The effect of dynamical scattering in off-axis holographic mean inner potential and inelastic mean free path measurements

ARTICLE IN PRESS Ultramicroscopy 110 (2010) 438–446

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The effect of dynamical scattering in off-axis holographic mean inner potential and inelastic mean free path measurements Axel Lubk , Daniel Wolf, Hannes Lichte Triebenberg Laboratory, Institute of Structure Physics, Technische Universit¨ at Dresden, 01062 Dresden, Germany

a r t i c l e in f o

Keywords: Electron holography Mean inner potential Inelastic mean free path

a b s t r a c t A detailed analysis of the influence of dynamical scattering in medium-resolution off-axis electron holography yields a systematic underestimation of mean inner potentials derived by means of the phase grating approximation. The deviations are categorized with respect to the specimen thickness and the scattering potential. Approximate and easy to use correction formulas valid for arbitrary materials of very small (several atomic layers) or large thicknesses are derived. It is additionally found that incorporation of thermal motion in elastic scattering theory leads to damping terms in the reconstructed amplitude, which have to be added to the previously considered damping due to inelastic scattering. & 2009 Elsevier B.V. All rights reserved.

1. Introduction The electrostatic potential V of a material as produced by the corresponding charge distribution is strongly coupled to the structural and electronic properties of the material. On the atomic length scale electrons can be regarded as moving in the mean field of the other constituents (Hartree–Fock approximation), thereby building up orbitals, which lead to chemical bonding. On a larger length scale potential differences are key elements of electric devices, like pn-junctions, thermoelements, capacitors, and piezoelements. A detailed knowledge of potential distributions thus yields important insight in the functionality of the materials. Within transmission electron microscopy (TEM), the total electrostatic potential of a material is probed by the beam electrons subsequently building up the image. By applying off-axis electron holography, the whole wave function of the elastically scattered electrons can be reconstructed [1,2]. The removal of inelastically scattered electrons from reconstructed waves leads to an amplitude damping, which correlates to the total strength of the inelastic scattering characterized by the inelastic mean free path lin . However, the strong interaction of the electron wave with the specimen potential makes an interpretation of reconstructed electron waves in terms of quantitative potentials and inelastic mean free paths complicated (e.g. [3]). These difficulties are less severe at dedicated imaging conditions, i.e. imaging of light and/or thin materials tilted away from a low-index zone axis [4]. The list of combined electrostatic potential and inelastic mean free path measurements by means of off-axis electron holography is still  Corresponding author.

E-mail address: [email protected] (A. Lubk). 0304-3991/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2009.09.009

limited, though. McCartney and Gajdardziska-Josifovska conducted one of the first comprehensive studies on different crystalline materials [5], thereby establishing a quantitative relationship between the reconstructed amplitude and lin . Harscher and Lichte [6] included the information from the conventional intensity in their investigations on amorphous carbon and vitrified ice, extending the scope of these investigations to biological samples and cryo-microscopy. This paper analyzes the theoretic basis of medium-resolution off-axis holographic measurements as provided by elastic scattering theory. The considered quantities are the mean inner potential (Section 2) and the amplitude damping due to thermal motion. The signal reconstructed by means of off-axis electron holography is discussed in Section 3. Sections 4 and 5 contain a discussion of the scattering on single atoms and thicker specimen, respectively. A comparison and discussion of experimental findings in the light of the theoretical results is subject of the last two Sections 6 and 7.

2. Mean inner potential We will concentrate our investigations on a fingerprint quantity of the total electrostatic potential, its average over a certain volume O spanned by an area A and a thickness t ðO ¼ A  tÞ Z V ¼ 1=O Vð~ r Þ d3 r; ð1Þ O

which will be referred to as averaged electrostatic potential (AEP). It can be regarded as a coarsened (or macroscopic) potential, which depends only the position of the integration volume O. By dividing the spatial domain into several subvolumes O, an arbitrary electrostatic potential Vð~ r Þ can be characterized by

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position dependent AEPs. If considering a perfect crystal and choosing one unit cell as the integration volume O, the AEP becomes a constant measure independent from the center point of O, which corresponds to the first term in the Fourier series expansion and is usually referred to as mean inner potential (MIP) [7]. Typical MIP values for unit cells of crystalline materials are listed in Table 1. In the following, we will concentrate our investigations on perfect crystals in order to simplify the notation. The findings, however, can be transferred to inhomogeneous materials characterized by non-uniform AEPs as will be pointed out below. Lower case letters will indicate 3D vectors with the z-axis pointing along the optical axis of the microscope (see Fig. 2), whereas 2D vectors perpendicular to the optical axis are indicated by capital letters. If a spherically symmetric charge distribution within a radius rr , e.g. a simple atomic model, is assumed, the evaluation of (1) in radial coordinates yields V¼ 

3 e0 2rO3

Z

rO

r 2 rðrÞ

0

 2  r  rr2 dr: 3

ð2Þ

The second term in the integral, corresponding to a non-zero overall charge, can significantly contribute to V (Fig. 1). This charging effect is usually removed in the experiment by carbon coating the specimen [8,9]. If the charge distribution is neutral, Rr i.e.) dr 0R r 2 rðrÞ dr ¼ 0 only the first term in the bracket ‘‘survives’’ and can be considered as a measure for the spatial extension of the charge density (e.g. [10]). This interpretation of the MIP can be less accurately extended to non-spherical charge distributions like unit cells of crystalline materials. Thus, the MIP is sensitive to

Table 1 Extract of experimental and theoretical MIP values V . mat

V exp (V)

V theo (V)

C Si MgO Ge GaAs PbS Cu Au

10.7 [6],8.8 [21 ] 12.5 [23] 13 [13] 14.3 [24] 14.5 [13], 14 [25] 17.2 [13] 21.2 [26] 22 [28], 21.4 [29]

11.4 [22] 12.57 [23]

0.18 0.26 0.14 0.51 0.98

14.67 [23] 14.19 [23]

[23] [13] [24] [13] [13]

24.35 [27] 30.26 [30]

The differences in the dynamical corrections DVdyn can occur due to different calculation methods (MS, Blochwaves), zone axis, acceleration voltages, etc.

