TY - JOUR AU1 - Mori,, Shigeo AU2 - Nakajima,, Hiroshi AU3 - Kotani,, Atsuhiro AU4 - Harada,, Ken AB - Abstract We describe small-angle electron diffraction (SmAED) and Lorentz microscopy using a conventional transmission electron microscope. In SmAED, electron diffraction patterns with a wide-angular range on the order of 1 × 10−2 rad to 1 × 10−7 rad can be obtained. It is demonstrated that magnetic information of nanoscale magnetic microstructures can be obtained by Fresnel imaging, Foucault imaging and SmAED. In particular, we report magnetic microstructures associated with magnetic stripes and magnetic skyrmions revealed by Lorentz microscopy with SmAED. SmAED can be applied to the analysis of microstructures in functional materials such as dielectric, ferromagnetic and multiferroic materials. Lorentz microscopy, small-angle electron diffraction, magnetic stripe domain, magnetic skyrmion Introduction Small-angle scattering refers to scattering in a region with a small scattering angle in reciprocal space [1]. In general, neutrons and x-rays are used to perform small-angle scattering experiments, which are used to observe and analyze microdomain structures, proteins and particle distributions [2–6]. Small-angle diffraction methods using an electron beam have been applied to latex structures, metal thin films, superconducting fluxons and electromagnetic field analysis in magnetic materials since the 1960s [7–11]. Recently, it has further been applied to the analysis of the helical magnetic structure of the chiral magnet Cr1/3NbS2 [12–14]. The magnetic domains and electromagnetic field inside a material can be analyzed using Lorentz microscopy [15–19], differential phase contrast microscopy [20–24], phase contrast imaging with a hole-free phase plate [25–27] and electron holography [28–33]. These analytical methods are based on the principle that the electron beam propagates as a wave and the phase of the electron beam changes under the effect of electric and magnetic fields. Such imaging methods have the advantage that they can directly observe the magnetic fine structure inside a material in real space. However, it is important to directly measure how much the electron beam is deflected in order to obtain quantitative knowledge of the electric and magnetic fields. Small-angle electron diffraction (SmAED) can quantitatively measure the angle of deflection of the electron beam due to the electric and magnetic fields in reciprocal space. In this paper, we review SmAED optics and selected research results obtained by SmAED and Lorentz transmission electron microscopy (hereafter, Lorentz microscopy), such as the Fresnel and Foucault imaging methods, for various functional magnetic materials [34–42]. Small-angle electron diffraction Electro-optics for small-angle electron diffraction SmAED is a method of observing diffraction spots due to magnetic deflection and long-period structures using a long camera length. Fig. 1 is a schematic diagram showing SmAED. When a crystalline specimen having a magnetic domain structure is irradiated with a parallel electron beam, both small-angle diffraction β1 due to magnetization and Bragg reflections β2 corresponding to the crystalline lattice are formed in reciprocal space. A Bragg reflection typically has a deflection angle of β2 = 10–2 to 10–3 rad, whereas magnetic deflection has a deflection angle of β1 = 10–3 to 10–7 rad. As a result, SmAED spots due to magnetic deflection are observed around the 000 direct spot. To detect the SmAED spots, a long camera length exceeding ∼ 100 m is required. Fig. 1. Open in new tabDownload slide Schematic illustration of small-angle electron diffraction. The angles |${\beta _1}$| and |${\beta _2}$| represent the scattering angle due to the magnetic deflection and Bragg-reflection angle corresponding to the crystalline lattice, respectively. Fig. 1. Open in new tabDownload slide Schematic illustration of small-angle electron diffraction. The angles |${\beta _1}$| and |${\beta _2}$| represent the scattering angle due to the magnetic deflection and Bragg-reflection angle corresponding to the crystalline lattice, respectively. The Foucault method delineates magnetic domains by selecting magnetically deflected spots. This method is useful in SmAED because it visualizes magnetic domains that cause particular SmAED spots. However, the electro-optical system has some restrictions because an angle limiting diaphragm (usually an objective diaphragm) is required in reciprocal space. Therefore, unlike other Lorentz microscopy methods, such as the Fresnel method, the Foucault method is not widely adopted by materials researchers. Recently, a lens-less