocus depends strongly on the spherical aberra-tion constant. An empirical relation has beenfound between the optimal defocus e; the optimalaperture radius gap and the spherical aberrationconstant Cs:

eE� 12Csl

2g2ap: ð51Þ

In this case, the transfer function is nearly equal toone over the whole angular range of the objectiveaperture. The optimal transfer function for a Si½1 0 0 column is presented in Fig. 6, where thevertical line represents the optimal aperture radius

and Cs is set to 0:5 mm: Eq. (51) is derived fromEq. (12) by setting the phase shift wðgÞ exactly tozero for g ¼ gap:Then, the potential merit of Cs-correctors in

quantitative STEM applications is studied. Figs. 7and 8 show the ratio s2CR=s

2CR ðCs ¼ 0 mmÞ as a

function of the spherical aberration constant, foran annular as well as an axial detector. This isdone for a Si ½1 0 0 column, as well as for a Au½1 0 0 column. The aperture radius has been set tothe value that is optimal for Cs ¼ 0:5 mm: FromFig. 7, it follows that the precision that can begained by reducing Cs is only marginal for a Si

Table 2

Initial microscope parameter values

E0ðkeVÞ

CsðmmÞ

K L Dx

ðpmÞDy

ðpmÞt ðsÞ Br

A

m2srV

!

300 0.5 99 99 20 20 8� 10�8 2� 107

-400 -200 0 200 4000.000

0.002

0.004

0.006

Cs = 0 mmCs = 0.5 mm

σ2 C

R(Å

)2

Defocus (Å)

Fig. 4. The criterion s2CR; which is defined in Eq. (49),computed as a function of defocus for a Si ½1 0 0 column,using an annular detector (for Cs ¼ 0 and 0:5 mm). Theaperture and detector radius are set to their optimal value

0:28 (A�1: All other parameters are fixed (see Tables 2 and 3).

-400 -200 0 200 4000.00

0.01

0.02

0.03

0.04

Cs = 0mmCs = 0.5 mm

σ2 C

R(Å

2 )

Defocus (Å)

Fig. 5. The criterion s2CR; which is defined in Eq. (49),computed as a function of defocus for a Si ½1 0 0 column,using an axial detector (for Cs ¼ 0 and 0:5 mm). The apertureand detector radius are set to their optimal values 0:28 and0:22 (A

�1; respectively. All other parameters are fixed (see

Tables 2 and 3).

Table 3

Parameters of an isolated atom column

bx ð (AÞ by ð (AÞ z ð (AÞ

0 0 �E0

E1sk


½1 0 0 column, whereas it follows from Fig. 8 thatsuch a reduction would be more likely to pay offfor a Au ½1 0 0 column. For an annular detector,the precision, expressed in terms of the variance, atCs ¼ 1 mm is about 1:008 and 1:752 times worsethan the precision at Cs ¼ 0 mm for a Si ½1 0 0 anda Au ½1 0 0 column, respectively. In terms of thestandard deviation, which is defined as the squareroot of the variance, these fractions are 1:004 and1:324; respectively. This result may be explained bythe fact that the optimal aperture setting isstrongly dependent on the atom column. Theoptimal aperture radius for a Au ½1 0 0 column ismuch larger than for a Si ½1 0 0 column. Becausespherical aberration is observable only for non-

paraxial rays, correction is only necessary forobjective lenses working with larger apertures. Inother applications, a Cs-corrector may be moreworthwhile: a larger pole-piece gap may bepossible, allowing greater access for X-ray detec-tors and sample holders for in situ experiments ora larger probe current is possible in a probe of agiven size, which is of importance in microanalysis.The optimal experimental settings described in

this part are derived for single isolated atomcolumns. One should keep in mind, that theprecision with which the position of a singleisolated column can be estimated is a goodperformance measure as long as neighboringcolumns are clearly separated in the image. In thiscase, the precision with which the position of anatom column is estimated is independent of thepresence of neighboring columns. However,images of atom columns taken under experimentalsettings that are optimal for isolated atom columnsmay show strong overlap for realistic materials,for example, for a Si ½1 0 0 crystal. Then, theprecision with which the position of an atomcolumn can be estimated is affected unfavorablyby the presence of neighboring columns [24]. Tofind out if the optimal experimental design changesin the case of neighboring atom columns, theparametric model for the intensity distributionderived in Section 2 will be extended from one totwo atom columns. Then, another interesting

0.0 0.2 0.4 0.6 0.8 1.01.0

1.5

2.0

2.5Annular detectorAxial detector

σ2 CR

/σ2 CR(C

s=0

mm

)

Cs (mm)

Fig. 8. The ratio s2CR=s2CR ðCs ¼ 0 mmÞ computed as a func-

tion of the spherical aberration constant, for a Au ½1 0 0 column, using an annular as well as an axial detector. The

aperture and detector radius are set to 0:50 (A�1: The defocus is

determined by Eq. (51). All other parameters are fixed (see

Tables 2 and 3).