2.5

MIP [V]

2

1.5 ρ = 1e/Ω

1

0.5

0 0

20

40

60

chemical bonding on an atomic length scale, where outer-shell electrons are redistributed [11].

3. Medium resolution off-axis holography Numerous investigations using various methods have been conducted in the past to accurately determine the MIP in different materials (e.g. [12–14]). Medium resolution off-axis electron holography proved to be particularly useful due to the following rather simple relationship between the reconstructed wave C, which is approximately equivalent to the zero component in ~ , to the MIP V : Fourier space K Z ~ ¼ 0Þ ¼ jC0 jeisV t : 1=A Cð~ RÞ d2 R  CðK ð3Þ A

The normalization factor 1=A results from the convention used for the Fourier transformation, t denotes the thickness of the sample and s is the so-called interaction constant. The size of the averaging area A in (3) determines the coarsening, i.e. experimentally the resolution, of the wave function. The approximation introduced in (3) becomes exact, if perfect crystals are considered and A equals one face of the unit cell. In the case of inhomogeneous materials, one would get coarsened waves, which depend on the position of A (similar to the AEP). Our restriction to perfect crystals is therefore equivalent to concentrating on one spatially resolved element in the reconstructed medium resolution wave. The amplitude jC0 j of the forward scattered wave is attenuated by scattering into non-zero scattering angles, inelastic scattering and destructive interference introduced by the thermal motion of atoms [15]. For the effect of inelastic interaction McCartney et al. [5] introduced the following expression: jC0 j ¼ et=ð2lin Þ

DVdyn (V)

80

100

rρ [nm] Fig. 1. Contribution of a non-zero homogeneously distributed positive spherical charge to the mean inner potential V .

439

ð4Þ

relating the damping of the reconstructed normalized amplitude to the inelastic mean free path lin . A derivation of (3) and (4) from the basic equations governing the electron scattering, however, is restricted to approximations, e.g. the WKB approximation [3] or the incorporation of inelastic scattering as imaginary absorption potential, which are often violated to some extend (depending on imaging conditions, specimen, etc.). One typical source of deviations from (3) and (4) is illustrated in Fig. 2. The electron beam impinging along a low-index zone axis is intermediately scattered into non-zero scattering angles before being rescattered into the zero beam again. The superposition of these different electron paths cannot be generally described by a potential V or inelastic mean free path lin integrated along z. Accurate MIP and inelastic mean free path measurements reported in literature therefore incorporated dynamical scattering simulations to take into account those effects (e.g. [12,13]). The deviations of the simple relationship (3) from the experimentally measured phase of the zero beam can be roughly divided into two categories depending on the sample properties: 1. If the sample is sufficiently thin, the scattered wave can be described by the phase grating approximation (PGA) (e.g. [3]) R is V ð~ R;zÞ dz Cð~ RÞ ¼ e t ; ð5Þ i.e. the wavefront is modulated by the projected potential R ~ t VðR; zÞ dz of the material, whereas the amplitude remains constant. This approximation is restricted to thin objects because the modulations of the wave due to interference of e.g. waves scattered at different atoms are neglected, The zero beam of such a wave reads Z Z R 1 1 is Vð~ R;zÞ dz 2 Cð~ RÞ d2 R ¼ e t d R: ð6Þ A A A A

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mathematically described as a slice-wise iteration x e-

Cð~ R; zN Þ ¼

y

Fig. 2. Diagram illustrating dynamical scattering. The arrows denote different scattering angles occurring at the scattering centers (atoms). Scattering at subsequent atoms leads to zero beam contributions incorporating an elongation of the optical path due to non-zero intermediate scattering angles (red arrows). The black arrows denote non-zero final scattering angles. The coordinate system with the z-axis pointing along the beam direction will be used throughout the paper. (For interpretation of the references to the color in this figure legend, the reader is referred to the web version of this article.)

If the object is additionally a weak phase object (6), it can be further approximated by a Taylor expansion of the exponential function , i.e. Z Z Z R 1 is is V ð~ R;zÞ dz 2 e Dz d R  1þ Vð~ R; zÞ dz  eisV t : ð7Þ A A Dz A A The same approximation holds, if the projected potential does not depend on ~ R within the region A, i.e. it is homogeneous over the area A of the reconstructed wave: Z Z Vð~ R; zÞ dz  VðzÞ dz ð8Þ

is

e

R t

V ð~ R;zÞ dz 2

d R  eisV t :

