Ultramicroscopyematweb.cmi.ua.ac.be/emat/pdf/1534.pdf · the crystal. This overestimated ratio of the thickness of the amorphous layer to the crystal is considered as an extreme scenario

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Ultramicroscopy 109 (2009) 237–246

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Ultramicroscopy

0304-39

doi:10.1

� Corr

Univers

Tel.: +1

E-m

journal homepage: www.elsevier.com/locate/ultramic

Effect of amorphous layers on the interpretation of restored exit waves

S. Van Aert a, L.Y. Chang b,�, S. Bals a, A.I. Kirkland b, G. Van Tendeloo a

a Electron Microscopy for Materials Science (EMAT), University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgiumb Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

a r t i c l e i n f o

Article history:

Received 26 March 2008

Received in revised form

20 October 2008

Accepted 29 October 2008

PACS:

02.70.Rr

07.05.Pj

07.78.+s

Keywords:

High-resolution transmission electron

microscopy (HRTEM)

Exit wave restoration

Model-based fitting

91/$ - see front matter & 2008 Elsevier B.V. A

016/j.ultramic.2008.10.024

esponding author. Canadian Center for Elec

ity, 1280 Main Street West, Hamilton, Canada

9055259140x21269; fax: +19055212773.

ail addresses: [email protected], sherychan

a b s t r a c t

The effects of amorphous layers on the quality of exit wave restorations have been investigated. Two

independently developed software implementations for exit wave restoration have been used to

simulated focal series of images of h0 0 1i SrTiO3 with amorphous carbon layers incorporated. The

restored exit waves have been compared both qualitatively and quantitatively. We have shown that

amorphous layers have a strong impact on the quantitative measurements of atomic column positions,

however, the error in the position measurements is still in the picometer range.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Transmission electron microscopy (TEM) is firmly establishedas one of the most powerful tools for the study of many materialssystems. However, interpretation of conventional high-resolutionTEM (HRTEM) images is far from intuitive and image simulationsusing trial structural models are often required. Recently,developments in TEM instrumentation including aberrationcorrectors [1] and monochromators [2] have improved resolutionto levels below 0.1 nm at intermediate voltages [3]. In addition toobtaining better resolution using aberration-corrected micro-scopes, which provide directly interpretable structural informa-tion over a wider range of spatial frequencies, improvedinterpretability can also be achieved using advanced indirecttechniques including exit wave restoration. The principle of thelatter is to restore the complex electron wave function at the exitplane of the object from a suitably conditioned data set ofconventional images. Ideally, this exit wave is free from imagingartifacts (i.e. objective lens aberrations), which considerablyenhances interpretability compared to the conventional experi-mental images for reasonably thin objects. One commonly applied

ll rights reserved.

tron Microscopy, McMaster

L8S 4M1.

[email protected] (L.Y. Chang).

experimental geometry for exit wave restoration is to use a seriesof differently aberrated images, for example, images taken atdifferent defocus values [4,5]. This method yields a restored wavecontaining all information up to the information limit of themicroscope, except for very low spatial frequencies which are notrecovered. More importantly the phase and amplitude arerecovered whereas the phase is lost in a conventional TEM image.Recently, focal series exit wave restoration has been successfullyapplied to the determination of the positions of atomic columns inmany materials [6–11].

Several exit wave restoration methods are currently available,each of them having their own independent software implemen-tation. In order to understand the reliability of the qualitative andquantitative interpretation of the structural information obtained,it is therefore important to critically compare their outcomesunder experimentally realistic conditions. In this paper, we havecompared two different programs: TrueImage [12] and Focal-TiltSeries Restoration (FTSR) [13]. TrueImage is a commerciallyavailable package [12] and FTSR, originally developed by Meyeret al. using the Semper image processing script language [13–15]has recently become available as commercial software [16].1

These two programs differ in the methods used for aberration

1 It should be mentioned that the FTSR restorations in this paper were carried

out using the Semper version rather than the commercial software.

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S.V. Aert et al. / Ultramicroscopy 109 (2009) 237–246238

determination and also in their treatment of TEM imageintensities in the recovery of the exit wave.

Several experimental factors, such as image alignments,specimen contaminations and residual aberrations present inthe restored exit wave, may affect the quality of the restorationsand the reliability of quantitative interpretation of exit waves. Inthis paper, we have focused on theoretical investigation ofamorphous layers on a TEM specimen caused by specimenpreparation procedures, specimen storage or electron beaminduced deposition of carbonaceous materials. Amorphous layersare indeed difficult to avoid or control experimentally, whereaserrors due to image alignment can be easily estimated andresidual aberrations can be removed a posteriori. This has beenstudied by simulating images of a crystalline SrTiO3 h0 0 1i samplewhere an amorphous carbon layer has been incorporated.SrTiO3 h0 0 1i was chosen as a model system since it containsboth heavy and light element columns, providing a reasonablediversity in atomic species. Moreover, the atomic columns areclearly separated in a h0 0 1i orientation. Several different speci-men geometries have been considered: a specimen with anamorphous layer on the top surface, one with an amorphous layeron the bottom surface, and one with amorphous layers on bothbottom and top surfaces of crystalline SrTiO3. In addition a purecrystalline sample (i.e. no amorphous layers) has been simulatedas a reference for quantitative measurements. For these geome-tries the exit wave restorations from both codes are initiallycompared in a qualitative, i.e. visual, manner and subsequently inmore detail using statistical parameter estimation theory [17] forquantitative measurements of the atomic column positions.

