Ultramicroscopy 136 (2014) 42–49

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Ultramicroscopy

0304-39http://d

n CorrDresden

E-m

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

Dynamic scattering theory for dark-field electron holographyof 3D strain fields

Axel Lubk a,b,c,n, Elsa Javon a,c, Nikolay Cherkashin a, Shay Reboh a,d,Christophe Gatel a, Martin Hÿtch a

a CEMES-CNRS 29, rue Jeanne Marvig B.P. 94347 F-31055 Toulouse Cedex, Franceb Institute of Structure Physics, Technische Universität Dresden, 01062 Dresden, Germanyc TEMAT, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgiumd CEA-Leti, 17rue des Martyrs, 38054 Grenoble, France

a r t i c l e i n f o

Article history:Received 15 January 2013Received in revised form3 July 2013Accepted 14 July 2013

In fond memory of Professor David Cockayne.

strained materials. They are used in the following to quantify the influence of various experimental

Available online 29 July 2013

Keywords:Electron holographyDynamic scatteringStrain engineering

91/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.ultramic.2013.07.007

esponding author at: Institute of Structure Ph, 01062 Dresden, Germany. Tel.: +49 3512150ail address: Axel.Lubk@yahoo.de (A. Lubk).

a b s t r a c t

Dark-field electron holography maps strain in crystal lattices into reconstructed phases over large fieldsof view. Here we investigate the details of the lattice strain–reconstructed phase relationship by applyingdynamic scattering theory both analytically and numerically. We develop efficient analytic linear projectionrules for 3D strain fields, facilitating a straight-forward calculation of reconstructed phases from 3D

parameters like strain magnitude, specimen thickness, excitation error and surface relaxation.& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Dark-field electron holography (DFEH) is a recently developedtechnique for measuring strain in nanostructures, in particularover wide fields of view [1,2]. It has been applied to the study ofstrained-silicon transistors [3–5] and epitaxial thin films [6,7].Different aspects of the technique itself have been investigatedover this period. Precision has been studied as a function ofexperimental parameters such as exposure time, biprism voltageand sample thickness [8,9]. The methodology has been extendedto correct for thickness variations by taking conjugate bright-fieldelectron holograms [2]. The range of imaging conditions, notablymagnification and spatial resolution, has been enlarged by adjust-ing lens configurations [10,11]. However, one particular basicassumption remains unchallenged.

The current assumption when using DFEH is that either (a) thestrain is uniform over the thickness of the foil, or that (b) themeasured strain corresponds to the average strain over thethickness of the foil. Whilst the former poses no problems withinthe other assumptions of the method such as the column approx-imation, such specimens do not exist in practice. Any specimen

ll rights reserved.

ysics, Technische Universität8911.

that had this characteristic in the bulk (indeed, the vast majority ofcurrently studied examples) will have lost it in the process ofsample preparation. The two new free surfaces introduced by thethinning process will have relaxed some of the strain through thewell-known thin-film effect [12]. More importantly, the strain willnow vary over the viewing direction, which we will definethroughout as the z-axis. Furthermore, there is a tendency to lookat specimens which have z-dependent strain, even in the “bulk”.Two cases in hand are quantum dot structures [8] and modern 3Dmicroelectronic devices such as FinFets [13]. It is therefore vital toknow what the measured strain corresponds to exactly.

The problem of z-dependent strain is not new and is inherentto any electron microscopy technique designed to measure strain.The difficulty is always how to evaluate, compensate and correctfor it in the analysis. Convergent-beam electron diffraction (CBED),the first technique used to study strained-silicon devices [14],breaks down in the presence of significant column bending due tothin-film relaxation [15]. The only solution is to model therelaxation with an assumed strain field, perform simulations andcompare with the experimental data [16]. To avoid brute-forceatomistic multislice calculations [17], a Feynman diagram techniqueapplied to dynamic theory was developed [18]. In this approach, thestrain is introduced as a perturbation to the full Bloch-wave calcula-tion within the column approximation, and integrated numericallyslice by slice through the specimen thickness. A more analyticaltheory does not currently exist.

Fig. 2. DFEH setup and coordinate system used in the text. Note that generalvectors are denoted by small bold letters, e.g. r, and vectors in ðx; y; z¼ const:Þ-planes will be denoted by capital bold letters, e.g. R.

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–49 43

The evaluation of z-dependent strain is perhaps even more difficultfor zone-axis techniques such as high-resolution electron microscopy(HRTEM) [3] or nano-beam electron diffraction (NBED) [19]. On onehand, specimens tend to be thinner than for CBED, thus reducingdynamic effects, but on the other hand, the number of beams involvedis prodigious. Beyond the woefully inadequate weak-phase objectapproximation, the only alternative is atomistic multislice simulations,coupled with image formation in the case of HRTEM [20]. Surprisingly,high-angle annular dark-field imaging (HAADF) has seen the mostprogress towards an analytical approach [21], following on the earlieranalysis in terms of strain-induced inter-band scattering [22].

