Simulation of atomic-scale high-angle annular dark ...

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Full-length paper

Simulation of atomic-scale high-angle annular dark field scanning transmission electron microscopy images Takashi Yamazaki1 , Kazuto Watanabe 2 , Aleksander Recnik3 , Miran Ceh3 , Masahiro Kawasaki4 and Makoto Shiojiri5,6* 1

Department of Physics, Science University of Tokyo, Shinjuku, Tokyo 162-8601, 2 Tokyo Metropolitan College of Technology, Shinagawa, Tokyo 140-0011, Japan, 3 Ceramics Department, J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 4 Electron Optics Division, JEOL Ltd., Akishima, Tokyo 196-8558, 5 Department of Anatomy, Kanazawa Medical University, Ishikawa 920-0293, and 6 Kyoto Institute of Technology, Kyoto 606 -8585, Japan 5 Present address: 1-297 Wakiyama, Enmyoji, Ohyamazaki, Kyoto 618-0091, Japan To whom correspondence should be addressed . E-mail: shiojiri@pc4. so-net.ne.jp ............................................................................................................................................................................

Abstract

Image simulations for high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) based on the Bethe’s eigen-value method are presented. The effects of aperture size and defocus of a probeforming lens, both of which determine the shape of the probe, and the effect of the distortion, influencing the Bloch wave field on atomic columns channeled by electrons, on the HAADF-image intensity are discussed in terms of dynamical effect. These effects are illustrated by our experimental atomic -scale HAADF-STEM images, detected in a detector range of 50110 mrad. It is emphasised that the image simulations are indispensable for quantific ation of experimental HAADF-STEM images and as such provide a valuable compositional analysis for every atomic column along the incident beam.

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Keywords HAADF-STEM, ZnO, SrTiO 3 , Bethe method, high-resolution image

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Received

9 May 2000, accepted 2 September 2000

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Introduction Atomic resolution high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) images, also known as Z-contrast images, have become available using a field emission gun electron microscope attached with a high angle annular detector [1-8]. Intensive HAADF-STEM investigations have been performed for analysis of crystal structure, impurities, precipitates and grain boundaries in advanced materials [8-20]. It is generally believed that HAADF-STEM provides incoherent images without any phase problem, and can be therefore directly inverted to the object without additional image simulations [8]. Recently, we have studied an As-doped Si crystal by atomic -resolution HAADF-STEM [21]. Observed HAADFSTEM images showed a characteristic excess brightness depending on the number of As atoms in the atomic column s. Quantification of these images allowed us to

obtain the distribution of As atoms at atomic resolution. We have also performed atomic -scale quantitative elemental analysis of boundary layers in Bi-doped SrTiO3 ceramic by HAADF-STEM [22]. We illustrated that Sr sites in the diffusion layer were replaced by Bi atoms with 14 at. %. In addition, we measured that the amount of Bi increased abruptly on outermost lattices near the grain boundary. The concentration of Bi atoms has been evaluated for every [001] Sr or Ti(O) column along the incident electron beam. In these investigations, the quantitative analysis was done with the aid of HAADF image simulations, and we showed that simulations are essential for interpretation of HAADF-STEM images. The image simulations are indispensable in such cases as quantitative elemental analysis, especially for crystals with distorted lattice. In the present work we present a method for HAADFSTEM image simulation. It is shown how the intensity

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754 JOURNAL OF ELECTRON MICROSCOPY, Vol. 49, No. 6, 2000

in HAADF-STEM images is influenced by the imaging conditions, such as the size of objective aperture and defocus of the probe-forming lens of STEM, and also by specimen structure, such as the lattice parameter, illustrated on previously observed HAADF images of ZnO [19, 20] and SrTiO3 crystals [21-24].

