ARTICLE IN PRESS

0304-3991/$ - se

doi:10.1016/j.ul

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Ultramicroscopy 106 (2006) 525–538

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Projected potential profiles across interfaces obtained by reconstructingthe exit face wave function from through focal series

Somnath Bhattacharyya�, Christoph T. Koch, Manfred Ruhle

Max-Planck-Institut fur Metallforschung, HeisenbergStrasse 3, Stuttgart-70569, Germany

Received 23 September 2005; received in revised form 19 January 2006; accepted 26 January 2006

Abstract

An iterative method for reconstructing the exit face wave function from a through focal series of transmission electron microscopy

image line profiles across an interface is presented. Apart from high-resolution images recorded with small changes in defocus, this

method works also well for a large defocus range as used for Fresnel imaging. Using the phase-object approximation the projected

electrostatic as well as the absorptive potential profiles across an interface are determined from this exit face wave function. A new

experimental image alignment procedure was developed in order to align images with large relative defocus shift. The performance of this

procedure is shown to be superior to other image alignment procedures existing in the literature. The reconstruction method is applied to

both simulated and experimental images.

r 2006 Elsevier B.V. All rights reserved.

PACS: 61.16.B; 42.30.R; 61.14.N

Keywords: Transmission electron microscopy; Exit face wave function; Projected potential profile; Interfaces; Through focal series; Phase retrieval; Fresnel

imaging; In-line holography

1. Introduction

Interfaces play an important role in material science.Intergranular glassy films in polycrystalline ceramics,quantum well layers in semiconductors, precipitate matrixinterfaces, etc. control many important mechanical, phy-sical, chemical and electrical properties of the materialscontaining them. Charge defects are often accumulated atinterfaces between two phases in electrically active systems.Such charge defects and consequently accumulated spacecharge clouds at the interface lead to spatially varyingelectrostatic potentials across them [1]. For metal–semi-conductor interfaces, the potential profile gives rise to theformation of a Schottky barrier [2]. The formation of aparticular specially shaped potential profile across GaAsdevices is probably caused by charge trapping at theinterface between the epitaxial layer and the substrate [3].Potential profiles across intergranular glassy films in

e front matter r 2006 Elsevier B.V. All rights reserved.

tramic.2006.01.007

ng author. Tel.: +49 711 689 3685; fax: +49 711 689 3522.

ss: somnath@mf.mpg.de (S. Bhattacharyya).

polycrystalline ceramics may provide information regard-ing structural and compositional changes across that grainboundary [4]. In general, potential profiles across interfacescan provide useful information about their structure,composition, density, ionicity of their constituents, thepresence of space charge layers, etc., showing that theability to measure potential profiles across interfaces is ofgreat importance in the field of material science.Transmission electron microscopy (TEM) techniques,

based on phase contrast imaging, can determine localvariations in electrostatic potential. Electrostatic potentialvariation and accumulated space charge across an interfaceproduce a phase shift of the transmitted electron beamwave function. Apart from off-axis holography, in whichthe phase of the transmitted wave function is measured byreconstructing its interference with a reference wave [1],microscope aberrations, such as defocus and sphericalaberration may also be used to encode local variations inthe phase of the exit face wave function in the imageintensity. Stobbs and coworkers [5–7] developed anapproach to use Fresnel contrast (contrast in largely

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538526

defocused images of objects containing local phasedifferences) to analyze potential profiles across interfacesby fitting a few parameter model of the mean innerpotential profile to the fringe contrast. This approach wasreviewed and extended further by Dunin-Borkoski [8]. Forthis method line profiles across an interface from a series ofexperimental images with varying defocus are required.Using a least squares optimization routine, the interfacewidth, its potential magnitude and the diffuseness of itsboundaries are fitted to the experimental image intensityprofiles by comparing them with simulated ones at everydefocus for the entire experimental focal series. Thisprocedure thus requires a global non-linear optimizationproblem to be solved, which is the reason for the low limiton the number of parameters that can be used to describethe potential profile.

Using the phase-object approximation (POA), electronholographic methods can be used to map the electrostaticpotential of a given specimen and in particular, as will bethe focus of this work, across interfaces. Off-axis holo-graphy requires the microscope to be equipped with abiprism. In this work we will report on the use of in-lineholography, i.e. the reconstruction of the complex exit-facewave function from a Fresnel image series, to determinelocal variations in the mean inner potential of thescattering specimen.

In order to extract the contribution due to pure electriccharge from a map of the projected specimen potential, themean inner potential due to the local composition must beknown. Because of the monotonic correlation of high-angle(Rutherford) scattering and atomic number (Z), annulardark field scanning transmission electron microscopy(ADF-STEM) has good sensitivity to local composition.We will show that in addition to the phase of the scatteredelectrons, information about relative local compositionmay be extracted from the focal series of zero-loss energyfiltered Fresnel image series recorded with an objectiveaperture. This allows, in principle, space charge andcomposition contributions to the projected potential tobe separated.

To retrieve the complex electron wave function from afocal series of experimental images, numerical phase retrievalmethods must be applied. These methods reconstruct thewave function at the exit surface of the specimen. Variousapproaches exist in the literature to retrieve the exit-facewave function using images of varying defocus [9–15]. In oneof the approaches, the wave function is reconstructediteratively by projecting between different focal planes[12,13]. Vincent [14] modified the iterative approach,introducing conjugate gradient optimization for globalconvergence. This work is based on the iterative wavefunction reconstruction (IWFR) method by Allen et al. [15].

