- Email: [email protected]
Effect of small crystal tilt on atomic-resolution high-angle annular dark field STEM imaging
Ultramicroscopy 92 (2002) 181–189
Effect of small crystal tilt on atomic-resolution high-angle annular dark field STEM imaging T. Yamazakia,*, M. Kawasakib, K. Watanabec, I. Hashimotoa, M. Shiojirid,e a
Department of Physics, Tokyo University of Science, 1-3 Kagurazaka-Shinjuku-ku, Tokyo 162-8601, Japan b JEOL USA inc., 11 Dearborn Rd., Peabody MA 01960, USA c Tokyo Metropolitan College of Technology, Shinagawa-ku Tokyo 140-0011, Japan d Department of Anatomy, Kanazawa Medical University, Ishikawa 920-0293, Japan e Kyoto Institute of Technology, Kyoto 606-8585, Japan Received 8 October 2001; accepted 15 December 2001
Abstract Using a slightly tilted convergent electron beam, high-angle annular dark field scanning transmission electron microscopy observations have been performed of a [0 1 1]-oriented Si crystal. A small tilt of the crystal zone axis with respect to the coma-axis of the probe-forming lens causes a difference in intensity between bright spots of a Si dumbbell. The semiangle of the beam probe and the tilting angle with respect to the specimen normal were determined by means of convergent beam micro-diffraction. The simulation using these parameters accounts for the image contrasts satisfactorily. r 2002 Elsevier Science B.V. All rights reserved. PACS: 68.37.Lp; 61.66.f Keywords: HAADF; STEM; Crystal tilt; Micro-diffraction; Si; Bethe method
1. Introduction In this decade, high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) has become one of the most powerful tools to visualize materials structures at atomic resolution [1–10]. Intensive HAADF STEM investigations have reported its potential for crystal and defect structure analysis (see for example, Refs. [11–19]). Generally, HAADF
*Corresponding author. E-mail address: [email protected] (T. Yamazaki).
STEM provides incoherent images without any phase problem, and can be therefore directly inverted to the object without additional image simulations [19]. In a series of HAADF STEM investigations [20–22], we have shown that the image simulation is indispensable for quantification of experimental HAADF STEM images and as such provides a valuable composition analysis for every atomic column along the incident beam. We have also found artificial bright spots, on no atomic columns along the electron beam, in some of a through-focal HAADF STEM images of a [0 1 1]-orientated Si [23]. They are ascribed to the subsidiary maximum of the incident probe which
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Fig. 1. (a)–(c) Experimental through-focal HAADF STEM images of a Si crystal recorded with a probe of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad: (d)–(f) Noise-filtered images of (a)–(c), respectively. (g)–(i) Calculated HAADF STEM images of a [0 1 1]-oriented Si crystal 35 nm thick, at Df ¼ 40; 60 and 70 nm for the tilted probe. (j)–(l) Experimental and calculated intensity line profiles along the dumbbells, corresponding to (d)–(f) and (g)–(i), respectively. Experimental intensities are indicated with error bars. The positions of dumbbells are shown by circles.
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forms wave field along the neighboring atomic columns to scatter incoherent electrons [24]. Effects of the defocus and aperture size of the probe-forming lens, the angular range of a detector and the thickness of a specimen, on ADF STEM imaging, have been investigated, in order to understand the HAADF STEM image formation and to get high-resolution images [4,9,21,22,25–30]. It has been known that the channeling of electrons in the crystal plays an important role in HAADF STEM imaging. Therefore, the axis of an incident probe was implicitly assumed to be exactly along the crystal zone axis, in spite of practical difficulty in microscope operation. McGibbon et al. [11] described that a HAADF STEM image is robust and insensitive to small specimen tilt, and any notice has not been taken of the effect of the tilting so far. This paper investigates the effect of small crystal tilt on atomic-resolution HAADF STEM images using a [0 1 1]-orientated Si crystal, which is a common test object for high-resolution electron microscopy. Experimental images recorded with probes at different probe-forming lens defoci are analyzed by image simulation based on the Bethe method.