Foucault method using the irradiation optical system for imaging has been realized [17]. However, there is a limitation in controlling the range of the irradiation electron dose delivered to the specimen. Accordingly, we constructed an electro-optical system in a conventional transmission electron microscope for SmAED and Lorentz microscopy, such as the Fresnel and Foucault imaging methods [38, 40, 43]. External magnetic fields can be applied normal to a specimen by using the objective lens, and the irradiation electron dose can be adjusted by controlling lens currents. Electro-optical system for the Foucault imaging method Fig. 2 shows an electro-optical system for SmAED and the Foucault method [38]. In this optical system, the condenser lens is used with strong excitation to maintain the divergence angle of the electron beam below 10–5 rad. Although the objective lens is turned off, it is possible to apply a magnetic field to the specimen parallel to the optical axis by weakly exciting the objective lens. The objective mini-lens is adjusted so that a crossover is formed in the imaging plane at the selected area aperture. In the SmAED optical system, intermediate lens 1 is focused on the crossover at the selected area aperture plane. The camera length is controlled by intermediate lenses 2 and 3, which are located below intermediate lens 1. These lenses are represented by the image forming lens in Fig. 2. In the Foucault method shown in Fig. 2b, the irradiation system is common to the SmAED optics, but intermediate lens 1 is weakly excited and a real image is obtained by focusing on the image of the specimen (virtual image of the objective mini-lens). The Foucault image is obtained by selecting a specific magnetic-deflection spot. Fig. 2. Open in new tabDownload slide (a) Optical system for small-angle electron diffraction. (b) Optical system for Foucault imaging. The image forming lens is composed of intermediate lenses 2 and 3 and a projection lens. Figure reproduced with permission from Ref [38]. Fig. 2. Open in new tabDownload slide (a) Optical system for small-angle electron diffraction. (b) Optical system for Foucault imaging. The image forming lens is composed of intermediate lenses 2 and 3 and a projection lens. Figure reproduced with permission from Ref [38]. The optical system in Fig. 2 was evaluated with a 500-nm carbon grating. Fig. 3a and b shows a real-space image of the carbon grating and the corresponding SmAED pattern. The diffraction angle is approximately 5.0 × 10–6 rad, corresponding to a diffraction spot spacing of (500 nm)−1. Fig. 3c and d shows the values of the camera length and magnification with respect to the current values of intermediate lenses 2 and 3. These values were estimated from the diffraction grating. The camera length increases as the current value of intermediate lens 2 increases. When the current values of intermediate lenses 2 and 3 are both 8.11 A, the maximum camera length is approximately 1300 m. In this optical system, the maximum magnification is approximately 3000 × because the objective lens current is almost zero. Fig. 3. Open in new tabDownload slide (a) 500-nm carbon grating. (b) Small-angle electron diffraction pattern of the grating taken with a camera length of 360 m. (c) Change in camera length and (d) magnification for current values of intermediate lenses 2 and 3, indicated as I2 and I3, respectively. Figure reproduced with permission from Ref [43]. Fig. 3. Open in new tabDownload slide (a) 500-nm carbon grating. (b) Small-angle electron diffraction pattern of the grating taken with a camera length of 360 m. (c) Change in camera length and (d) magnification for current values of intermediate lenses 2 and 3, indicated as I2 and I3, respectively. Figure reproduced with permission from Ref [43]. Figure 4 displays the camera length when the objective lens is used. The camera length can be increased by weakly exciting the objective lens. When the objective lens is excited to 0.36 A, the camera length changes from 1300 to 2500 m. By controlling the current value of the objective lens, external magnetic fields ranging from 0 to 200 mT can be applied to the thin specimen. In this manner, by controlling the current values of each magnetic lens, the strength of the external magnetic field, irradiation area and parallelism of the incident electron beam can be adjusted independently. Fig. 4. Open in new tabDownload slide Camera length values as a function of current values (I3) of intermediate lens 3. The notation IOL, I2 and I3 are the current values of the objective lens, intermediate lens 2 and intermediate lens 3, respectively. Figure reproduced with permission from