The IWFR algorithm was originally designed forreconstructing the exit face wave function using high-resolution images of a small defocus range. In this paperthis algorithm is modified to reconstruct the exit face wavefunction using images of a large defocus range produced by

a partially coherent electron beam in a medium resolutiontransmission electron microscope. Numerical instabilitiesin the IWFR algorithm due to the suppression of spatialfrequencies by the spatial coherence envelope at largedefoci, as well as the difficulty of aligning images ofstrongly different defocus made these modifications neces-sary. Because of the focus of this work on potential profilesacross interfaces we limited ourselves to the one-dimen-sional case by performing the reconstruction on integratedline scans from experimental images recorded withdifferent objective lens defocus.Finally, the projected potential profile across the inter-

face is derived from the reconstructed wave function usingthe POA:

cðrÞ ¼ expðistV ðrÞÞ, (1)

where the average projected potential V(r) (per unitthickness) may be complex, s is the electron interactionconstant and t the specimen thickness. From here on theaverage projected potential profile will be referred to‘potential profile’.The real part of the reconstructed potential profile

contains information regarding the phase shift, while itsimaginary part is produced by inelastic and high-angle(scattering angle greater than the objective aperture)scattering events and provides therefore mass-thickness,or, to some extend, chemical, information. We will showthat, when the information regarding the phase shift acrossan interface is scaled with respect to a reference phase (e.g.near the specimen edge, where vacuum may be used forreference), absolute mean inner potential values may beretrieved by this method.The reconstruction method is tested for both simulated

as well as experimental examples. To test it for experi-mental data, a specimen edge of a polycrystalline Si3N4

specimen, i.e. a vacuum–Si3N4 interface is used. Point-to-point resolution of the reconstruction method, in theframework of the present investigation, is also derivedexperimentally using a specimen containing a nominallyabrupt Al–Al2O3 interface. Using simulated intensityprofiles instead of experimental ones as input, the accuracyof the new image alignment and reconstruction procedureis demonstrated. A comparison of this new alignmentprocedure with other image alignment methods based ondifferent correlation functions is also presented.The limitations of the method, half of which are generally

true for any holographic method, are listed below:

�

While the use of an objective aperture allows the high-angle scattering to be separated and the chemicalinformation to be retrieved, it also limits the resolution.However, the use of the POA for retrieving the projectedpotential is most valid for low spatial frequencyinformation anyway. � The electrostatic potential may only be retrieved across

distances of the order of the coherence length of theincident electron beam.

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 527

�

A large defocus range is necessary for retrieving lowspatial frequency components of the electrostaticpotential. � Dynamic scattering effects are taken into account by the

POA only in the zero wavelength and zero excitationerror limit.

2. Reconstruction procedure

2.1. The phase retrieval algorithm

As already mentioned, the iterative exit face wavefunction reconstruction algorithm, used in the presentinvestigation, is a modified version of the work presentedby Allen et al. [15]. In addition to the new image driftcorrection method, the modifications also include the useof weighted averaging of the wave functions instead of anunweighted one.

The reciprocal space wave function at focal plane n maybe obtained from the exit face wave function by multi-plying it with the exponential of the phase distortionfunction exp½iwðk;Df nÞ� [16] (k is the reciprocal spacecoordinate and Df n the defocus of image n) as well as theenvelope functions related to temporal coherence ðEDðkÞÞ

and spatial coherence ðEsðk;Df nÞÞ. Considering onlysymmetric lens aberrations up to third order, the above-mentioned functions are given by

wðk;Df nÞ ¼ plDf nk2þ

1

2pl3Csk

4, (2)

EDðkÞ ¼ exp �1

4p2D2

f l2k4

� �, (3)

Esðk;Df nÞ

¼ exp �p2b2 k2Df 2n þ 2Df nl

2Csk4þ l4C2

s k6� �� �

. ð4Þ

Here l is the electron wavelength, Df is the defocus spreadand b represents the beam convergence semi-angle. Whilepropagating from image plane n back to the Df ¼ 0 planewe must divide the wave function by these terms inreciprocal space again (or, in real space, deconvolute bytheir Fourier transforms). The spatial coherence envelopeEsðk;Df nÞ vanishes for large ranges of k at large defoci (forthe present study the selected defocus range is;Df ¼ �4mm . . . þ 4mm), making division by this termnumerically unstable. The IWFR algorithm only requiresthe computation of an average exit face wave function andnot the individual back-propagated wave functions ob-tained from each focal plane. Numerical stability maytherefore be gained by computing a weighted average of thewave functions, where more weight is given to those wavefunctions whose envelope function is large at a givenspatial frequency k:

cjavgðkÞ ¼

PNn¼1c

jn;0ðkÞwnðk;Df nÞPN

n¼1wnðk;Df nÞ(5)

with wnðk;Df nÞ ¼ Esðk;Df nÞEDðkÞ. The original IWFRalgorithm uses wn k;Df n

� �� 1.

An outline of the modified algorithm is presented inFig. 1 and described in detail below:

1.

From a through focal series comprising N experimentalimages line profiles across the interface in each focalplane n (denoted by InðrÞ) are extracted. From priorwork in literature [17], NX3 should suffice for obtaininga unique result.

2.

The complex electron wave function at focal plane n isgiven by

CjnðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiInðrÞ

peif

jnðrÞ, (6)

where j denotes the number of the iteration step, r

denotes the real space coordinate and fjnðrÞ is the phase

of the wave function from the previous iteration. Thealgorithm starts with an initial guess of f1

nðrÞ ¼ 0, i.e.c1

nðrÞ ¼ffiffiffiffiffiffiffiffiffiffiInðrÞ

p.

3.

The wave function in each focal plane is propagated tothe nominal exit surface plane ðDf ¼ 0Þ. This is done bydividing the Fourier transform Cj

nðkÞ of CjnðrÞ by

exp½iwðk;Df nÞ�Esðk;Df nÞEDðkÞ and multiplying it by theweighting factor wnðk;Df nÞ ¼ Esðk;Df nÞEDðkÞ, so that

cjn;0;wðkÞ ¼ cj

nðkÞ exp½�iwðk;Df nÞ�, (7)

where, cjn;0;wðkÞ is the wave function at Df ¼ 0

propagated from plane n weighted by the correspondingweighting function wnðk;Df nÞ. So this can be expressedas cj

n;0;wðkÞ ¼ cjn;0ðkÞwnðk;Df nÞ.

4.

A new estimate of the exit face wave function is obtainedby averaging over the individual back-propagatedweighted wave functions cj

n;0;wðkÞ and renormalizing:

cjavgðkÞ ¼

XN

n¼1cj

n;0;wðkÞ.XN

n¼1wnðk;Df nÞ. (8)

5.