2. Experimental procedure P-type silicon wafers produced by the Czochraski method were used in the present experiment. Specimens were prepared by mechanical polishing and two-step ion milling, as previously reported [20,23]. HAADF STEM observations were performed with a JEM-2010F-TEM/STEM having an annular detector, operated at 200 keV; together with the corresponding convergent beam microdiffraction. The spherical aberration Cs of the probe-forming lens was 1:0 mm and the angular range of the annular detector was 60–160 mrad: The illumination system of the microscope was precisely aligned so as to get coma-free condition. Image processing was performed by Fourier filtering, where a mask of 2 nm1 diameter was used for each spot in a diffractogram of the image. Altering mask size, from 2 to 3 nm1 ; does not have much of an effect on the final conclusion.
Fig. 2. Microdiffraction patterns of the [0 1 1]-Si crystal recorded with an aperture used for HAADF STEM (a) and with a different aperture (b), observed at the same camera length. The aperture in (a) is 4 times as large as that in (b). The semiangle of the aperture in (b) is estimated to be 3:4 mrad from the diffraction pattern.
Using the electron beam that had passed through the detector aperture, parallel electronenergy-loss spectroscopy (PEELS) was simultaneously carried out to estimate the sample thickness.
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Fig. 3. Schematic diagram of the non-tilted (a) and crystal tilted (b) with respect to the [0 1 1] zone axis. a is the semiangle of the probes. The crystal is tilted about the ½0 1% 1 axis toward the [1 0 0] direction. The tilting angle y is the angle between the probe axis and the [0 1 1] axis.
The PEELS was acquired with a Gatan Digi PEELS model 766 and provides an estimate of sample thickness to be 40 nm at the observed area. The simulations of HAADF STEM images and intensity line profiles were carried out by a method developed by us [31]. The method provides ADF STEM images formed by coherent Bragg scattering and incoherent TDS, simultaneously or separately, in a short time. In the present simulation, we calculated only the contribution from TDS because the coherent scattering is negligible to images recorded with the 60–160 mrad detector [31].
3. Results and discussion Figs. 1(a)–(c) show a through-focal series of experimental HAADF STEM images of a [0 1 1]oriented Si crystal, and Figs. 1(d)–(f) show the corresponding noise-filtered images. The processing decreases drastically noises and thereby provides the characteristics of these atomic images. Bright spots, indicating Si atomic columns, in dumbbells are not resolved in Fig. 1(d), but they are clearly resolved in Figs. 1(e) and (f). Artificial spots, which were reported in our previous paper
[23], also appear in images taken with the small tilting probe, as seen in Figs. 1(e) and (f) (see arrowheads). It is seen in Fig. 1(f) that the dumbbells are asymmetrical or that the two spots in each dumbbell are appreciably different in intensity. The experimental intensity profiles, along the dumbbells in Figs. 1(d)–(f), are shown by curves with error bars in Figs. 1(j)–(l), respectively. The curves make the asymmetry of the dumbbells clear. The aperture size of the probe-forming lens or the semiangle of the incident convergent beam greatly influences the HAADF STEM images [22], because it assigns the probe function together with the defocus. Figs. 2(a) and (b) show diffraction patterns taken using different apertures with the same camera length. The aperture of the probe for Fig. 2(a) was used to the present HAADF STEM observations and has a diameter indicated by a circle corresponding to the 000 disk. The semiangle a of the probe (see Fig. 3) was evaluated to be 13:5 mrad using Fig. 2(b) as a measure. The tilt angle of the crystal was measured from a micro-diffraction pattern taken with the probe which provided the images in Figs. 1(a)–(c). The simulation of this diffraction pattern, shown in Fig. 4(a), was carried out, varying the tilt angle and the film thickness. Some of calculated patterns
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Fig. 4. (a) Experimental convergent beam micro-diffraction pattern of the area shown in Fig. 1(a)–(c). (b)–(d) Calculated convergent beam micro-diffraction patterns of a [0 1 1]-oriented Si crystal 35 nm thick, taken with crystal tilted by y ¼ 0; 1:6; and 2:5 mrad about the ½0 1% 1 axis toward the [1 0 0] direction as seen in Fig. 2(b).
