Scattering of Å-scale electron probes in silicon

Ultramicroscopy 96 (2003) 343–360

Scattering of (A-scale electron probes in silicon

C. Dwyera,*, J. Etheridgeb

aDepartment of Materials Science and Metallurgy, University of Cambridge, Pembroke St., Cambridge CB2 3QZ, UKbSchool of Physics and Materials Engineering, P.O. Box 69M, Monash University, Victoria 3800, Australia

Received 10 September 2002; accepted 21 November 2002

Abstract

We use frozen phonon multislice calculations to examine the scattering behaviour of (A-scale electron probes in

/0 0 1S and /1 1 0S silicon. For each crystal orientation, we consider the distribution of scattered intensity in realspace as a function of crystal thickness, probe size and probe position. The scattered intensity distribution is found to

vary drastically for different probe sizes. For a given probe size, the scattered intensity distribution is also significantly

influenced by the crystal orientation. We discuss the implications for the simultaneous acquisition of an annular dark-

field image and electron energy loss spectra in the scanning transmission electron microscope, with specific reference to

the spatial resolution with which electron energy loss spectra can be related to local atomic structure.

r 2003 Elsevier Science B.V. All rights reserved.

PACS: 61.14.�x

Keywords: Image simulation; Scanning transmission electron microscopy; Electron energy loss spectroscopy

1. Introduction

Recent developments in Cs-corrected scanningtransmission electron microscopes (STEMs) sug-gest it will soon be possible to form a 100 keVelectron probe having full-width at half-maximum(FWHM) smaller than 1 (A with enough currentfor acquisition of electron energy loss (EEL)spectra [1,2]. Such an instrument would permitthe simultaneous acquisition of an atomic resolu-tion annular dark-field (ADF) image and EELspectra—the ADF image serving as a reference forthe probe position in EELS. This offers the

prospect of correlating measurements of localelectronic structure with measurements of atomicstructure. However, if such correlations are to bemeaningful, a precise knowledge of the scatteringbehaviour of the electron probe is required. Inparticular, we need to know from which atoms inthe specimen the EEL and ADF signals derive. Asa first step towards this goal, we have calculatedthe scattering behaviour of (A-scale electron probesin silicon and examined its dependence on probesize, probe position and crystal orientation.Whilst there have been several excellent studies

[3–8] simulating the scattering of small probes,these preceded the advent of Cs-corrected probesand mainly considered the scattering behaviour inrelation to ADF imaging. More recently, Ishizuka[9] has considered the propagation of both

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*Corresponding author. Tel.: +44-1223-334563; fax: +44-

1223-334437.

E-mail address: [email protected] (C. Dwyer).

0304-3991/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0304-3991(03)00100-1

Cs-corrected and non-corrected probes, positionedon and between atomic columns using a phenom-enological model of phonon scattering.In the present study, the scattering behaviour of

(A-scale electron probes in silicon was examined asfollows. The distribution of scattered intensity inreal space was calculated in order to identify whichparts of the specimen are sampled by the probe.The intensity within a 0:2 (A radius of each atomiccolumn vs. crystal thickness was considered in orderto identify which parts of the specimen mightcontribute to the EELS signal. ADF intensities forselected probe positions were calculated in order toassess whether the ADF image can provide anaccurate measure of the probe position.All calculations were performed for three

different probe sizes (2.0, 1.4 and 0:7 (A FWHM)and different probe positions (for example, on anatomic column and midway between two nearestneighbour columns) in silicon. Furthermore,calculations were performed for two crystalorientations /0 0 1S and /1 1 0S: In /0 0 1S theminimum inter-column spacing of 1:92 (A is suchthat there is minimal overlap of projected columnpotentials. In /1 1 0S; the minimum spacing is1:36 (A and there is considerable overlap of theprojected column potentials. There is a furtherdifference between the two orientations in that thelinear density of the atomic columns is greater in/1 1 0S than /0 0 1S:For the purposes of the present work, the above

calculations need to be accomplished with as muchprecision as possible, with computation time beinga secondary concern. We therefore chose a frozenphonon multislice method which is the mostrigorous semi-classical method currently available[10,11]. The method incorporates several featurescritical for our tasks. For example, multi-phononand multiple phonon scattering events are in-cluded, as is multiple elastic scattering before andafter each phonon scattering event. Many treat-ments of phonon scattering ignore subsequentelastic scattering of phonon-scattered waves,thereby treating the phonon-scattered waves asfree waves. We do not wish to make thisapproximation in this work, as it would prohibitthe accurate calculation of scattered intensity inreal space.

We note that this level of precision can beredundant in the computation of ADF images,since the annular detector integrates over thestructure in the high-angle intensity distribution,reducing the sensitivity of the ADF signal to thescattering processes that give rise to this structure.However, our primary concern is not with ADFintensities, but with accurate calculations of realspace intensity distributions, the latter requiring arigorous treatment of phonon scattering.For the calculations in the present work, both

the Einstein and Born-von Karman phononmodels have been used. However, we find thatthere is negligible difference between the calculatedscattered intensity distribution for each model.This appears consistent with the failure to observediffuse streaks due to correlated atomic motions insmall probe experimental diffraction patterns of/0 0 1S silicon [12]. We emphasize, however, thatphonon scattering, with or without correlations,has a significant impact on the scattered intensitydistribution.

