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Microscopy
, Volume 67 (4) – Aug 1, 2018

12 pages

/lp/ou_press/annular-dark-field-scanning-confocal-electron-microscopy-studied-using-tn7QLrjCMH

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0022-0744
- eISSN
- 1477-9986
- D.O.I.
- 10.1093/jmicro/dfy023
- Publisher site
- See Article on Publisher Site

Abstract Annular dark-field scanning confocal electron microscopy (ADF-SCEM) has been studied using multislice simulations. Thermal diffuse scattering was considered in the calculations. Geometric aberrations of the lenses were introduced. A finite-sized pinhole was taken into consideration, in addition to an ideal point pinhole. ADF-SCEM images of Al crystals aligned along a zone-axis exhibit elongated contrast along the optic axis. Results of simulations suggest that if geometric aberrations of an imaging lens are corrected, depth resolution in ADF-SCEM can be improved by employing a large collection semi-angle of an annular aperture, even with a finite pinhole. scanning confocal electron microscopy (SCEM), 3D characterization, multislice simulation, electron channeling, Al–Cu alloy, Guinier–Preston zone Introduction Scanning confocal electron microscopy (SCEM) is a novel technique that can enable three-dimensional (3D) microstructural characterization of materials with atomic horizontal resolution. Fig. 1 depicts a schematic diagram of the optic configuration of SCEM. This diagram can be understood as the combination of scanning transmission electron microscopy (STEM) for the pre-sample geometry and conventional transmission electron microscopy (TEM) for the post-sample geometry. The condenser lens is used to form an electron probe. The image of the electron probe is magnified using the imaging lens. Either a circular aperture or an annular aperture is inserted in the back focal plane of the imaging lens and these two confocal modes are referred to as bright-field (BF) SCEM and annular dark-field (ADF) SCEM, respectively. The ADF-SCEM mode allows electrons scattered by the sample to be collected, as illustrated in Fig. 1. A pinhole is placed in front of the detector. The pinhole plays a key role in blocking electrons under an out-of-focus condition improving the image resolution along the optic axis. Both the pinhole and the detector can be considered to be on the imaging plane of the imaging lens. The optic configuration for SCEM can be realized even in a TEM equipped with two spherical aberration correctors [1]. Energy-filtered SCEM (EF-SCEM) is also available by placing an energy filter in the SCEM configuration and collecting inelastically scattered electrons [2]. The EF-SCEM has a potential of determining the locations of isolated impurity atoms embedded within a matrix [3]. Fig.1. View largeDownload slide Schematic diagram of the configuration of the annular dark-field scanning confocal electron microscopy (ADF-SCEM). Fig.1. View largeDownload slide Schematic diagram of the configuration of the annular dark-field scanning confocal electron microscopy (ADF-SCEM). SCEM observations have been first implemented by Frigo et al. [4] and Zaluzec [5]. They developed an instrument equipped with coils to raster scan the electron probe over the specimen as a common STEM, but also other coils to descan the transmitted electrons. The researchers demonstrated that sub-surface structure of semiconductor devices can be revealed with a horizontal resolution of <100 nm. Hashimoto et al. [6] and Takeguchi et al. [7] have established an alternative technique to implement SCEM observations, using a fixed electron-optics configuration without beam-scanning, where the sample is scanned instead using a piezo-driven specimen stage. With this stage-scanning system, SCEM images of lattice fringes with 0.235 nm spacing in gold nanoparticles were observed. In addition, 3D SCEM observations have been performed by translating the sample in both parallel and perpendicular directions to the optic axis [8–16]. Xin et al. [17] also used a piezo-driven specimen stage and revealed that valence electron energy losses can be used for signals of a SCEM mode. More recently, Zheng et al. [18] have established the off-axis SCEM as a technique to conduct chemical mapping without a spectrometer. However, it is known that both BF- and ADF-SCEM modes suffer from the elongation of the images along the optic axis. The problem of the elongation has been investigated from the viewpoint of point spread functions (PSFs) of the condenser lens and imaging lens, and also optical transfer functions (OTFs) as a 3D Fourier transform of the PSFs [19–22]. It has been revealed that there is a missing cone of information transfer in the OTFs and this can cause the elongation of the images in the BF- and ADF-SCEM modes. If the sample is an isolated atom or a nanoparticle embedded in an amorphous matrix, the