are shown in Figs. 4(b)–(d). From the agreement of Fig. 4(c) with 4(a), it was deduced that the crystal was tilted by y ¼ 1:6 mrad about the ½0 1% 1 axis with respect to the coma-free axis of the probe-forming lens, as schematically shown in Fig. 3(b), and that the film thickness is 35 nm nearly equal to that estimated by PEELS. Then, we carried out simulations for the images in Figs. 1(d)–(f) using these parameters. The defocus values were experimentally estimated from Ronchigram [32] and the steps of the probe-
forming lens current knob, and confirmed by simulation. The simulated images are shown in Figs. 1(g)–(i), and the intensity profiles from these simulated images are also shown in Figs. 1(j)–(l). They account for the experimental images quite well. The small disagreement between the experimental and calculated intensities may be ascribed to the source size and energy spread of the field emission beam, which were not taken into account in the present calculation. We also calculated the images and the line profiles at the same condition
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Fig. 5. Calculated probe functions of the incident beam of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad on the (0 1 1) surface of a Si crystal. (a) Df ¼ 40 nm: (b) Df ¼ 60 nm: (c) Df ¼ 70 nm: (d) Projection of atoms in the Si crystal along the [0 1 1] direction. Filled circles in (a)–(c) denote atomic column positions for the beam located at A and open circles those for the beam located at C.
using the non-tilted crystal. The intensity of the two Si spots became exactly the same. Hence, it was interpreted that the small crystal tilt makes a difference in intensity between the spots in the dumbbells, but gives no influence on spot positions. Next, we examine the influence of the crystal tilt on the probe function, which is the effective probe intensity on the crystal surface. The probe functions used for imaging of Fig. 1 were calculated and are shown in Figs. 5(a) and (c), where they are displayed as the intensity on the (0 1 1) surface of the Si. The line profiles along X–X0 in these images
are shown in Fig. 6(a). Strong subsidiary peaks which appear around the main peak at Df ¼ 70 nm cause artificial spots on no atomic columns (see arrowheads in Fig. 1). This was interpreted in our previous paper [23] and can be seen from geometry that the subsidiary peaks are placed on neighboring atomic columns when the probe is centered at D (see Figs. 5 and 6). The wider probe at Df ¼ 40 nm yields unresolved dumbbells (see Fig. 1(d)) when the probe is centered at B (also see Figs. 5(d) and 6). The tilting probe function can thus give a simple explanation for these characteristic images [23].
T. Yamazaki et al. / Ultramicroscopy 92 (2002) 181–189
Fig. 6. Calculated probe functions for the incident beam of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad (a) and non-tilted (b). (Df ¼ 40; 60 and 70 nm). The functions are indicated along the [1 0 0] direction. A–D denote the positions on the surface of the Si crystal indicated in Fig. 5(d).
Fig. 6(b) shows probe functions for non-tilted incident beams, which are almost similar to those for the tilted beams in Fig. 6(a) but are completely
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symmetric. In any case, the asymmetrical dumbbells, or the difference in scattered TDS intensity between A and C where the relatively tilted probe is located, cannot be predicted intuitively from the probe function of the tilted crystal, in a similar manner where the artificial spots are explained. The probe function of an ideal lens of Cs ¼ 0 at Df ¼ 0 was calculated on the tilted crystal and is shown in Fig. 7(a). This probe function is symmetrical, which can be easily proved. The dumbbells, however, are asymmetrical in a HAADF STEM image calculated for this probe, as shown in Figs. 7(b) and (c). This confirms that the asymmetry of the dumbbell spots cannot be explained by the probe function, because it is an effective intensity losing the information about Kz on the crystal surface which is shown in Fig. 3. The wave fields of the channeling electrons in the crystal were calculated for the tilted beam that is centered at column L or column R in Fig. 7(b), and their intensities are shown along the depth in Fig. 7(d). The excited wave field is stronger along column R than column L, as seen in Fig. 7(e) that shows the line profile correspond to Fig. 7(d). This causes the HAADF STEM image contrast as shown in Figs. 7(b) and (c). Hence, the only way to explain these asymmetric contrast is the image simulation based on the dynamical diffraction theory and the electron optics taking account of defocus and spherical aberrations, as presented in this paper. A small crystal tilt, which makes the dumbbell spots asymmetrical, may occur during usual microscope operation, so that special care to the beam alignment and the crystal setting needs for quantitative HAADF STEM applications such as the determination of crystallographic polarity or the compositional analysis of impurity atoms. We have not dealt with the beam tilt, where the principle ray of an incident probe is tilted against the optical axis or the coma-free axis of probeforming lens. In this case, even if the crystal zone axis is parallel to the optical axis, the beam tilt might cause more complicate effects on HAADF STEM images in the presence of the comaaberration. The convergent beam micro-diffraction is helpful for the check of a beam tilt, as seen in Fig. 4.