The structure of this article is as follows. Firstly, the simulationof image focal series with amorphous layers on the crystallinesample is described. Secondly, the two different exit waverestoration methods are briefly discussed and the quantitative,model-based method for atomic column position measurement isoutlined. Finally, the restored exit waves for the different speci-men geometries are interpreted and compared.

2 It should also be noted that the PAM method is related to the linear Wiener

filter outlined in the previous section and is explained in more detail elsewhere

[22,27,28].

2. Simulation of SrTiO3 h0 0 1i focal series

The exit waves of SrTiO3 h0 0 1i including amorphous carbonlayers were calculated using the established multi-slice method[18,19]. The thickness of the SrTiO3 crystal was chosen to be1.965 nm (i.e. 5 unit cells in projection) corresponding to a thinregion in a realistic SrTiO3 sample. Amorphous carbon layers werechosen to have the same thickness as the crystal. Four specimengeometries were considered: (a) pure crystalline SrTiO3 withoutany amorphous layers, (b) SrTiO3 with an amorphous carbon layeron top of the SrTiO3 crystal, (c) SrTiO3 with an amorphous carbonlayer on the bottom of the SrTiO3 crystal and (d) SrTiO3 withamorphous carbon layers on both the top and on the bottom ofthe crystal. This overestimated ratio of the thickness of theamorphous layer to the crystal is considered as an extremescenario for the estimation of the atomic column positions. Torepresent a more realistic specimen condition, an additional caseusing a thicker SrTiO3 crystal of 9.825 nm (i.e. 25 unit cells inprojection) with the same thickness of amorphous carbon layersas before (1.965 nm) on both top and bottom of the crystalwas also simulated. Coordinates for bulk SrTiO3 were taken fromRef. [20]. The amorphous layers were constructed using a simplemodel with randomized atomic positions, having the atomdensity of a diamond unit cell. Although this randomized atomicconfiguration is not the true amorphous carbon structure, it issufficient for the purpose of this investigation.

Focal series of 20 images were calculated from simulated exitwaves of the four specimen geometries, calculated from the full

non-linear image intensity [21,22]. The defoci were chosen withequally spaced intervals of �7 nm, starting from a defocus value ofþ5 nm. An accelerating voltage of 200 kV, a spherical aberrationcoefficient of 0.4 mm, a beam divergence angle of 0.15 mrad and afocal spread parameter of 4 nm were used in all cases.

3. Exit wave restoration

3.1. FTSR

The FTSR program restores complex specimen exit waves byfirstly measuring the objective lens aberrations due to defocusand twofold astigmatism together with specimen drift for allmembers of the input focal series, using a phase compensatedphase correlation function (PCF) and subsequently a phasecontrast index (PCI) function [13]. Given these aberrations, alinear Wiener filter is then applied to a series of images to restorethe complex exit wave function [4,13]. Full details of this methodare described elsewhere [4,13,23].

As noted above, the relative defoci and displacements betweenimages in a focal series are initially measured using a phasecompensated PCF. Although this is suitable for crystalline speci-mens where small amounts of amorphous materials are present, itcannot be applied to perfectly crystalline specimens. Frommeasurements of the relative defoci and displacements betweenall images in the focal series, a reference image wave is thenrestored using a Wiener filter. The PCI is then used to determinethe absolute values of defocus and two fold astigmatism (withrespect to the specimen exit plane) under the assumption that thespecimen acts as a weak phase object for most spatial frequencies.This is valid for a thin amorphous area at the edge of thespecimen. Importantly, measuring lens aberrations in this wayallows exit wave restorations to be calculated in the presence ofdefocus fluctuations and specimen drift. However, a sufficientphase difference between neighbouring images is essential for theaccurate determination of relative defocus.

The Wiener filter used in the restoration step is based on thelinear imaging approximation in the presence of noise. The filterfinds the best estimate of the exit wave function using a leastsquares criterion. In principle this approach only requires threeimages to redundantly restore the exit wave. However, in practice,7–20 images with a focal step of typically 7–12 nm are required toachieve a restoration with a good signal to noise ratio and faithfulinformation transfer over a wide range of spatial frequencies.