ð11Þ

with RÞ ¼ pf n ð~

Z

pf n ð~ RÞ  tf n ð~ RÞCð~ R; z0 Þ;

n¼1

z

1 A

N Y

ð9Þ

A

It is now useful to introduce the quotient rdyn between the phase of the zero beam and the MIP V multiplied with the interaction constant s and the thickness t  Z  R 1 is Vð~ R;zÞ dz 2 e t d R sV t; ð10Þ rdyn ¼ arg A A which measures the deviations of the measured phase from the MIP and which is well defined even in the limit of infinitely large integration volumes O (see Appendix A.1). We can conclude, that in thin samples the simple relationship (3) is rather well fulfilled ðrdyn ¼ 1Þ, if the object is either a weak phase object or thick enough to provide a homogeneous projected potential. The latter condition leads, however, to the breakdown of the PGA, because the interference effects introduced by the propagation of the electron wave cannot be ignored anymore. Therefore, thick objects have to be considered in a second category. 2. In a sufficiently thick sample, the PGA is violated, i.e. the propagation of the wave within the sample has to be considered. The equations describing the scattering become more involved (for an overview see [16,17]), e.g. the widely spread multislice (MS) method consists of slice-wise propagation and modulation R; z0 Þ by the potential V of the initial wave function Cð~

is ik ~2 eik=2ðzn zn1 ÞjRj tf n ð~ RÞ ¼ e 2pðzn  zn1 Þ

R zn zn1

Vð~ R;zÞ dz

:

ð12Þ

Here, pf is a Fresnel propagator or propagation function, which describes the propagation of the electron wave as a convolution integral denoted by . tf is referred to as transmission function and takes into account the effect of the electrostatic potential equivalent to the PGA . n, k, s, zn and zn1 are the slice number, the wave number, the so-called interaction constant, the initial and final z-coordinate of the slice, respectively. The complicated alternating application of transmission and propagation functions along the beam direction z makes an analytical analysis of the deviations of the zero beam phase from the MIP as predicted by (3) difficult. Perturbation theory yields (see A.2) that, if a small part of the potential Vs can be separated from the total scattering potential V, Eq. (3) may hold for Vs, i.e. R ~ dz is V ðR;zÞ Cð~ RÞ ¼ C0 ð~ RÞe t s : ð13Þ Here, the wave C0 scattered by the potential V  Vs is not necessarily described by the PGA R is ðV ð~ R;zÞVs ð~ R;zÞÞ dz C0 ð~ RÞ a jC00 je t : ð14Þ In case of a sufficiently homogeneous Vs the zero beam of (13) reads Z R ~ 1 C0 ¼ eis t V s ðR;zÞ dz C0 ð~ RÞ d2 R: ð15Þ A A This relationship allows the measurement of Vs under dynamical scattering conditions, i.e. moderate tilt angles with respect to low-index zone axis. Therefore, it is frequently exploited when investigating materials with functional potentials, like pnjunctions, which are usually much weaker than the electrostatic potentials of the atomic constituents (e.g. the MIP) and homogeneous with respect to the resolution of the microscope [4]. Additionally, we propose a rather rough approximation applicable to amorphous or sufficiently out-of-zone-axis oriented crystals, i.e. materials with a structure, which appears randomized in projection. This approach is based on a configurational average denoted by /S of the reconstructed zero beam over completely randomized lattice configurations, i.e. + Z Z *Y Z Y N N 1 1 1 2 ~ CðR; zN Þ d R ¼ pf n  tf n Cðz0 Þ d2 R ¼ pf A A A A n¼1 A A n¼1 n  /tf n SCðz0 Þ d2 R ¼

N Y

/tf n S:

n¼1

ð16Þ In the second transformation, a homogeneous distribution of the atom positions (denoted by Rm ) in slice n Y Z is R zn Vm ð~R~R m ;zÞ dz Y Z is R zn Vð~R m ;zÞ dz /tf n S ¼ d2 Rm ¼ d2 Rm e zn1 e zn1 m

Y ~ ¼ 0Þ; ¼ tf nm ðK

m

ð17Þ

m

was inserted, which reduces the propagation function to a unity transformation, if the initial wave Cðz0 Þ is a plane wave

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normalized to 1. The zero beam of the single atom transmission ~ ¼ 0Þ is equivalent to the medium resolution functions tf nm ðK wave obtained from a single atom (see 6). Thus, the total ~ ¼ 0; zN Þ obtained in this random medium resolution wave CðK object approximation (ROA) is just the product of all single atom waves as obtained from the PGA, i.e. Y PGA ~ ¼ 0; zN Þ ¼ ~ ¼ 0Þ: CðK Cnm ðK ð18Þ n;m

Since the whole crystal structure has been randomized, each slice n can be considered to be identical yielding Y PGA ~ ¼ 0; zN Þ ¼ ~ ¼ 0ÞÞN : CðK ðCnm ðK ð19Þ m

This expression provides a considerable simplification to the Multislice formula describing fully dynamic scattering (second line of (16)) and can be used to quickly estimate the influence of dynamic scattering in medium resolution holographic MIP measurements if the PGA approximation of single atoms is known. The latter is, however, very easy to calculate and a list of all elements in the periodic table will be given below. It will be demonstrated in Section 5 that the ROA provides a sufficiently accurate and convenient way of estimating systematic deviations of the zero beam phase from the MIP as expressed by the initial formula (3). The ROA (16) predicts for instance a constant increase of the phase with growing thickness t (number of slices N) as observed experimentally, i.e. Y PGA Y ~ ¼ 0; zN Þ ¼ ~ ¼ 0ÞÞN ¼ CðK ðCm ðK jCm jN eisrdyn V NðtÞ : ð20Þ m

m

The correction factor rdyn for one slice is obtained by a weighted average of single atomic correction factors 1X rdyn ¼ V m rdyn;m ; ð21Þ V m where V is the MIP of the material under consideration and V m is the AEP of a single atom within one unit cell. The slope of the phase, however, is, due to the correction factor rdyn r 1, decreased with respect to the MIP, which is in good agreement with experimental findings in particular on heavy elements, where dynamical scattering effects are particularly strong (see Table 1). The main corrections to the exponential damping (4) of the zero beam amplitude due to inelastic scattering (4) arise from the configurational average of the reconstructed wave over the lattice vibrations [16,15]. The configurational average for a reconstructed wave scattered at a single atom with a position distribution f ð~ RmÞ reads R zn Z ~m ;zÞ dz is Vm ð~ RR Cð~ RÞ ¼ f ð~ R m Þe zn1 d2 Rm ð22Þ