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Fig. 7. (a) Calculated probe function of a beam of a ¼ 13:5 mrad tilted at y ¼ 1:6 mrad; focused at Df ¼ 0 with an ideal lens of Cs ¼ 0: (b) Calculated HAADF STEM image of a [1 1 0]-oriented Si crystal 35 nm thick, using the probe shown in (a). (c) Intensity line profile along the dumbbells in (b). Column R is brighter than column L in a dumbbell. (d) Calculated intensity distribution of the wave field along columns L or R where the incident probe is focused. (e) Line profile of the intensity in (d).
4. Conclusions A thorough-focal series of atomic resolved HAADF STEM images of a [0 1 1]-orientated Si crystal was experimentally obtained using a relatively small-tilted crystal. The determination of important experimental parameters, such as the crystal tilt, semiangle of the incident convergent beam, annular detector angle, defocus and thickness, allowed us to account for the experimental images by the simulation. As a result, it was found that a small crystal tilt causes a difference in intensity between bright spots in a dumbbell but does not change the spot positions. It is empha-
sized that special care to the beam alignment and the crystal setting needs for quantitative HAADF STEM applications such as the determination of crystallographic polarity or the compositional analysis.
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[19] S.J. Pennycook, P.D. Nellist, in: D.G. Rickerby, G. Valdre, U. Valdre (Eds.), Impact of Electron and Scanning Probe Microscopy on Materials Research, Kluwer Academic Publishers, Dordrecht, 1999, pp. 161–207. [20] T. Yamazaki, K. Watanabe, Y. Kikuchi, M. Kawasaki, I. Hashimoto, M. Shiojiri, Phys. Rev. B 81 (2000) 13 833. [21] M. Kawasaki, T. Yamazaki, S. Sato, K. Watanabe, M. Shiojiri, Philos. Mag. A 81 (2001) 245. [22] T. Yamazaki, K. Watanabe, A. Recnik, M. Ceh, M. Kawasaki, M. Shiojiri, J. Electron Microsc. 49 (2000) 753. [23] K. Watanabe, T. Yamazaki, Y. Kikuchi, Y. Kotaka, M. Kawasaki, I. Hashimoto, M. Shiojiri, Phys. Rev. B 63 (2001) 85 316. [24] T. Yamazaki, M. Kawasaki, K. Watanabe, I. Hashimoto, M. Shiojiri, J. Electron Microsc. 50 (2001) 517. [25] E.J. Kirkland, R.F. Loane, J. Silcox, Ultramicroscopy 23 (1987) 77. [26] R.F. Loane, E.J. Kirkland, J. Silcox, Acta Crystallogr. Sect. A: Found. Crystallogr. 44 (1988) 912. [27] R.F. Loane, P. Xu, J. Silcox, Ultramicroscopy 40 (1992) 77. [28] S. Hillyard, R.F. Loane, J. Silcox, Ultramicroscopy 49 (1993) 14. [29] S. Hillyard, R.F. Loane, J. Silcox, Ultramicroscopy 52 (1993) 325. [30] S. Hillyard, J. Silcox, Ultramicroscopy 58 (1995) 6. [31] K. Watanabe, T. Yamazaki, I. Hashimoto, M. Shiojiri, Phys. Rev. B 64 (2001) 115 432. [32] E.M. James, N.D. Browning, Ultramicroscopy 78 (1999) 125.
Effect of small crystal tilt on atomic-resolution high-angle annular dark field STEM imaging T. Yamazakia,*, M. Kawasakib, K. Watanabec, I. Hashimotoa, M. Shiojirid,e a
Department of Physics, Tokyo University of Science, 1-3 Kagurazaka-Shinjuku-ku, Tokyo 162-8601, Japan b JEOL USA inc., 11 Dearborn Rd., Peabody MA 01960, USA c Tokyo Metropolitan College of Technology, Shinagawa-ku Tokyo 140-0011, Japan d Department of Anatomy, Kanazawa Medical University, Ishikawa 920-0293, Japan e Kyoto Institute of Technology, Kyoto 606-8585, Japan Received 8 October 2001; accepted 15 December 2001
Abstract Using a slightly tilted convergent electron beam, high-angle annular dark field scanning transmission electron microscopy observations have been performed of a [0 1 1]-oriented Si crystal. A small tilt of the crystal zone axis with respect to the coma-axis of the probe-forming lens causes a difference in intensity between bright spots of a Si dumbbell. The semiangle of the beam probe and the tilting angle with respect to the specimen normal were determined by means of convergent beam micro-diffraction. The simulation using these parameters accounts for the image contrasts satisfactorily. r 2002 Elsevier Science B.V. All rights reserved. PACS: 68.37.Lp; 61.66.f Keywords: HAADF; STEM; Crystal tilt; Micro-diffraction; Si; Bethe method
1. Introduction In this decade, high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) has become one of the most powerful tools to visualize materials structures at atomic resolution [1–10]. Intensive HAADF STEM investigations have reported its potential for crystal and defect structure analysis (see for example, Refs. [11–19]). Generally, HAADF
*Corresponding author. E-mail address: [email protected] (T. Yamazaki).