Indeed, we have to return to simpler scattering conditions, such asthose prevalent in a DFEH experiment, to find an analytical theorywhich can incorporate a z-dependent strain field, exemplified by2-beam dynamical theory [23,24]. Within this theory, analyticalsolutions were found for some special cases, such as Moiré contrastand stacking fault contrast. These represent a single step in latticeparameter (or strain) or displacement, respectively, within the foilthickness. Other cases have been implemented by a slice by sliceapproach with transmissionmatrices (see, e.g., the description in [25]).In the following we will show that the theory can be extended toinclude a varying z-dependent strain field in a more analytical way.

The organization of the paper follows closely the different levelsof approximations used to incorporate strained lattices into scat-tering theory. After a short introduction to the optical setup of DFEH(Section 2) and high-energy electron scattering (Section 3.1), wediscuss the notion of the geometric phase (Section 3.2) as anapproximate way to describe weakly deformed lattices. The next stepconsists of contracting many-beam theory to the experimentally used2-beam case (Section 3.3). Subsequently, Section 3.4 is devoted to thediscussion of special analytic solutions of the 2-beam case. Finally,perturbation theory is applied to analyze the influence of a weaklydeformed lattice on scattering under 2-beam conditions (Section 3.5).We use a Si-lattice uniaxially strained along ½001�-direction by meansof H-ion implantation as model system (see Fig. 1), which is suffi-ciently simple for our purposes but also technologically importantwithin the so-called Smart Cut™ technology (SOITEC, France). Accord-ingly, the ½004�-diffracted beam has been used for analyzing the strain.

2. Optical setup

To generate a dark field electron interference pattern, a stronglyexcited diffracted beam is generated by deliberately tilting thespecimen into 2-beam conditions. Subsequently, the transmittedbeam is blocked by an aperture and diffracted beams originatingfrom an undisturbed and strained specimen region are superimposedwith the help of a Möllenstedt biprism to form a hologram in theimage plane. This optical setup is illustrated in Fig. 2. The slightlychanging diffraction angle within the strained region translates into a

Fig. 1. Strain field exxðrÞ generated by Hþ-implantation in a Si matrix as calculatedby finite element elastic strain theory. Note the significant effect of surfacerelaxation due to the small TEM specimen thickness and the symmetry of thestrain with respect to the middle plane of the specimen. The linescan along x in themiddle plane z¼50 nm approaches the bulk strain of an infinitely thick specimenand will be denoted by ebulkxx in the text.

phase shift in the reconstructed wave which is currently directlyinterpreted in terms of a z-independent displacement field uðRÞ, i.e.ϕGðRÞ ¼ �2πG � uðRÞ with G being the reciprocal lattice vector of thediffracted beam [1,26]. From the displacement field one usuallyderives the components of the physically more significant (infinite-simal) strain tensor eij ¼ ð∂ui=∂rj þ ∂uj=∂riÞ=2. Additional phase termsare due to thickness variations and misorientation of the sample incombination with dynamic scattering; it has been argued that thesephases are small compared to the geometric term [1]. In the followingwe will refer to reconstructed displacement or strain, when describingthe quantity measured by DFEH, in order to distinguish it from thephysical displacement or strain of the lattice. Furthermore, we neglectany effect introduced by the aberration of the microscope sincemodern TEM is equipped with hardware correctors [27], whichsuppress the influence of aberrations for spatial resolutions in thenm range considered here.

3. Scattering theory

3.1. High-energy electron scattering

We begin our discussion with defining some notation and basicconcepts, which will be used throughout the paper: the stationaryelectron wave function will be denoted by ψðx; y; zÞ, with z beingparallel to the optical axis of the microscope. Planes conjugate tothe object exit plane will be described by magnification indepen-dent Seidel coordinates [28] R¼ ðx; yÞT . The according reciprocalspace coordinates are denoted by K or G. Consequently, the 2DFourier decomposition of the wave function reads

ψðR; zÞ ¼ ei2πK0 �R∑G~ψ ðG; zÞei2πG�R ; ð1Þ

where K0 ¼ sin θ=λ is the in-plane component of the electronwave with wave length λ and wave vector k0 spanning an angle θwith the z-axis. Electrons are scattered by the electrostaticpotential VðR; zÞ with the according 2D Fourier decomposition

VðR; zÞ ¼∑G

~V ðG; zÞei2πG�R : ð2Þ

Furthermore, it is useful to define the so-called interaction con-stant depending only on the total electron energy E and somefundamental physical constants

CE ¼E

c2ℏ2k0z: ð3Þ

Making use of this notation, the well-known Howie–Whelan (HW)equation, describing the propagation of an electron wave in thesmall-angle scattering approximation, reads (e.g. [25])

∂ ~φðG; zÞ∂z

¼�i2πG2 þ 2K0 � G

2k0zþ iCE

~V G; zð Þ⊗ ~φ G; zð Þ: ð4Þ

10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

x [nm]

e xx

reconstructed strainlattice strain

10 x ebulkxx

ebulkxx

20 x ebulkxx

Fig. 3. Comparison of reconstructed strain obtained from the ½004�-beam simulatedby means of many-beam MS on 3 increasingly strong ebulkxx -uniaxially strainedlattices and the respective input lattice strain ebulkxx .