0.0 0.2 0.4 0.6 0.8 1.01.000

1.005

1.010

σ2 CR/σ

2 CR(C

s= 0

mm

) Annular detectorAxial detector

Cs (mm)

Fig. 7. The ratio s2CR=s2CR ðCs ¼ 0 mmÞ computed as a func-

tion of the spherical aberration constant, for a Si ½1 0 0 column,using an annular as well as an axial detector. The aperture and

detector radius are set to 0:28 (A�1: The defocus is determined

by Eq. (51). All other parameters are fixed (see Tables 2 and 3).

0.1 0.2 0.3 0.4 0.50.0-1.0

-0.5

0.0

0.5

1.0

Real partImaginary partT

rans

fer

func

tion

Å-1

Fig. 6. Transfer function for a spherical aberration constant of

0:5 mm and defocus of �8 nm: The vertical line represents theaperture radius, which is equal to 0:28 (A

�1:


question that can be answered is if an increase ofthe diameter of the source image, accompaniedwith larger probe currents, results in higherprecisions. Thus far, the diameter of the sourceimage has been determined by the diameter of thediffraction-error disc, following Eq. (50). Increas-ing this diameter results in a higher precision in thecase of a single isolated atom column. The‘optimum’ value of s; and therefore of dI50; wouldbe infinity. However, this is not a realistic value,since neighboring columns will then stronglyoverlap. This overlap will be taken into accountin the following part.

5.3.2. Two neighboring atom columns

In this part, the experimental design is evaluatedand optimized for the special case of a parametricmodel consisting of two neighboring atom col-umns. The atom column types are similar to thoseconsidered in Section 5.3.1 and the intercolumndistances are given in Table 1. The criterion s2CRwhich is defined in Eq. (49), is now equal to thesum of the lowest variances with which the x- andy-coordinates of the two atom column positionscan be estimated. The FOV is chosen centeredabout the atom column positions, whose coordi-nates are given in Table 4. In this case, it may beshown that the variances on the x-coordinates aswell as the variances on the y-coordinates areequal.First, it has been assumed that the diameter of

the source image is given by Eq. (50). From theevaluation of the criterion s2CR; it follows that theoptimal design for two neighboring atom columnsis almost equal to the one for an isolated atomcolumn. This means that neighboring columns ofsome atom types may show strong overlap inimages taken under the optimal conditions.Compared to the results given in Section 5.3.1,

changes in the optimal aperture radius are only inthe order of 5 percent. The optimal detector radiusis still equal to or slightly smaller than the optimalaperture radius for annular and axial detectors,respectively. The attainable precision is againhigher for annular than for axial detectors.Furthermore, the optimal defocus is still given byEq. (51) and a Cs-corrector is more likely to payoff for heavy than for light atom columns,although the precision that can be gained is onlymarginal.Second, the diameter of the source image has

been taken variable. In practice, this is possible byadjusting the settings of the condenser lenses,allowing the demagnification of the source to becontinuously varied. It is well known that anincrease of the diameter of the source image isaccompanied with two side effects: an increase ofthe source size and of the probe current, having anunfavorable and a favorable effect on the preci-sion, respectively [6]. The potential merit ofincreasing the diameter of the source image isstudied by the evaluation of the criterion s2CR: Forannular detectors, it has been found that theoptimal diameter of the source image is abouttwice as large as the one defined in Eq. (50) for theatom column types and distances given in Table 1.For example, Fig. 9 shows the computed criterions2CR as a function of the ratio dI50=ð0:54l=aÞfor two neighboring Si ½1 1 0 columns. Theaperture radius has been set to 0:36 (A