STEM provides incoherent images without any phase problem, and can be therefore directly inverted to the object without additional image simulations [19]. In a series of HAADF STEM investigations [20–22], we have shown that the image simulation is indispensable for quantification of experimental HAADF STEM images and as such provides a valuable composition analysis for every atomic column along the incident beam. We have also found artificial bright spots, on no atomic columns along the electron beam, in some of a through-focal HAADF STEM images of a [0 1 1]-orientated Si [23]. They are ascribed to the subsidiary maximum of the incident probe which
0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 2 ) 0 0 1 3 1 - 6
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Fig. 1. (a)–(c) Experimental through-focal HAADF STEM images of a Si crystal recorded with a probe of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad: (d)–(f) Noise-filtered images of (a)–(c), respectively. (g)–(i) Calculated HAADF STEM images of a [0 1 1]-oriented Si crystal 35 nm thick, at Df ¼ 40; 60 and 70 nm for the tilted probe. (j)–(l) Experimental and calculated intensity line profiles along the dumbbells, corresponding to (d)–(f) and (g)–(i), respectively. Experimental intensities are indicated with error bars. The positions of dumbbells are shown by circles.
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forms wave field along the neighboring atomic columns to scatter incoherent electrons [24]. Effects of the defocus and aperture size of the probe-forming lens, the angular range of a detector and the thickness of a specimen, on ADF STEM imaging, have been investigated, in order to understand the HAADF STEM image formation and to get high-resolution images [4,9,21,22,25–30]. It has been known that the channeling of electrons in the crystal plays an important role in HAADF STEM imaging. Therefore, the axis of an incident probe was implicitly assumed to be exactly along the crystal zone axis, in spite of practical difficulty in microscope operation. McGibbon et al. [11] described that a HAADF STEM image is robust and insensitive to small specimen tilt, and any notice has not been taken of the effect of the tilting so far. This paper investigates the effect of small crystal tilt on atomic-resolution HAADF STEM images using a [0 1 1]-orientated Si crystal, which is a common test object for high-resolution electron microscopy. Experimental images recorded with probes at different probe-forming lens defoci are analyzed by image simulation based on the Bethe method.
2. Experimental procedure P-type silicon wafers produced by the Czochraski method were used in the present experiment. Specimens were prepared by mechanical polishing and two-step ion milling, as previously reported [20,23]. HAADF STEM observations were performed with a JEM-2010F-TEM/STEM having an annular detector, operated at 200 keV; together with the corresponding convergent beam microdiffraction. The spherical aberration Cs of the probe-forming lens was 1:0 mm and the angular range of the annular detector was 60–160 mrad: The illumination system of the microscope was precisely aligned so as to get coma-free condition. Image processing was performed by Fourier filtering, where a mask of 2 nm1 diameter was used for each spot in a diffractogram of the image. Altering mask size, from 2 to 3 nm1 ; does not have much of an effect on the final conclusion.
Fig. 2. Microdiffraction patterns of the [0 1 1]-Si crystal recorded with an aperture used for HAADF STEM (a) and with a different aperture (b), observed at the same camera length. The aperture in (a) is 4 times as large as that in (b). The semiangle of the aperture in (b) is estimated to be 3:4 mrad from the diffraction pattern.
Using the electron beam that had passed through the detector aperture, parallel electronenergy-loss spectroscopy (PEELS) was simultaneously carried out to estimate the sample thickness.
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Fig. 3. Schematic diagram of the non-tilted (a) and crystal tilted (b) with respect to the [0 1 1] zone axis. a is the semiangle of the probes. The crystal is tilted about the ½0 1% 1 axis toward the [1 0 0] direction. The tilting angle y is the angle between the probe axis and the [0 1 1] axis.