PSF/OTF model can be applied since the incident probe intensity may not change significantly during propagation within the sample. On the contrary, the PSF/OTF model breaks down when the sample is a crystal aligned along a zone-axis because electron channeling occurs along atomic columns. This makes it impossible to express image formation as a convolution between the PSF and a function describing the structure. Hence, the elongation of BF- and ADF-SCEM images of an aligned crystal should be investigated by considering the electron channeling, rather than from the viewpoint of the PSFs and OTFs. BF- and ADF-SCEM under channeling effects have been studied using a multislice method [23–26] and a Bloch wave-based method [27] while considering electron propagation within a sample properly. It should be noted that an infinitesimal pinhole is assumed in most of the previous studies. However, the pinhole used in experimental observations had a finite diameter such as 0.32 nm [15], 0.38 nm [12] and 0.60 nm [10]. A measurable diameter of a pinhole may be chosen experimentally to ensure that the electron probe is kept inside the pinhole during a prolonged observation. In addition, the use of a pinhole with a large diameter can increase the SCEM signal detected. Zhang et al. [13] has studied effects of different diameters of the pinhole on ADF-SCEM images of an amorphous Ge/C film with gold nanoparticles. Also, in the previous studies, aberration-free lenses were assumed as in an ideal case, or only third- and fifth-order spherical aberrations were taken into consideration. Inelastically scattered electrons were considered only in a few simulations [24,27]. In this paper, we conduct multislice simulations using a pinhole with a finite diameter to study ADF-SCEM images of face-centered cubic (fcc) Al crystals under channeling effects. Geometric aberrations up to fifth-order and thermal diffuse scattering (TDS) are taken into consideration. Besides, effects of difference in the collection semi-angle of the annular aperture are investigated. Furthermore, the possibility of determination of the depths of nanometer-sized Guinier–Preston 1 (GP1) zones in an Al crystal is investigated. GP1 zones are viewed as coherent Cu-rich plates, a single plane thick, and parallel to the {100} planes of the α-Al matrix of a fcc structure [28–32]. Results of this study reveal that if the geometric aberrations of the imaging lens are corrected to some extent, the elongation of the ADF-SCEM images along the optic axis can be reduced by employing a large collection semi-angle of the annular aperture even when a finite diameter of the pinhole is used. The remnant of this paper is organized as follows. First, we introduce a formalism of the multislice simulations for ADF-SCEM intensity. Next, simulation conditions are described in detail. Results of simulations are then presented and discussed. Finally, concluding remarks are presented. A formalism of the multislice simulations for ADF-SCEM intensity The wave function of an electron probe as a function of a 2D position vector r in a perpendicular plane to the optic axis, ψp(r, rp), can be expressed for a given probe position rp as follows: ψp(r,rp)=FT−1{AC(k)exp[−iχ(α,ϕ)+2πik⋅rp]} (1) here FT−1 represents a 2D inverse Fourier transform. Ac(k) is the function representing a condenser aperture and k is the 2D wave vector in reciprocal space. The magnitude of the wave vector k, k is associated with a polar angle α from the optic axis and the electron wavelength λ, as k = α/λ. Ac(k) is one when α ≤ αmax, which is the convergence semi-angle, and otherwise zero. χ(α, ϕ) in Eq. (1) denotes a phase shift caused by geometric aberrations of a lens. In this work, the following definition of χ(α, ϕ) is adopted [33]: χ(α,ϕ)=2πλ∑m,n1n+1Cnmαn+1cos[m(ϕ−ϕnm)] (2) here Cnm denotes the aberration coefficient, n the order of aberration, m the azimuthal symmetry around the optic axis, ϕ the azimuthal angel, and ϕnm the azimuthal orientation of aberration. ϕ is defined as tan−1(ky/kx), where kx and ky denote the orthogonal components of k. For each n, m takes values from 0 to n + 1. If n is odd, m is even and vice versa [34]. This notation of aberrations is similar to that introduced by Krivanek [35]. C10 stands for defocus. Δfc in Fig. 1 is the defocus for the condenser lens. Positive values of C10 and Δfc are related as C10 = −Δfc. The wave function of the electron probe, ψp(r, rp) in Eq. (1), interacts with the sample according to the common multislice method for STEM [33]. As a result, a transmitted wave function after passing though the sample, ψt(r, rp), is obtained. ψt(r, rp) is then Fourier-transformed and its phase is modulated by the geometric aberrations of the imaging lens. A wave function for the phase-modulated wave, Ψm(k, rp), is expressed in the back focal plane of the imaging lens as follows: Ψm(k,rp)=FT[ψt(r,rp)]exp[−iχ(α,ϕ)] (3) FT denotes a 2D Fourier transform. The exponential function in Eq. (3) is calculated using Eq. (2), although Cnm and ϕnm of the imaging lens may be different from those of the condenser lens. As for the imaging lens, C10 = Δfc – t where t is the thickness of the sample. Ψm(k, rp) in Eq. (3) is then multiplied by a function Aa(k) representing an annular aperture. Given that βin and βout are the inner and outer collection semi-angles of the annular aperture, respectively, Aa(k) is one when βin ≤ kλ ≤ βout, and otherwise zero. A wave function on the detector plane, ψi(r, rp), is calculated with the following equation: ψi(r,rp)=FT−1{Ψm(k,rp)Aa(k)} (4) The intensity distribution of the ψi(r, rp) is limited by a pinhole, which is represented by a function Ap(r). Given that r is the magnitude of the position vector r and dp is the pinhole diameter, Ap(r) is one when r ≤ dp/2, and otherwise zero. As proposed in the literature [13], ADF-SCEM intensity I(rp) at the probe position rp is calculated by integrating the squared magnitude of ψi(r, rp) multiplied by Ap(r): I(rp)=∫|ψi(r,rp)|2Ap(r)dr (5) Simulation details A program for the multislice simulations for ADF-SCEM has been developed in this work. Elastic scattering factors parameterized by Peng et al. [36] are employed up to a scattering vector of 6.0 Å−1. The frozen phonon approach [33] is employed to include the contributions of TDS to the ADF-SCEM intensity. Calculations are performed for 20 different configurations of atoms. For every calculation, each atom is moved from its equilibrium position according to a Gaussian distribution with a standard deviation defined as the average of thermal vibration amplitude, (B/(8π2))0.5. Here, B indicates the Debye–Waller factor at temperature of 295 K [37]. An incoherent average of |ψi(r, rp)|2 in Eq. (5) over 20 different atom configurations is estimated to obtain the ADF-SCEM intensity, I(rp). The acceleration voltage was set at 80 kV and the convergence semi-angle to 20 mrad. Three collection semi-angles of the annular aperture, namely 22–32 mrad, 42–52 mrad and 62–72 mrad are employed. Both ADF-SCEM and ADF-STEM intensities are simulated using the same collection semi-angles. Various defocus values of the condenser lens are adopted. The defocus value of the condenser lens is defined as zero when the electron probe is focused at the entrance surface of a supercell. A positive defocus value of the condenser lens indicates underfocus. Defocus values of the imaging lens are changed according to values of the condenser lens, as mentioned in the previous section. Unless otherwise specified, geometric aberrations except defocus of both condenser and imaging lenses are ignored. Three pinhole diameters, 0.01 nm, 0.4 nm and 4.0 nm, are used. Pinhole diameter of 0.01 nm is used when an ideal point detector is assumed. Pinhole diameter of 0.4 nm represents one that has been employed in experimental SCEM observations [6–16]. A pinhole with diameter of 4.0 nm allows nearly all electrons (a fraction of ~0.99) to pass through the pinhole and reach the detector. A unit cell of fcc structure containing four Al atoms with lattice parameter of 0.404 nm is considered. The supercells used for multislice simulations consist of 10 × 10 unit cells (4.04 nm × 4.04 nm) along the [100] and [010] axes of the fcc structure and have thickness of either one unit cell (0.404 nm) or 50 unit cells (20.2 nm) along the [001] axis. The 20.2 nm thick supercell is divided into 50 slices so that each slice has thickness of 0.404 nm. In some calculations, GP1 zones are introduced into the 20.2 nm thick supercell as Cu clusters on the (010) plane. The sampling number for both real space and reciprocal space is 512 × 512 pixels. ADF-SCEM images perpendicular to the optic axis are obtained by changing the probe position rp in Eq. (1) over the supercell at a particular defocus value of the condenser lens Δfc. On the other hand, the electron probe can be scanned in a vertical plane parallel to the optic axis by changing Δfc and moving the probe in a horizontal direction. In this paper, images obtained in the former way are referred to as horizontal images whereas the latter vertical images. Through this paper, each horizontal and vertical image is displayed in eight-bit grayscale with a linear display range as the minimum for black and the maximum for white. On the other hand, intensities of electron waves are displayed on a logarithmic scale [33] to adapt their large dynamic ranges. In the present work, chromatic aberration of the lenses is not considered. The chromatic aberration may affect the depth resolution of ADF-SCEM images. Hence, its effects on ADF-SCEM deserve to be investigated in a separate study. Results and discussion Effects of the pinhole diameter Figure 2(a), (b) and (c) show ADF-SCEM horizontal images of a 0.404-nm