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–4944

They constitute a set of coupled first order differential equations,which can be integrated by various methods starting with theunperturbed electron wave at the entrance face of the crystal.A straightforward method consisting of a numerical integrationwith a predefined stepsize, the well-known Multislice algorithm[29], will be used in the following to provide numerical referencefor less accurate analytical approximations describing the influ-ence of strained lattices on electron scattering.

3.2. Geometric phase

One important approximation to describe the influence ofstrained crystal lattices on electron scattering is the so-calledgeometric phase.1 That approximation is based upon the pre-sumption that an otherwise perfectly periodic crystal structure ismodified by a displacement field, which changes slowly on thelength scale of the lattice constant a, i.e. δu=δa⪡1: then one candivide the total scattering volume into n subvolumes centeredaround rn and much larger than the volume of one unit cell, i.e.Ωn⪢Ωuc, where the displacement un is approximately constant.According to the shift property of the Fourier-transformation thescattering potential reads

VnðrÞ≈∑g

~V ucðgÞei2πg�ðr�rn�unÞ: ð5Þ

By noting that electron scattering within the high-energy regimealso possesses a limited correlation length (i.e. the lateral distanceover which an atomic potential influences the wave at the exitface), we can apply the so-called column approximation [30] todescribe scattering within a cylindrical subvolume centeredaround Rn

∂ ~φðG; z;RnÞ∂z

¼�i2πG2 þ 2K0 � G

2k0~φ G; z;Rnð Þ

þiCE~V ðG; zÞe�iG�uðz;RnÞ⊗ ~φðG; z;RnÞ ð6Þ

Note that the only difference to HW consists of an additionalmultiplication of the Fourier components of the potential with thegeometric phase term expð�i2πG � uðz;RnÞÞ. Exactly the sameintegration schemes applicable to HW remain valid due to thesame mathematical structure. The complete exit wave of astrained lattice is obtained by patching together all columnsolutions, which is also computationally much less demandingthan calculating the exit wave of the complete lattice, i.e. allcolumns, at once.

To illustrate the limitations of the geometric phase approxima-tion, determined mainly by δu=δa⪡1, we compare the recon-structed strain obtained from DFEH imaging many-beam MSsimulation, performed on 3 increasingly strong ebulkxx -uniaxiallystrained lattices (Fig. 3, for details of the simulation see Appendix C).As expected the geometric phase approximation becomes less validwith increasingly large strain. Since lattice strain usually stays wellbelow 10%, however, the deviations can be neglected with respect toother factors discussed further below.

3.3. 2-Beam equation

It was mentioned above that in DFEH the crystal is orientedsuch that the intensity of the diffracted beam, carrying theinformation about the strained lattice, is maximized. This corre-sponds to a 2-beam condition, where the original beam and onediffracted beam are significantly stronger than all other beams.Note that the validity of the 2-beam approximation depends onvarious factors such as thickness, scattering potential, acceleration

1 Do not confuse with the notion of a geometric phase as a Berry phase.

voltage, etc., and has to be verified experimentally or numericallyby more accurate n-beam calculations in reality [31]. In the2-beam approximation, Eq. (6) reads

∂∂z

~φ0

~φG

!¼ i

0 CE~V�Gei2πG�uðzÞ

CE~VGe�i2πG�uðzÞ 2πsG

!~φ0

~φG

!ð7Þ

with the excitation error

sG≡�G2 þ 2K0 � G

2k0z; ð8Þ

i.e., any influence of beams other than the two strong ones isneglected. Note that the potential was projected into z-direction inEq. (7), i.e. ~V ðG; zÞ- ~VG ¼ 1=a

R ~V ðG; zÞ dz, which is referred to asneglecting higher-order Laue zones in the context of the Blochwave formalism (e.g. [25]). Furthermore, the factor due to themean potential ~V 0 was already incorporated into the wave func-tion, i.e. ~φ0- ~φ0 expðiCE

~V 0Þ, in order to simplify notation. Note alsothat by adding a small imaginary absorption potential, i.e.

V-V þ iV ′-F ~V�G~VG ¼ j ~VGj2�j ~V ′

Gj2 þ 2iRf ~V n

G~V′Gg; ð9Þ

one can approximately incorporate the outflow of intensity intoother beams either due to elastic (dynamical absorption potential)or inelastic scattering (inelastic absorption). Typical absorptionpotentials are two orders of magnitude smaller than the electro-static part (see e.g. [30]).

In order to facilitate a further discussion, we transform Eq. (7)into a second order differential equation for the diffracted (detailsin Appendix A)

d2 ~φG

dz2¼�i2π

∂ðG � uðzÞÞ∂z

�sG

� �d ~φG

dz

� C2E~VG

~V�G þ 4π2∂ðG � uðzÞÞ

∂zsG

� �~φG ð10Þ

and the transmitted beams (see Appendix A).Those equations have the well-known shape of a dampened

harmonic oscillator with one important modification, a generallyz-dependent parameter, i.e., the displacement term ∂uðzÞ=∂z. Dueto this term closed analytical solutions can be only found for somespecial choices of ∂uðzÞ=∂z (see next Section). Since, however,∂ðG � uðzÞÞ=∂z is usually small compared to the other terms, wecan expand the general solution into a von-Neumann seriesincluding only the zero and first order terms (see Section 3.5).Within this approximation, we find closed analytic expressionsgreatly simplifying the interpretation of dark-field holography interms of z-dependent strain fields. Note furthermore that there isno explicit dependency on u, i.e. any initial displacement u

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–49 45

introduces but a constant phase shift, which corresponds to a freechoice of the reference area in DFEH.