2. Background theory

2.1. Multislice approach for focused electron probes

In this section we outline the multislice ap-proach [13] as it applies to dynamical scattering offocused electron probes. For details regarding themultislice approach the reader should consultRef. [14] or Ref. [15].We use the standard Cartesian co-ordinate

system where the x-axis and y-axis define a planethat coincides with the entrance surface of thecrystal (assumed to be a parallel-sided foil), andthe z-axis points along the direction of beampropagation. Symbols in upper-case bold, such asR and K ; denote two-dimensional vectors in thex–y plane.In finite difference form, the multislice approach

can be written as a recurrence relation

cjðRÞ ¼ PjðRÞ* ðQjðRÞcj�1ðRÞÞ; ð1Þ

where cjðRÞ is the electron wave function after thejth slice of the specimen. In Eq. (1), the multi-plication by QjðRÞ describes the phase change

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C. Dwyer, J. Etheridge / Ultramicroscopy 96 (2003) 343–360344

produced by the projected electrostatic potentialVjðRÞ of the jth slice, and the convolution withPjðRÞ describes the propagation of the wavefunction, a la Huygen’s principle, from the jthslice to the ðj þ 1Þth slice. For a slice of thicknessDz; the explicit form of QjðRÞ is

QjðRÞ ¼ exp �isZ zjþDz

zj

dz0V ðR; z0Þ

!

¼ expð�isVjðRÞÞ; ð2Þ

where s ¼ 2pmel=h2: For an electron havingwavelength l; PjðRÞ is given by

PjðRÞ ¼1

ilDzexp

pilDz

R2� �

: ð3Þ

The recurrence relation (1) is invoked for each sliceof the specimen to calculate the effect of the totalspecimen potential on the electron wave.The multislice approach can be used to calculate

dynamical scattering of a plane wave or a focusedprobe. In the case of a focused probe centered atthe point R0; the initial wave function can bedescribed as a coherent sum of plane waves, eachhaving an appropriate phase relationship [16]

c0ðR � R0Þ ¼ZdK OðKÞexpð2piwðKÞÞ

� expð2piK � ðR � R0ÞÞ: ð4Þ

In this expression OðKÞ is equal to 1 for transversewave vectors K admitted by the probe-formingaperture and zero otherwise, and wðKÞ describesthe phase change due to a probe-forming lenswith first-, third- and fifth-order aberrationsC1; C3 and C5; respectively. The form of wðKÞ isgiven by

wðKÞ ¼ 12C1lK2 þ 1

4C3l3K4 þ 1

6C5l5K6: ð5Þ

One of the main advantages of the multisliceapproach over other approaches to dynamicalscattering is that it makes no assumptions regard-ing the form of the specimen potential. It istherefore valid for both periodic and non-periodicpotentials. However, if the FFT algorithm is usedin a multislice program then periodic continuationof the supercell (in both R- and K-space) isimplicit. Even so, the multislice approach, whenused with the FFT algorithm, provides an

extremely efficient way of calculating electronscattering in a supercell large enough to minimizeaffects arising from the periodic boundary condi-tions. It is therefore suitable for the frozen phononalgorithm described in Section 2.2.

2.2. Frozen phonon algorithm

In this section we outline the so-called frozenphonon model [17] used to describe electron–phonon scattering. In this model, electron–phononscattering is approximated as elastic scatteringfrom static atoms displaced from their equilibriumpositions. Further, the observed electron intensityis assumed to result from an ensemble averageover atomic positions (only). The justificationsfor this approximation are [18]: (i) the energyexchanged in a fast electron–phonon interaction isnegligible compared to the initial kinetic energy ofthe fast electron, (ii) the high velocity of fastelectrons means the duration of the interactionwith an atom in the specimen is much less than theperiod of vibration of the atom, and (iii) arrivaltimes of fast electrons at the specimen areessentially uncorrelated with atomic vibrations.Wang [10] has considered the validity of thefrozen phonon approximation and established itsrigour.Mathematically, the frozen phonon approxima-

tion can be summarized by the following expres-sion for the observed electron intensity IðKÞ at apoint K in the diffraction plane:

IðKÞ ¼ /IðK ; uÞSu ¼Zd3NuIðK ; uÞProbðuÞ: ð6Þ

In this expression, u is a 3N-dimensional vectordenoting the instantaneous displacements of the N

atoms that make up the crystal (referred to as aphonon configuration), IðK ; uÞ is the electronintensity resulting from the phonon configurationu; and ProbðuÞ is the probability density of such aconfiguration.In our calculations, we have used the algorithm

of Loane et al. [18] to evaluate the ensembleaverage (6). This is essentially a Monte Carloalgorithm whereby atomic displacements u aregenerated using a (quasi-)random number sourcewith the required probability distribution ProbðuÞ:

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C. Dwyer, J. Etheridge / Ultramicroscopy 96 (2003) 343–360 345

The electron intensity resulting from this phononconfiguration is then calculated using the multi-slice approach. This process is repeated fordifferent phonon configurations. The intensitiesresulting from each phonon configuration areadded to obtain a final intensity that approximatesthe ensemble average in Eq. (6).The main advantages of the frozen phonon

algorithm are: (i) it can achieve any desiredaccuracy, (ii) it includes all combinations of elasticscattering and electron–phonon scattering events,and (iii) any phonon model can be adopted tocalculate the phonon configurations.The main disadvantages of the frozen phonon

algorithm are: (i) long computation times, and(ii) the lack of distinction between elastic- andphonon-scattered electrons. A consequence of thelatter is that the algorithm does not permit anindependent analysis of, say, the phonon-scatteredelectrons. This is contrary to a quantum mechan-ical treatment of electron–phonon scatteringwhere there is a natural distinction between elastic-and phonon-scattered electrons due to theirmutual incoherence [19]. However, for the presentwork the frozen phonon algorithm is adequatebecause we are interested in the total electronintensity.

3. Calculations performed and representation of the

data

In this section we detail the parameters usedin our calculations and describe the formatsin which the resulting data is presented. Theresults themselves are presented and discussed inSection 4.