thick Al sample at a defocus value of the condenser lens Δfc = 0 nm. A collection semi-angle of an annular aperture ranging 22–32 mrad is used. A pinhole diameter of (a) 0.01 nm, (b) 0.4 nm and (c) 4.0 nm is applied. Since the same structure exists repeatedly in the supercell, results for a unit cell are presented here. Fig. 2(d) is an ADF-STEM image simulated using the same collection semi-angle for an annular detector. Al atoms can be observed in all the ADF-SCEM and ADF-STEM images in a similar manner. However, it is noted that the ranges of the intensities displayed in Fig. 2(a)–(d) are different: (a) 2.0 × 10−6–2.3 × 10−4, (b) 2.0 × 10−3–3.2 × 10−2, (c) 3.7 × 10−3–4.4 × 10−2 and (d) 3.9 × 10−3–4.4 × 10−2 (in arb. units). These numerical values indicate that when a large pinhole is used, the ADF-SCEM intensity becomes large and close to the ADF-STEM intensity. Fig. 2. View largeDownload slide (a–c) ADF-SCEM and (d) ADF-STEM horizontal images of a 0.404 nm thick Al sample at a defocus value of the condenser lens Δfc = 0 nm. The ranges of the displayed intensities are (a) 2.0 × 10−6–2.3 × 10−4, (b) 2.0 × 10−3–3.2 × 10−2, (c) 3.7 × 10−3–4.4 × 10−2 and (d) 3.9 × 10−3–4.4 × 10−2 (in arb. units). The dashed line indicates the projection of the vertical plane where ADF-SCEM and ADF-STEM vertical images in Fig. 3 are simulated. Simulation conditions: collection semi-angle: 22–32 mrad; pinhole diameter: (a) 0.01 nm, (b) 0.4 nm and (c) 4.0 nm. Fig. 2. View largeDownload slide (a–c) ADF-SCEM and (d) ADF-STEM horizontal images of a 0.404 nm thick Al sample at a defocus value of the condenser lens Δfc = 0 nm. The ranges of the displayed intensities are (a) 2.0 × 10−6–2.3 × 10−4, (b) 2.0 × 10−3–3.2 × 10−2, (c) 3.7 × 10−3–4.4 × 10−2 and (d) 3.9 × 10−3–4.4 × 10−2 (in arb. units). The dashed line indicates the projection of the vertical plane where ADF-SCEM and ADF-STEM vertical images in Fig. 3 are simulated. Simulation conditions: collection semi-angle: 22–32 mrad; pinhole diameter: (a) 0.01 nm, (b) 0.4 nm and (c) 4.0 nm. Figure 3(a–d) present ADF-SCEM and ADF-STEM vertical images obtained at a location indicated by the dashed line in Fig. 2(a–d), respectively. Defocus value of the condenser lens changes from −40 nm (overfocus) to +40 nm (underfocus). The images of the Al columns exhibit elongated contrast along the optic axis. Full width at half maximum (FWHM) of the intensity distribution along the optic axis is estimated to be approximately (a) 12.1 nm, (b) 13.8 nm, (c) 16.1 nm and (d) 16.0 nm. Thus, the ADF-SCEM image with a pinhole diameter of 4.0 nm (Fig. 3(c)) has a similar intensity distribution to the ADF-STEM (Fig. 3(d)). This can be ascribed to the fact that the 4.0 nm pinhole encompassed 99% of the electron in the image plane, and if all electrons are collected in the image plane, ADF-SCEM reduces precisely to ADF-STEM. The elongation of the images can be attributed to the fact that the electron probe has an intensity distribution along the optic axis with a FWHM of ~18.3 nm. Fig. 3. View largeDownload slide (a–c) ADF-SCEM and (d) ADF-STEM vertical images of a 0.404 nm thick Al sample. Simulation conditions: collection semi-angle: 22–32 mrad; pinhole diameter: (a) 0.01 nm, (b) 0.4 nm and (c) 4.0 nm. Fig. 3. View largeDownload slide (a–c) ADF-SCEM and (d) ADF-STEM vertical images of a 0.404 nm thick Al sample. Simulation conditions: collection semi-angle: 22–32 mrad; pinhole diameter: (a) 0.01 nm, (b) 0.4 nm and (c) 4.0 nm. In the literature, SCEM intensities have been studied by a convolution between a function describing a sample and the PSF for the SCEM geometry with an ideal point pinhole [19–22]. The pinhole diameter of 0.01 nm in the present study corresponds to the ideal situation. The sample considered in the present study is only 0.404 nm thick and therefore the ADF-SCEM intensity with a pinhole diameter of 0.01 nm could be adequately calculated by the PSF formulation. By contrast, to the best of the authors’ knowledge, the PSF with a practical pinhole diameter (e.g. 0.4 nm and 4.0 nm) has not been formulated so far. Hence, multislice simulations have to be employed instead of the PSF formulation to investigate ADF-SCEM images with a finite pinhole diameter, as demonstrated in this section. Effects of the collection semi-angle of the annular aperture As shown in Fig. 1, an electron wave that passes through the sample is refocused on the detector plane. The wave function of the refocused wave is expressed as ψi(r, rp) in Eq. (4) and its intensity is calculated as |ψi(r, rp)|2. Fig. 4(a–b) and (c–d) present |ψi(r, rp)|2 simulated using the annular aperture ranging 22–32 mrad and 62–72 mrad, respectively. The electron probe is focused on an Al column in a 0.404-nm thick sample with a defocus value of the condenser lens Δfc = 0 nm and +10 nm. The circle in Fig. 4(d) represents a pinhole with 0.4 nm in