3.2. TrueImage

The TrueImage Focal Series restoration Package [24–26] makesuse of the PAM-MAL method, in which an approximate, linearParabolic algorithm (PAM) is followed by a non-linear MaximumLikelihood algorithm (MAL). The PAM method, devised by VanDyck and Op de Beeck [5], aims to preferentially select linearimage contributions in a focal series,2 whereas the MAL methoduses the full non-linear contributions [24,29]. Using the result ofthe PAM algorithm as a starting point for the MAL method, goodrestoration results together with high computational performancehave been obtained [24]. In practice the TrueImage code needs10–20 high resolution images as an input, taken at equidistantdefocus steps, and the focal range is positioned symmetricallyaround the Lichte defocus [30]. The instrumental as well as the

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S.V. Aert et al. / Ultramicroscopy 109 (2009) 237–246 239

experimental parameters such as the starting defocus and thefocal increment, must be accurately known as Trueimage requiresnominal values as part of the restoration process.

Although the experimental parameters, such as, the startingdefocus during acquisition, should be determined with highaccuracy, this is not always straightforward in practice. Tocalibrate the defocus value, one usually determines the minimalcontrast condition (i.e. Gaussian defocus) using an amorphousedge of the specimen. However, for wedge shaped samples, it islikely that the area of interest is thicker than the edge, yielding ahigher underfocus value. Furthermore, this procedure is carriedout visually and therefore, a defocus offset might be expected.Using TrueImage, it is possible to correct for deviations from theexact starting defocus after initial restoration of the exit wave. Toachieve this, the preliminary restored wave function is propagatednumerically to the specimen exit plane. More specifically, in orderto obtain the correct value for the starting defocus the wavefunction can be propagated by an amount such that the amplitudecontrast is minimized for a specific region, for an object that obeysthe phase object approximation. This method, however, is notadequate for studying crystalline objects. In that case, the defocusoffset can be determined by comparing the preliminary restora-tion to simulated exit waves. In practice, we have noted thatdetermination of the defocus offset using the minimization ofamplitude contrast does not always show good visual agreementwith the amplitude of the input exit wave. Therefore, in this studywe have also used an alternative approach to estimate the offsetby visually matching the amplitude of the preliminary restoredexit wave and the input exit wave.

4. Estimation of atomic column positions

Although the restored phase of the exit wave can beinterpreted directly without the aid of image simulations, it doesnot directly provide quantitative measurements of the atomiccolumn positions. In order to obtain this, statistical parameterestimation theory is a very useful technique [17,31]. Using thisapproach, a precision in the estimated atomic column positionsthat is orders of magnitude better than the information limit ofthe electron microscope can be achieved. To apply statisticalparameter estimation theory, a parametric model is required. Inthis study the model used describes the expectations of the pixelvalues in the restored phase. The physics behind the electron–object interaction for thin objects is sufficiently well understoodto give a relatively simple expression describing the phase of anelectron exit wave. For thin specimens acting as pure phaseobjects, the phase of the exit wave can be modelled as asuperposition of Gaussian peaks [32]. The expectation at a pixelðk; lÞ at the position ðxk ylÞ is given by the following function:

f klðyÞ ¼ zþXI

i¼1

XMi

mi¼1

amiexp

�ðxk � bxmiÞ2� ðyl � bymi

Þ2

2r2i

0@

1A (1)

where z is a constant background, ri is the width of the Gaussian

peak of a particular column type i (e.g. Sr, TiO or O), amiis the

height of the Gaussian peak and bxmiand bymi

are the x- and y-

coordinate of the mith atomic column, respectively. The unknown

parameters in this model i.e., y ¼ ðbx11. . .bxMI

by11. . .byMI

a11

. . .aMIr1 . . .rI zÞT, are estimated in the least squares sense. The

uniformly weighted least squares estimates by are given by the valuesof t that minimize the uniformly weighted least squares criterion:

by ¼ arg mint

XK

k¼1

XL

l¼1

ðwkl � f klðtÞÞ2 (2)

with wkl the value of the restored phase at the pixel ðk; lÞ. We notethat for pixel values which are independent and identicallynormally distributed about the expectations, the uniformlyweighted least squares estimator is identical to the maximumlikelihood estimator, which is known to have optimal statisticalproperties [17]. As shown in Ref. [33], this assumption about thestatistical distribution of the pixel values is valid.

The method as described above was applied to the phases ofexit waves restored from simulated focal series in order tomeasure the atomic column position coordinates. These measuredpositions were subsequently compared with the true atomiccolumn position coordinates used in the image simulations. Auseful criterion for this purpose is the root mean-square error,measuring the spread of the estimated values about the truevalues. For the different atomic column types, labelled with indexi, this root mean-square error is estimated as follows:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Mi

XMi

mi¼1

bd2mi

vuut (3)

with bdmithe distance between the estimated and true position of

the mth column of a particular column type i.Finally, we note that in this study only the pixel values of the

restored phase are used to estimate the unknown structuralparameters. In principle, more accurate and precise results mightbe expected if the restored amplitude is also taken into account.This requires a parametric model describing both the restoredphase and modulus. Channelling theory [34], which provides aclosed analytical expression for the complex exit wave, is suitablefor this purpose. However, it is known that the modulus is moresensitive to residual aberrations in comparison to the phase andthis is not yet fully included in the present form of channellingtheory. Therefore, only the phases are used in this study. Thevalidity of this approach will be demonstrated in the followingsection.