Ref [38]. Fig. 4. Open in new tabDownload slide Camera length values as a function of current values (I3) of intermediate lens 3. The notation IOL, I2 and I3 are the current values of the objective lens, intermediate lens 2 and intermediate lens 3, respectively. Figure reproduced with permission from Ref [38]. Application of SmAED to Bragg reflection and dark-field imaging The SmAED optical system can be extended to obtain Bragg reflections with the objective lens turned off [40]. In SmAED, the camera length can be changed from several tens to several hundreds of meters by controlling the current value of intermediate lens 1. However, in obtaining Bragg reflection spots due to the crystalline lattice, the current value of intermediate lens 1 is reduced and the camera length can be set to several tens of centimeters. Further, an external magnetic field can be applied normal to the thin specimen by weakly exciting the objective lens. This optical system was used to obtain information about the crystal and magnetic structure of M-type hexaferrite (BaFe10.35Sc1.6Mg0.05O19; BFSMO) [40]. Fig. 5 shows a dark-field image obtai-ned using reflection 220 and the electron diffraction pattern. In the dark-field image, defocus is applied to observe the magnetic domain structure. In such an image, both the magnetic domain and the crystal domain structures can be observed. In an ordinary electro-optical system, it is not possible to obtain both the magnetic domain and crystal domain structures simultaneously because it is necessary to use an objective lens and select a diffraction spot with an objective diaphragm. Diffraction spots around the direct beam spot, indicated by blue arrows in the inset, are due to the periodicity of the magnetic domains. Further, the diffuse streaks are due to Bloch-type domain walls. Fig. 5. Open in new tabDownload slide (a) Dark-field image of the c plane of an M-type hexaferrite. The inset is an enlarged view of (a). (b) Electron diffraction pattern. The arrowhead indicates reflection 220 used for the dark-field image of (a). The camera length is approximately 1 m. The inset is a small-angle electron diffraction (SmAED) pattern of the direct beam spot with a camera length of 360 m obtained from the dotted area indicated in (a). Because the SmAED pattern was captured from unidirectional domains, the SmAED spots appear in one direction. The scale bar is 1 × 10–5 rad. Figure reproduced with permission from Ref [40]. Fig. 5. Open in new tabDownload slide (a) Dark-field image of the c plane of an M-type hexaferrite. The inset is an enlarged view of (a). (b) Electron diffraction pattern. The arrowhead indicates reflection 220 used for the dark-field image of (a). The camera length is approximately 1 m. The inset is a small-angle electron diffraction (SmAED) pattern of the direct beam spot with a camera length of 360 m obtained from the dotted area indicated in (a). Because the SmAED pattern was captured from unidirectional domains, the SmAED spots appear in one direction. The scale bar is 1 × 10–5 rad. Figure reproduced with permission from Ref [40]. Application of SmAED and Lorentz microscopy to functional magnetic materials Magnetic textures in M-type hexaferrite Here, we describe the magnetization distribution of magnetic textures of BaFe12O19 (BFO) and BFSMO by using SmAED and Fresnel imaging [41]. The magnetic domains of BFO and BFSMO have been investigated [44–47], and the magnetization is considered to be oriented in the perpendicular direction in magnetic domains and in the in-plane direction in magnetic domain walls, as shown in Fig. 6a. Fig. 6b and c shows the Fresnel image and SmAED pattern of the magnetic striped domains of BFO. The directions of the magnetization estimated from the Fresnel image are indicated by arrows in Fig. 6b. The magnetization is oriented in the in-plane direction at the domain walls, and the domain walls are observed in stripes with a width of approximately 250 nm. The domain wall magnetization period is four times the inter-domain wall spacing. The SmAED spots derived inter-domain wall spacing, with a diffraction angle of 2.6 µrad, corresponding to (980 nm)–1, is also four times the wall spacing period. To minimize the magnetostatic energy of the magnetic dipoles, it is necessary for the magnetizations in the adjacent domain walls to be opposed. However, as indicated by the arrows in Fig. 6b, the magnetizations are parallel to the nearest domain wall and antiparallel to the next-nearest domain wall. It is suggested that this arrangement is due to competition between exchange energy and magnetostatic energy. These results suggest that the magnetostatic interaction