diameter. The intensity distributions in Fig. 4(a)–(d) depend on the employed defocus values and collection semi-angles. The intensity values in Fig. 4(a)–(b) and (c)–(d) are azimuthally summed and the results are plotted in Fig. 4(e) and (f), respectively, as a function of radial distance. In Fig. 4(e) and (f), the pinhole with 0.4 nm in diameter (0.2 nm in radius) is indicated by the shaded region. Integration of the intensities within the shaded region provides the ADF-SCEM intensities at Δfc = 0 nm and +10 nm when collection semi-angles of 22–32 mrad and 62–72 mrad are used. In Fig. 4(f), the intensities within the region of the pinhole at Δfc = +10 nm are weak but have a peak at radial distance of ~0.65 nm. This result implies that scattered electrons under an out-of-focus condition can be blocked by the finite-sized pinhole. Fig. 4. View largeDownload slide (a–d) Intensity distributions of ψi(r, rp) in Eq. (4) when the electron probe is focused on an Al column in a 0.404-nm thick sample with a defocus value of the condenser lens Δfc of (a, c) 0 nm and (b, d) + 10 nm. The collection semi-angles are (a, b) 22–32 mrad and (c, d) 62–72 mrad. The circle in (d) represents the pinhole with 0.4 nm in diameter. (e) and (f) Azimuthal summation of the intensities in (a, b) and (c, d). Fig. 4. View largeDownload slide (a–d) Intensity distributions of ψi(r, rp) in Eq. (4) when the electron probe is focused on an Al column in a 0.404-nm thick sample with a defocus value of the condenser lens Δfc of (a, c) 0 nm and (b, d) + 10 nm. The collection semi-angles are (a, b) 22–32 mrad and (c, d) 62–72 mrad. The circle in (d) represents the pinhole with 0.4 nm in diameter. (e) and (f) Azimuthal summation of the intensities in (a, b) and (c, d). Figure 5 presents ADF-SCEM vertical images of a 0.404 nm thick sample simulated using a pinhole diameter of 0.4 nm and an annular aperture with various collection semi-angles: (a) 22–32 mrad, (b) 42–52 mrad and (c) 62–72 mrad. The results show that the elongation along the optic axis is reduced by using a large collection semi-angle of the annular aperture. Note that the ADF-SCEM intensity decreases with increasing collection semi-angle, as indicated by the following minimum and maximum of the intensities of each image: (a) 5.9 × 10−5–3.2 × 10−2, (b) 9.1 × 10−6–1.5 × 10−2 and (c) 3.8 × 10−6–7.8 × 10−3 (in arb. units). Fig. 5. View largeDownload slide ADF-SCEM vertical images of a 0.404-nm thick Al sample simulated using various collection semi-angles of the annular aperture: (a) 22–32 mrad, (b) 42–52 mrad and (c) 62–72 mrad. Pinhole diameter of 0.4 nm is used. The ranges of the displayed intensities are (a) 5.9 × 10−5–3.2 × 10−2, (b) 9.1 × 10−6–1.5 × 10−2 and (c) 3.8 × 10−6–7.8 × 10−3 (in arb. units). Fig. 5. View largeDownload slide ADF-SCEM vertical images of a 0.404-nm thick Al sample simulated using various collection semi-angles of the annular aperture: (a) 22–32 mrad, (b) 42–52 mrad and (c) 62–72 mrad. Pinhole diameter of 0.4 nm is used. The ranges of the displayed intensities are (a) 5.9 × 10−5–3.2 × 10−2, (b) 9.1 × 10−6–1.5 × 10−2 and (c) 3.8 × 10−6–7.8 × 10−3 (in arb. units). Effects of geometric aberrations In this section, effects of the geometric aberrations of the condenser lens and imaging lens on ADF-SCEM intensity are investigated. Table 1 lists the values of aberration coefficients Cnm and azimuthal orientations ϕnm in Eq. (2) that will be used for simulations. This set of the geometric aberrations is applied for either the condenser lens or the imaging lens, or both. Two-fold astigmatism (C12) and axial coma (C21) are assumed to be fully corrected and the other aberrations up to third-order to be corrected to some adequate degree. In practical situations, C12 and C21 can be decreased by observation of Ronchigram and a focused image of the electron probe [1]. Geometric aberrations up to third-order can be corrected routinely using an auto-correction system. The values of Cnm and ϕnm can be evaluated experimentally up to fifth-order. Realistic values are assumed for the fourth- and fifth-order Cnm. As for the defocus (C10), various values will be applied and they are not presented in Table 1. The values of the azimuthal orientations are selected arbitrarily. Table 1. Aberration coefficients (Cnm) and azimuthal orientations (ϕnm) used in simulations. Order of aberration, n Azimuthal symmetry, m Cnm ϕnm 1 2 0 nm 0° 2 1 0 nm 0° 2 3 50 nm 0° 3 0 3 μm – 3 2 0.3 μm 20° 3 4 0.5 μm 40° 4 1 30 μm 60° 4 3 30 μm 80° 4 5 30 μm 100° 5 0 5 mm – 5 2 5 mm 120° 5 4 5 mm 140° 5 6 5 mm 160° Order of aberration, n Azimuthal symmetry, m Cnm ϕnm 1 2 0 nm 0° 2 1 0 nm 0° 2 3 50 nm 0° 3 0 3 μm – 3 2 0.3 μm 20° 3 4 0.5 μm 40° 4 1 30 μm 60° 4 3 30 μm 80° 4 5 30 μm 100° 5 0 5 mm – 5 2 5 mm 120° 5 4 5 mm 140° 5 6 5 mm 160° Table 1. Aberration coefficients (Cnm) and azimuthal orientations (ϕnm) used in