The use of an objective aperture of radius kaperture limitsthe spatial resolution of the reconstruction to spatialfrequencies within a circle defined by kaperture. Thedivision of cj

avgðkÞ byPN

n¼1wnðk;Df nÞ in the previousstep amplifies spatial frequencies outside this range. Theapplication of a sharp numerical aperture, to limit thespatial frequency content of cj

avgðkÞ, may introduceartifacts. We therefore decided to use a smooth aperturefunction exp½�ðAjkjÞq�, which we multiply by the currentestimate of the wave function cj

avgðkÞ. This suppresseshigh frequency noise at kXkaperture. The value of A ischosen to match the amplitude of the sum of theweighting functions at k ¼ kaperture. Fromexp½�ðAjkjÞq� ¼

PNn¼1wnðk;Df nÞ we obtain

A ¼1

kaperture

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ln

1

N

XN

n¼1

wnðkaperture;Df nÞ

!" #q

vuut . (9)

Fig. 2 shows plots of exp½�ðAjkjÞq� for several valuesof q. The value of q should be chosen according to the

ARTICLE IN PRESS

Experimental profiles Starting phase=0

Smoothening of selected region of profiles Equation (15 & 16)

Determination of �2 at each focal plane Equation (17)

Aligning experimental profiles

Wave Function constructionEquation (6)

Fourier transformation (reciprocal space) and propagation Equation(7)

Propagation to each focal plane Equation (10)

No

Yes

Inverse Fourier transformation (real space) To obtain reconstructed phase

Equation (11)

After specified

number of steps

Calculation of SSE at each plane Equation (12)

Averaging of SSE Equation (13)

If j>1

Exit face wave function Equation (14)

Reconstructed phase

Ye

No

jj-1SSEavg - SSEavg < �

V'(r) Vabs(r)Equation (24) V''(r)

V´´´(r)

Exit face w

ave function reconstructionR

etrieval of the

projected potential

No

Alignment of experimental images

Yes

No

Phase Equation (21) using POA

V (r) Equation (23)

Averaging Equation (8)

Fig. 1. Outline of the iterative exit face wave function reconstruction algorithm (shaded boxes indicate the modification to the IWFR algorithm) combined

with the outline of the projected potential retrieval from a complex exit face wave function.

S. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538528

noise level in the experimental data. Low values of q

might be used for noisy data, while larger values may beused for less noisy data.

6.

The averaged wave function in reciprocal space ispropagated to each of the n image planes to obtain~c

j

nðkÞ. The estimated wave functions are now given by

~cj

nðkÞ ¼ cjavgðkÞ exp½iwðk;Df nÞ�Esðk;Df nÞEDðkÞ. (10)

7.

The real space wave function is obtained by inverseFourier transformation:

cjnðrÞ ¼ FT�1ð ~c

j

nðkÞÞ. (11)

The phase of cjnðrÞ is now used as the input phase for

the next iteration beginning at step 2.

8. As in the original algorithm by Allen et al. convergence

is reached when SSEj�1avg � SSEj

avgox for a chosen

convergence criterion x, where

SSEjn ¼

P ffiffiffiffiffiffiffiffiffiffiInðrÞ

p� cj

nðrÞ�� ��� �2P

InðrÞ(12)

and

SSEiavg ¼

1

N

XN

n¼1

SSEjn. (13)

The real space exit face wave function is now obtainedby inverse Fourier transformation:

cðrÞ ¼ FT�1 cfinal stepavg ðkÞ

h i. (14)

The reconstruction starts with the normalized experi-mental image intensity profiles across the same area along

the interface. While the position along the interface at

ARTICLE IN PRESS

Reciprocal space distance (Å-1)

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

N

N

n=1Σ wi (k)e

- (A1k)4

e- (A2k)6

e- (A3k)8

e- (A4k)10 e

- (A5k)16

kaperture

Fig. 2. Plots of the smoothening functions used in the averaging step (5) of the exit face wave function in reciprocal space. Values of A1, A2, A3, A4 and A5

calculated using Eq. (6) are 10.55, 9.62, 9.19, 8.94 and 8.58, respectively.

S. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 529

which to extract the line scan is not very critical, itsposition normal to it is crucial for the quality of thereconstruction. However, the changes in the (Fresnel-)images with defocus make this alignment normal to theinterface very difficult, requiring a new image alignmentprocedure. After starting the reconstruction with thepossibly misaligned images and completing a specifiednumber of iteration steps (e.g. 1000), we have a first guessof what the exit face wave function may look like. Now aregion around the interface of the experimental profiles andreconstructed profiles is selected where maximum changesin image intensity with defocus occur (e.g. up to 50 A oneither side of the interface). Then the selected portions ofall the experimental and simulated profiles are smoothenedby convoluting them with a Gaussian function, obtaining

AnðJÞ ¼ FT�1 FTffiffiffiffiffiffiffiffiffiInðJ

pÞ

n oexpð�Bk2

Þ

h i(15)

and

BnðJÞ ¼ FT�1 FT cjnðJÞ

�� �� expð�Bk2

Þ� �

. (16)

Here J denotes the selected region and B is a constant.Instead of aligning images with different defocus, each ofthe experimental (An) line profiles is now aligned with thecorresponding simulated one (Bn) by minimizing

w2n ¼XjBnðJÞ � AnðJ þ pxÞj2, (17)

where px indicates the number of pixels that AnðJÞ isshifted with respect to the corresponding BnðJÞ. After

aligning all the experimental profiles, the aligned profilesare taken as input to step 1 of the reconstruction algorithm.

2.2. Retrieval of the projected potential

Retrieval of the projected potential from the exit facewave function is outlined in the bottom part of Fig. 1 anddescribed in more detail here.According to the POA the exit face wave function

produced by the scattering potential of the specimen can bedescribed as

cðrÞ ¼ eif, (18)

where

f ¼ sV ðrÞt. (19)

Here, t is the specimen thickness parallel to the beamdirection, V ðrÞ is the (complex) projected scatteringpotential, and s is the relativistic electron interactionconstant given by [18]

s ¼2pl

E0 þ E

2E0 þ Ejej, (20)

where E0 is the electron’s rest energy, E is electron’s kineticenergy and e is the charge of an electron.For the reconstructed in-focus exit-face wave function,

regions of vacuum within the image have the largest exitface wave function amplitude and may be used for

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538530

normalization, so that f may be obtained by

f ¼ �i lncðrÞ

max jcðrÞj

� �. (21)

If the imaged region contains no vacuum, this normal-ization makes at least sure that we will have positiveabsorption everywhere, which is physically sensible. Theinformation regarding the phase shift and the composi-tional changes across the interface may then only beinterpreted in relation to a given location within the imagedregion.

The specimen thickness along the direction of theelectron beam can be determined using energy-filteredTEM (EFTEM). Only two images, a zero-loss filtered oneðI0Þ and an unfiltered one ðI lÞ, both recorded with the sameexposure time, are needed for this purpose. Extracting lineprofiles from the same area of the images as that used in thefocal series reconstruction, the local specimen thickness, t,is calculated according to [19]

t ¼ linel lnI l

I0

� �, (22)

where linel is the inelastic mean free path (IMFP) of theelectrons in the given material. It can be calculated asdescribed by Egerton [19].