The PEELS was acquired with a Gatan Digi PEELS model 766 and provides an estimate of sample thickness to be 40 nm at the observed area. The simulations of HAADF STEM images and intensity line profiles were carried out by a method developed by us [31]. The method provides ADF STEM images formed by coherent Bragg scattering and incoherent TDS, simultaneously or separately, in a short time. In the present simulation, we calculated only the contribution from TDS because the coherent scattering is negligible to images recorded with the 60–160 mrad detector [31].
3. Results and discussion Figs. 1(a)–(c) show a through-focal series of experimental HAADF STEM images of a [0 1 1]oriented Si crystal, and Figs. 1(d)–(f) show the corresponding noise-filtered images. The processing decreases drastically noises and thereby provides the characteristics of these atomic images. Bright spots, indicating Si atomic columns, in dumbbells are not resolved in Fig. 1(d), but they are clearly resolved in Figs. 1(e) and (f). Artificial spots, which were reported in our previous paper
[23], also appear in images taken with the small tilting probe, as seen in Figs. 1(e) and (f) (see arrowheads). It is seen in Fig. 1(f) that the dumbbells are asymmetrical or that the two spots in each dumbbell are appreciably different in intensity. The experimental intensity profiles, along the dumbbells in Figs. 1(d)–(f), are shown by curves with error bars in Figs. 1(j)–(l), respectively. The curves make the asymmetry of the dumbbells clear. The aperture size of the probe-forming lens or the semiangle of the incident convergent beam greatly influences the HAADF STEM images [22], because it assigns the probe function together with the defocus. Figs. 2(a) and (b) show diffraction patterns taken using different apertures with the same camera length. The aperture of the probe for Fig. 2(a) was used to the present HAADF STEM observations and has a diameter indicated by a circle corresponding to the 000 disk. The semiangle a of the probe (see Fig. 3) was evaluated to be 13:5 mrad using Fig. 2(b) as a measure. The tilt angle of the crystal was measured from a micro-diffraction pattern taken with the probe which provided the images in Figs. 1(a)–(c). The simulation of this diffraction pattern, shown in Fig. 4(a), was carried out, varying the tilt angle and the film thickness. Some of calculated patterns
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Fig. 4. (a) Experimental convergent beam micro-diffraction pattern of the area shown in Fig. 1(a)–(c). (b)–(d) Calculated convergent beam micro-diffraction patterns of a [0 1 1]-oriented Si crystal 35 nm thick, taken with crystal tilted by y ¼ 0; 1:6; and 2:5 mrad about the ½0 1% 1 axis toward the [1 0 0] direction as seen in Fig. 2(b).
are shown in Figs. 4(b)–(d). From the agreement of Fig. 4(c) with 4(a), it was deduced that the crystal was tilted by y ¼ 1:6 mrad about the ½0 1% 1 axis with respect to the coma-free axis of the probe-forming lens, as schematically shown in Fig. 3(b), and that the film thickness is 35 nm nearly equal to that estimated by PEELS. Then, we carried out simulations for the images in Figs. 1(d)–(f) using these parameters. The defocus values were experimentally estimated from Ronchigram [32] and the steps of the probe-
forming lens current knob, and confirmed by simulation. The simulated images are shown in Figs. 1(g)–(i), and the intensity profiles from these simulated images are also shown in Figs. 1(j)–(l). They account for the experimental images quite well. The small disagreement between the experimental and calculated intensities may be ascribed to the source size and energy spread of the field emission beam, which were not taken into account in the present calculation. We also calculated the images and the line profiles at the same condition
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Fig. 5. Calculated probe functions of the incident beam of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad on the (0 1 1) surface of a Si crystal. (a) Df ¼ 40 nm: (b) Df ¼ 60 nm: (c) Df ¼ 70 nm: (d) Projection of atoms in the Si crystal along the [0 1 1] direction. Filled circles in (a)–(c) denote atomic column positions for the beam located at A and open circles those for the beam located at C.