Theory of HAADF Image Simulations

∫

(1)

Probe

where Ro is the centre of the beam probe, and k || is transverse component of each plane wave having a wavelength of λ within the incident electron beam [1, 25]. W(k || ) is the transfer-function in the objective lens with a spherical aberration Cs and defocus ∆f, and is given by: (2)

Generally, HAADF-STEM images are produced by the thermal diffuse scattering (TDS) and less by the general elastic scattering (or Bragg reflection) of electrons, when the detector is set in a high angle range, as mentioned later. The intensity of the TDS electrons is proportional to the intensity of the wave field at all sites where atoms scatter the electrons. We may calculate the wave field in the specimen crystal using Bethe’s method, which describes a wave function of multiply scattered electrons by three-dimensional Bloch waves. By the use of Beth’s method we can evaluate the integrated convergent beam diffraction intensity over the annular detector for each probe. Then, HAADF image intensity I HA (Ro ) for a specimen of thickness t can be deduced from the following equation [1, 5] I HA (Ro ) =

∑ ∫ i

t 0

σ X |ψ(R i − Ro , z) |2 d z

(3)

where ψ (R i − Ro , z) is the wave function at a depth of z in i-column, due to the probe located at the surface (Ro , 0), and is given by: ψ (R i − Ro , z) =

∑ j

and

A j (R i − Ro , z)

σ TDS [×10 -6 nm 2 ]

Bi 0.398 253 ....................................................................................................................... Sr 0.544 62.0 ....................................................................................................................... Ti 3.88 18.4 ....................................................................................................................... O 0.955 2.38

∫

φ j (R, z, k || ) exp [- i k | | Ro +iW(k | | )] dk | |

Probe

=

∫

ε j (k || ) τ j (Ri , k ||) exp [i (κz +γ j ) z ] exp(−µ j z )

Probe

exp [-2 i k | | Ro +iW (k | | ) ] dk | |

exp [ i k | | (R − Ro ) + iW(k | | )] dk | |

W(k | | ) = π λk || 2 (Cs λ2 k ||2 / 2 + ∆f )

σ els [×10 -6 nm 2 ]

Aj (R i − Ro , z) =

In HAADF-STEM a very fine convergent beam is focused by objective lens or probe-forming lens on the specimen. The wave function of an incident-probe ϑ (R, Ro ) at R on the object surface can be described by: ϑ (R, Ro ) =

...................................................................................................,................. Table 1 Calculated cross section of elastic scattering σ els and TDS σTDS, detected in a range of 50-110 mrad of HAADF detector.

(4)

(5)

where Aj (R i − Ro , z) is amplitude contribution from each Bloch state τ j (Ri , k ||) integrated over all angles comprising the incident probe, κz +γj transverse energies, ε j (k ||) excitation amplitudes and µj absorption coefficient for each branch j. φ j (R, z, k || ) is the wave function of branch j in three-dimensional Bloch waves. σX in eq. 3 is the high-angle cross-section of TDS of an atom X for the HAADF detector and is given from the form factor fX (s), having a good approximation by the Einstein model [6]: σx = [4π(m/mo)/(2π/λ)]2

∫

|fX(s)|2 [1−exp{−2Mx (s)}]ds 2 (6)

Detector

where m and mo are the mass and the rest mass of electron, and s = sinθ /λ corresponding to scattering angle 2θ. MX (s) is the Debye-Waller factor for atom X. The HAADF image due to TDS may be regarded as an incoherent image since the image intensity depends on the wave field intensity |ψ |2 as shown in eq. (3), although the TDS is a result of the elastic and coherent scattering ascribed to relative transverse shifts between thermally vibrating atoms. The integration in eq. (6) is made over the detection. In our previous papers [21,22], intensities of elastic scattering and TDS for atoms that are dealt with in this paper were shown against scattering angle 2θ. Table 1 shows calculated cross-sections of the general elastic scattering and the TDS for Bi, Sr, Ti and O atoms, in Bi-doped SrTiO3 crystal, detected in a range of 50-110 mrad, which is the detection range of the annular detector used in the present experiment. The TDS provides the dominant contrast mechanism in this detection range. When smaller angle scattering is recorded or a crystal having smaller DebyeWaller factor is observed, the elastic scattering appreciably influences the image formation. Numerical algorithms for

T. Yamazaki et al. Simulation of atomic-scale HAADF-SEEM images 755

Fig. 1 Effective intensity profiles for an incident probe with a semi- angle of 12 mrad calculated at defoci of ∆ f = −70, −50, and −30 nm. ∆f = −50 nm corresponds to the Scherzer focus of a JEM-2010F-TEM/STEM with an objective lens of Cs=1.0 mm. Full circles denote Zn [120]-column positions along the a axis and open circles Zn or O [120]-column positions along the c axis of ZnO crystal. (see Fig. 4b).