simulations. Order of aberration, n Azimuthal symmetry, m Cnm ϕnm 1 2 0 nm 0° 2 1 0 nm 0° 2 3 50 nm 0° 3 0 3 μm – 3 2 0.3 μm 20° 3 4 0.5 μm 40° 4 1 30 μm 60° 4 3 30 μm 80° 4 5 30 μm 100° 5 0 5 mm – 5 2 5 mm 120° 5 4 5 mm 140° 5 6 5 mm 160° Order of aberration, n Azimuthal symmetry, m Cnm ϕnm 1 2 0 nm 0° 2 1 0 nm 0° 2 3 50 nm 0° 3 0 3 μm – 3 2 0.3 μm 20° 3 4 0.5 μm 40° 4 1 30 μm 60° 4 3 30 μm 80° 4 5 30 μm 100° 5 0 5 mm – 5 2 5 mm 120° 5 4 5 mm 140° 5 6 5 mm 160° Figure 6 displays intensity distributions of ψi(r, rp) when a collection semi-angle of 42–52 mrad is used under various conditions with regard to geometric aberrations. The images in Fig. 6(a) is for the aberration-free condition for both condenser and imaging lenses whereas in Fig. 6(b), the geometric aberrations listed in Table 1 are applied for the condenser lens only. The intensity distribution of ψi(r, rp) is not altered markedly even when the aberrations of the condenser lens are applied. On the other hand, when the aberrations for the imaging lens are considered, the image of the refocused wave is distorted, as shown in Fig. 6(c). These findings can be interpreted by the fact that the convergence semi-angle is 20 mrad while the collection semi-angle of the annular aperture is 42–52 mrad, so that the geometric aberrations of the imaging lens affects the refocused wave because of the larger angle used. Fig. 6(d) shows the intensity distribution simulated with the geometric aberrations for both condenser and imaging lenses applied and the result is similar to that in Fig. 6(c). Fig. 6. View largeDownload slide Intensity distributions of ψi(r, rp) in Eq. (4) when the electron probe is focused on an Al column in a 0.404-nm thick sample with a defocus value of the condenser lens Δfc = 0 nm. (a) Aberration-free condition. (b–d) Geometric aberrations listed in Table 1 are applied for (b) condenser lens, (c) imaging lens and (d) both condenser and imaging lenses. The collection semi-angle is 42–52 mrad. Fig. 6. View largeDownload slide Intensity distributions of ψi(r, rp) in Eq. (4) when the electron probe is focused on an Al column in a 0.404-nm thick sample with a defocus value of the condenser lens Δfc = 0 nm. (a) Aberration-free condition. (b–d) Geometric aberrations listed in Table 1 are applied for (b) condenser lens, (c) imaging lens and (d) both condenser and imaging lenses. The collection semi-angle is 42–52 mrad. Figure 7(a–c) show ADF-SCEM vertical images with the geometric aberrations in Table 1 applied for the condenser lens but not for the imaging lens. Simulation conditions except the aberrations are the same as those of Fig. 5(a–c) under the aberration-free condition. The images in Fig. 7(a–c) are practically identical to the corresponding images in Fig. 5(a–c). Thus, the aberrations of the condenser lens have no significant effect on the ADF-SCEM images when the aberrations up to third-order are corrected and the convergence semi-angle of 20 mrad is selected. On the other hand, Fig. 7(d–f) display ADF-SCEM vertical images when the geometric aberrations are applied for the imaging lens but not for the condenser lens. Fig. 7(d) that is simulated with a collection semi-angle of 22–32 mrad is similar to Fig. 7(a) except that the intensity peak is shifted by approximately −3 nm along the optic axis. By contrast, the images in Fig. 7(e) and (f) simulated using higher collection semi-angles of the annular aperture differ markedly from the aberration-free cases in Fig. 5(b) and (c). Fig. 7. View largeDownload slide ADF-SCEM vertical images of a 0.404-nm thick Al sample simulated when the geometric aberrations listed in Table 1 are applied for (a–c) condenser lens and (d–f) imaging lens. Simulation conditions: collection semi-angle: (a, d) 22–32 mrad, (b, e) 42–52 mrad and (c, f) 62–72 mrad; pinhole diameter: 0.4 nm. Fig. 7. View largeDownload slide ADF-SCEM vertical images of a 0.404-nm thick Al sample simulated when the geometric aberrations listed in Table 1 are applied for (a–c) condenser lens and (d–f) imaging lens. Simulation conditions: collection semi-angle: (a, d) 22–32 mrad, (b, e) 42–52 mrad and (c, f) 62–72 mrad; pinhole diameter: 0.4 nm. Although the present study has clarified that the elongation of ADF-SCEM images is shortened by employing a large collection semi-angle of the annular aperture, this fails when geometric aberrations of the imaging lens are introduced. However, it has been reported that newly-developed aberration correctors enable higher-order aberration correction. For example, the Delta-corrector applied for an imaging system has enabled that the angle of uniform phase can be extended up to ~70 mrad at an acceleration voltage of 60 kV [38,39]. With this type of aberration correctors, ADF-SCEM is likely to possess a good depth resolution in practical situations. Effects of TDS on images of a thick sample Figure 8(a–c) show