5. Results and discussion

In this section, the phases and moduli of the four differentsimulated specimen geometries, with the crystal and eachamorphous layer having a thickness of 2 nm, restored using FTSRand TrueImage, are compared both qualitatively and quantita-tively with the phases and moduli of the input exit waves. Anadditional geometry with a crystal thickness of 10 nm and 2 nmthick amorphous layers on both top and bottom of the crystal isalso investigated.

For restorations using FTSR, defocus values are determinedusing the PCI function, based on the weak phase object criterium.For the restorations using TrueImage, two different criteria havebeen used to determine the defocus offset: minimization of theamplitude contrast of the exit wave, which is carried outautomatically in TrueImage, and visual matching the restoredmodulus with the modulus of the input exit wave, since themodulus is apparently most sensitive to changes in defocus.

Another major difference in the FTSR and TrueImage restora-tion algorithms is that the former uses the Wiener filter, which isbased on the linear imaging approximation, whereas the latteralso treats the non-linear image intensities. For the specimenthickness we investigated here, the linear imaging approximationis valid and hence both restoration algorithms can be applied.

5.1. Purely crystalline specimen of 2 nm thickness

A purely crystalline specimen is an idealistic case, asamorphous layers are extremely difficult to avoid entirely.

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0.82 1.8 Sr Ti O

Fig. 2. Modulus of exit waves for a specimen of SrTiO3 h0 0 1i of 2 nm thickness,

without an amorphous layer. (a) Input exit wave, (b) FTSR restoration with forced


However, it is essential for this study as it serves as a referencefor the model-based statistical estimation of atomic columnpositions.

Qualitatively, good agreement has been found between thephases and moduli of the input exit wave and the exit waverestored using TrueImage, as shown in Figs. 1 and 2. Therestoration using the FTSR, however, disagrees with the inputexit wave, as shown in the inset to Figs. 1 and 2(b). However,it is important to note that this arises from a failure in theaberration determination within FTSR for this purely crystallineexample (as explained in Section 3.1), rather than in the useof a linear Wiener filter. If the defocus values used in FTSR areforced to the input values (as is the case for TrueImage), therestoration agrees with the input exit wave, as shown in Figs. 1and 2(b).

Comparison of the results at the quantitative level (Table 1)shows that the root mean-square error in the estimated atomiccolumn positions lies in the sub-picometer range for the(resolution limited) input exit wave (row a). These extremelysmall errors demonstrate the validity of the model-basedestimation used in this paper. The root mean-square errors forboth the FTSR (using the input defocus values, row b) andTrueImage restorations (rows c and d) for the heavier Sr and TiOcolumns are also in the sub-picometer range. The error in thelighter O column from the FTSR is still in the sub-picometer rangewhereas the errors from Trueimage are slightly higher. It shouldalso be noted that, when comparing the results between the(resolution limited) input (row a) and restored exit waves (rows cand d), there is an increase in the root mean-square error by afactor of approximately 10.

-0.28 1.31Sr Ti O

Fig. 1. Phases of exit waves for a specimen of SrTiO3 h0 0 1i of 2 nm thickness,

without an amorphous layer. (a) Input exit wave, (b) FTSR restoration with forced

input defocus values and with defocus values determined using PCF/PCI (top right

inset), (c) TrueImage restoration optimized by minimization of amplitude contrast,

(d) TrueImage restoration optimized by visual matching with the input exit wave.

input defocus values and with defocus values determined using PCF/PCI (top right

inset), (c) TrueImage restoration optimized by minimization of amplitude contrast,

(d) TrueImage restoration optimized by visual matching with the input exit wave.

Table 1Estimated root mean-square errors (RMSE) for Sr, TiO and O atomic columns

obtained from the phase of the exit waves for a specimen of 2 nm thickness

without any amorphous layers.

RMSE (pm) Sr TiO O

a 0.05 0.05 0.05

b 0.05 0.05 0.17

c 0.97 0.50 1.03

d 0.97 0.50 1.03

Values are obtained from the phase of (a) the input exit wave, (b) the FTSR

restoration using input defocus values, (c) the TrueImage restoration optimized by

minimization of amplitude contrast and (d) the TrueImage restoration optimized

by visual matching with the input exit wave.

5.2. Amorphous layer on the bottom of a 2 nm crystalline sample

Figs. 3 and 4 show the phase and modulus of input andrestored exit waves for a specimen with an amorphous layer onthe bottom surface of a crystalline sample of 2 nm thickness.Comparison of Figs. 3 and 4 with Figs. 1 and 2, where noamorphous layer is present, shows that the effect of theamorphous layer is immediately obvious both in the phase aswell as in the modulus. Visual comparisons between therestorations show that both the phase (Fig. 3(b)) and the modulus(Fig. 4(b)) restored using the FTSR program show a closeresemblance to the input exit wave. Examining the resultsobtained from TrueImage, it can be seen that the restored phaseoptimized by amplitude contrast minimization (Fig. 3(c)) issimilar to the restored phase where the defocus offset is

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-0.4 0.12 Sr Ti O

Fig. 3. Phase of the exit wave for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with an amorphous layer on the bottom surface of the sample. (a) Input exit wave,

(b) FTSR restoration, (c) TrueImage restoration optimized by minimization of

amplitude contrast, (d) TrueImage restoration optimized by visual matching with

the simulated input exit wave.