using the non-tilted crystal. The intensity of the two Si spots became exactly the same. Hence, it was interpreted that the small crystal tilt makes a difference in intensity between the spots in the dumbbells, but gives no influence on spot positions. Next, we examine the influence of the crystal tilt on the probe function, which is the effective probe intensity on the crystal surface. The probe functions used for imaging of Fig. 1 were calculated and are shown in Figs. 5(a) and (c), where they are displayed as the intensity on the (0 1 1) surface of the Si. The line profiles along X–X0 in these images
are shown in Fig. 6(a). Strong subsidiary peaks which appear around the main peak at Df ¼ 70 nm cause artificial spots on no atomic columns (see arrowheads in Fig. 1). This was interpreted in our previous paper [23] and can be seen from geometry that the subsidiary peaks are placed on neighboring atomic columns when the probe is centered at D (see Figs. 5 and 6). The wider probe at Df ¼ 40 nm yields unresolved dumbbells (see Fig. 1(d)) when the probe is centered at B (also see Figs. 5(d) and 6). The tilting probe function can thus give a simple explanation for these characteristic images [23].
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Fig. 6. Calculated probe functions for the incident beam of a ¼ 13:5 mrad; tilted at y ¼ 1:6 mrad (a) and non-tilted (b). (Df ¼ 40; 60 and 70 nm). The functions are indicated along the [1 0 0] direction. A–D denote the positions on the surface of the Si crystal indicated in Fig. 5(d).
Fig. 6(b) shows probe functions for non-tilted incident beams, which are almost similar to those for the tilted beams in Fig. 6(a) but are completely
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symmetric. In any case, the asymmetrical dumbbells, or the difference in scattered TDS intensity between A and C where the relatively tilted probe is located, cannot be predicted intuitively from the probe function of the tilted crystal, in a similar manner where the artificial spots are explained. The probe function of an ideal lens of Cs ¼ 0 at Df ¼ 0 was calculated on the tilted crystal and is shown in Fig. 7(a). This probe function is symmetrical, which can be easily proved. The dumbbells, however, are asymmetrical in a HAADF STEM image calculated for this probe, as shown in Figs. 7(b) and (c). This confirms that the asymmetry of the dumbbell spots cannot be explained by the probe function, because it is an effective intensity losing the information about Kz on the crystal surface which is shown in Fig. 3. The wave fields of the channeling electrons in the crystal were calculated for the tilted beam that is centered at column L or column R in Fig. 7(b), and their intensities are shown along the depth in Fig. 7(d). The excited wave field is stronger along column R than column L, as seen in Fig. 7(e) that shows the line profile correspond to Fig. 7(d). This causes the HAADF STEM image contrast as shown in Figs. 7(b) and (c). Hence, the only way to explain these asymmetric contrast is the image simulation based on the dynamical diffraction theory and the electron optics taking account of defocus and spherical aberrations, as presented in this paper. A small crystal tilt, which makes the dumbbell spots asymmetrical, may occur during usual microscope operation, so that special care to the beam alignment and the crystal setting needs for quantitative HAADF STEM applications such as the determination of crystallographic polarity or the compositional analysis of impurity atoms. We have not dealt with the beam tilt, where the principle ray of an incident probe is tilted against the optical axis or the coma-free axis of probeforming lens. In this case, even if the crystal zone axis is parallel to the optical axis, the beam tilt might cause more complicate effects on HAADF STEM images in the presence of the comaaberration. The convergent beam micro-diffraction is helpful for the check of a beam tilt, as seen in Fig. 4.
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Fig. 7. (a) Calculated probe function of a beam of a ¼ 13:5 mrad tilted at y ¼ 1:6 mrad; focused at Df ¼ 0 with an ideal lens of Cs ¼ 0: (b) Calculated HAADF STEM image of a [1 1 0]-oriented Si crystal 35 nm thick, using the probe shown in (a). (c) Intensity line profile along the dumbbells in (b). Column R is brighter than column L in a dumbbell. (d) Calculated intensity distribution of the wave field along columns L or R where the incident probe is focused. (e) Line profile of the intensity in (d).
4. Conclusions A thorough-focal series of atomic resolved HAADF STEM images of a [0 1 1]-orientated Si crystal was experimentally obtained using a relatively small-tilted crystal. The determination of important experimental parameters, such as the crystal tilt, semiangle of the incident convergent beam, annular detector angle, defocus and thickness, allowed us to account for the experimental images by the simulation. As a result, it was found that a small crystal tilt causes a difference in intensity between bright spots in a dumbbell but does not change the spot positions. It is empha-
sized that special care to the beam alignment and the crystal setting needs for quantitative HAADF STEM applications such as the determination of crystallographic polarity or the compositional analysis.
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