The projected potential V ðrÞ, is now given by

V ðrÞ ¼fst

(23)

and may be split up into [20]

V ðrÞ ¼ V 0ðrÞ þ i½V 00ðrÞ þ V 000ðrÞ�, (24)

where potential V0(r) is due to the elastic interaction of theelectrons with the specimen refereed to as the ‘electrostaticpotential’, V00(r) describes the loss of electrons due toinelastic scattering events involving electronic excitationsand V 000ðrÞ describes the elastic (and quasi-elastic thermaldiffuse scattering (TDS)) scattering outside the objectiveaperture which can be described as an objective aperturedependent ‘pseudo-absorptive’ scattering potential. Thesum of both the terms V00(r) and V 000ðrÞ will from now on bereferred to as ‘absorptive potential’:

VabsðrÞ ¼ V 00ðrÞ þ V 000ðrÞ. (25)

In addition to enhancing Fresnel contrast, the objectiveaperture thus allows to extract high-angle scatteringinformation from the experimental data.

3. Application to simulated images

At first the reliability of the new alignment procedurewas tested using 14 simulated images (at defocus values of1–4 mm with a step size of 0.5 mm in both over and underfocus directions) which had intentionally been misaligned.From the model potential profile shown in Fig. 3a, the exitface wave function was calculated using Expression (1). Aseries of defocused images was calculated by forward

propagation to different focal planes according to

cnðrÞ ¼ FT�1fFTðcðrÞÞ

� exp½iwðk;Df nÞ�Esðk;Df nÞEDðkÞf apertureg, ð26Þ

where f aperture is the objective aperture function in reciprocalspace. All other terms are the same as in Section 2.1. Allparameters used in the simulation are listed in Table 1.Poisson noise was added to the image intensities using

I nðrÞ ¼ InðrÞ þ1ffiffiffiffiffiffiffiffiffiffiffiffi

Ne�

pixel

q ffiffiffiffiffiffiffiffiffiffiInðrÞ

pGðrÞ, (27)

where I nðrÞ is the intensity including noise, Ne�

pixel is themean number of electrons per pixel averaged over therelevant specimen area and G(r) are normally distributedrandom numbers with the mean being zero, variance oneand standard deviation one.In order to realistically simulate images recorded on

CCD, the point spread function of the slow scan CCDcamera of the Zeiss 912 [21] was convoluted with I nðrÞ.Four simulated profiles at different defocus values withNe�

pixel ¼ 1000 and without focus/image shift are shown inFig. 3b.At first the simulated profiles of each focal plane were

shifted by different but fixed numbers of pixels from theiroriginal positions. In addition to the image shift, we alsoadded uncertainty in defocus. This was done in order toreflect inaccurate measurements of objective lens current(random focus shifts for each defocus) as well as inaccurateestimates of the Df ¼ 0 focal plane (a constant focus offsetin all focal planes). The reliability of the focus/image shiftreconstruction algorithm was quantified using the follow-ing figure of merit:

SSE2 ¼

PNn¼1Dx2

N, (28)

where Dx is the difference between the initial imposed shiftand the final shift after alignment in pixels for each of thesimulated intensity profiles, and N stands for the number ofprofiles. Table 2 shows how Ne�

pixel and the focus shiftaffects the profile alignment procedure. The alignment wasperformed twice for every case to achieve maximumaccuracy. Table 2 also illustrates the rigidity of the presentalignment method. The choice of smoothening function (instep 5 of the reconstruction algorithm) does not have anyeffect on the alignment procedure. Image noise also doesnot influence the alignment procedure (as seen from Table2), because SSE2 ¼ 0.0714 corresponds to 1 pixel misalign-ment of only 1 image within 14 images.For comparison, three different correlation functions,

described by Meyer et al. [22], were tested for the samemisaligned simulated intensity profiles, giving very poorresults. The contrast reversal of images at overfocus andunderfocus is the main reason for the failure of these imagealignment methods based on different correlation functionsin the framework of the current study. Table 3 compares thealignment method of the current study with cross-mutual

ARTICLE IN PRESS

Real space distance (A)

-100 -50 0 50 100 150-7

-6

-5

-4

-3

-2

-1

0

2

3

V´(r)

Vabs (r)

-150

1

-100-0.5

0

0.5

1

1.5

2

2.5

3

-5

-4

-3

-2

-1

0

1

2

3

-100-7

-6

-5

-4

-3

-2

-1

0

1

2

Ele

ctro

stat

ic p

oten

tial (

V)

A1

A3

A2

A4A5

-80 -60 -40 -20 0 20 40 60 80Distance

Distance (A) Distance (A)

Distance (A)

Inte

nsity

4µm

2µm

-2µm

-4µm

-80 -60 -40 -20 0 20 40 60 80 100 -80 -60 -40 -20 0 20 40 60 80

-80 -60 -40 -20 0 20 40 60 80 100(b) (c)

Abs

orpt

ive

pote

ntia

l (V

)E

lect

rost

atic

pot

entia

l (V

)

(e)(d)

Pot

entia

l (V

)

∆Vab

s (r

)∆V

′ (r)

A5

A2A1

A3A4

(a)

˚(A)

Fig. 3. Testing the algorithm with simulations (a) assumed electrostatic and absorptive potential profile, (b) four simulated image line profiles with

Ne�

pix ¼ 1000 at different defoci (corresponding defocus values are indicated in the figure), (c) reconstructed absorptive and (d) electrostatic potentials using

the five smoothening functions (A1, A2, A3, A4, A5 represent the smoothening functions e�ð10:5452A:KÞ4, e�ð9:62A:KÞ

6, e�ð9:19A:KÞ

8, e�ð8:94A:KÞ

10and e�ð8:58A:KÞ

16

respectively) presented in Fig. 2. Profiles are shifted intentionally with respect to one another to show the effect of different smoothening functions.

(e) Electrostatic potential after correcting for wrongly reconstructed low spatial frequency components (with smoothening function e�ð10:5452 KÞ4). Here the

distance chosen for fitting the low-k components were from �107.1 to �20.5 A at left side and from 21.4 to 96.8 A to the right of the interface.

S. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 531

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538532

and phase correlation functions only for the overfocusintensity profiles. Here also the correlation function-basedimage alignment methods do not work properly. Instead ofusing high-resolution images of relatively small defocusrange (as described in previous works [15,22]), images acrossa very large defocus range were used in the presentinvestigation, explaining the result shown in Table 3.

Fig. 3c and d show the reconstructed absorptive andelectrostatic potentials respectively for the image profilesshown in Fig. 3b, using all the smoothing functionspresented in Fig. 2. For reconstructing the exit face wavefunction from a through focal series of simulated images,SSEavg of the last iteration step was chosen to be of thesame order of magnitude in the reconstruction of theexperimental data in the present study. The functionexp(�(10.5452A.k)4) produces minimum high frequencynoise for both electrostatic and the absorptive potential.Fig. 3c also shows that the algorithm can reconstructVabsðrÞ quite accurately, as the reconstructed absorptivepotential has the same difference in the V absðrÞ valueðDV absðrÞ ¼ 1:95VÞ between the two sides of the interfaceas the input absorptive potential.

Table 1

Parameters used in simulation

Parameter Value

Cell length 150 pixel

Sampling density 1.875 A

Specimen thickness 30 nm

Accelerating voltage (V0) 120 kV

Wavelength (l) 0.0335 A

Spherical aberration (Cs) 2.7mm

Chromatic aberration (Cc) 2.7mm

Beam convergence semi-angle (b) 0.25mrad

Root mean square value of high voltage

fluctuation ðffiffiffiffiffiffiffiffiffis2Vp

Þ

1.5V

Objective aperture size Radius ¼ 0.1248 A�1

Interaction constant (s) 8.6361� 103V�1 nm�1

Table 2

Effect of mean number of electrons per pixel integrated over sample thickness a

Focus shift SSE2

Using Ne�

pix ¼ 2000 Using Ne�

pix ¼ 1000

pfs nfs fu pfs nfs Fu

500 0 0 0 0 0 0

1000 0 0 0 0 0 0

2000 0 0.0714 0 0 0.0714 0

3000 0 0.0714 0 0 0.0714 0

4000 0 0.0714 0 0 0.0714 0

5000 0 0.0714 0 0 0.0714 0

6000 0.0714 0.0714 0 0.0714 0.0714 0.0

Columns pfs, nfs and fu represent the SS2 values related to the focus shift in p

focus shifts (focus uncertainty) respectively. For example, SSE2 for a syste

�ðnfsÞ2000 it is 0.0714, while random focus shifts (fu) with a maximum amplitu

Focus offsets in the focus uncertainty (fu)-columns are determined by the fo

numbers in the interval [�0.5y 0.5].

3.1. Retrieval of very low spatial frequency components of

V0(r)

The shape of the reconstructed electrostatic potential(Fig. 3d) suggests that while the potential drop has beenreconstructed rather faithfully, its low spatial frequencycomponents are wrong. This is due to the fact that themicroscope transfer function, MTF(k) (Expression (2) forsmall k), in relation to the noise present in the data, doesnot change very much between images for low values of k,despite the very large range of defocus values used here.The DC component (k ¼ 0), for example, cannot bereconstructed at all, since the absolute phase is not encodedin the images. In addition, the experimental data presentedin this study suffers from the fact that the partiallycoherent illumination used in our experiments forbidsany coherent interaction between points on the specimenthat are more than about 100 A apart.A plot of

PðkÞ ¼ ln jjFTðinput V 0ðrÞÞj � jFTðreconstructed V 0ðrÞÞjj

(29)

serves to identify the range of k-values that have beenreconstructed wrongly. The noise introduced in the

Table 3

Comparison of intensity profile alignment procedure of current study with

cross, mutual and phase correlation method for a series of overfocus

images ranging from 1 to 4mm with a step size of 0.5 mm using Ne�

pix ¼ 1000

and without focus shift

Image alignment procedureSSE2 ¼

PNn¼1Dx2

N

This work 0

Using cross correlation 846.67

Using mutual correlation 499.17

Using phase correlation 45.17

nd focus shift on intensity profiles alignment procedure of the current study

Using Ne�

pix ¼ 600 Using Ne�

pix ¼ 300

Pfs nfs fu pfs nfs fu

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0.0714 0 0 0 0

0 0.0714 0 0 0 0

0 0.0714 0 0 0 0

714 0.0714 0.0714 0.0714 0.0714 0 0

ositive (overfocus) direction, negative (underfocus) direction and random

matic focus offset of+(psf) 2000 at Nepix ¼ 2000 is 0, but for df offset ¼

de 1000 for each image produce again perfectly aligned images (SSE2 ¼ 0).

cus shift value in column 1 multiplied by uniformly distributed random

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 533

simulation makes a perfect reconstruction impossible, sothat PðkÞ40 for most values of k. However, a few lowspatial frequency components in P(k) are significantlyabove average and have therefore not been sufficiently wellencoded in the image series, i.e. they are free parameterswhich must be fixed using additional information about thespecimen. For the present investigation, the number of freeparameters obtained in this way did not exceed 4 (i.e. for atotal length of line profiles of 218 A, k � 218 A ¼ �2,�1, 1, 2)and turned out to be independent of the value of Ne�

pixel.For samples investigated in the present study it is safe to

assume that the grains on both sides of the interface arehomogeneous. This means that in regions where thecorrectly reconstructed absorptive potential profile Vabs(r)is constant, V 0ðrÞ must also be constant, i.e.

qV 0ðrÞ

qrr2R

¼qV absðrÞ

qrr2R

. (30)

Here R defines the range (ideally, one region in each grain)of real space coordinate r over which Vabs(r)Econst. Usingthe derivative in expression (30) has the advantage thatknowledge of the proportionality factor (which depends onmaterial properties as well as the local diffraction condi-tions) between V 0ðrÞ and Vabs(r) is not required. Acomputer program equipped with a graphical user interface(GUI) for very fast definition of the range R by the user,has been written to determine the low spatial frequencycomponents of V 0ðrÞ automatically by first performing anexhaustive search and then a least squares refinement. Thereconstructed electrostatic potential profile (using a smoothaperture function of exp½�ð10:5452kÞ4�) corrected in theway described above is shown in Fig. 3e. Apart from someresidual noise stemming from the Poisson noise added tothe simulated images, it agrees very well with the input. Acomparison of Figs. 3b and e shows that feeding thealgorithm with 14 images was able to significantly reducethe level of residual noise in the reconstruction. Thedifference between both sides of the interface in the finalreconstructed electrostatic potential is the same as in theinput (DV 0ðrÞ ¼ 6:65V).