441

scattering simulations, based on two different methods to quantify the approximations leading to (3) and (4).

4. Zero beam of single atoms Since the MIP of a single atom is a function of the averaging volume O (see Eq. 1), it approaches zero as O goes to infinity. ~ ¼ 0Þ is approaching 0 Likewise, the phase of the zero beam jðK according to (6), if A tends to infinity. The ratio rdyn is a finite quantity though (see Appendix A.1), hence can be used to quantify deviations of the zero beam phase from the MIP. For instance, it has been shown above that rdyn approaches 1 for weak phase objects. We will use the forward scattering approximation with predefined integration steps as implemented in the MS algorithm to evaluate the scattered waves. In Fig. 3 the phase calculated by means of MS of both a single Si and a single Au atom is depicted. Contrary to the Si case, the Au atom cannot be treated as a weak phase object and both potentials are not homogeneous in projection. Therefore, both approximations (7), (8) of (6) cannot be applied in case of a single gold atom, and consequently also not for atoms with similar core charges. In Fig. 5, rdyn as defined by (10) of the whole periodic table up to the sixth period is shown. With increasing violation of the weak phase object approximation, i.e. growing core charge and decreasing acceleration voltage, rdyn becomes smaller. The smooth transition is interrupted by jumps due to the shell structure of the electronic shell as explained by expression (2). It is pointed out that the single atomic correction factors rdyn depicted in Fig. 4 in combination with AEP from single atoms within one unit cell provide the basis for the calculation of correction factors rdyn for arbitrary materials (see Eq. 21) within the ROA derived above.

5. Zero beam of arbitrary specimens The investigation of the general behavior of the zero beam for thicker specimen against the background of millions of different crystal structures and orientations is complicated. We therefore found it useful to apply two different approaches: The first one is the PGA, which has the advantage that it can be extrapolated to arbitrary structures. It does, however, not take into account the Fresnel propagation of the electron wave and is therefore only

1.5

Au Si

1 phase [rad]

according to the PGA (5) and contains position dependent amplitudes jCð~ RÞj smaller than one, i.e. the reconstructed wave is dampened by destructive interference. The magnitude of the damping depends strongly on the magnitude of the projected potential of the atom, i.e. strong scatterers like a gold atom will produce a rather strong damping, whereas the amplitude of light atoms stays rather unaffected. As it is explained in Ref. [15], this damping effect must not be misunderstood as electrons removed from the elastic channel by inelastic scattering, it is solely an effect of destructive interference in the reconstructed elastic wave due to a moving lattice. Consequently, this particular damping is not contained in a mean free inelastic path describing the inelastic scattering power and has to be added to the latter to describe the total damping in the reconstructed amplitude. In the following, we will solve the above expressions by numerical integration of expression (3) and perform accurate

0.5

0 −4

−3

−2

−1

0

1

2

3

4

distance from atom position [Å] Fig. 3. Phase shift of object exit wave as produced by scattering on a single silicon and a single gold atom at a distance of 1 A˚ behind the atom. Acceleration voltage: ˚ Weickenmeier and Kohl atomic potentials [18]. 200 kV, sampling: 16/A,

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maximally violates the homogeneity condition for the projected potential (8), hence showing the strongest deviation from the MIP. With increasing number of projected atoms, the homogeneity is restored and the phase approximates the MIP. This behavior serves as a hand-waving argument that out-of-zone-axis conditions are, by increasing the projected homogeneity, important to ensure proportionality of the MIP and the zero beam phase.

1

0.98

rdyn

Si 0.96

Kr

0.94

5.2. Zero beam at moderate and large thicknesses

Xe 300kV 0.92

200kV Au

0.9 20

40 60 atomic number

80

Fig. 4. Ratio rdyn of phase of the zero beam to MIP multiplied with s and t for the periodic table of elements with 1 atoms per unit cell. Integration volume O: (1  1  1 nm), acceleration voltage: 200 and 300 kV.