�1; beingoptimal for Cs ¼ 0:5 mm and the radius of theannular detector is set equal to the aperture radius.For axial detectors, it has been found that theoptimal diameter of the source image is aboutequal to the one defined in Eq. (50) for the atomcolumn types and distances given in Table 1.Furthermore, it has to be noticed that theevaluation of the diameter of the source imagehas no effect on the optimal settings of the othermicroscope parameters.Finally, it has to be mentioned that thermal

diffuse scattering is not taken into account in theparametric model for the image intensity distribu-tion used in this paper. This is justified by the factthat the inner radius of the annular detector mustnot be taken too large in a quantitative experi-ment, where high-dose efficiency is important in

Table 4

Parameters of two neighboring atom columns

bx1 ð (AÞ by1 ð (AÞ bx2 ð (AÞ by2 ð (AÞ z ð (AÞ

dA�A

20 �

dA�A

20 �

E0

E1sk


order to provide high precision. In this case, the1s-states dominate the scattering.In the following subsection, an intuitive inter-

pretation of the described results is given.

5.4. Interpretation of the results

Proportionality relations for s2CR for dark-fieldand bright-field images consisting of an isolatedatom column have been derived. In dark-fieldimaging, where the non-interacting electrons areeliminated from detection, it has been found that

s2CRBd250

Dint:; ð52Þ

where Dint: is equal to the total number ofinteracting electrons and d50 represents the radiusof the image intensity distribution of the interact-ing electrons containing 50% of its total intensity.In particular, relation (52) holds for annulardetectors, having an inner radius greater than orequal to the aperture radius. In bright-fieldimaging, where the non-interacting electrons con-tribute to the background intensity in the image, ithas been found that

s2CRBDn:int:d

450

D2int:KLDxDy; ð53Þ

where Dn:int: is equal to the total number of non-interacting electrons. Relation (53) holds for axial

detectors as well as for annular detectors having aninner radius smaller than the aperture radius. Thevalidity of these proportionality relations areillustrated in Figs. 10 and 11, where its right-handand left-hand side are shown as a function of theaperture radius for a Sr ½1 0 0 column, using anannular and an axial detector, respectively. Thisrelation allows us to get deeper insight into thenumerical results, derived in Section 5.3. It showsthat, in order to obtain a higher precision, one hasto balance the width of the image intensity

0 1 2 3 4 50.000

0.008

0.016

0.024

σ2 CR

(Å2)

dI50 /(0.54λ/α )

Fig. 9. The criterion s2CR computed as a function of the ratiodI50=ð0:54l=aÞ for two neighboring Si ½1 1 0 columns for anannular detector. The aperture and detector radius are set to

their optimal value 0:36 (A�1: The defocus is determined by

Eq. (51). All other parameters are fixed (see Tables 2 and 4).

0.3 0.4 0.50.000

0.002

0.004

0.006

σ 2CR

d250/Dint.

σ2 CR

(Å2 )

Aperture radius (Å-1)

Fig. 10. The left-hand and right-hand side members of Eq. (52)

computed as a function of the aperture radius for a Sr ½1 0 0 column, using an annular detector. The radius of the annular

detector is equal to the aperture radius. The defocus is


Tables 2 and 3).

0.3 0.4 0.50.00

0.04

0.08

0.12σ 2

CR

(Dn.int.d450)/(D2

int.KL ∆x∆y)

σ2 C

R(Å

2)

Aperture radius (Å-1)

Fig. 11. The the left-hand and right-hand side members of

Eq. (53) computed as a function of the aperture radius for a Sr

½1 0 0 column, using an axial detector. The radius of the axial

detector is equal to the aperture radius. The defocus is


Tables 2 and 3).


distribution and the number of interacting andnon-interacting electrons.For example, at the optimal design, the optimal

aperture radius gap strongly depends on the atomcolumn. Furthermore, an annular detector is to beused of which the inner radius gdet is equal to gap:How can this be explained? On the one hand, d50will become smaller if the probe becomes smaller,that is, if the aperture radius increases. However,the decrease of d50 will become less important ifthe probe width is about equal to the width of the1s-state. This is due to the fact that d50 is mainlydetermined by the excitation of the 1s state,described by Eq. (3). On the other hand, theaccompanied increase of the detector radius resultsin an enormous loss of interacting electrons. As aconsequence, the optimal design balances theexcitation of the atom column-dependent 1s-stateand the loss of electrons in the detector. In thisway, the dependence of the optimal apertureradius on the atom column can be explained.