3.1. Calculation parameters

Using the theory outlined in Section 2, wecalculated the scattered intensity of focused probesin /0 0 1S and /1 1 0S silicon. The calculationparameters are summarized in Table 1. Calcula-tions were performed for three different probessizes (2:0; 1:4 and 0:7 (A FWHM) at 100 keV: Theprobe sizes range from those available in currentSTEMs equipped with a high-resolution pole piece(2:0 (A probe), to those that might be available inCs-corrected STEMs in the near future (0:7 (Aprobe) [1]. The parameters for each probe size areshown in Table 2.

3.2. Maps of intensity in real space

To examine the probe dispersion, we havemapped the scattered intensity in real space forprobe positions A1, B1, A2 and B2 (see Fig. 1).The intensity was mapped in the x–y plane forcrystal thicknesses up to 500 (A at 100 (A intervals.The area of each map is 32:6 (A� 32:6 (A for the/0 0 1S orientation and 30:7 (A� 32:6 (A for the/1 1 0S orientation. The intensity was mappedusing a log scale to enhance detail. The maximumvalue on each plot, corresponding to white, is

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Table 1

Parameters for frozen phonon multislice calculations

Orientation No. of pixels Supercell dim. ( (A) ajKmaxj ð (A�1Þ Dz ( (A) bph. model cxrms ( (A) No. ph. configs.

/0 0 1S 512� 512 32:6� 32:6 5.2 1.36 BvK E0.08 24

/1 1 0S 512� 512 30:7� 32:6 5.2 1.92 Ein 0.078 24

a jKmaxj is the maximum transverse wave vector in the calculation.b ‘BvK’ and ‘Ein’ refer to the Born-von Karman and Einstein phonon models.cxrms is the atomic root-mean-square displacement along the x-axis.

Table 2

Probe parameters ð100 kevÞ

FWHM ( (A) aa (mrad) C1 ( (A) C3 (mm) C5 (mm)

2.0 10.0 �700 1.3 0

1.4 13.0 �430 0.5 0

0.7 30.0 +90 �0.053 50

aa is the convergence semi-angle.


normalized with respect to the relevant incidentprobe intensity. The minimum value, correspond-ing to black, is always zero and is therefore notshown. The minimum and maximum values werecalculated before log scaling.

3.3. Intensity within 0.2 (A of the atomic columns

As mentioned in the introduction, a potentialapplication of the new generation of Cs-correctedSTEMs is the simultaneous collection of ADF andEELS signals. As the electron probe is scannedacross the specimen, the ADF signal is used to forma high-resolution reference image, and for eachpixel in the image there is a corresponding EELspectrum. If such a technique is to be used to obtaina high-resolution map of the local electronicstructure of the specimen, we must first answer thequestion: For a given probe position, which parts ofthe specimen contribute to the EELS signal?A rigorous answer to this question requires the

incorporation of the matrix element for atomicionization into an electron scattering calculation(see, for example Ref. [20]). Such an approach isnot attempted in this paper. However, a roughanswer can be obtained by considering the squaredmodulus of the fast electron wave function insidethe crystal, i.e. the scattered intensity in real space.This approach can be justified as follows.The matrix element for K-shell atomic ioniza-

tion, considered as a function of the position of thefast electron, is only significant in the region closeto the atomic nucleus. This is because the matrixelement consists of an overlap integral involvingthe 1s atomic wave function, which itself has a

significant value only in the region close to thenucleus. Furthermore, the amplitude of the in-elastic fast electron wave function also depends onthe amplitude of the elastic fast electron wavefunction. For example, if the amplitude of theelastic fast electron wave function is very small inthe region close to the nucleus, then the amplitudeof the inelastic wave function will also be small.The detected EELS signal is given by the inelasticfast electron intensity in the diffraction plane. Ifwe assume that the EELS collector aperture is verylarge, then the transition rate for the ionization ofa particular atom is given by (see Appendix A)

wa0-a ¼m

_3ka;zL

Zd2Rj/ajV ðRÞja0Sj2j/Rjc0Sj2;

ð7Þ

where /Rjc0S is the projected elastic amplitude atthe atom site, and /ajV ðRÞja0S is the projectedmatrix element for the transition of the atom fromits ground state a0 to a continuum state a: Eq. (7)states that the transition rate wa0-a is given by theintegral of the projected elastic intensity at theatom site, weighted by the squared modulus ofthe projected matrix element. Since the projectedmatrix element is a highly localized functioncentered at the nucleus of the atom, only theelastic intensity in the immediate vicinity ofthe atom is important. Although this result forthe transition rate is rigorous only for the case ofa very large collector aperture, we feel that it issufficient to estimate which parts of the specimenwill contribute to the EELS signal when typicalcollector apertures are used.

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Fig. 1. Projected structure of /0 0 1S (left) and /1 1 0S silicon (right) showing the probe positions considered.


The extent of the volume from which the EELSsignal originates has been considered in somedetail by Rafferty and Pennycook [21]. Using theinelastic matrix element given by Maslen andRossouw [22], and assuming incoherent condi-tions, Rafferty and Pennycook [21] calculated K-shell EELS object functions for various atomicnumbers and collector apertures. Using thesecalculations as a guide, we estimated the width ofthe Gaussian K-shell EELS object function forsilicon to beE0:25 (A for a 100 keV electron beamand a 20 mrad collector aperture. In our approach,we make the further simplification of assuming theobject function to be a top-hat function, not aGaussian, and we choose the width of this functionto be 0:4 (A: Thus, we consider the fast electronintensity within 0:2 (A of the atomic column coresin order to estimate which parts of the specimengenerate the EELS signal. This approach effec-tively replaces the squared modulus of theprojected matrix element in Eq. (7) by a top-hatfunction of width 0:4 (A:

3.4. ADF intensity

If an ADF image is to give an accurateindication of the probe position, then the atomiccolumns must be well-resolved. In order to assessthe resolution of the ADF image and its depen-dence on thickness for each probe size, the ADFintensity vs. crystal thickness was calculated forprobes at positions A1, B1, and C1 in the /0 0 1Sorientation, and positions A2, B2 and C2 in the/1 1 0S orientation (see Fig. 1). In our calcula-tions, the ADF detector inner and outer angleswere assumed to be 50 and 150 mrad; respectively.The ADF intensities are the result of 8 phononconfigurations, and in this respect they aredifferent from the other calculations presented inthis work where 24 configurations were used.