HAADF image simulation taking account of both the elastic scattering and TDS are to be reported els ewhere. For an atomic column including impurity atoms, we use an averaged cross-section σXav and an averaged form factor fX av : σ

X

av

= (1−x) σ X + xσ Y

fX av =(1−x) fX + x fY

(7) (8)

where x is the concentration of impurity atoms Y. These replacements are allowed unless the specimen is so thin that the ad-atom position effect, or top-bottom effect, takes place [25, 26].

Results and discussion Probe function and its effect on HAADF-STEM image The effective incident-probe intensity or probe function P eff (R, Ro ) is obtained from eq. 1 as: P eff (R, Ro ) = |ϑ (R, Ro ) | 2 =|

∫

exp [ i k | | (R − Ro ) + iW(k | | )] dk | | | 2

(9)

Probe

In our computations of P eff (R, Ro ) we used beam probes with a convergence semi-angle of β =12 mrad, using Cs =1.0 mm, that corresponds to a JEM-2010FTEM/STEM microscope, operated at 200 kV, which was used in our HAADF-STEM experiments [19-24]. Fig. 1 shows theoretical intensity profiles of the probes at Ro = 0 calculated at defoci of ∆ f = − 30, −50, and −70 nm. The defocus of ∆ f = −50 nm is the Scherzer focus condition for this microscope, and gives a sharp profile with very small subsidiary maxima.

Fig. 2 (a) An experimental HAADF-STEM image of SrTiO3 crystal. (b) A theoretical HAADF-STEM image of [001]-oriented SrTiO3 crystal 35 nm thick, together with its projection along the [001]-axis. (c) Experimental (circles with error bars) and theoretical intensity profiles corresponding to (a) and (b), respectively. Profiles are along the (200) Sr-O-Sr layer and the (200) O-Ti(O)-O layer. The theoretical calculations were made using 73 partial waves and 149 partial waves, at ∆ f = -45.6 nm for the convergent incident beam of β=12 mrad.

Numerically, the integral over the probe in eq. (9) or eq. (5) is replaced by the sum of partial plane waves within β =12 mrad. The summation for Fig. 1 was made using partial incident waves whose vectors pass through 149 points subdividing the lens-aperture. These profiles agreed with those calculated using partial waves from 1249 points and 7837 points in the same aperture, and were almost the same as those calculated with only 73 points. Figs. 2a and 2b show an experimental HAADF image of a [001]-oriented SrTiO3 crystal and the corresponding theoretical image calculated at t = 35 nm and ∆ f =-45.6 nm, which were determined by the simulations in our previous paper [22]. In Figs. 2c theoretical intensity line profiles are shown along the (200) Sr-O-Sr layer and (200) O-Ti(O)-O layer, together with the experimental line profiles from the image in Fig. 2a. The theoretical line profiles were calculated using 73-point summation (dotted curves) and 149-point summation (solid curves) for the integral from eq. (5). Two calculated curves look almost the same but there are very small differences, that is, the 73 point summation exhibits narrower contrast features at atomic columns than those predicted by 149point summation. This discrepancy implies that 73-point simulation is not sufficient to appropriately simulate the experimental contras t features. The small difference between the experimental and theoretical profiles might be ascribed to effects of the source size and energy agitation of the field emission beam. The 149-point su mmation,

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Fig. 3 (a) Effective intensity profiles for incident probes at ∆f = − 50 nm calculated at β=6 and 12 mrad. (b) and (c) Theoretical HAADF-STEM images of a [110]-oriented Si crystal 20 nm thick for the probes at β= 6 and 12 mrad, respectively. (d) and (e) The corresponding line intensity profiles along the dumbbells in (b) and (c), respectively.