0.75 1.67Sr Ti O

Fig. 4. Modulus of the exit wave for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with an amorphous layer on the bottom surface of the sample. (a) Input exit wave,

(b) FTSR restoration, (c) TrueImage restoration optimized by minimization of




determined by visually matching with the input exit wave(Fig. 3(d)). However, this is not the case for the restored modulus(Figs. 4(c) and (d)) where a clear difference in contrast is observed.The relative difference in defocus offset between the twoalternative TrueImage restorations is found to be 1.1 nm.

Quantitative comparison of the root mean-square errorsobtained from the input phase for this geometry (Table 2(a)),with those for the crystalline sample without amorphous layers(Table 1(a)) shows an overall increase by a factor of approximately100, demonstrating the impact of amorphous contamination onthe reliability of quantitative atomic column position measure-ments. However, although the effect of the amorphous layer issignificant, the root mean-square errors are still in the picometerrange. The data in Table 2 show that the root mean-square errorsfor both the heavier atomic columns and the oxygen columns,including those from the corresponding input phase, all have thesame order of magnitude. This trend is different from thatobserved for the specimen where no amorphous layer is presentin which the root mean-square errors obtained from the restoredphases are significantly higher than those of the input phase.3 Thistrend suggests that quantitative atomic column position determi-nation from the restored phase of a specimen with the presence ofan amorphous layer can be as reliable as that from the input phaseimplying that the process of exit wave restoration has noquantitative effect on this geometry. Moreover the qualitativedifferences observed between the FTSR and TrueImage restora-tions are not reflected in the quantitative comparison.

3 With the exception of the RMSE of the Sr and TiO atomic columns from the

phase restored by FTSR.

5.3. Amorphous layer on the top of a 2 nm crystalline sample

The phases and moduli of input and restored exit waves for thespecimen geometry with an amorphous layer on the top surface ofthe crystalline sample of 2 nm thickness are shown in Figs. 5and 6. As was found for the specimen geometry with anamorphous layer on the bottom surface of the crystalline sample,the effect of the amorphous layer is again clearly visible in boththe phases and the moduli. The phases for this geometrycompared to those where the amorphous layer is located on thebottom of the crystalline sample (Figs. 3 and 5) appear similar,whereas the moduli appear different (Figs. 4 and 6). This effectmight be attributed to a different local path followed by theincident electrons in the different specimen geometries. In thespecimen geometry with an amorphous layer on the bottomsurface of the crystalline sample, well-defined electron channel-ling within the crystalline region of the sample is followed byinteraction with a weak phase amorphous object. In the casewhere the amorphous layer is present on the top of the crystallinesample, the order of the scattering processes is reversed andhence the ideal channelling condition (i.e. an electron beamperfectly aligned with the atomic columns) is no longer fulfilled[35]. Comparison of the restorations between the two programsshows that the FTSR restoration (Figs. 5(b) and 6(b)) again gives acloser visual resemblance to the input exit wave, compared to theTrueImage restorations (Figs. 5(c,d) and 6(c,d)). In particular thecontrast of the oxygen columns in the modulus (optimized byminimization of amplitude contrast) restored using TrueImage,shown in Fig. 6(c) has completely vanished and the contrast in thecorresponding phase (Fig. 5(c)) is less pronounced. The relativedifference in defocus offset between the two alternative True-Image restorations is found to be 3 nm.

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obtained from the phase of the exit waves for a specimen with an amorphous layer

on the bottom of the crystalline sample of 2 nm thickness.

RMSE (pm) Sr TiO O

a 3.4 3.2 11.7

b 4.9 4.8 15.6

c 5.6 4.9 9.5

d 6.0 5.0 9.1

Values are obtained from the phase of (a) the input exit wave, (b) FTSR restoration,

(c) TrueImage restoration optimized by minimization of amplitude contrast and

(d) TrueImage restoration optimized by visual matching with the simulated input

exit wave.

-0.4 0.51Sr Ti O

Fig. 5. Phase of the exit waves for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with an amorphous layer on the top surface of the sample. (a) Input exit wave, (b)

FTSR restoration, (c) TrueImage restoration optimized by minimization of



0.52 1.38Sr Ti O

Fig. 6. Modulus of the exit wave for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with an amorphous layer on the top surface of the sample. (a) Input exit wave, (b)

FTSR restoration, (c) TrueImage restoration optimized by minimization of




obtained from the phases of the exit waves with an amorphous layer on the top of

the crystalline sample of 2 nm thickness.

RMSE (pm) Sr TiO O

a 2.3 2.2 13.1

b 3.3 3.1 16.0

c 7.1 7.9 23.9

d 5.0 4.3 9.7


restoration, (c) the TrueImage restoration optimized by minimization of amplitude

contrast and (d) the TrueImage restoration optimized by visual matching with the

simulated input exit wave.