4. Results and discussion

We studied the scattering of electron probes intwo orientations of silicon, /0 0 1S (Section 4.1)and /1 1 0S (Section 4.2). As previously men-tioned, the two orientations have distinct mini-

mum inter-column spacings. In the /0 0 1Sorientation, the nearest neighbour column separa-tion of 1:92 (A is such that there is minimal overlapof the individual projected column potentials (seeFig. 2). In the /1 1 0S orientation, the nearestneighbour column separation of 1:36 (A; corre-sponding to the so-called dumbbell, is such thatthere is significant overlap of the individualprojected column potentials (see Fig. 3). In the

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−1 0 1 2 3distance (Å)

0

500

1000

1500

proj

ecte

d po

tent

ial (

volt_

Å)

Fig. 2. Projected potential (solid line) of two silicon atoms

separated by 1:92 (A; the smallest inter-column spacing in/0 0 1S silicon. The projected potential of the individual atomsis also shown (dashed lines). The inter-column spacing is such

that there is minimal overlap of individual atomic potentials.

−1 0 1 2distance (Å)

0

500

1000

1500

proj

ecte

d po

tent

ial (

volt_

Å)

Fig. 3. Projected potential (solid line) of two silicon atoms

separated by 1:36 (A; the smallest inter-column spacing (thedumbbell spacing) in /1 1 0S silicon. The projected potential ofthe individual atoms is also shown (dashed lines). The inter-

column spacing is such that there is significant overlap of

individual atomic potentials (compare with Fig. 2).


/1 1 0S case the projected potential midwaybetween nearest neighbour columns is significant.A further distinction between the two orientationsis that the linear density of atomic columns in/0 0 1S is 0.18 silicon atoms per (A, whilst in/1 1 0S it is 0.26 silicon atoms per (A.

4.1. Silicon /0 0 1S

4.1.1. Position A1

The results for probes at position A1 (centeredon an atomic column) are presented in Figs. 4 and5. It is evident from these plots that the scatteringbehaviour is complicated, however, there areconsistent features that can be identified:

(i) For all probe sizes, the probe intensityscatters beyond the column of interest (CI)i.e. the column beneath the focused probe(Fig. 4). The dispersion is greater andquicker for smaller probes. This is due tothe greater convergence angles required forthe smaller probes, which therefore compriselarger components of transverse momentum.

(ii) The majority of the total intensity liesbeyond 0:2 (A of the CI (Fig. 5).

(iii) For the 2.0 and 1:4 (A probes, the globalmaximum in the intensity distribution al-ways lies on the CI (Fig. 4). This is not thecase for the 0:7 (A probe, for example, at adepth of 80 (A; the intensity on the CI isnearly zero (Fig. 5).

(iv) For the 2.0 and 1:4 (A probes, more intensitylies on the CI than the 1st and 2nd nearestneighbour (NN) columns combined at alldepths (Fig. 5). This is not the case for the0:7 (A probe for depths greater than about400 (A; and for certain depths such as 80 (A:

(v) The intensity peak that develops on the CIpersists to greater depths for larger probes,as indicated by the maximum value in eachintensity map in Fig. 4 and by Fig. 5.

(vi) The depth dependence for the 0:7 (A probe iscomplex. A very strong intensity peak lies onthe CI for the first 50 (A of the crystal. By adepth of 80 (A there is no intensity left on theCI. For depths beyond 100 (A the intensitypeak on the CI returns but is much weaker.

(vii) For the 0:7 (A probe, the intensity propagatesaway from the CI, preferentially along theclose packed directions (see Fig. 4). Thisbehaviour is less obvious for the larger probes.

4.1.2. Position B1

The maps of intensity in real space for probes atposition B1 (centered between nearest neighbouratomic columns) are shown in Fig. 6. Thescattering behaviour is again complex. Specificfeatures can be identified:

(i) For all probe sizes, the dispersion is greaterand quicker than for probes at position A1.

(ii) The extent of dispersion of the 0:7 (A probe isfar greater than for the other probes. (Again,this can be attributed to the much largerconvergence angle ð30 mradÞ for this probe.)

(iii) More intensity lies between atomic columnsthan for probes at position A1. This is mostapparent for the 0:7 (A probe.

4.2. Silicon /1 1 0S

4.2.1. Position A2

The results for probes at position A2 (centeredon an atomic column) are shown in Figs. 7 and 8.Whilst there are many similarities between thescattering behaviour of probes centered on atomiccolumns in /0 0 1S (Figs. 4 and 5) and /1 1 0Ssilicon (Figs. 7 and 8), there are also severaldifferences that are observed here:

(i) For all probe sizes, the global intensitymaximum does not always reside on the CIbut transfers to the 1st NN (the othercolumn in the dumbbell). The transfer takesplace at a depth of about 300–400 (A;depending on the probe size.