potential 1

Si

0.96

homogeneity

rdyn

0.98

1 atom 4 atoms

0.94

8 atoms 0.92

16 atoms 32 atoms

0.90

20

Au

40 60 atomic number

We calculate the thickness t dependent dynamical scattering on thick objects for both a weakly scattering material Si (Fd3m, a ¼ 5:431 A, V ¼ 13:9 [18]) and a strong scatterer Au (Fm3m, ˚ a ¼ 4:078 pffiffiffi ˚A, V ¼ 29:8 [18]) tilted around the rotation axis ½1 þ 5=2; 1; 0 into different orientations away from the zone axis ½001 (see Fig. 6). The orientation of the tilt axis along the Golden Ratio (‘‘most irrational number’’) ensures a large distance to other low-index zone axes upon rotation. It is pointed out that the MIP values for Si and Au are derived for an assembly of free atoms, hence do not agree with the exact MIP of the real material (see Table 1). This difference is, however, not important for the following analysis, since we are only interested in the deviations of the measured phase to the input MIP of the calculation due to dynamic effects rather than to the real MIP itself. If neglecting influences of the microscope which depend on the position of the optical axis, like objective lens aberrations, etc., it is in principle not possible to distinguish between a tilted beam and a tilted specimen. Making use of beam tilt in the calculations, however, has the advantage of preserving the periodicity in the xy-plane (see Fig. 6) and facilitates the use of a single unit cell within the framework of dynamical scattering simulations. We will use a numerical forward integration of the approximated Klein–Gordon equation within the forward scattering approximation [19] and the MS formalism derived from the former by applying the small angle approximation and fixing the integration steps. The

80

Fig. 5. Ratio rdyn of phase of the zero beam at 200 kV to MIP multiplied with s and t for the periodic table of elements with 1, 42 , 82 , 162 and 322 atoms per unit cell ð1  1  1 nmÞ.

suited to illustrate the behavior of the zero beam at thicknesses up to a couple of monoatomic layers (thickness t eOðnmÞ). The second one is based on fully dynamic scattering calculations incorporating properly the wave propagation, and is therefore valid up to large thicknesses. The drawback is that only particular crystal lattices can be considered. Furthermore, the fully dynamic calculations are compared to the ROA derived above to test its validity.

beam direction [0,0,1]

Au unit cell

beam tilt (bt)

α

[-1.62,1,0]

ZOLZ FOLZ SOLZ x

a aAu = 4.078 Å aSi = 5.431 Å crystal tilt (ct) supercell

y z bt α [mrad]

ct t

0 5.1. Zero beam at moderate thickness To simulate the increase of the thickness t of an out-of-zone axis oriented specimen, a reference lattice with 1  1  t nm spatial extension and a sampling of 128  128  128 was generated and filled with an increasing number of the same atom species such that the projected atomic density becomes increasingly homogeneous. The zero beam phase is then calculated by means of the PGA. In Fig. 5 the ratio of the zero beam phase to the MIP for an increasing number of atoms in the box, corresponding to an increasing thickness t is depicted. A single atomic layer

10 50 87.3 174.5

fill 8xa

8xa

Fig. 6. Scattering geometry of the dynamic calculations. The scattering angle and rotation axis are oriented with respect to the diffraction pattern of Au as given by the crystal coordinate system, i.e. the x and y axis of the microscope point along the a and b axis of the crystal respectively. The MS method is applied to the crystal tilt (ct) and the numerical forward integration to the beam tilt (bt) geometry. The calculated tilt angles a are tabulated.

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443

14

22

MIP 20

14

13.9

φ /(σ⋅t) [eV ]

φ/ (σ⋅t) [eV]

ROA 18

0 10

16

50

ROA

13.8

13.5

0

25

50

13.7

14

5°

13.6

10°

12

0

10

20 30 thickness t [nm]

40

50

0

50

100

150

200

250

thickness t [nm]

30

Fig. 8. Retrieved MIP from the zero beam phase at tilt of 53 (87.3 mrad) and 103 (174.5 mrad) according to numerical forward integration (shown in the subimage) and MS (large graph) for Si at 200 kV acceleration voltage and scattering factors according to Weickenmeier and Kohl [18]. The small differences between numerical forward integration and MS are due to the incorporation of thermal motion in MS leading to a phase damping. The nominal MIP of 13.9 V (ROA of 13.8 V) is depicted as a green dashed line (black dotted line). (For interpretation of the references to the color in this figure legend, the reader is referred to the web version of this article.)

20 φ/ (σ⋅t) [eV]

13.5

10 0 0

10 50

−10 0

10

20 thickness t [nm]

30

40

Fig. 7. Zero beam phase at 0, 10 and 50 mrad according to numerical forward integration for (a) Si and (b) Au at 200 kV acceleration voltage and scattering factors according Weickenmeier and Kohl [18]. The dynamical effects decrease with increasing tilt angle.

computationally demanding numerical forward integration has the advantage of accepting very large beam tilt angles sufficient for our purposes. We therefore use the numerical forward integration in combination with tilting the beam. The major drawback of the method is its incompatibility with a proper treatment of the thermal motion of the atoms, because the separate calculation of different lattice configurations and a subsequent averaging is too time consuming and an analytical averaging could not be incorporated into the numerical solver. To properly treat the thermal motion and to check the results obtained by the forward integration, we therefore additionally apply a standard approach, the MS algorithm. The MS algorithm, however, fails already at moderate beam tilt angles larger than 63 [20], thus requiring a rotation of the whole crystal instead of tilting the beam. The rotation in turn destroys the periodicity within the xy-plane, which is a requirement for the MS method. To still obtain sufficiently accurate results, a supercell geometry was set up (see Fig. 6) incorporating the tilted crystal for the MS calculations. Although the violation of the periodicity cannot be completely removed due to the limited extension of the supercell, its influence could be reduced. One additionally has to keep in mind that the dominating influence on the zero beam is the MIP V , hence, when setting up the supercell, one has to ensure that the MIP of the whole supercell equals the MIP of the bulk material. The modified periodicity now shows up as a secondary effect changing the diffracted beams in particular at the boundary of the