3.4. Special (analytic) cases

In order to discuss some general aspects of the scattering, wedeliberately choose a constant displacement (i.e. ∂u=∂z¼ 0, wavefunctions denoted by ð0Þ) and analyze the well-known solutions ofthe dampened harmonic oscillator (Eq. (10)). The two elementarysolutions ~φð0Þ

1;2 ¼ expð2πik1;2zÞ read

k1;2 ¼sG2

7κG ð11Þ

κG ¼ 712π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2s2G þ C2

E~VG

~V�G

qð12Þ

Note that the parameter κG is almost purely real due to thegenerally small absorptive part of the potential. By taking intoaccount the boundary conditions

~φð0ÞG 0ð Þ ¼ 0 and

1iCE

~VG

∂ ~φð0ÞG

∂z0ð Þ ¼ ~φð0Þ

0 0ð Þ ¼ 1 ð13Þ

at the entrance face, the diffracted beam reads

~φð0ÞG ¼ CE

~VG

2πðk1�k2Þexp 2πik1zð Þ�exp 2πik2zð Þð Þ ð14Þ

Similarly, with

~φð0Þ0 0ð Þ ¼ 1;

1iCE

~V�G

∂ ~φð0Þ0

∂z0ð Þ ¼ ~φð0Þ

G 0ð Þ ¼ 1 ð15Þ

the transmitted beam reads

~φð0Þ0 ¼ 1

k1�k2k1 exp 2πik2zð Þ�k2 exp 2πik1zð Þð Þ ð16Þ

Here, we observe the well-known Pendellösung with the effectiveextinction length

ξG≡R π=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2s2G þ C2

E~VG

~V�G

q� �ð17Þ

defined as the propagation distance between two zeros in thediffracted beam.2 Note that the extinction length is increased bynon-zero excitation errors sG. In the experiment it is thereforepossible to influence the amplitude of the diffracted beam bydeliberately changing sG. We furthermore observe a damping ofthe electron wave if absorption ~V

′G≠0 introduces imaginary com-

ponents in k1;2. The phase of the diffracted beam increases linearlywith z and the increase is proportional to the mean inner potential~V 0 and the excitation error sG. These phase terms have to besubtracted, when analyzing the reconstructed phase in terms ofthe geometric phase, which is most elegantly achieved experi-mentally by choosing a reference region with the same thicknessand orientation like the strained region.

3.5. von-Neumann expansion

As it was mentioned above, in case of a general z-dependentdisplacement field no closed analytic form can be derived. How-ever, one can exploit that usually ∂uðzÞ=∂z⪡1, for applying aperturbation expansion. We use the well-known approximationscheme based on the von-Neumann series expansion of thedifferential equation (10) with respect to the smallness parameter∂uðzÞ=∂z⪡1. When stopping the expansion after the first order term,i.e. ~φG ¼ ~φð0Þ

G þ ~φð1ÞG þOðð∂uðzÞ=∂zÞ2Þ, we obtain (see Appendix B) for

2 Usually, taking the real part is dropped because the imaginary part from theabsorptive potential can be safely neglected here.

the diffracted beam (sG ¼ 0 for the moment)

~φG ¼ πCE~VG

κG

14π2

e2πiκGt�e�2πiκGt� ��

þκG

Z t

0ðe2πiκGðt�2zÞ þ e�2πiκGðt�2zÞÞG � uðzÞ dz

�ð18Þ

If we now keep in mind that absorption is a second order effectyielding a comparatively small imaginary component of thediffracted wave vector RfκGg⪢IfκGg and therefore

sin 2πκGtð Þ≈ e2πiκGt�e�2πiκGt

2i

cos 2πκG t�2zð Þð Þ≈ e2πiκGðt�2zÞ þ e�2πiκGðt�2zÞ

2ð19Þ

and use again that the first order term is small compared to thezero order one, i.e.

~φG ¼ ~φð0ÞG ð1þ ~φð1Þ

G = ~φð0ÞG Þ≈ ~φð0Þ

G expð ~φð1ÞG = ~φð0Þ

G Þ ¼ ~φð0ÞG expðiϕGÞ; ð20Þ

Eq. (18) could be further simplified to yield an expression for thereconstructed phase in DFEH

ϕG ¼�2πZ t

0f Gu ðzÞG � uðzÞ dz ð21Þ

with

f Gu z; tð Þ≡R 2πκG cos ð2πκGðt�2zÞÞsin ð2πκGtÞ

� �ð22Þ

defining the weighting function f Gu for the displacement projec-tion. This integral expression constitutes the main result of ouranalytic analysis. We first note that in the leading order thestrained lattice produces an effect in the phase and not theamplitude of the diffracted beam, which explains why phasesreconstructed with DFEH can be used for strain measurements.The mathematical structure, i.e. a line integral of u with aweighting kernel, facilitates a straight-forward interpretation oftypical strain profiles observed experimentally without perform-ing large scale scattering simulations. By comparison to “bruteforce” MS simulations, we will show below that the error of thisfirst order approximation is small if strain fields are small. Theweighted integral is a linear projection rule for the originally 3Dgeometric phase (or distortion) field (see Section 3.2) and there-fore applies equally to all values linearly depending on the recon-structed phase, in particular the strain tensor.