ADF-SCEM vertical images of a 20.2-nm thick sample simulated with various collection semi-angles of the annular aperture and a pinhole diameter of 0.4 nm. As seen in Fig. 8(a) that was produced with a collection semi-angle ranging 22–32 mrad, the ADF-SCEM intensity is high when the electron probe is focused near the entrance surface and slightly under the exit surface. The length of the elongated contrast near the entrance surface is comparable to that observed in a 0.404 nm thick sample (Fig. 5(a)). The white arrows in Fig. 8(a) indicate artifacts appearing between Al columns. Fig. 8. View largeDownload slide ADF-SCEM vertical images of a 20.2-nm thick Al sample simulated (a–c) with TDS and (d–f) without TDS consideration. Simulation conditions: collection semi-angle: (a, d) 22–32 mrad, (b, e) 42–52 mrad and (c, f) 62–72 mrad; pinhole diameter: 0.4 nm. Fig. 8. View largeDownload slide ADF-SCEM vertical images of a 20.2-nm thick Al sample simulated (a–c) with TDS and (d–f) without TDS consideration. Simulation conditions: collection semi-angle: (a, d) 22–32 mrad, (b, e) 42–52 mrad and (c, f) 62–72 mrad; pinhole diameter: 0.4 nm. So far, we have presented the results of the simulations with considering contributions of TDS by the frozen phonon approach. On the contrary, Fig. 8(d–f) show ADF-SCEM vertical images produced in the absence of TDS. In this simulation, all Al atoms were located at their lattice points and this single atomic configuration was employed to calculate |ψi(r, rp)|2 in Eq. (5). Comparison in Fig. 8(a) and (d) with using a collection semi-angle of 22–32 mrad indicate that the ADF-SCEM images are virtually independent of the inclusion of TDS. By contrast, with a collection semi-angle of 42–52 mrad and 62–72 mrad used, the ADF-SCEM images are altered remarkably by the inclusion of TDS. These results are related directly to the intensity distributions of FT[ψt(r, rp)] in Eq. (3), the diffraction patterns, which indicates that contributions of elastic scattering are significant in a range of 22–32 mrad when the electron probe is focused near the entrance and exit surfaces of the sample. On the other hand, TDS contributions are predominant in the range more than 42 mrad, regardless of the defocus values. Ruben et al. [40] has reported that in their study of low-angle ADF-STEM (LAADF-STEM) image simulations of GaAs layered structure, elastic scattering contributes to the LAADF intensities significantly when the electron probe is focused near the entrance and exit surfaces of the specimen, similarly to the present study. They suggested that the intensity peak of the elastic scattering at the entrance surface is understood based on an s-state channeling model [41], while the intensity peak at the exit surface occurs because electrons scattered into the LAADF region no longer undertake further scattering. On the other hand, Fig. 8(b) and (c) indicate that TDS signals in the polar angle more than 42 mrad are large when the electron probe is focused at depths slightly under the entrance surface. On the basis of the discussion provided by Ruben et al. [40], this result can be attributed to channeling effects, where electron waves propagating within the sample exhibit a peak with the largest intensity when the defocus value of the condenser lens is ~+5 nm. GP1 zones in an Al crystal GP1 zones are introduced into a 20.2-nm thick supercell to investigate the possibility of determining their depths in an Al crystal. Fig. 9(a) depicts a cluster of Cu atoms with ~2.4 nm in diameter, single plane thick and parallel to the (010) plane of the fcc structure. Here, big spheres represent the Cu atoms and small spheres Al atoms. This cluster of Cu atoms can be considered as an atomic model of a GP1 zone. Five GP1 zones were constructed in the same atomic model and arranged at different depths in a 20.2 nm thick supercell, as illustrated in Fig. 9(b) and (c). The centers of the GP1 zones denoted by L1, L2, L3, L4 and L5 are at depths of 1.6 nm, 5.7 nm, 9.7 nm, 13.7 nm and 17.8 nm, respectively. Fig. 9. View largeDownload slide Atomic configurations of a supercell containing five GP1 zones. (a) Model of a GP1 zone. Big spheres represent Cu atoms and small spheres Al atoms. (b–c) Projections of the supercell viewed along (b) the [001] axis and (c) the [100] axis. Fig. 9. View largeDownload slide Atomic configurations of a supercell containing five GP1 zones. (a) Model of a GP1 zone. Big spheres represent Cu atoms and small spheres Al atoms. (b–c) Projections of the supercell viewed along (b) the [001] axis and (c) the [100] axis. Vertical images at the location indicated by the dashed line in Fig. 9(b) were simulated with considering TDS. Fig. 10(a)–(c) display ADF-STEM images using collection semi-angles of the annular detector of (a) 22–32 mrad, (b) 42–52 mrad and (c) 62–72 mrad. The rightmost column draws diagrams