unit cell. Both simulations provide thickness dependent information by extracting intermediate results at the desired thickness. Both scatterers exhibit a pronounced transition between zoneaxis and completely out-of-zone-axis scattering conditions, as predicted by scattering theory. The zone-axis waves show the typical oscillations in both amplitude and phase (Fig. 7), violating heavily any monotonic thickness dependence like the one predicted by (3), or the exponential damping (4). However, Eq. (3) serves as an upper boundary for the phase of the zero beam in the transition regime between dynamical scattering in zone-axis conditions and kinematic scattering. This behavior can be explained by the elongated optical path of the scattered beams with respect to the zero beam, leading to a retarded phase, which, after being backscattered into the zero beam, is attenuating the latter (see Fig. 2). At small tilt angles (10, 50 mrad) the oscillations become smaller in both materials, whereupon the magnitude of the dynamic effects in Au is always larger than in silicon. The oscillations vanish almost completely at a tilt angle of 103 in both silicon and gold, thus marking these angles as out-of-zone axis conditions, which will be discussed in the following. In the case of Si a systematic attenuation of the by MS calculated zero beam phase from the projected MIP is observed (Fig. 8). The attenuation starts at around 13.76 eV (rdyn ¼ 0:99) in the very thin region and goes up to the nominal value of 13.9 eV, before it drops again to a value around 13.76 eV (rdyn ¼ 0:99). This is in good agreement with both the correction graph calculated with the help of the PGA in the previous section and the expression derived from the ROA (16). In case of a strong scatterer like gold, the scattering behavior becomes more involved (Fig. 9). Again, at thin regions the behavior is in good agreement with the correction displayed in Fig. 4. However, the subsequent maximum is much less pronounced than in the Si case, i.e. the input MIP is not attained. Subsequently, the phase drops again to a value which is predicted by the ROA (rdyn ¼ 0:92). The reason for not attaining the nominal MIP value at a moderate thickness is the strong dynamical scattering of Au. To illustrate the behavior of the zero beam amplitude jC0 j, it is useful to make a comparison to the total intensity I of all scattered beams in the sideband. In spite of

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30

0

20

40

26 5°

25

0.95

0.9

0.9

I1/2

0.85

0.8

24 100 thickness t [nm]

150

200

Fig. 9. Retrieved potential from the zero beam phase at 53 (87.3 mrad) and 103 (174.5 mrad) according to numerical forward integration (shown in the subimage) and MS (large graph) for Au at 200 kV acceleration voltage and scattering factors according to Weickenmeier and Kohl [18]. The small differences between numerical forward integration and MS are due to the incorporation of thermal motion in MS leading to a phase damping. The nominal MIP of 29.8 V (ROA of 27.4 V) is depicted as a green dashed line (black dotted line). (For interpretation of the references to the color in this figure legend, the reader is referred to the web version of this article.)

having performed a fully elastical scattering simulation, an overall damping of the total intensity I in the sideband (Fig. 10) was obtained, which is due to the destructive interference introduced by the thermal motion. A loss of the total intensity in the sideband naturally leads to a decrease of the zero beam amplitude jC0 j as well, which is much larger in the Au case, due to the larger scattering potential (Fig. 10).

6. Comparison to experimental results MIP measurements on various specimen are reported in literature. Recently, they have been combined with ab-initio calculations in order to relate the measured values to effects stemming from chemical bonding, etc. The experimental results are obtained using different microscopes, sample geometries (thin film, cleaved wedge, etc.) prepared by various preparation methods, hence a certain scatter in the reported values has to be taken into account. From the incomplete comparison of reported measurements to ab-initio calculations in Table 1, a trend towards underestimating the actual MIP (the ab-initio value) can be extracted which increases towards stronger scatterers. The reported dynamical corrections DVdyn ¼ ð1  rdyn ÞV calculated by means of the MS or Bloch wave algorithm support this attenuation in the phase and agree well with the approximate correction formula (10) of the ROA. Experimental determinations of the exponential damping constant of the reconstructed amplitude and systematic comparison to inelastic mean free path lengths obtained from Energy Electron Loss Spectroscopy (EELS) are unfortunately still rare. The investigations summed up in Table 2 compare the holographically obtained values with reported theoretic calculations valid for EELS. Although the holographic values seem to be systematically smaller than the EELS values, the database is too small to validate the additional amplitude damping due to thermal motion derived above. Furthermore, the magnitude of the deviations found for the light materials C and MgO is too large to be explained by the influence of thermal motion.

0

50

100 150 thickness t [nm]

0.8 250

200

1

1 I1/2 normalized amplitude |Ψ0|

50

0.85

|Ψ0|

10° 0

normalized I1/2

26

27

0.95

0.8

|Ψ0|

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

50

100 thickness t [nm]

150

normalized I1/2

φ/ (σ⋅t) [eV]

ROA

ROA

27

normalized amplitude |Ψ0|

28

28

1

1

29

MIP 29

0 200

Fig. 10. Total intensity (I) damping in the sideband due to thermal motion and zero beam amplitude damping at 103 tilt angle as calculated by MS (200 kV) for (a) Si and (b) Au. The difference between the total intensity damping and the zero beam amplitude damping stems from intensity scattered into non-zero angles.

Table 2 Extract of experimental and theoretical inelastic mean free path lin values. Mat

lin;exp ðnmÞ

lin;theo ðnmÞ

C Si MgO Cu

51.5 [6] 92 [5] 71 [5] 96 [26]

70 [31] 90 [5] 98 [5] 111 [26]

The investigations have been performed at different acceleration voltages.