(ii) The intensity peak that develops on the CI isinitially greater but then decays more rapidlythan in the /0 0 1S case (compare Figs. 5and 8). Further, the oscillations of intensityon the CI are more pronounced than thosefor /0 0 1S:

(iii) For all probe sizes, the intensity on the single

1st NN is greater than the combinedintensity on the four 1st NNs in the

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max=2.22

500 Åmax=2.10

max=3.30

max=4.45

max=5.40

100 Å

2.0 Å probe

max=1.53

500 Åmax=1.97

max=2.49

max=2.86

max=4.03

100 Å

1.4 Å probe

max=0.09

500 Åmax=0.19

max=0.36

max=0.41

max=0.17

100 Å

0.7 Å probe

Fig. 4. Real space intensity maps vs. crystal thickness for 2:0 (A (left column), 1:4 (A (middle column) and 0:7 (A (right column) probes

at position A1 in /0 0 1S silicon. The intensity is mapped for crystal thicknesses up to 500 (A at 100 (A intervals (increasing from top to

bottom). The size of each cell is 32:6 (A� 32:6 (A:


/0 0 1S case. The intensity on the 2nd NNsis less than in the /0 0 1S case.

4.2.2. Position B2

Fig. 9 shows the real space intensity maps forprobes at position B2, i.e. at the center of thedumbbell. It is useful to consider the differencesbetween this case and Fig. 6 where the probes arecentered between nearest neighbour columns in/0 0 1S silicon.

(i) For all probe sizes, the dispersion tends to beless extensive than the /0 0 1S case, asindicated by the maximum value on eachintensity map in Fig. 9. A significant propor-tion of the intensity is confined to thedumbbell.

(ii) The separation of the global intensity maximain Fig. 9 varies with depth, so that they do notalways coincide with the column positions.The behaviour has been noted by Nellist andPennycook [7].

Perhaps the most striking feature when comparingFigs. 9 and 6 is the ability of the dumbbellpotential to trap electrons from the 0:7 (A probe(even though the probe is placed between thedumbbell). It seems the dumbbell potential doesnot present two distinct potential wells for the fastelectron, but acts, to some extent, like a singlepotential well. This behaviour can be attributed tothe significant value of the projected potential atthe center of the dumbbell (see Fig. 3). Suchbehaviour has been predicted using approximateanalytic expressions by Anstis and Cockayne [23].

5. Significance for EELS and ADF signals

We now consider the implications of thescattering behaviour summarized in Section 4 forthe spatial resolution with which EELS data canbe related to local atomic structure.If the EELS signal is to be directly and simply

related to local atomic structure, then (i) the EELSsignal should derive predominantly from theatoms within the column beneath the probe,(ii) the EELS signal should derive with equalstrength from each atom within this column, and

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0 200 400 600thickness (Å)

0.00

0.10

0.20

0.30

0.40

0.50

norm

aliz

ed in

tens

ity

CI

1st NNs 2nd NNs

0.7 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

0.25

norm

aliz

ed in

tens

ity

CI

1st NNs2nd NNs

1.4 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

norm

aliz

ed in

tens

ity

CI

1st NNs2nd NNs

2.0 Å probe

Fig. 5. Intensity within a 0:2 (A radius of CI (column of

interest), 1st NNs (1st nearest neighbour) and 2nd NNs (2nd

nearest neighbour) columns vs. crystal thickness for 2:0 (A (top),

1:4 (A (middle) and 0:7 (A (bottom) probes at position A1 in

/0 0 1S silicon. Note that there are four 1st NNs and four 2ndNNs for this probe position.


(iii) each atomic column in the ADF image shouldbe clearly resolved, with the intensity peaks in theimage corresponding to the centers of the columns.As discussed in Section 3.3 and Appendix A, the

contribution to the core-loss EELS signal from agiven atom can be related to the elastic fastelectron intensity in the immediate vicinity of thatatom, providing that the EELS collector aperture

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max=1.34

500 Åmax=1.38

max=1.66

max=2.48

max=3.00

100 Å

2.0 Å probe

max=0.71

500 Åmax=0.84

max=0.89

max=1.25

max=1.55

100 Å

1.4 Å probe

max=0.02

500 Åmax=0.02

max=0.03

max=0.05

max=0.15

100 Å

0.7 Å probe


at position B1 in /0 0 1S silicon. The intensity is mapped for crystal thicknesses up to 500 (A at 100 (A intervals (increasing from top to



is large. We can therefore use our calculations ofthe elastic intensity within a 0:2 (A radius of theatomic column as a guide to the spatial origin, andhence resolution, of the core-loss EELS signal. We

emphasize, however, that for typical collectoraperture sizes, the elastic intensity cannot beused to quantify the EELS signal, only toestimate from where it is generated. We consider

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max=1.80

500 Åmax=1.42

max=2.76

max=1.84

max=6.64

100 Å

2.0 Å probe

max=1.37

500 Åmax=1.13

max=1.86

max=1.84

max=4.72

100 Å

1.4 Å probe

max=0.30

500 Åmax=0.31

max=0.29

max=0.57

max=0.85

100 Å

0.7 Å probe


at position A2 in /1 1 0S silicon. The intensity is mapped for crystal thicknesses up to 500 (A at 100 (A intervals (increasing from top to



below whether, and under what circumstances,the three requirements mentioned above aresatisfied.Let us first consider the spatial resolution in the

x–y plane with which we can relate the EELSsignal to local atomic structure. For all probesizes, when the probes are placed at positions A1and A2, the majority of the EELS signals willcome from the CI (see Figs. 5 and 8). However,there will also be a small but significant signalgenerated by the neighbouring columns, except forthe 0:7 (A probe for very thin crystals (o50 (A).For all probe sizes, and for thicknesses greaterthan about 100 (A; the CI contribution to theEELS signal will tend to decrease, whilst that ofthe neighbouring columns will tend to increase.For the case of /1 1 0S; the contribution to theEELS signal from the 1st NN is larger and occursearlier. For example, after a depth of about 400 (A;most of the EELS signal will originate from atomsin the nearest neighbour column, not the columnbeneath the probe.In summary, requirement (i) above is satisfied