hence, turned out to be necessary and satisfactory criterion for the HAADF image simulation using the probe having a semi-angle of β=12 mrad. The size of objective aperture also influences the resolution of the HAADF-STEM image [3]. Fig. 3a shows theoretical intensity profiles of probes calculated at a defocus of ∆ f = − 50 nm using β =6 and 12 mrad. The theoretical HAADF-STEM images of a [110]-oriented Si crystal for the probes at β =6 and 12 mrad are shown in Figs. 3b and 3c, together with the corresponding line intensity profiles along the dumbbells, shown in Figs. 3d and 3e. Every atom in the dumbbell is resolved in the image in Fig. 3c. The use of a larger aperture, hence, increases the spatial resolution. This is the main reason for using the 12 mrad aperture in our experiments [2224]. This, however, brings a risk that the elastically scattered electrons (due to Bragg reflections from HOLZ) may contribute to the HAADF image formation. Detailed discussion on atomic resolved HAADF-STEM images of the Si crystal is to be reported els ewhere [27].

Fig. 4 (a) An experimental HAADF-STEM image of ZnO grains containing a grain boundary (indicated by arrowhead) in an Sb 2 O3-doped ZnO ceramic. (b) An enlarged image of a part in (a). (c) The projection of ZnO structure along the [120] axis. (d-i) Theoretical HAADF-STEM images of the [120]-oriented ZnO crystals of t= 20 nm calculated with probes focused at ∆f= −30 (d), −50 (e), and −70 nm (f) and of t= 40 nm calculated with probes focused at ∆f= −30 (g), −50 (h), and −70 nm (i).

Figure 4a shows an experimental HAADF image of ZnO crystal containing an Sb-rich inversion boundary [19,20]. Figure 4b is an enlarged image of a part in Fig. 4a, and Fig. 4c displays the projection of ZnO structure along the [120] axis. Simulated HAADF-STEM images of the [120]-oriented ZnO crystal are illustrated in Figs. 4d-4i, which demonstrate the specimen thickness (t) and defocus (∆f) dependence of the image contrast. The images exhibit an excess intensity at Zn columns, while the contrast is influenced rather by the defocus than by thickness. In the images with a probe at ∆ f = −30 nm, which is slightly above the Sherzer focus ( ≈ −50 nm), the bright dots of Zn columns are elongated to form bands along the a axis, as seen in Figs. 4d and 4g. On the contrary, the bright spots elongate along the c axis in the images with the defocus of ∆f = −70 nm, as seen in Figs. 4f and 4i.


The probe at ∆ f = −30 nm is rather wide as shown in Fig. 1. When the probe is located at position 1 in Fig. 4c, the neighbouring Zn columns are included in its peak (see the probe profile at ∆ f = −30 nm and full circles (Zn atoms) in Fig. 1) so that the Bloch waves are standing along these columns to cause TDS. This is the reason of the elongation of the bright spots along the a axis. When the probe at ∆ f = −70 nm is located at the position 2 in Fig. 4c, the nearest Zn atom is under the maximum peak and also the second nearest is illuminated by the subsidiary peak (see the probe profile at ∆ f = −70 nm and open circles (Zn or O atoms) in Fig. 1). When the probe is located at position 1 in Fig. 4c , the neighbouring Zn columns are near the minimum position of this probe. This could produce the elongation of the bright spots not along a axis but along c axis. The optimum focus, where a sharp and fine probe is obtained and the image can in vert directly to the object, is the Scherzer focus, as illustrated in Figs. 1 and 4. The experimental images shown in Figs. 4a and 4b seem to have been observed with a probe focused near ∆f = −30 nm. In the above case the simulation is necessary to interpret the HAADF image. A through-focal observation may also be recommended even for HAADF-STEM. In our previous work [21], we evaluated the defocus and the specimen thickness shown in Fig. 2a to be ∆ f = −45.6 nm and t = 35 nm from the simulation varying defocus and thickness. The Bi concentration of the diffusion layer in the SrTiO3 was also estimated from the simulation varying x in both eqs. (7) and (8), and the effect of the use of eq. (8) on the concentration estimation was discussed. In any case, the simulation has completed the quantitative elemental analysis. The structure analysis of the boundary indicated by an arrowhead in Fig. 4a, which might have Sb atoms, is being performed by HAADF-STEM.