Similar trends within the quantitative comparisons for thisspecimen geometry and those with the amorphous layer on thebottom of the sample have been observed (Tables 2 and 3). Theroot mean-square errors in the Sr and TiO columns are all of thesame order of magnitude and still within the picometer rangewhereas the root mean-square errors in the oxygen columns arehigher. It is also observed that the root mean-square error in theoxygen column, obtained from the Trueimage restoration opti-mized by amplitude contrast minimization, is much higher thanthe other errors in the oxygen column. This particularly high errorreflects the qualitative observation that the contrast of the oxygencolumn in the phase restored using TrueImage, optimized byamplitude contrast minimization (Fig. 5(c)) is very weak and hascompletely vanished in the restored modulus. This extremelyweak contrast has an unfavourable effect on the errors in theestimated atomic column positions.

5.4. Amorphous layers on both the top and bottom of a 2 nm

crystalline sample

For this specimen geometry of amorphous layers on both thetop and bottom of a 2 nm crystalline sample, the combinedthickness of the amorphous layers is twice the amorphous layerthickness of the previous two geometries. However, the restoredphases still show good agreement with the phase of the input exitwave, as shown in Fig. 7. The modulus restored using FTSR showsa closer visual match to that of the input exit wave, comparedwith that restored using TrueImage, as shown in Fig. 8.

Table 4 shows that the estimated root mean-square errors inthe heavier Sr and TiO columns are still in the picometer range,whereas those of oxygen columns are an order of magnitudehigher. In addition the overall root mean-square errors for

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-0.7 Sr Ti O0.36

Fig. 7. Phase of the exit waves for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with amorphous layers at both the top and bottom surfaces of the sample. (a)

Input exit wave, (b) FTSR restoration, (c) TrueImage restoration optimized by

minimization of amplitude contrast, (d) TrueImage restoration optimized by visual

matching with the simulated input exit wave.

0.6 1.8 Sr Ti O

Fig. 8. Modulus of the exit waves for a specimen of SrTiO3 h0 0 1i of 2 nm thickness

with amorphous layers at both the top and bottom surfaces of the sample. (a)

Input exit wave, (b) FTSR restoration, (c) TrueImage restoration optimized by

minimization of amplitude contrast, (d) TrueImage restoration optimized by visual

matching with the simulated input exit wave.

Table 4Estimated root mean-square errors (RMSE) for the Sr, TiO and O atomic columns


this specimen geometry are higher compared to those for theprevious three specimen geometries, indicating that the rootmean-square errors increase with increasing thickness of theamorphous layers.

obtained from the phases of the exit waves with an amorphous layer on the top

and bottom of the crystalline sample of 2 nm thickness.

RMSE (pm) Sr TiO O

a 5.3 5.7 21.8

b 6.2 7.1 24.8

c 9.7 10.8 20.5

d 10.6 8.8 14.9





5.5. Amorphous layers on both the top and bottom of a 10 nm

crystalline sample

This specimen geometry corresponds to a more realisticsample condition with a crystal thickness of 10 nm and eachamorphous layer having a thickness of 2 nm as in the previouscases.

It can be seen immediately from Figs. 9 and 10 that thecontrast in the phases and moduli for both the Sr and TiO sites areweaker compared to that of the oxygen sites, which is contrary tothe previous cases of thinner crystalline specimens. This isdue to the dynamical scattering which causes contrast reversalwithin a certain thickness range. However, the restored phasesfor this thicker specimen still show good agreement with thephase of the input exit wave, as shown in Fig. 9. It should bementioned that in this case the exit wave restored usingTrueImage cannot be optimized by minimization of amplitudecontrast due to the specimen thickness, therefore only theoptimization by visually matching with the simulated input exitwave has been applied. The phase and modulus restored usingTrueImage show a closer visual match to those of the input exitwave, compared to those restored using FTSR, as shown in Figs. 9and 10. This may be due to a less than ideal specimen thickness(although still applicable) for the linear imaging approximationused in the FTSR restoration.

Table 5 shows that the estimated root mean-square errors forall atomic column sites are still in the picometer range for thisthicker specimen. However, the errors in the heavier Sr and TiOcolumns are higher compared to the lighter O columns as thecontrast in O sites is stronger, which differs from the resultsobtained for thinner specimens.

Comparison of the root mean-square errors for this thickerspecimen with those obtained for the thin specimen withequally thick amorphous layers on both sides shows that theoverall error range for the thick crystal does not exceed that of thethin crystal.

Based on the qualitative and quantitative comparisons(summarized in Fig. 11) for the five specimen geometriesconsidered, we have identified the following trends:

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-0.68Sr Ti O

0.93

Fig. 9. Phase of the exit waves for a specimen of SrTiO3 h0 0 1i of 10 nm thickness with amorphous layers at both the top and bottom surfaces of the sample. (a) Input exit

wave, (b) FTSR restoration, (c) TrueImage restoration optimized by minimization of amplitude contrast is not applicable in this case, (d) TrueImage restoration optimized by

visual matching with the simulated input exit wave.