for thin crystals but caution is required in theinterpretation of EELS data from thicker crystals(> 500 (A).Let us further consider the variation of intensity

on the columns with depth. The variation dependson two separate influences: the probe size and thecrystal orientation. First consider the effect ofprobe size. With the exception of extremely thincrystals, the larger probes have an advantagebecause the intensity on the CI varies more slowlywith thickness (see Figs. 5 and 8). Therefore, thefraction of the EELS signal arising from eachatom in the CI would be comparable for theseprobes. On the other hand, the quickly varyingintensity on the CI for the 0:7 (A probe implies thatthe fraction of the EELS signal arising from eachatom in the CI would vary drastically. Inparticular, the atoms in the first 60 (A of the CIwould contribute far more to the EELS signal thanother atoms in the CI, and the atoms at about80 (A would contribute almost nothing. It seemsthat a dopant atom in the CI at a depth of around80 (A would be very difficult to detect using the0:7 (A probe. The situation is less complex for verythin ðo50 (AÞ crystals.

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0 200 400 600thickness (Å)

0.00

0.10

0.20

0.30

0.40

0.50

norm

aliz

ed in

tens

ity CI

1st NN2nd NNs

0.7 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

0.25

norm

aliz

ed in

tens

ity CI

1st NN

2nd NNs

1.4 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

norm

aliz

ed in

tens

ity CI

1st NN

2nd NNs

2.0 Å probe

Fig. 8. Intensity within a 0:2 (A radius of CI (column of

interest), 1st NN (1st nearest neighbour) and 2nd NNs (2nd

nearest neighbour) columns vs. crystal thickness for 2:0 (A (top),

1:4 (A (middle) and 0:7 (A (bottom) probes at position A2 in

/1 1 0S silicon. Note that there are two 2nd NNs for this probeposition.


Now consider the influence of the crystalorientation on the variation of intensity on thecolumns. As noted previously, for a given probesize, the intensity on the CI in /1 1 0S develops

and then decays more rapidly with depth, and theoscillations of this intensity with depth are morepronounced. Further, the intensity on the 1st NNis greater for /1 1 0S: The characteristics of the

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max=1.72

500 Åmax=1.43

max=2.56

max=1.65

max=4.34

100 Å

2.0 Å probe

max=1.18

500 Åmax=0.96

max=1.53

max=1.31

max=2.43

100 Å

1.4 Å probe

max=0.10

500 Åmax=0.08

max=0.15

max=0.13

max=0.30

100 Å

0.7 Å probe


at position B2 in /1 1 0S silicon. The intensity is mapped for crystal thicknesses up to 500 (A at 100 (A intervals (increasing from top to



intensity distribution in /1 1 0S; as compared to/0 0 1S; arise from the smaller 1st NN separationand the greater linear density of atomic columns in/1 1 0S: Therefore, the crystal orientation itselfcan significantly effect the homogeneity of inten-sity along the CI.In summary, requirement (ii) above is not

satisfied. In all cases studied here, the variationof intensity with depth on the atomic columns iscomplex and varies significantly with both theprobe size and the crystal orientation. Thissuggests it will be difficult to interpret quantita-tively EEL spectra from specimens where thecontent of the atomic column is not homo-geneous. The situation is most complex for thesmallest probe.In addition to requiring a localized and homo-

geneous distribution of intensity on the CI (points(i) and (ii) above), we also need to accurately inferthe position of the probe from the ADF image(point (iii)). To asses this we have calculated theADF image intensities as a function of thicknessand probe position. For brevity, we only presentthe results for three probe positions in eachorientation (see Fig. 1). The results for probepositions other than those indicated in Fig. 1(which are not presented in this paper) indicatethat, in all cases where the atomic columnsare resolved, the intensity peaks in the ADFimage correspond to the centers of the atomiccolumns.First consider the ADF image resolution for the

/0 0 1S orientation. The ADF image generatedfrom the 2:0 (A probe does not resolve the columnsin the /0 0 1S orientation, however, the imagesgenerated from the 1.4 and 0:7 (A probes do.Therefore, ADF images generated from the 2:0 (Aprobe cannot provide a reference for the probeposition, whereas those from the 1:4 (A or 0:7 (Aprobes can (see Fig. 10).Now consider the /1 1 0S orientation. Neither

the 2:0 (A or 1:4 (A probes generate ADF imagesthat resolve the dumbbells (see Fig. 11). Again,there will be ambiguity in determining the probeposition within the dumbbell. However, the ADFimage from the 0:7 (A probe clearly resolves thedumbbells and therefore provides an accuratereference for the probe position. The above results

are in agreement with the notion that the FWHMof the probe determines the resolution achievablein ADF imaging [24].

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0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

AD

F in

tens

ity

A1

B1

C1

0.7 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

AD

F in

tens

ity

A1

B1

C1

1.4 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

AD

F in

tens

ity

A1 B1

C1

2.0 Å probe

Fig. 10. Calculated ADF intensities vs. crystal thickness for

2:0 (A (top), 1:4 (A (middle) and 0:7 (A (bottom) probes in

/0 0 1S silicon. Each graph shows the calculated ADF intensityfor probe positions A1, B1 and C1. The ADF detector inner

and outer angles are 50 and 150 mrad; respectively.