5.5. Comparison with conventional approach

In the conventional approach, which is based ondirect visual interpretability, the Scherzer condi-tions for incoherent imaging are usually applied[3,4]

gap ¼1

l4lCs

� �1=4;

e ¼ �ðCslÞ1=2: ð54Þ

Table 5 compares these Scherzer conditions andthe optimal conditions for a Sr ½1 0 0 column. In

the column of the Scherzer conditions, the value ofgdet has been taken two times larger than gap;which is representative for a typical Crewedetector [26]. As can be noticed clearly, theScherzer conditions differ significantly from theoptimal conditions. The precision, expressed interms of the variance, at Scherzer conditions isabout 62 times worse than the precision that couldbe reached at the optimal design. In terms of thestandard deviation, which is defined as the squareroot of the variance, this fraction is about 8: This isnot astonishing and can be explained fromEq. (52). Due to the large hole in the detector,the dose efficiency is very low at Scherzerconditions, thus affecting the precision in anunfavorable way. Due to the smaller probe size,the width of the intensity distribution is slightlysmaller at Scherzer conditions than at the optimalconditions, thus affecting the precision in afavorable way. However, the extremely lownumber of detected electrons is the dominantfactor, resulting in a low precision.From this comparison, it may be concluded that

there is a world of difference between the Scherzerconditions and the optimal conditions. Although,one has to keep in mind that both conditions arederived for different purposes: direct visualinterpretability on the one hand and precisemeasurement of the atom column positions onthe other hand. However, both purposes may gohand in hand. As explained earlier, quantitativestructure determination is done by numericallyfitting the parametric model to the experimentaldata. The fit is evaluated using a criterion ofgoodness of fit. In practice, the search for theglobal optimum of the criterion of goodness of fitis an iterative numerical procedure. At eachiteration, the coordinates are slightly changed inorder to improve the fit. In order to guaranteeconvergence to the global optimum of the good-ness of fit, good initial conditions are required.This means that it is important to find a reason-able trial structure. Trial positions for the atomcolumns may be obtained from experimentalimages that are optimized for qualitative inter-pretation, whereas the refinement may result fromexperimental images that are optimized for quan-titative interpretation.

Table 5

Comparison between the optimal conditions (for an isolated Sr

[1 0 0] column) and the Scherzer conditions for an annular

detector, with Cs ¼ 0:5 mm

Optimal conditions

(Sr [1 0 0])

Scherzer

conditions

e ðnmÞ �16 �32gap ð (A

�1Þ 0.40 0.56

gdet ð (A�1Þ 0.40 1.12

s2CR ð (A2Þ 0.0037 0.2304

DE ð%Þ 2.9 0.034

d50 ð (AÞ 0.94 0.75


6. Conclusions

Conventionally, the design of a STEM experi-ment is based on qualitative image interpretation.However, in terms of image interpretation, thefuture lies in quantitative measurement of struc-tural parameters. Since this is a different purpose,the design has to be reconsidered. A quantitativemeasure has been proposed to evaluate andoptimize the design of a high-resolution STEMexperiment. It is based on the statistical precisionwith which the positions of atom columns can beestimated.In the resulting optimal design, the aperture

radius has been found to be mainly determined bythe object under study. More specifically, it isproportional to the weight of the atom column.The optimal inner or outer radius of an annularor an axial detector turns out to be equal toor slightly smaller than the optimal apertureradius, respectively. However, an annular detectorresults in a higher precision than an axialdetector. The resulting optimal defocus is the onefor which the transfer function comes closeto unity over the whole angular range of theaperture. The merit of Cs-correctors in quantita-tive STEM applications depends on the objectunder study. It pays off more for heavy atomcolumns, although the precision that can be gainedis only marginal. For annular detectors, increasingthe size of the source image beyond the size of thediffraction-error disc, which increases the probecurrent at the expense of resolution, has afavorable effect on the attainable precision. Foraxial detectors, the optimal size of the sourceimage is about equal to the size of the diffraction-error disc.

Acknowledgements

The authors would like to thank Dr. J.E. Barthand Dr. M.A.J. van der Stam for fruitful discus-sions related to this work. The research of Dr. A.J.den Dekker has been made possible by a fellow-ship of the Royal Netherlands Academy of Artsand Sciences.

References

[1] S.J. Pennycook, Scanning transmission electron micro-

scopy: Z-contrast, in: S. Amelinckx, D. Van Dyck, J. Van

Landuyt, G. Van Tendeloo (Eds.), Handbook of Micro-

scopyFApplications in Materials Science, Solid-StatePhysics and Chemistry, Methods II, pages 595–620,

Weinheim, 1997. VCH.