We want to point out that one can alternatively derive theweighting function in a less formal way based on a z-dependentMaster equation for the 2-beam case. Accordingly, one writesdown the intensity of the diffracted beam in the exit plane as asum of beams created at a certain depth in the crystal phase-shifted with the corresponding geometric phase. Both the originaltransmitted (Eq. (16)) and diffracted beams (Eq. (14)) contributelocally to the diffracted beam. The respective weight in the finalresult is determined by the value of the transmitted (diffracted)beam at z multiplied with the value in the exit plane t of thediffracted (transmitted) beam created at z, i.e.

f Gu;1∼ cos ð�2πκGzÞ cos ð�2πκGðt�zÞÞ ð23Þand

f Gu;2∼ sin ð�2πκGzÞ sin ð�2πκGðt�zÞÞ: ð24ÞUsing

cos ð�2πκGðt�2zÞÞ ¼ cos ð�2πκGðt�zÞÞ cos ð�2πκGzÞþ sin ð�2πκGðt�zÞÞ sin ð�2πκGzÞ ð25Þ

and normalizing correctly, the sum of the two contribution yieldsthe weighting function (21). Similar to the more stringent pertur-bation approach it was assumed that the original transmitted and

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–4946

diffracted beams are not modified by the scattering, i.e., theperturbation condition was used implicitly.

The corresponding result for the transmitted beam reads

ϕ0 ¼�2πZ t

0f 0uðzÞG � uðzÞ dz ð26Þ

with the weighting function

f 0u z; tð Þ≡�R2πκG sin ð2πκGðt�2zÞÞ

cos ð2πκGtÞ

� �: ð27Þ

We want to highlight a connection between Fourier transforma-tion and the weighting integrals (21) and (26). κG is almost purelyreal, hence ϕG is approximately proportional to a windowed cosinetransform and ϕ0 to a windowed sine transformwithin a z-intervalbetween 0 and t. It could therefore be possible to combine darkand bright field holography under 2-beam conditions to obtaininformation on both symmetric and antisymmetric parts of thedisplacement.

We will now discuss the general shape of the weightingfunction f Gu of the diffracted beam and its influence on themeasured phase in more detail. f Gu depends on two variables, thethickness and the integration variable z (see Fig. 4). It is readilyobserved, that

R t0 f

Gu ðzÞdz¼ 1, i.e. in case of a constant displacement

0 0.25 0.5 0.75 1 1.25 1.5

0

0.25

0.5

0.75

1

1.25

1.5

depth z [ξG

]

thic

knes

s t [ξ G

]

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0 0.25 0.5 0.75 1 1.25 1.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

depth z [ξG

]

wei

ghtin

g fu

nctio

n fG u

t = 0.5 ξG

t = 0.75 ξG

t = 1.25 ξG

t = 1.5 ξG

weighting function f Gu

1D linescans

Fig. 4. Weighting function f Gu ðz; tÞ (color) with 4 1D-cuts at special thicknessest ¼ 0:5;0:75;1:25;1:5ξG . Note that f Gu ð0ozo0:5ξG ;0:5ξGÞ ¼ f Gu ð0ozo0:5ξG ;1:5ξGÞ.The black dashed/dotted line indicates the point of symmetry at z¼ t=2. (Forinterpretation of the references to color in this figure caption, the reader is referredto the web version of this article.)

field in z-direction (uðzÞ ¼ u) the reconstructed geometric phasecorresponds exactly to the 2D displacement field of the crystal lattice,i.e. ϕG ¼�2πG � u. Thus, reconstructed strains from dark field EH aremost easily obtained if strain fields are constant along z, i.e. uðzÞ≈u.To that end one has to suppress surface relaxations which is onlypartly possible by dedicated specimen preparation techniques. Theevaluation of strains in the general 3D case is much more involved:the weighting kernel f Gu ðz; tÞ is G-dependent, hence different dif-fracted waves ~φG measure differently projected parts of the strain.For instance a ½004�- and a ½008�-beam in our test object are subjectto very different weighting functions (κG is changing) resultingeventually in different reconstructed strains even though the diffrac-tion direction is the same. Also general strain tensor componentsreconstructed from linearly independent diffraction directionsbelonging to different lattice plane families, e.g. ½004� and ½220�,have to be interpreted with caution since their weighting kernel isdifferent. In Si that problem can be avoided if using ⟨111⟩-diffractedbeams for reconstructing the strain tensor.