In summary, the three requirements listed abovecannot be simultaneously satisfied and so theexperimental conditions must be tailored to suit

the materials problem at hand. If the specimen ishomogeneous with respect to the beam direction,and the features of interest are inhomogeneities inthe x–y plane only, then the optimum probe sizewould appear to be the smallest probe size. In thiscase the best ADF image resolution is achievedand the majority of the EELS will derive from thecolumn of interest (at least for crystal thicknessesup to those studied here ð600 (AÞ).If the specimen is not homogeneous with respect

to the beam direction, then the requirementsregarding EELS and ADF images mentionedabove seem conflicting. Whilst the intensitydistribution on the CI is more homogeneous forlarger probes, ADF images from these probes donot provide any indication of the probe position.Therefore, it seems that a compromise should bemade, the optimum probe size being the largestprobe size that still provides the necessary resolu-tion in ADF imaging. Even for this optimumprobe size, it will be difficult to interpret the EELSsignal from the column of interest because of thecomplex variation in intensity.

6. Summary of conclusions

Using frozen phonon multislice calculations, thescattering behaviour of (A-scale electron probes in/0 0 1S and /1 1 0S silicon was examined as afunction of probe size, probe position and crystalthickness. The detailed conclusions are complexand were presented in earlier sections. We sum-marize the main conclusions here as follows.The dispersion of (A-scale probes with increasing

depth is greater and quicker for smaller probes.This can be attributed to the larger transversemomentum components of such probes. In addi-tion, the dispersion is greater in /0 0 1S than/1 1 0S silicon. This can be attributed to the shapeof the dumbbell potential which appears to trapthe probe within it, limiting dispersion.Focusing a probe on an atomic column in

/0 0 1S silicon results in more intensity being onthat column than all its neighbour columnscombined for thicknesses up to at least 600 (A forthe 2.0 and 1:4 (A probes, and for thicknesses up to400 (A for the 0:7 (A probe (with the exception of

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0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

0.25

AD

F in

tens

ity

A2

B2

C2

0.7 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

0.25

AD

F in

tens

ity

A2B2

C2

1.4 Å probe

0 200 400 600thickness (Å)

0.00

0.05

0.10

0.15

0.20

0.25

AD

F in

tens

ity

A2

B2

C2

2.0 Å probe

Fig. 11. Calculated ADF intensities vs. thickness for 2:0 (A

(top), 1:4 (A (middle) and 0:7 (A (bottom) probes in /1 1 0Ssilicon. Each graph shows the calculated ADF intensity for

probe positions A2, B2 and C2. The ADF detector inner and

outer angles are 50 and 150 mrad; respectively.


certain thicknesses such as 80 (A for the 0:7 (Aprobe). For probes focused on a column in/1 1 0S silicon, the intensity peak transfers tothe other column in the dumbbell at around300–400 (A depending on the probe size. Thedifferent scattering behaviour in the two orienta-tions can be attributed to their distinct minimuminter-column separations and the different lineardensity of atomic columns.When a probe is focused on an atomic column,

the intensity peak that develops on this columngrows and then decays more rapidly with depth forsmaller probes. This is similarly the case fornearest neighbour and next nearest neighbourcolumns. The variation of the intensity on thecolumn beneath the probe is also influenced by thecrystal orientation, the variation being greater andmore complex for the /1 1 0S case than for the/0 0 1S case.In considering the effect of this scattering

behaviour on the EELS signal, we first note that,for large collector apertures, the core-loss EELSsignal is related to the elastic intensity inside thecrystal (Eq. (7)). For typical collector aperturesizes, the elastic intensity provides an estimate ofthe EELS signal. Thus, a useful qualitativemeasure of the spatial origin of the EELS signalis provided by the elastic intensity in the immedi-ate vicinity of the atoms.With this in mind, we note that the complex

intensity variation on the column beneath smallerprobes means the contributions of different atomsto the EELS signal may vary greatly. This makes itdifficult to determine quantitatively the columncomposition from the EELS signal. In particular,defects such as dopant atoms at certain depths inthe column can fail to be detected with either theEELS or ADF signals.The optimum experimental conditions for the

simultaneous acquisition of an ADF image andEEL spectra will depend on the nature of thematerials problem. If the specimen is known to behomogeneous in composition parallel to the beam,the complex intensity variation on the column ofinterest is not likely to present any difficulties inthe analysis of EEL spectra. For such a specimen,the smallest probe is likely to offer the strongestEELS signal from the column of interest relative to

the neighbouring columns, in addition to provid-ing the highest resolution in the ADF image.However, if the composition of the specimen varieswith depth, the differing contributions of theatoms in the column could make a straightforwardanalysis of EEL spectra difficult. In this case, inorder to minimize the difficulties in the analysis ofthe EEL spectra, the optimum probe may be thelargest probe that can still generate an ADF imagewith adequate resolution to provide a reference forthe probe position.

Acknowledgements

This work was supported by EPSRC GrantGR/R42276/01. CD is grateful to the CambridgeCommonwealth Trust for a Packer Scholar-ship and to Universities UK for an OverseasResearch Students Award. JE is grateful to TheRoyal Society for a University Research Fellow-ship.

Appendix A

In this appendix, we derive an expression for thetransition rate for atomic ionization by fastelectrons. We show that, if the EELS collectoraperture is very large, the transition rate is givenby the integral of the projected elastic fast electronintensity inside the crystal, weighted by thesquared modulus of the projected matrix elementfor atomic ionization. The derivation given here issimilar to that given by Muller and Silcox [25]. Thekey approximations for obtaining the final resultare: (i) the first Born approximation for inelasticscattering, (ii) subsequent elastic scattering of thefast electron after an inelastic scattering event doesnot occur, and (iii) the small-angle approximation.Since the inelastic amplitudes resulting from the

ionization of different atoms can be regarded asbeing incoherent [26], we may consider theinelastic intensity resulting from the ionization ofeach atom separately, and then simply add theinelastic intensity contributions for all atoms. Tothis end, we will consider the transition rateassociated with the ionization of a single atom.