7. Conclusion/outlook A detailed analysis of the phase and amplitude reconstructed by means of medium resolution off-axis electron holography revealed systematic deviations from the phase grating approximation and an previously not considered amplitude damping due to the thermal motion of the atoms. The magnitude of the deviations increases with the scattering power of the specimen. We proposed simple correction formulas for mean inner potential measurements, which can be applied to arbitrary materials below and above certain thicknesses. More accurate calculations of dynamical scattering influences have been performed by dynamic scattering simulations on the weak scatterer Si and the strong

ARTICLE IN PRESS A. Lubk et al. / Ultramicroscopy 110 (2010) 438–446

scatterer Au. The influence of the thermal motion of the atoms in attenuating the sideband amplitude by destructive interference leads to a systematic misconception in the interpretation of amplitude damping as an exclusively inelastic effect. Indeed, the thermal attenuation adds to the inelastic damping, leading to a larger damping of the sideband than that measured in EELS experiments. Since this effect is particularly large at strong scatterers and large specimen thicknesses, we propose the use of cryo-electron microscopy to reduce the thermal motion and consequently increase the reconstructed amplitude, if necessary. Combined EELS and off-axis holographic measurements to quantify the effects of thermal motion experimentally are underway.

445

The formal solution R ½EV þ c2 ‘ 2 DR~  i dz 2c2 ‘ 2 k0z Cð~ R; zf Þ ¼ e Cð~ R; zi Þ

ð26Þ

is now expanded with respect to a perturbation parameter l as occurring in a series expansion of the electrostatic potential P n V ¼ n l Vn . The zero order term C0 is i

C0 ð~ R; zf Þ ¼ e

R ½EV 0 þ c2 ‘ 2 DR~  2c2 ‘ 2 k0z

dz

Cð~ R; zi Þ:

The first order term is C1 R ½EV 0 þ lEV 1 þ c2 ‘ 2 DR~  i dz 2c 2 ‘ 2 k0z C1 ð~ R; zf Þ ¼ e Cð~ R; zi Þ

Acknowledgments i

e

R

lEV 1 dz 2c2 ‘ 2 k0z

C0 ð~ R; zi Þ

ð27Þ

ð28Þ

ð29Þ

The authors acknowledge financial support from the European Union under the Framework 6 program under a contract for an Integrated Infrastructure Initiative. Reference 026019 ESTEEM.

For the last transformation, the Baker–Campbell–Haussdorf formula was used.

Appendix A

References

A.1. The dynamical correction factor

¨ ¨ [1] G. Mollenstedt, H. Duker, Beobachtungen und messungen an biprismainterferenzen mit elektronenwellen, Zeitschrift fur Physik A: Hadrons and Nuclei 145 (3) (1956) 377–397 10.1007/BF01326780 / http://dx.doi.org/ 10.1007/BF01326780http://dx.doi.org/10.1007/BF01326780 S. ¨ [2] E. Volkl, L. Allard, D.C. Joy, (Eds.), Introduction to Electron Holography, Kluwer Academic Publishers/Plenum Publishers, 1999. [3] E. Kasper, P.W. Hawkes, Principles of Electron Optics Wave optics, vol. 3, Academic Press, New York, 1995. [4] P. Formanek, E. Bugiel, On specimen tilt for electron holography of semiconductor devices, Ultramicroscopy 106 (2006) 292–300. [5] M.R. McCartney, M. Gajdardziska-Josifovska, Absolute measurement of normalized thickness, t/[lambda]i from off-axis electron holography, Ultramicroscopy 53 (3) (1994) 283–289. [6] A. Harscher, H. Lichte, in: Proceedings of the International Conference on Electron Microscopy ICEM14, vol. 1, 1998. [7] H.A. Bethe, theorie der beugung von elektronen an kristallen, Annalen der Physik 87 (1928) 55–128. [8] R.E. Dunin-Borkowski, S.B. Newcomb, T. Kasama, M.R. McCartney, M. Weyland, P.A. Midgley, Conventional and back-side focused ion beam milling for off-axis electron holography of electrostatic potentials in transistors, Ultramicroscopy 103 (1) (2005) 67–81. [9] M.R. McCartney, Characterization of charging in semiconductor device materials by electron holography, Journal of Electron Microscopy 54 (2005) 239–242. [10] M. O’Keeffe, J.C.H. Spence, On the average coulomb potential ðS0 Þ and constraints on the electron density in crystals, Acta Crystallographica Section A 50 (1) (1994) 33–45 /http://dx.doi.org/10.1107/S010876739300474XS. [11] J. Spence, Quantitative electron microdiffraction, Journal of Electron Microscopy 45 (1996) 19–26. [12] P. Kruse, A. Rosenauer, D. Gerthsen, Determination of the mean inner potential in iii–v semiconductors by electron holography, Ultramicroscopy 96 (1) (2003) 11–16. [13] M. Gajdardziska-Josifovska, M.R. McCartney, W.J. Ruijter, D.J. de Smith, J.K. Weiss, J.M. Zuo, Accurate measurements of mean inner potential of crystal wedges using digital electron holograms, Ultramicroscopy 50 (1993) 285–299. ¨ [14] G. Mollenstedt, M. Keller, Zeitschrift fur Physik 148 (1957) 34. [15] A. Rother, T. Gemming, H. Lichte, The statistics of the thermal motion of the atoms during imaging process in transmission electron microscopy and related techniques, Ultramicroscopy 109 (2) (2009) 139–146. [16] Z. Wang, Elastic and Inelastic scattering in Electron Diffraction Imaging, Plenum Press, New York, 1995. [17] E.J. Kirkland, Advanced Computing in Electron Microscopy, Plenum Press, New York, 1998. [18] A. Weickenmeier, H. Kohl, Computation of absorptive form factors for highenergy electron diffraction, Acta Crystallography A 47 (1991) 590–597. [19] A. Rother, K. Scheerschmidt, Relativistic effects in elastic scattering of electrons in TEM, Ultramicroscopy 109 (2) (2009) 154–160. [20] J.H. Chen, D. Van Dyck, M. Op de Beeck, Multislice method for large beam tilt with application to Holz effects in triclinic and monoclinic crystals, Acta Crystallographica Section A 53 (5) (1997) 576–589 10.1107/ S0108767397005539. [21] D. Shindo, T. Musashi, Y. Ikematsu, Y. Murakami, N. Nakamura, H. Chiba, Characterization of DLC films by eels and electron holography, Journal of Electron Microscopy, 54. [22] M. Schowalter, J.T. Titantah, D. Lamoen, P. Kruse, Ab initio computation of the mean inner coulomb potential of amorphous carbon structures, Applied Physics Letters 86 (11) (2005) 112102 10.1063/1.1885171.