It is now useful to rescale the z-coordinate in terms of theextinction length, i.e. z-z=ξG and t-t=ξG, which facilitates amaterial and G-independent discussion of the weighting function.It is observed that fu

G depends in an oscillating and symmetricmanner on the distance from the specimen center. The conse-quences of this behavior are very important and we will illustratethem at the hand of some remarkable and partially unexpectedeffects: (i) since thin TEM specimen usually relax in a rathersymmetric manner at both the exit and entrance faces (see e.g. Fig.1), the phase of the diffracted beam (sensitive to symmetric strain)is strongly affected, whereas the zero beam phase, which mea-sures the antisymmetric displacement, is only weakly modified.This explains, why bright-field EH yields only a very weak strainsignal in the reconstructed phase. Experimentally, one exploitsthis property by using the bright field phase for measuringthickness profiles [2]. (ii) Reconstructed strains can have invertedsigns if they are dominated by contributions stemming fromdepths, which are weighted by the negative part of the weightingkernel. For instance, the ubiquitous surface relaxation in thin TEMspecimen is weighted negatively if toξG (see Fig. 4). More detailedinvestigations on this particular effect will be presented else-where. (iii) The reconstructed phase is “blind” with respect toabrupt displacement field changes, such as occurring at abruptinterfaces, if they occur at particular depths zϵfðnþ 1=2ÞξG; nϵNg. Inthat case the changed displacement field is weighted by an antisym-metric function (see e.g. f Gu ð0:5ξGozot; t ¼ f1:25ξG; 1:5ξGgÞ inFig. 4), hence its integral vanishes. To illustrate this behavior andverify the accuracy of the weighting function formalism as promisedabove, we calculated the ½004�-diffracted beam within a multiple-beam MS simulation at a sample containing a sharp change ofebulkxx -uniaxial strain at different depths (see Appendix C for details ofthe simulation). Fig. 5 shows that the change at exactly 0:5ξG is notvisible in the diffracted wave which is in line with the analyticprediction of the perturbative 2-beam theory.3 Notably, such achange in lattice strain becomes visible in the phase of the trans-mitted beam, which can be understood by the following intuitiveargument (or by directly evaluating Eq. (26)): starting with amaximal (minimal) diffracted (transmitted) beam at 0:5ξG exchangesthe usual role of transmitted and diffracted beams at the entranceface. Consequently the phase information about a strain starting at0:5ξG is now attached to the transmitted beam.

We finally note the weighting function for the recon-structed phase of the diffracted beam with non-vanishing

3 Small deviations close to ξG are caused by the influence of multiple beamsand the limited sampling in the MS simulation.

Fig. 5. Reconstructed strain from ½004�-diffracted beam (ξG ¼ 157:1 nm) calculated bymeans of many-beam MS scattering simulations at a specimen containing the modelstrain profile ebulkxx , sharply beginning at (a) the entrance face, (b) ξG=2 and (c) ξG .

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–49 47

excitation error sG

f Gu z; tð Þ≡2πRκG cos 2πκG t�2zð Þð Þ�i

sG2

sin 2πκG t�2zð Þð Þsin ð2πκGtÞ

8><>:

9>=>;: ð28Þ

Note that the additional antisymmetric weighting is predomi-nantly imaginary, hence it introduces an amplitude (and no phase)modulation proportional to the antisymmetric part of the distor-tion. Consequently, the effect of a non-zero excitation error to thereconstructed phase mainly reduces to changing the extinctionlength ξG (see Eq. (17)). One can therefore fine-tune the extinctionlength in order to maximize the diffraction amplitude and thus thesignal-to-noise ratio in the reconstructed phase [32]. A positiveside effect of maximizing the diffracted beam amplitude is that theweighting function assumes one of the particularly simple shapesat f Gu ðz; tϵfðnþ 0:5ÞξGgÞ, i.e., the weighting becomes independentfrom the real thickness of the specimen. Indeed maximizing thediffraction amplitude by adjusting the excitation error sG is

common practice, when performing DFEH. Unfortunately, theprecision of the goniometer and beamtilt provided by the instru-ment is limited which puts some restrictions on the latter method.

4. Summary and outlook

Based on dynamical scattering and perturbation theory wederived closed and easy to use analytical expressions for phasesreconstructed by means of dark field EH. Accordingly, generallyz-dependent strain fields are weighted with a z-dependent weightingfunction, when projected by the electron beam. This weightingfunction is predominantly symmetric with respect to the middleplane of the specimen and depends on the diffracted beam, thecrystal potential, the specimen thickness and the deviation of thecrystal orientation from ideal Bragg conditions. As a consequence,depending on the particular shape of the z-dependent strain fieldsand the diffracted beam, reconstructed phases might be moresensitive to surface strain or bulk strain. This is of particularimportance since surface relaxation is difficult to avoid whenpreparing thin TEM specimen. We furthermore point out that thisformalism, i.e. linear projection of strain or displacement fields with aweighting function, constitutes an important prerequisite towardsthe tomography of 3D strain fields, since it provides a link to thecommonly used projection transformations (Radon transformation).In future publications we will present experimental evidence for theabove presented theory.

Acknowledgments

The authors acknowledge financial support from the EuropeanUnion under the Seventh Framework Program under a contract foran Integrated Infrastructure Initiative (Reference 312483 –

ESTEEM2) and the French National Agency (ANR) in the frame ofits program in Nanosciences and Nanotechnologies (HD STRAINProject no ANR-08-NANO-0 32).