The effects of lattice distortions Figure 5a shows an experimental HAADF-STEM image near the strain boundary of SrTiO3 grain with Bi-rich diffusion layer [23, 24]. Figures 5b and 5c illustrate averaged, observed intensity profiles (with error bars) along the (200) Sr-O-Sr and O-Ti(O)-O layers, like X-Y and X’Y’ in Fig. 5a, respectively. For comparison, the intensity profiles (open circles) obtained by the simulations with no lattice distortion are also given. They reveal the most pronounced expansion of the layers closed to the grain boundary, which is caused by Bi substitution. In the simulation which takes into account the lattice expansion the provided theoretical profiles (full circles) closely match the experimental profiles. The obtained Bi concentrations remarkably differ from those (indicated in brackets in Figs 5b and 5c) used for simulations without any lattice distortion, that is, they were underestimated. In Fig. 6 are illustrated Bloch wave fields |ψ(R i − Ro , z) |2 in a hypothetical SrTiO3 crystal where the lattice spacing is expanded by 10 % and in the normal SrTiO3 crystal.

Fig. 5 (a) An experimental HAADF-STEM image near the edge of a [001]-oriented grain in a boundary-layer type semiconducting SrTiO3 ceramic. (b) The experimental intensity profiles along X-Y on the (200) Sr-O-Sr layer in (a) and theoretical ones calculated at t=35 nm and ∆ f= -45.6 nm, by taking account and no account of the lattice distortion. The Bi concentrations used in the calculation are indicated. (c) The experimental intensity profiles along X’-Y’ on the (200) Sr-O-Sr layer in (a) and theoretical ones.

Figure 6a is the field intensity at a depth of z = 5 nm for the Sr column at probe position Ro =(0,0) in the latticeexpanded SrTiO3 . Figure 6c is for the Ti columns in the same crystal. Figures 6b and 6d are the corresponding intensities for the Sr and Ti columns in the undistorted SrTiO3 , respectively. The fields in the normal lattice are sharper and stronger (by about 5% at this depth) than those in the expanded lattice. Since TDS scattering is proportional to the field intensity by eq. (3), the HAADF

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Fig. 6 Theoretical wave field at a depth of 5 nm in SrTiO3 crystals caused by an incident convergent beam probe at ∆ f = −50 nm and β =12 mrad. (a) and (b) Sr site; (c) and (d) Ti site. The crystal lattice for (a) and (c) are elongated by 10 % in comparison with the lattice for (b) and (d).

image of the ideal SrTiO3 crystal might exhibit stronger contrast than the image of the realistic crystal. Similarly, the field might weaken in the SrTiO3 crystal with expanded lattice by substituted Bi atoms. As a result, this leads to the underestimation of Bi concentration by the simulation not taking into account the lattice distortions, as shown in Figs. 5b and 5c.

conclusion Some examples of the simulations of the probe function and the HAADF-STEM image are presented, following a brief description of the HAADF-STEM imaging theory based on the Bethe’s eigen-value method. The effects of the number of waves representing the convergent probe beam on the image are discussed, and it has been found


that 149 partial incident waves in the lens aperture are necessary and satisfactory condition for the accurate HAADF image simulation. The effects of aperture size and the defocus of the probe-forming lens, both of which determine the shape of the probe, are also discussed. The HAADF-STEM image contrast is greatly influenced by the lattice distortion, which is interpreted on the basis of the Bloch wave field in the crystal. The dynamical effect of the incident plane-wave in crystals causes an appreciable Bloch wave field intensity not only along the atomic column at the probe position but also along the neighbouring atomic columns, so that the HAADF-STEM image contrast does not always depend on the number or atomic number of atoms in the column. Therefore, the image simulations are indispensable for accurate interpretation of HAADF-STEM images, and provides a unique tool for the analysis of atomic column composition along the incident beam.

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