[2] J.M. Cowley, Scanning Transmission Electron Micro-

scopy, in: S. Amelinckx, D. van Dyck, J. Van Landuyt,

G. Van Tendeloo (Eds.), Handbook of MicroscopyFAp-plications in Materials Science, Solid-State Physics and

Chemistry, Methods II, VCH, Weinheim, 1997,

pp. 563–594.

[3] O. Scherzer, J. Appl. Phys. 20 (1949) 20.

[4] S.J. Pennycook, D.E. Jesson, Ultramicroscopy 37 (1991)

14.

[5] P.D. Nellist, S.J. Pennycook, Phys. Rev. Lett. 81 (19)

(1998) 4156.

[6] P.D. Nellist, S.J. Pennycook, The principles and inter-

pretation of annular dark-field Z-contrast imaging, in:

P.W. Hawkes (Ed.), Advances in Imaging and Electron

Physics, Vol. 113, Academic Press, San Diego, 2000,

pp. 147–199.

[7] D. Van Dyck, Prospects of quantitative high resolution

electron microscopy, in: L. Frank, F. $Ciampor (Eds.),

Proceedings of the 12th European Congress on Electron

Microscopy, Instrumentation and Methodology, Vol. III,

The Czechoslovak Society for Electron Microscopy, Brno,

2000, pp. 13–18.

[8] A. van den Bos, Measurement errors, in: J.G. Webster

(Ed.), Encyclopedia of Electrical and Electronics Engineer-

ing, Vol. 12, Wiley, New York, 1999, pp. 448–459.

[9] A. van den Bos, Parameter estimation, in: P.H. Sydenham

(Ed.), Handbook of Measurement Science, Vol. 1, Wiley,

Chicester, 1982, pp. 331–377.

[10] A.J. den Dekker, J. Sijbers, D. Van Dyck, J. Microsc. 194

(1999) 95.

[11] J. Broeckx, M. Op de Beeck, D. Van Dyck, Ultramicro-

scopy 60 (1995) 71.

[12] D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 (1996)

99.

[13] S.J. Pennycook, D.E. Jesson, Acta Metall. Mater. 40

(1992) S149–S159.

[14] P. Geuens, J.H. Chen, A.J. den Dekker, D. Van Dyck, An

analytic expression in closed form for the electron exit

wave, Acta Crystallogr. Section A, 55 Supplement,

Abstract P11.OE.002, 1999.

[15] J.M. Cowley, Ultramicroscopy 2 (1976) 3.

[16] C. Mory, M. Tence, C. Colliex, J. Microsc. Spectrosc.

Electron. 10 (1985) 381.

[17] J.E. Barth, P. Kruit, Optik 3 (1996) 101.

[18] L. Reimer, Elements of a transmission electron micro-

scope, in: Transmission Electron Microscopy, Physics of

Image Formation and Microanalysis, Springer, Berlin,

Heidelberg, 1993, pp. 86–135.


[19] A. van den Bos, A.J. den Dekker, Resolution reconsider-

edFconventional approaches and an alternative, in:P.W. Hawkes (Ed.), Advances in Imaging and Electron

Physics, Vol. 117, Academic Press, San Diego, 2001, pp.

241–360.

[20] A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the

Theory of Statistics, 3rd Edition, McGraw-Hill, Tokyo,

1974.

[21] M.G. Kendall, A. Stuart, The Advanced Theory of

StatisticsFInference and Relationship, Volume 2, 2ndEdition, Charles Griffin and Company Limited, London,

1967.

[22] V.V. Fedorov, Theory of Optimal Experiments, Academic

Press, New York, London, 1972.

[23] A. P!azman, Foundations of Optimum Experimental De-

sign, D. Reidel Publishing Company, Dordrecht, Boston,

Lancaster, Tokyo, 1986.

[24] E. Bettens, D. Van Dyck, A.J. den Dekker, J. Sijbers, A.

van den Bos, Ultramicroscopy 77 (1999) 37.

[25] D. Van Dyck, J.H. Chen, Solid State Commun. 109 (8)

(1999) 501.

[26] S.J. Pennycook, D.E. Jesson, M.F. Chisholm, N.D.

Browning, A.J. McGibbon, M.M. McGibbon, J. Microsc.

Soc. Am. 1 (6) (1995) 231.


Documents

Optimal experimental design of STEM measurement of atom column positions