Appendix A. Detailed derivation of second order differentialequation

In order to shorten notation and facilitate a better under-standing of the transformations given below we redefined allreciprocal lattice vectors, i.e. G and sG, to include the factor 2π.Only in the final result, we have drawn out the factor again to becoherent with the results shown in the main text. The transforma-tion steps leading to a second order differential equation for thediffracted beam read

∂2 ~φG

∂z2¼ iCE

~VGe�iG�uðzÞ �i∂ðG � uðzÞÞ

∂z~φ0 þ

∂ ~φ0

∂z

� �þ isG

∂ ~φG

∂z

¼Eq:ð7Þ iCE~VGe�iG�uðzÞ �i

∂ðG � uðzÞÞ∂z

∂ ~φG

∂z�isG ~φG

iCE~VGe�iG�uðzÞ

0B@

1CA

0B@

þi CE~V 0

∂ ~φG

∂z�isG ~φG

iCE~VGe�iG�uðzÞ

0B@

1CAþ CE

~V�GeiG�uðzÞ ~φG

0B@

1CA1CAþ isG

∂ ~φG

∂z

¼�i∂ðG � uðzÞÞ

∂z∂ ~φG

∂z�isG ~φG

� �þ iCE

~V 0∂ ~φG

∂z�isG ~φG

� ��

�C2E~V�G

~VG ~φG

�þ isG

∂ ~φG

∂z

¼ �i∂ðG � uðzÞÞ

∂z�sG

� �∂∂z

�C2E~VG

~V�G�∂ðG � uðzÞÞ

∂zsG

� �~φG:

ðA:1Þ

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–4948

Using the reciprocal lattice vector definition without the prefactor2π one obtains Eq. (10)

∂2 ~φG

∂z2¼ �i2π

∂ðG � uðzÞÞ∂z

�sG

� �∂∂z

�C2E~VG

~V�G�4π2∂ðG � uðzÞÞ

∂zsG

� �~φG:

ðA:2ÞThe same procedure carried out for the transmitted beam

yields a slightly modified expression

∂2 ~φ0

∂z2¼ iCE

~V�GeiG�uðzÞ i∂ðG � uðzÞÞ

∂z~φG þ ∂ ~φG

∂z

� �

¼Eq:ð7Þ iCE~V�GeiG�uðzÞ

0B@i

∂ðG � uðzÞÞ∂z

∂ ~φ0

∂ziCE

~V�GeiG�uðzÞ

þiðCE~VGe�iG�uðzÞ ~φ0 þ sG ~φGÞ

1CA

¼ i∂ðG � uðzÞÞ

∂z∂ ~φ0

∂z�C2

E~VG

~V�G ~φ0 þ isG∂ ~φ0

∂z

¼ i∂ðG � uðzÞÞ

∂zþ sG

� �∂∂z

�C2E~V�G

~VG

� �~φ0: ðA:3Þ

Again we note the result using the reciprocal lattice vectordefinition without the prefactor 2π

∂2 ~φ0

∂z2¼ i2π

∂ðG � uðzÞÞ∂z

þ sG

� �∂∂z

�C2E~V�G

~VG

� �~φ0: ðA:4Þ

In case of ∂u=∂z¼ 0 the two elementary solutions ~φ1;2 ¼ expð2πik1;2zÞ are equivalent to those of the diffracted beam, i.e.

k1;2 ¼sG2

712π

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2s2G þ C2

E~VG

~V�G

q: ðA:5Þ

With ~φ0ð0Þ ¼ 1, ð1=iCE~V�GÞð∂ ~φ0=∂zÞð0Þ ¼ ~φGð0Þ ¼ 0 the transmitted

beam reads

~φ0 ¼1

k1�k2k1eik2z�k2eik1z� �

: ðA:6Þ

Appendix B. Detailed derivation for first-order von-Neumannapproximation

In order to shorten notation and facilitate a better under-standing of the transformations given below we redefined allreciprocal lattice and wave vectors, i.e. G, sG, k1;2 and κ, to includethe factor 2π. Only in the final result, we have drawn out the factoragain to be coherent with the results shown in the main text. Theintegral formulation of the second order differential Eqs. ((10) and A.4)reads

~φG ¼ ~φð0ÞG �

Z t

0Γ t�zð Þ ∂ðG � uðzÞÞ

∂zi∂ ~φG

∂zþ sG ~φG

� �dz ðB:1Þ

and

~φ0 ¼ ~φ0;hom þZ t

0Γ t�zð Þ ∂ðG � uðzÞÞ

∂zi∂ ~φ0

∂z~φ0

� �dz ðB:2Þ

respectively, where

Γ t�zð Þ ¼ ik2�k1

exp ik1zð Þ�exp ik2zð Þð Þ ðB:3Þ

denotes the Greens function of the dampened harmonic oscillator. Thefirst-order approximation in a von-Neumann type series expansion isnow obtained by replacing all expressions on the right hand sidecontaining ~φ with the corresponding zero-order expression ~φð0Þ, i.e.