The dynamical correction factor rdyn as defined in (10) approaches 1 for weak phase objects (see (7)). Here we demonstrate that rdyn is finite and becomes independent of the integration volume O in the limit O=OV a 0 b1 R isR V ð~R;zÞ dz 2 argð A e t d RÞ rdyn ¼ R R; zÞ d3 r i sA O Vð~ R R R is Vð~ R;zÞ dz 2 argð AðV a 0Þ e t d R þ AðV ¼ 0Þ dRÞ ¼ R i s Vð~ R; zÞ d3 r R nRA O o 0 1 ~ atan@ R R

¼

r

The

is

I

AðV a 0Þ is

AðV a 0Þ

e

R

t

e

t

~;zÞ dz VðR

VðR ;zÞ dz

d2 R þ

R

d2 R

A

R AðV ¼ 0Þ

dR

~ 3 A O VðR; zÞ d r R R ~ dz 2 is VðR;zÞ AI AðV a 0Þ e t d R R R R R ~ dz is VðR;zÞ ~ 3 t d2 R þ AðV ¼ 0Þ O VðR; zÞ d rðR AðV a 0Þ e s

s

last

expression

is

now

used

to

:

ð23Þ

dRÞ

obtain

the

limes

O=OV a 0 -1, i.e.

R nR o is Vð~ R;zÞ dz 2 I AðV a 0Þ e t d R : lim rdyn r R O=OV a 0 -1 s OðV a 0Þ Vð~ R; zÞ d3 r

ð24Þ

The result shows that rdyn becomes independent from the integration volume, since the remaining integrals depend only on the size of OðV a 0Þ, which is a constant value. It is furthermore important to note that the smallest integration volumes O considered in this work are unit cells of crystalline materials. ˚ hence are one They have an extension in the range of several A, order of magnitude larger than the extension of typical atomic potentials. Consequently, O=OV a 0 b1 is approximately fulfilled and the correction factors rdyn calculated for single atoms become independent from the integration volume O (unit cell volume) used in the calculation. A.2. Perturbation theory for small potentials The small angle approximation reads 2

i½EV þ c2 ‘ D~R  ~ @ Cð~ R; zÞ ¼ CðR; zÞ: 2 @z 2c2 ‘ k0z

ð25Þ

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[23] P. Kruse, M. Schowalter, D. Lamoen, A. Rosenauer, D. Gerthsen, Determination of the mean inner potential in iii–v semiconductors, Si and Ge by density functional theory and electron holography, Ultramicroscopy 106 (2) (2006) 105–113. [24] J. Li, M.R. McCartney, R.E. Dunin-Borkowski, D.J. Smith, Determination of mean inner potential of germanium using off-axis electron holography, ACA 55 (1999) 652–658. [25] D.J.S. Suk Chung, M.R. McCartney, Determination of the inelastic mean-freepath and mean inner potential for AlAs and GaAs using off-axis electron holography and convergent beam electron diffraction, Microscopy and Microanalysis 3 (2007) 329–335. [26] Y.G. Wang, Determination of IMFP by electron holography along with electron dynamic calculation, Chinese Physical Letters 20 (2003) 888. [27] D. Rez, P. Rez, I. Grant, Dirac–Fock calculations of X-ray scattering factors and contributions to the mean inner potential for electron scattering, Acta

[28]

[29]

[30]

[31]

Crystallographica Section A 50 (4) (1994) 481–497, doi:10.1107/S010876739 3013200. S. Ichikawa, T. Akita, M. Okumura, M. Haruta, K. Tanaka, M. Kohyama, Size dependence of the mean inner potential in gold catalyst, in: Proceedings of ICEM 15, Durban, 2002, pp. 311–312. A. Goswami, N.D. Lisgarten, The measurement of inner potentials for copper, silver and gold, Journal of Physics C: Solid State Physics 15 (19) (1982) 4217–4223. M. Schowalter, A. Rosenauer, D. Lamoen, P. Kruse, D. Gerthsen, Ab initio computation of the mean inner coulomb potential of wurtzite-type semiconductors and gold, Applied Physics Letters 88 (23) (2006) 232108 10.1063/1.2210453. R. Burge, Journal of Microscopy 98 (1973) 251.