4. Experimental details

The experimental images were obtained using a Zeiss 912microscope (Cs ¼ 2.7mm, Cc ¼ 2.7mm, Kohler illumina-tion, in-column Omega-type energy filter), operated at anaccelerating voltage of 120 kV. The experiments were doneusing an electron beam of 0.25mrad illumination semi-angle and an objective aperture of size 0.1248 A�1. All focalseries were recorded with the energy filter selecting zeroenergy loss, i.e. a 15 eV energy slit centered on the zero losspeak was used.

One experiment was done using a plan view polycrystal-line Si3N4 sample (La2O3 and MgO doped) to obtain athrough focal series of images from a vacuum–specimen(Si3N4) interface. The specimen (for TEM) was preparedby the standard techniques of grinding, dimpling and ion-

beam thinning (Precision Ion Polishing System, GatanInc.). In another experiment, a through focal series ofimages was captured from an Al–Al2O3 interface of across-sectional sample where on the Al2O3 substrate, layersof Al and Cu, each of 400 nm thickness, had been deposited(see [23] for further details about the Al–Al2O3 interface).This TEM specimen was provided by Dr. Gunther Richterand Ms. Limei Cha (Max-Planck-Institut fur Metall-forschung, Stuttgart, Germany). Both of the specimenswere coated with a thin layer of carbon to minimizecharging under the electron beam.Care was taken to keep the interfaces of interest parallel

to the incident electron beam. All the experimental imageswere captured onto a 1024� 1024 pixel CCD array using5 s exposure time at a magnification producing an imagescale of about 0.19 nm per pixel. The in-focus conditionwas calibrated as accurately as possible by minimizingFresnel contrast. The through focal series of images wastaken for a defocus range of 1–4 mm with a step size of0.5 mm in both over and under focus directions. Digitalmi-crographs software (Gatan Inc., Pleasanton, CA, USA)was used to extract line profiles across the same area of theinterface from each of the images (line scans wereintegrated over a width of 100 pixels). The focal seriesreconstruction and image alignment algorithm was pro-grammed in the Matlab language using MATLAB 7.0s

software.

5. Application to experimental images

To study the performance of the reconstruction methodfor experimental data, it was applied to a vacuum–Al2O3

interface and an Al–Al2O3 interface. The microscopic andexperimental parameters, which were used for reconstruc-tion, are listed in Table 1. Led by our results fromsimulations, we decided to use a smooth aperture functionwith q ¼ 4, i.e. exp[�(10.5452A.k)4] in step 5 of thereconstruction algorithm, since it introduces the leastamount of high frequency noise in the reconstructedpotential profiles. This aperture function is used for bothof the experiments.

5.1. Vacuum–Si3N4 interface

An image of the vacuum/Si3N4 interface at an overfocusof 1.1 mm is shown in the inset of Fig. 4a. The box showsthe area from where the integrated line scan was taken.Some of the aligned line profiles (of different defocus)across the interface are shown in this figure. As ademonstration of convergence of the algorithm, Fig. 4b

shows a plot of ln SSEjavg

�vs. the number of iterations j.

The specimen thickness starting from the edge to thespecimen along the electron beam is presented in Fig. 4c.To make the thickness variation smooth, the thicknessprofile was replaced by a fitted 2nd order polynomial asalso shown in Fig. 4c. Reconstructed electrostatic (after

ARTICLE IN PRESS

Distance (A)

-80 -60 -40 -20 0 20 40 60 80

Vacuum

Inte

nsity

20 nm20 nm

-4µm

4µm

-2µm

2µm20 nm20 nm20 nm20 nm

Si3N4

20 nm20 nm

-60 -40 -20 0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance (A)

Abs

orpt

ive

pote

ntia

l (V

)

Vacuum Specimen

-80 -60 -40 -20 0 20 40 60 80-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Distance (A)

Ele

ctro

stat

ic p

oten

tial a

fter

adju

stin

g lo

w s

peci

al fr

eque

ncie

s (V

)

VacuumSpecimen

A B

0 10 20 30 40 50 608

10

12

14

16

18

20

Distance from the edge of the specimen (A)

Spe

cim

en th

ickn

ess

alon

g th

e el

ectr

on b

eam

(nm

)

Specimen

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

10-5

10-4

Number of iterations

(a)

In (

SS

Ej av

g)

(b) (c)

(d) (e)

Fig. 4. (a) An image of vacuum–Si3N4 interface at an overfocus value of 1.1mm (inset) with four experimental 1D intensity profiles across this interface

(corresponding defocus values are indicated in the figure), (b) change of averaged sum squared value [ln(SSEavg)] with number of iterations, (c) thickness

change along the electron beam from the edge to the interior of the specimen (d) reconstructed electrostatic potential (after correcting the wrongly

reconstructed low-k components) and (e) reconstructed absorptive potential.

S. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538534

fitting the low-k components to the vacuum region) andabsorptive potentials are shown in Fig. 4d and erespectively. Numerical instabilities due to the vanishingspecimen thickness in the vacuum region, forced us to setV 0ðrÞ ¼ V absðrÞ ¼ 0 for ro0.

In Fig. 4d, Si3N4 shows a more negative electrostaticpotential than vacuum. This is because the electrons, beingnegatively charged particles, are decelerated by the repul-

sion of the electron clouds of the atoms of the materialwhile traveling through the specimen. Hence the specimenproduces a negative phase shift and therefore showsnegative electrostatic potential with respect to the vacuum.The absorptive potential is produced by the sum of the

electron scattering outside the objective aperture used inthese experiments and the inelastic scattering outside theenergy slit of 15 eV width centered at the zero loss peak.

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 535

The loss of electrons in both, the diffraction plane (by theobjective aperture) and the energy-dispersive plane (by theenergy slit) produces a positive absorptive potential insidethe sample relative to vacuum (Fig. 4e).

Just as in off-axis electron holography [1], the estimatesfor the electrostatic potential are absolute, since they aremade in reference to vacuum, where we know thatV 0ðrÞ ¼ 0.