~φð1ÞG ¼ ~φð0Þ

G �Z t

0Γ t�zð Þ ∂ðG � uðzÞÞ

∂zi∂ ~φð0Þ

G

∂z′þ sG ~φ

ð0ÞG

!dz ðB:4Þ

and

~φð1Þ0 ¼ ~φð0Þ

0 þZ t

0Γ t�zð Þ ∂ðG � uðzÞÞ

∂zi∂ ~φð0Þ

0∂z

!dz: ðB:5Þ

Repeating this procedure iteratively yields higher-order terms andeventually the complete von-Neumann series, which converges underthe well-known convergence conditions [33]. Inserting the zero ordersolutions (Eqs. (14) and (16)) and omitting the perturbation orderindex, one obtains

~φG ¼ CE~VG

k1�k2

�ðeik1t�eik2tÞ þ i

k1�k2

Zeik1ðt�zÞ�eik2ðt�zÞ� �

� ∂ðG � uðzÞÞ∂z

sG�k1ð Þeik1z� sG�k2ð Þeik2z� �

dz�

¼ CE~VG

k1�k2ðeik1t�eik2tÞ�

þ ik1�k2

ZsG�k1ð Þeik1t þ sG�k2ð Þeik2t� sG�k2ð Þeik1ðt�zÞþik2z

�

� sG�k1ð Þeik2ðt�zÞþik1z� ∂ðG � uðzÞÞ

∂zdz�

¼ CE~VG

k1�k2eiðsG=2Þt ðeiκGt�e�iκGtÞ

�

þ ik1�k2

ZsG2�κG

� �eiκGt þ κG þ sG

2

� �e�iκGt� κG þ sG

2

� �eiκGðt�2zÞ

�

� sG2�κG

� �e�iκGðt�2zÞ

� ∂ðG � uðzÞÞ∂z

dz�¼p:I: CE

~VG

2κGeiðsG=2Þt ðeiκGt�e�iκGtÞ

�

þZ

κG þ sG2

� �eiκGðt�2zÞ þ κG�

sG2

� �e�iκGðt�2zÞ

� �G � u zð Þ dz

�ðB:6Þ

Reinserting the prefactor 2π one obtains Eq. (18) if sG ¼ 0 or thegeneral Eq. (28) when applying the transformations leading from Eqs.(18)–(21).

The corresponding transformations for the transmitted beamread

~φ0 ¼1

k1�k2k1eik2t�k2eik1t�

1k1�k2

Zeik1ðt�zÞ�eik2ðt�zÞ� ��

� ∂ðG � uðzÞÞ∂z

�ik2k1eik2z þ ik1k2eik1z� �

dz�

¼ 1k1�k2

k1eik2t�k2eik1t�1

k1�k2

Zik1k2 eik1t�eik1zþik2ðt�zÞ

� ���

þik2k1 eik2t�eik2zþik1ðt�zÞ� �� ∂ðG � uðzÞÞ

∂zdz�

¼ eiðsG=2Þt

k1�k2

sG2þ κ0

� �e�iκ0t� sG

2�κ0

� �eiκ0t�

is2G4�κ20

� �k1�k2

0BBB@

eiκ0t þ e�iκ0t� �

G � u zð Þ�Z

e�iκ0ðt�2zÞ þ eiκ0ðt�2zÞ� � ∂ðG � uðzÞÞ

∂zdz

� ��:

ðB:7ÞSetting sG ¼ 0 and performing a partial integration yields

~φ0 ¼12

e�iκ0t þ eiκ0t� �

þ i2

ðeiκ0t þ e�iκ0tÞG � uðzÞ��

�Z

eiκ0ðt�2zÞ þ e�iκ0ðt�2zÞ� � ∂ðG � uðzÞÞ

∂zdz��

¼p:I: 12

ðe�iκ0t þ eiκ0tÞ þ κ0

Zðe�iκ0ðt�2zÞ�eiκ0ðt�2zÞÞG � uðzÞ dz

� �� �:

ðB:8ÞAfter applying the transformations leading from Eq. (18) to Eq. (21)and reinserting the prefactor 2π one obtains Eq. (26) including theweighting function f 0u for the transmitted beam phase.

A. Lubk et al. / Ultramicroscopy 136 (2014) 42–49 49

Appendix C. Multislice simulation parameters

We used the home-grown (S)TEM simulation software SEMI,implementing a numerical integration of the paraxial approxima-tion of the approximated Klein–Gordon wave equation withpredefined stepsize referred to as Multislice in the literature [29]to accurately propagate the electron probe through uniaxiallystrained Si lattices in Sections 3.2 and 3.5. Simulation para-meters separated into TEM (A), crystallographic (B) and numerical(C) parameters are summarized in the following: (A) Ua ¼ 200 kV,beam tilt (≅ dark-field conditions for ½004�-beam)¼arcsin ðjGj=jq0jÞ ¼ 9:2 mrad, diameter of dark-field aperture¼4.6 mrad; (B)aSi ¼ 5:431 Å, SG¼ Fd3m, t¼250 nm; (C) Δtslice ¼ 1 Å, sampling¼0.008 nm�1, supercell¼128�1 aSi ≅ 695:17� 5:43 Å, atomic scat-tering potentials from Ref. [34]. The Si-atoms within the supercell aredisplaced according to the strain fields mentioned in the main text.

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