0.43 2.15 Sr Ti O

Fig. 10. Modulus of the exit waves for a specimen of SrTiO3 h0 0 1i of 10 nm thickness with amorphous layers at both the top and bottom surfaces of the sample. (a) Input

exit wave, (b) FTSR restoration, (c) TrueImage restoration optimized by minimization of amplitude contrast is not applicable in this case, (d) TrueImage restoration

optimized by visual matching with the simulated input exit wave.

Table 5Estimated root mean-square errors (RMSE) for the Sr, TiO and O atomic columns

obtained from the phases of the exit waves with an amorphous layer on the top

and bottom of the 10 nm thick crystal.

RMSE (pm) Sr TiO O

a 9.14 7.89 4.55

b 11.66 9.45 8.12

c N/A N/A N/A

d 7.71 7.23 4.67






(1)
The root mean-square errors obtained from a purely crystal-line sample are significantly lower for all atomic column types(Sr, TiO and O in this case) than those obtained from a samplewith amorphous layers present, suggesting that the amor-phous layers have a significant effect on the errors in theestimated atomic column positions.
(2)
For a given crystal thickness, the root mean-square errorsincrease with increasing thickness of the amorphous layers.However, the errors show little dependence on the geometryof the amorphous layers relative to the crystalline sample.
(3)
The root mean-square errors measured from the restoredphases are lower for those sites with stronger contrast (i.e. Srand TiO sites for 2 nm crystalline specimen and O sites for10 nm crystalline specimen) than those sites with weakercontrast. This suggests that the presence of amorphous layersin a specimen will have a greater impact on the errors in thelocation determination of the atomic columns with weakcontrast.
(4)
The root mean-square errors for a thicker crystalline specimendo not exceed those of a thinner crystalline specimen withequally thick amorphous layers on both sides.
(5)
Due to the dynamical scattering, the atomic columncontrast oscillates with specimen thickness. Therefore, itcan be deduced that the root mean-square errors for aspecific element type would also oscillate with specimenthickness.
(6)
For a specimen with amorphous layers, the root mean-squareerrors obtained from the restored phases are of the same orderof magnitude as those obtained from the input phases,suggesting that the process of exit wave restoration has noimportant quantitative effect on the error in the estimatedatomic column positions.
(7)
The root mean-square errors are independent of the restora-tion implementations, despite qualitative differences in therestored exit waves.

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12a c

b

10

8

6

4

2

0

12

10

8

6

4

2

0

0

5

10

15

20

25

30

rms

erro

r (pm

)

rms

erro

r (pm

)rm

s er

ror (

pm)

A B C D E

A BSpecimen geometry

C D E

A B C D E

(resolution limited) input exit wave

TrueImage optiimised by visual comparisonwith the input exit wave

TrueImage optiimised by minimisation of theamplitude contrast

FTSR exit wave

Fig. 11. Estimated root mean-square errors for (a) Sr (b) TiO (c) O atomic columns for different specimen geometries: (A) a purely crystalline 2 nm sample; (B) a 2 nm

crystalline specimen with an amorphous layer on the bottom surface; (C) a 2 nm crystalline specimen with an amorphous layer on the top surface; (D) a 2 nm crystalline

specimen with an amorphous layer on both the top and the bottom surfaces; (E) a 10 nm crystalline specimen with an amorphous layer on both the top and the bottom

surfaces.


6. Conclusions

As amorphous layers are difficult to avoid for most samples, itis important to study theoretically their effect on the interpreta-tion of exit wave restorations both qualitatively and quantita-tively. This has been done using two independently developedsoftware implementations for exit wave restoration, FTSR andTrueImage, for a simulated SrTiO3 h0 0 1i focal series of imageswith various configurations of amorphous carbon layers incorpo-rated. We have shown that the presence of such amorphous layershas significant effects on both the quality of the exit waverestoration and the quantitative measurement of the atomiccolumn positions using model-based statistical parameter esti-mation theory. In particular, for a given crystalline specimenthickness, the root mean-square errors in the atomic columnposition measurements increase with increasing thickness of theamorphous layer. Generally, the root mean-square errors mea-sured from the restored phases are lower for those sites withstronger contrast (i.e. Sr and TiO sites for 2 nm crystallinespecimen and O sites for 10 nm crystalline specimen) than thosesites with weaker contrast. This suggests that the presence ofamorphous layers in a specimen will have a greater impact on theerrors in the location determination of the atomic columns withweak contrast. As the atomic column contrast oscillates due to thedynamical scattering it can be deduced that the root mean-squareerrors for a specific atomic column type will oscillate withspecimen thickness as well. Although the effect of amorphouslayers on the quantitative measurement of the atomic columnpositions is significant, picometer range accuracy and precision is

still feasible in this model system. Despite qualitative visualdifferences in some of the restorations from the two softwareimplementations our quantitative analysis of the atomic columnpositions shows that the root mean-square errors are independentof the restoration implementations. The results of this paperprovide an initial indication of the errors in the estimated atomiccolumn positions when applying this method to many importantmaterials problems such as grain boundary structures in ceramics.