All experiments were done using partially coherentelectron beam illumination. The coherence width ðX cÞ

[24] is the distance at the object over which the illuminatingradiation may be treated as perfectly coherent. Thecoherence width can be calculated as X c ¼ l=2pyc, wherel is the electron wavelength, and yc is the illuminationsemi-angle (the beam divergence). The illumination semi-angle was kept at 0.25mrad for the experiments presentedhere, resulting in a width of 21.3 A over which the electronscan be treated as being perfectly coherent. For a partiallycoherence beam, another criterion is the incoherence widthðX iÞ [24], which defines the distance at the object abovewhich the illumination is completely incoherent. Theillumination at the object over distances between X c andX i is partially coherent. X i is defined as X i ¼ l=yc, which,for the present investigation is 134 A. So it can be statedsafely that in the present experimental condition, for boththe vacuum–Si3N4 and the Al–Al2O3 interface the recon-struction of V 0ðrÞ is valid beyond distances of 21.3 A, butnot in excess of 134 A.

Fig. 4d shows that the change of V 0ðrÞ from vacuum toSi3N4 is not abrupt but gradual. This could be due to theformation of an amorphous layer just near the holeproduced during ion milling (the final TEM specimenpreparation step). A less dense amorphous materialconsisting of the same atomic species is expected to havelower amplitude of electrostatic potential than its crystal-line counter part Si3N4. The intentionally depositedCarbon layer on the specimen surface may also contributeto the lower mean inner potential at the specimen edge,since its relative contribution is greater at lower specimenthicknesses.

The average electrostatic potential of Si3N4 as deter-mined from the region AB of Fig. 4d is �16.7V. Thisexperimentally determined mean inner potential of Si3N4 isin good agreement with the calculated value (�17.4V [4]).The discrepancy between the calculated and reconstructedmean inner potential, depends on a good estimate of theIMFP of Si3N4 (essential for accurate specimen thicknessdetermination) as well as the scattering factors (for neutralor charged atoms) used for the evaluation of Eq. (31).Beyond point B (Fig. 4d) V 0ðrÞ again increases due to thelimited spatial coherence in the microscope used for thisinvestigation.

5.2. Al–Al2O3 interface

Fig. 5a shows an image of the Al–Al2O3 interface at anoverfocus of 2 mm containing two regions of opposite

contrast conditions, caused mainly by local bending.The dotted boxes show the regions (1 and 2) from wherethe integrated line scans were taken. Fig. 5b compares thereconstructed absorptive as well as the electrostaticpotentials (after correcting for wrongly reconstructedlow-k components) from both of the regions shown inFig. 5a. Although, because of the different diffractingconditions regions 1 and 2 have very different absorptivepotential profiles, the electrostatic potential profiles looksimilar and have the same difference at the interfacebetween Al and Al2O3. This demonstrates that the localdiffraction condition does not have any influence on thereconstructed electrostatic potential.The mean inner potential of an assembly of neutral

atoms is calculated using the equation,

V0 ¼�h0

2pm0jejO

XNatoms

j¼1

fðjÞel ðs ¼ 0Þ, (31)

where, O is the unit cell volume, m0 is the rest mass of theelectron, e is the charge of an electron, h0 is Planck’sconstant and

PNatomsj¼1 f

ðjÞel ðs ¼ 0Þ is the electron scattering

factor of an assembly of neutral atoms [25] at zeroscattering angle. Using the information that a-Al2O3 has10 atoms/unit cell [26] and O for Al2O3 is 84.929 A

3 [27], V 0

for Al2O3 becomes �19.95V. For Al (FCC unit cell, latticeparameter 4.0496 A) we obtain V0 ¼ �16:97V.Since an objective aperture is used in the present study,

the expected point-to-point resolution of the reconstructedpotential profiles from the size of the objective aperture islimited to 8 A. The Al–Al2O3 interface, which wasinvestigated in the present study, has been reported to beabrupt [23]. This interface may therefore be used todetermine the resolution of the reconstruction methodexperimentally. Using schematic diagrams, Fig. 5c demon-strates a way to determine the point-to-point resolution ofthe reconstructed potential profiles. An abrupt potentialdrop (A) at the interface becomes a gradual drop (B) due tolimited point-to-point resolution. In order to estimate thewidth of this potential drop, the upper portion of the curveB should be flipped about a horizontal mirror axis midwaybetween the potential of the left and right side of theinterface. The full-width at half-maximum (FWHM) of theresulting peak is expressed as the point-to-point resolutionwith which the sharp interface can be resolved.The resolution must be determined from the recon-

structed electrostatic potential since the reconstructedabsorptive potential depends mainly on local diffractingconditions which do not change as abruptly as the materialcomposition. Using the reconstructed electrostatic poten-tial of region 1, the point-to-point resolution determined isabout 8 A as shown in Fig. 5d. The experimentallydetermined resolution agrees with that expected from thesize of the objective aperture.The electrostatic potential obtained from both the

regions (as shown in Fig. 5b) show that the differenceDV 0ðrÞ at the interface between Al and Al2O3 stays close to

ARTICLE IN PRESS

20 nm20 nm20 nm20 nm

Region 2

Region 1

-1.5

-1

-0.5

0

0.5

1

2

Al2O3

Al

-60 -40 -20 0 20 40 60 80

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Distance (A) Distance (A)

Distance (A)Distance (A)

Al

Al2O3

Al

Al2O3

Al2O3

Al

Abs

orpt

ive

pote

ntia

l (V

)

Al

-100

1.5Al

-800

0.2

0.4

0.6

0.8

1

1.2

1.4

Ele

ctro

stat

ic p

oten

tial (

V)

Abs

orpt

ive

pote

ntia

l (V

)E

lect

rost

atic

pot

entia

l (V

)

Region 2Region 1

-80 -60 -40 -20 0 20 40 60 80 100-1.5

-1

-0.5

0

0.5

1

2

-100

1.5

-80 -60 -40 -20 0 20 40 60 80 100

-100 -80 -60 -40 -20 0 20 40 60 80 100

-80 -60 -40 -20 0 20 40 60 80 100

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

A-100

FWHM = 8 A

Mirror plane

A

B

V

(c)

(d)

(b)

(a)

Fig. 5. (a) An image of Al–Al2O3 interface at an overfocus value of 2mm, showing contrast variations due to local bending, (b) comparison of the

reconstructed absorptive as well as the electrostatic potentials (after correcting for wrongly reconstructed low-k components) from regions 1 and 2, (c)

schematic to demonstrate the way to determine the point-to-point resolution of the reconstructed profiles and (d) determination of point-to-point

resolution from the reconstructed electrostatic potential of region 1.

S. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538536

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538 537

the calculated value of DV0 (Alumina-Al)E3V. Errorsmay be present in both, the experimental, as well as thecalculated value of V0. Our computation of V0, forexample, assumed neutral atoms, and may for this reasonalone be incorrect.

6. General discussion

The accuracy of the retrieved potential profile, andtherefore an additional source of the discrepancy betweenthe calculated and reconstructed mean inner potential,depends on a good estimate of the IMFP of the materialspresent in the sample (essential for accurate specimenthickness determination).

One should also bear in mind that dynamic scatteringeffects are not fully taken into account by the POA and anew, more accurate method of retrieving the projectedpotential of a crystalline sample by dynamic inversion froman exit face wave function recently developed [28] may beapplied. However, Fig. 5b shows very clearly that while thereconstructed absorptive potential depends strongly on thelocal diffraction conditions and thus dynamical scatteringthe relative reconstructed phase does not. The effect ofdynamical scattering, in particular non-vanishing excita-tion errors not included in the otherwise to all ordersdynamical phase object approximation on the recon-structed electrostatic mean inner potential is thereforenegligible.

If no vacuum reference phase is available, the recon-structed electrostatic potential profile represents only therelative change of the potential across the interface, as inthe second example above, the Al–Al2O3 interface. How-ever, knowing the mean inner potential of at least one ofthe components next to the interface will allow an absolutescale to be given to the potential profile. The exit face wavefunction was normalized with respect to its maximum value(while retrieving the projected potential). This procedureeliminates negative values in the absorptive potential, i.e.the apparent production of electrons by the sample itself.The scale of the absorptive potential is therefore relative tothe least absorbing point within the field of view. A morerigorous normalization would be to divide by a flat fieldimage (an image recorded without a specimen in the field ofview) recorded under the same imaging conditions. Thiswould give the absorptive potential profile an absolutescale and may be important for quantitative measurementsinvolving both absorptive, as well as the electrostaticpotential, or their ratio.

It may also be mentioned that it is possible to checkwhether the reconstructed exit face wave function is reallyat the in-focus (Df ¼ 0) plane. Using expression (26), it ispossible to propagate the complex wave function in freespace until the contrast in its amplitude is minimized sincethe in-focus plane is the focal plane of minimum (Fresnel)image contrast.

Both the inelastic scattering described by V00(r) and thepseudo-absorptive by V000(r) increase with the atomic

number (Z) of the scattering element. In fact, high-angleannular dark-field scanning transmission electron micro-scopy (HAADF-STEM) technique, being mainly based onV000(r), is also called Z-contrast STEM for this very reason.If no energy slit is used, then the absorptive potentialrepresents only the scattering outside the objective aper-ture, i.e. scattering to relatively high angle. High values ofV000(r) therefore correspond to positions on the sample withlarge thickness and/or heavy atoms. If the thickness isknown, chemical information may be extracted from theabsorptive potential. However, since, as we have shown,the absorptive potential may also be affected by dynamicaldiffraction and changes in sample orientation, great caremust be taken in the interpretation of the data.The size of the objective aperture to be used is ultimately

determined by the number of images used for thereconstruction. The use of a larger objective aperture hasthe advantage that (a) the resolution of the reconstructedpotential is higher, and (b) in principle, the retrieval of lowspatial frequency components of the phase of the electronwave function is possible for a smaller range of defocusvalues. However, while the interference of low spatialfrequency components of the wave function with very highones reduces the necessary defocus range, it also reducesthe image contrast. If, in order to maximize the spatialcoherence, the electron beam intensity is already very low,the image contrast is reduced and longer effective exposuretimes are required, in order to achieve the signal-to-noiseand signal-to-background conditions necessary to performthe reconstruction. The increased spatial resolution alsoputs more stringent constraints on the specimen driftduring each exposure, thus reducing the exposure time andtherefore decreasing the signal-to-noise ratio even further.In principle, this problem can be overcome by recordingmore images and applying the drift compensation methodimplemented within this reconstruction algorithm. How-ever, the effect of image contrast has not been tested forwith rigor sufficiently high to make more quantitativestatements here.Again, each objective aperture size requires a minimum

range in defocus values in order to reconstruct theprojected potential down to the low spatial frequency limitimposed by the spatial coherence of the electron beam. Ifonly this lower limit of defocus range is used, an approachbased on the transport of intensity equation (TIE) [29–31]may be used as well. However, since the TIE is a linearapproximation to the non-linear image formation process,it is non-iterative, but its accuracy is expected to be inferiorto the method described here, especially, if the defocusrange is large.

7. Conclusions

A modified version of the IWFR method [15] forreconstructing the exit face wave function using focalseries has been presented. It was applied to reconstruct thewave function from image line profiles across interfaces.

ARTICLE IN PRESSS. Bhattacharyya et al. / Ultramicroscopy 106 (2006) 525–538538

The electrostatic and absorptive potential profiles acrossinterfaces are obtained from the reconstructed wavefunction using the POA. A new alignment procedure ofexperimental line profiles has been introduced and itsrobustness in the presence of high noise and uncertaintiesin defocus has been tested using simulated images. For thepresent study, it produces the best alignment in comparisonwith alignment procedures existing in the literature. Meaninner potentials retrieved by this method agreed well withliterature values. The resolution of the reconstruction isdetermined by the size of the objective aperture, as wasconfirmed by tests on an abrupt interface. The recon-structed electrostatic potential has been found to beinsensitive to the sample orientation, although, due to thepresence of an objective aperture, the absorptive potentialstrongly depends on the local orientation and thusdiffracting condition.

For surfaces, i.e. vacuum–specimen interfaces, theelectrostatic potential profile represents the absolute meaninner potential profile of the material next to the vacuum.For internal interfaces, only relative changes in potentialcan be measured. This method can prove to be very usefulfor determining phase shifts and compositional changesacross interfaces such as intergranular glassy films inceramics.

Acknowledgements

The authors are extremely thankful to Dr. GuntherRichter and Ms. Limei Cha for providing the TEM samplecontaining the Al–Al2O3 interface investigated in this workand much useful information about it. They are alsothankful to Dr. Raphaelle Satet and Prof. MichaelHoffmann (University of Karlsruhe) for providing thebulk Si3N4 ceramic sample used in the work. Heartfeltthanks are expressed for Mr. Kersten Hahn for hisconsistent support. The financial assistance from theEuropean Commission under Contract nos. G5RD-CT-2001-00586 (NANOAM) and NMP3-CT-2005-013862(INCEMS) is acknowledged.

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