of the supercell to indicate the depths of the GP1 zones. The GP1 zones can exhibit strong contrast in the Al crystal because of the larger atomic number of Cu than that of Al. As observed in Fig. 10(a–c), the GP1 zones appears with elongated contrast along Al columns for all collection semi-angles selected. It is noted that the five GP1 zones are observed in different defocus settings, depending on their individual depths in the supercell. However, the GP1 zones L4 and L5 are observed even when defocus values of ~0 nm are applied, as indicated in Fig. 10(a). This can be attributed to channeling effects. As pointed out in the literature [42], when the electron probe is focused at a defocus value of ~0 nm, electron waves propagating within the crystal has an intensity peak at depth of ~20 nm owing to channeling. This is close to the depths of the GP1 zones L4 and L5, leading to the appearance of the GP1 zones at a defocus value of ~0 nm. Fig. 10. View largeDownload slide Vertical images at the location of the dashed line in Fig. 9(b). (a–c) ADF-STEM under an aberration-free condition. (d–f) ADF-SCEM under an aberration-free condition. (g–i) ADF-SCEM with the geometric aberrations in Table 1 applied for both condenser and imaging lenses. The rightmost column draws diagrams of the supercell to indicate the depths of the GP1 zones. Simulation conditions: annular collection semi-angle: (a, d, g) 22–32 mrad, (b, e, h) 42–52 mrad and (c, f, i) 62–72 mrad; pinhole diameter: 0.4 nm for (d–i). Fig. 10. View largeDownload slide Vertical images at the location of the dashed line in Fig. 9(b). (a–c) ADF-STEM under an aberration-free condition. (d–f) ADF-SCEM under an aberration-free condition. (g–i) ADF-SCEM with the geometric aberrations in Table 1 applied for both condenser and imaging lenses. The rightmost column draws diagrams of the supercell to indicate the depths of the GP1 zones. Simulation conditions: annular collection semi-angle: (a, d, g) 22–32 mrad, (b, e, h) 42–52 mrad and (c, f, i) 62–72 mrad; pinhole diameter: 0.4 nm for (d–i). Figure 10(d–f) present ADF-SCEM images simulated using the same collection semi-angles as in Fig. 10(a–c). Compared with the corresponding ADF-STEM images, the ADF-SCEM images of the GP1 zones are short along the optic axis when a large collection semi-angle is applied. Besides, although electron channeling occurs identically for the ADF-STEM and ADF-SCEM, the channeling effect is suppressed, i.e. the GP1 zones L4 and L5 exhibit only a weak contrast at a defocus value of ~0 nm, as observed in Fig. 10(e) and (f). However, this cannot be realized when the geometric aberrations of the imaging lens are present. Fig. 10(g–i) are ADF-SCEM images simulated with the geometric aberrations in Table 1 for both condenser and imaging lenses. The images of the GP1 zones are elongated when a collection semi-angle more than 42–52 mrad is used. As we argued in the above text that the aberrations of the condenser lens have no significant effect on ADF-SCEM, the deterioration of the depth resolution observed in Fig. 10(h) an (i) is due to the geometric aberrations of the imaging lens. In recent years, depth sectioning technique using high-angle annular dark-field-STEM (HAADF-STEM) has been applied to 3D microstructural characterization in a variety of systems [2,19,21,43–54]. This technique has a potential to possess nanoscale depth resolution because the depth of focus in HAADF-STEM is inversely proportional to the square of the convergence semi-angle. However, electron flux in the probe increases with the convergence semi-angle, which may introduce severe irradiation damages to the sample. ADF-SCEM can be a promising alternative to HAADF-STEM for 3D imaging since the convergence semi-angle in ADF-SCEM can be as small as 20 mrad, so that the sample is observed under a low dosage. Concluding remarks ADF-SCEM images of Al crystals have been studied using multislice simulations. Image simulations were performed properly by including TDS. Similar to an ideal point pinhole, the pinhole with 0.4 nm in diameter can play a role in blocking electrons that are scattered under an out-of-focus condition. The use of a large collection semi-angle of the annular aperture can reduce the elongation of images along the optic axis. The elongation of images in ADF-SCEM is less sensitive to channeling effects than in ADF-STEM, although electron channeling occurs identically in ADF-SCEM and ADF-STEM. Geometric aberrations of the imaging lens deteriorate ADF-SCEM images when a large collection semi-angle is employed, whereas geometric aberrations of the condenser lens have no significant effect as long as small convergence semi-angle is used (<20 mrad). 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Microscopy – Oxford University Press

**Published: ** Aug 1, 2018

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