Acknowledgements

S. Van Aert and S. Bals gratefully acknowledge financialsupport from the Fund for Scientific Research—Flanders (FWO).Financial support from EPSRC (Grant GR/S83968/01) is gratefullyacknowledged. Financial support from the European Union forthe Framework 6 program under a contract for an IntegratedInfrastructure Initiative (Reference 026019 ESTEEM) is acknowl-edged.

References

[1] M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, Ultramicro-scopy 75 (1998) 53–60.

[2] K. Tsuno, Journal of Electron Microscopy 48 (1999) 801–820.[3] B. Freitag, S. Kujawa, P.M. Mul, J. Ringnalda, P.C. Tiemeijer, Ultramicroscopy

102 (2005) 209.[4] W. Saxton, Scanning Microscopy 2 (1988) 213–224.[5] D. Van Dyck, M. Op de Beeck, W. Coene, Optik 93 (1993) 103–107.[6] C.L. Jia, A. Thust, Physical Review Letters 82 (1999) 5052–5055.[7] C.L. Jia, M. Lentzen, K. Urban, Science 299 (2003) 870.

ARTICLE IN PRESS


[8] J. Ayache, C. Kisielowski, R. Kilaas, Journal of Materials Science 40 (2005)3091.

[9] X. Xu, S.P. Beckman, P. Specht, E.R. Weber, D.C. Chrzan, R.P. Erni, I. Arslan, N.Browning, A. Bleloch, C. Kisielowski, Physical Review Letters 95 (2005)145501.

[10] S. Bals, S. Van Aert, G. Van Tendeloo, D. Avila-Brande, Physical Review Letters96 (2006) 096106-1–096106-4.

[11] R.M. Wand, O. Dmitrieva, M. Farle, G. Dumpich, H.Q. Ye, H. Poppa, R. Kilaas, C.Kisielwski, Physical Review Letters 100 (2008) 017205.

[12] Program TrueImage: Focal-series Reconstruction Package (Version 1.0.2), FEIcompany, 2003.

[13] R.R. Meyer, A.I. Kirkland, W.O. Saxton, Ultramicroscopy 92 (2002) 89–109.[14] R.R. Meyer, A.I. Kirkland, W.O. Saxton, Ultramicroscopy 99 (2004) 115–123.[15] W.O. Saxton, T.J. Pitt, M. Horner, Ultramicroscopy 4 (1979) 343–353.[16] FTSR for DM (Focal and Tilt Series Reconstruction), HREM Research Inc., 2006.[17] A.J. den Dekker, S. Van Aert, A. van den Bos, D. Van Dyck, Ultramicroscopy 104

(2005) 83.[18] J.M. Cowley, A.F. Moodie, Acta Crystallographica 10 (1957) 609.[19] E.J. Kirkland, Advanced Computing in Electron Microscopy, Plenum Press,

New York, 1998.[20] W.G. Stirling, Journal of Physics C: Solid State Physics 5 (1972) 2711–2730.[21] K. Ishizuka, Ultramicroscopy 5 (1980) 55.

[22] L.Y. Chang, A.I. Kirkland, Microscopy and Microanalysis 12 (2006)469–475.

[23] A.I. Kirkland, R.R. Meyer, Microscopy and Microanalysis 10 (2004) 401–413.[24] W.M.J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64

(1996) 109–135.[25] M. Op de Beeck, D. Van Dyck, W. Coene, Ultramicroscopy 64 (1996) 167–183.[26] A. Thust, W.M.J. Coene, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64

(1996) 211–230.[27] W.O. Saxton, Ultramicroscopy 55 (1994) 171–181.[28] R.R. Meyer, Quantitative automated object wave restoration in high-

resolution electron microscopy, Ph.D. Thesis, Technischen Universitat Dres-den, 2002.

[29] E.J. Kirkland, Ultramicroscopy 15 (1984) 151–172.[30] H. Lichte, Ultramicroscopy 38 (1991) 13–22.[31] S. Van Aert, A.J. den Dekker, A. van den Bos, D. Van Dyck, J.H. Chen,

Ultramicroscopy 104 (2005) 107.[32] D. Van Dyck, Advances in Imaging and Electron Physics 123 (2002) 105.[33] M.A.O. Miedema, A. van den Bos, A.H. Buist, IEEE Transactions on

Instrumentation and Measurement 43 (1994) 181.[34] D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 (1996) 99–107.[35] S. Van Aert, P. Geuens, D. Van Dyck, C. Kisielowski, J.R. Jinschek, Ultramicro-

scopy 107 (2007) 551.

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Ultramicroscopyematweb.cmi.ua.ac.be/emat/pdf/1534.pdf · the crystal. This overestimated ratio of the thickness of the amorphous layer to the crystal is considered as an extreme scenario