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Within the validity of the first Born approxima-tion, the inelastic amplitude associated with theionization of a particular atom is given by

/rjcaS ¼ �m

2p_2

Zd3r0/ajV ðr0Þja0S/r0jc0S

�expðikajr � r0jÞ

jr � r0j; ðA:1Þ

where m is the electron mass (corrected forrelativistic effects). V represents the Coulombinteraction between the fast electron and theatomic electron. /ajV ðr0Þja0S is the matrix elementof V involving the atomic ground state a0 and theatomic excited state a: The matrix element isconsidered as a function of the position r0 of thefast electron. /r0jc0S is the elastic fast electronamplitude. ka is the magnitude of the fast electronwave vector after inelastic scattering.Due to the localized nature of the matrix

element, the significant contributions to theintegral in Eq. (A.1) come from the region in theimmediate vicinity of the atom. Therefore, weassume that the elastic fast electron amplitude inthis region can be described to sufficient accuracyby using plane waves of energy E � Ea0 ; where E isthe total energy available to the fast electron andthe atom, and Ea0 is the energy of the atomicground state a0: Thus, we write (using boxnormalization)

/rjc0S ¼L3=2

ð2pÞ3

Zd3k/kjc0Sexpðik � rÞ; ðA:2Þ

where /kjc0S ¼ dk;k0/kykfjc0S and k0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mðE � Ea0 Þ

p=_:

At large distances from the atom, the inelasticamplitude has the form of an outward movingspherical wave

/rjcaS ¼ �mL3=2

2p_2/kaajV ja0c0S

expðikarÞr

; ðA:3Þ

where the origin of the spherical coordinate systemis situated at the atom, so that the directions #r and#ka coincide. The matrix element /kaajV ja0c0S;which describes the angular variation of the

inelastic amplitude, is given by

/kaajV ja0c0S ¼1

ð2pÞ3

Zd3k/kjc0S

�Zd3r/ajV ðrÞja0S

� expð�iðka � kÞ � rÞ: ðA:4Þ

By Fermi’s golden rule, the transition rate wa0-a

for an inelastic scattering event in which the finalelectron wave vector lies within the element d3ka ofk-space situated at ka is given by

dwa0-a ¼2p_j/kaajV ja0c0Sj2rðE � EaÞ; ðA:5Þ

where rðE � EaÞ represents the density of finalstates available to the fast electron.We now introduce the small-angle approxima-

tion. This approximation has the effect of neglect-ing the curvature of the spheres in k-space definedby the possible values of the initial and final wavevectors k0 and ka; respectively. Thus, instead oftwo spheres in k-space, the possible values of k0and ka define two planes perpendicular to the opticaxis #z: Adopting this approximation, the elasticfast electron amplitude (A.2) becomes

/Rzjc0S ¼L1=2

ð2pÞ2expðik0zÞ

�Zd2K/K jc0SexpðiK � RÞ; ðA:6Þ

where upper case bold symbols denote vectors inthe x–y plane perpendicular to the optic axis.Within the small-angle approximation, conserva-tion of energy imposes a condition only on the z

component ka;z of the wave vector ka:Thus, the transition rate (A.5) becomes

wa0-a ¼2pm

_3ka;z

L

2p

� �3j/Kaka;zajV ja0c0Sj2d2Ka;

ðA:7Þ

where the value of ka;z fulfills the requirement ofconservation of energy. Eq. (A.7) represents theprobability per unit time that the fast electron willbe inelastically scattered with an accompanyingatomic transition a0-a; such that its finaltransverse wave vector lies within the elementd2Ka of K-space situated at Ka: In order to obtain

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the transition rate relevant to EELS in a STEM,we must integrate (A.7) over all values of Ka thatcontribute to the EELS signal. This integrationcan be performed by defining an aperture function*DðKaÞ that is equal to unity for values of Ka

admitted by the aperture, and zero otherwise.Performing the integral of Eq. (A.7) with respectto Ka; and making use of Eqs. (A.4) and (A.6), weobtain

wa0-aðDÞ ¼m

_3ka;zL

Zd2R1 dz1 d

2R2 dz2DðR2 � R1Þ

�/ajV ðR1z1Þja0S/a0jV ðR2z2ÞjaS

�/R1jc0S/c0jR2S

� expð�iðka;z � k0Þðz1 � z2ÞÞ; ðA:8Þ

where DðRÞ is the aperture function in R-space,and

/Rjc0S ¼L

ð2pÞ2

Zd2K/K jc0SexpðiK � RÞ: ðA:9Þ

The integrand in Eq. (A.8) represents the inter-ference of the inelastic amplitude originating fromtwo points ðR1z1Þ and ðR2z2Þ in the crystal,weighted by the R-space aperture functionDðR2 � R1Þ: We may then carry out the integra-tions with respect to z1 and z2 to obtain

wa-a0ðDÞ ¼m

_3ka;zL

Zd2R1d

2R2DðR2 � R1Þ

�/ajV ðR1Þja0S/a0jV ðR2ÞjaS

�/R1jc0S/c0jR2S; ðA:10Þ

where /ajV ðRÞja0S is the ‘projected’ matrixelement, i.e. the matrix element integrated withrespect to z:As a final step, we consider the case of a very

large collector aperture, so that the R-spaceaperture function DðR2 � R1ÞEdðR2 � R1Þ: In thiscase, we obtain the total transition rate wa0-a; i.e.the rate for an atomic transition a0-a with noregard for the final transverse wave vector of thefast electron

wa-a0 ¼m

_3ka;zL

Zd2Rj/ajV ðRÞja0Sj2j/Rjc0Sj2:

ðA:11Þ

Eq. (A.11) is the desired result, stating that for avery large collector aperture the transition rate foratomic ionization is given by the integral of theprojected elastic fast electron intensity, weightedby the squared modulus of the projected matrixelement.

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