between magnetic dipoles affects this long-range ordered structure. Fig. 6. Open in new tabDownload slide (a) Schematic of magnetization distribution in the magnetic stripe domains. Change in magnetic stripe domains as a function of the tilt angle. The tilt angles are (b) 0° and (d) 30°, respectively. The arrows indicate the direction of magnetization. Small-angle electron diffraction patterns obtained from the regions (b) and (d) are shown in (c) and (e), respectively. The camera length was approximately 300 m. Figure reproduced with permission from Ref [41]. Fig. 6. Open in new tabDownload slide (a) Schematic of magnetization distribution in the magnetic stripe domains. Change in magnetic stripe domains as a function of the tilt angle. The tilt angles are (b) 0° and (d) 30°, respectively. The arrows indicate the direction of magnetization. Small-angle electron diffraction patterns obtained from the regions (b) and (d) are shown in (c) and (e), respectively. The camera length was approximately 300 m. Figure reproduced with permission from Ref [41]. It is difficult to determine the magnetic deflection component from the SmAED pattern of Fig. 6c. Therefore, we investigated changes in Fresnel images and SmAED patterns by tilting the specimen. Fig. 6d and e shows the obtained Fresnel image and SmAED pattern after tilting the specimen by 30°. Bright or dark lines can be seen at a domain wall in Fig. 6d. By comparison, the contrast at a domain wall was depicted as a pair of bright and dark lines before tilting. This suggests that the in-plane component of the magnetization increases in the magnetic domains because of the tilting, and the resulting contrast appeared at the domain walls. The increase of in-plane magnetization in the magnetic domains due to the tilting of the specimen is also observed in the SmAED pattern. At the tilt angle of 30°, the intensity of the diffraction spots is increased, suggesting that the magnetic deflection is superimposed on these spots. Therefore, it is considered that the projection component of the magnetization oriented in the observation direction should be detected in the Fresnel image and the SmAED pattern: The in-plane magnetization in the domains was increased by tilting. In Fig. 6e, the domain period estimated from the spots (5.3 µrad), corresponding to (470 nm)–1. This corresponds to twice the magnetic domain width (250 nm). Thus, the domain period is observed in the SmAED patterns. From the above results, the diffraction spots in Fig. 6e show the projected components of magnetization in domains, which exhibit in-plane components after the tilt. Skyrmion formation process in FeGe In this section, we explain the formation process of skyrmions over a wide area [48]. Fig. 7 shows the magnetic-field dependence of the FeGe helical domain at 260 K. When no external magnetic field is applied, stripe-shaped contrast in Fig. 7a exists because of helical magnetism. Grain boundaries exist on the left side and bottom, and the propagation vector of the helix is perpendicular to the grain boundaries. An SmAED pattern acquired from the area indicated by A is shown in Fig. 8a. There are superlattice reflections near the transmitted spot and the magnetic structure is periodically modulated. The superlattice reflection split angle θ ∼ 3.3 × 10–5 rad corresponds to (76 nm)−1, which is equal to the distance between the bright lines in the Fresnel image (Fig. 7a). Therefore, these superlattice reflections are considered to be magnetic diffraction spots derived from a helical magnetic structure. When a magnetic field was applied, the magnetic domain structures changed as shown in Fig. 7d and e. Fig. 8. Open in new tabDownload slide Small-angle electron diffraction patterns at 260 K in FeGe. For applied fields of (a) 0 T, (b) 60 mT, (c) 60 mT and (d) 65 mT, respectively. The diffraction spots in (b) and (c) are indicated by red and blue arrowheads, respectively. Camera lengths are (a) 93 m, (b) 100 m, (c) 100 m and (d) 95 m. The deflection angle of 3.6 × 10–5 rad corresponds to (70 nm)−1. Scale bar: 4.0 × 10–5 rad. Figure reproduced with permission from Ref [48]. Fig. 8. Open in new tabDownload slide Small-angle electron diffraction patterns at 260 K in FeGe. For applied fields of (a) 0 T, (b) 60 mT, (c) 60 mT and (d) 65 mT, respectively. The diffraction spots in (b) and (c) are indicated by red and blue arrowheads, respectively. Camera lengths are (a) 93 m, (b) 100 m, (c) 100 m and (d) 95 m. The deflection angle of 3.6 × 10–5 rad corresponds to (70 nm)−1. Scale bar: 4.0 × 10–5 rad. Figure reproduced with permission from Ref [48]. Fig. 7. Open in new tabDownload slide Formation process of skyrmion lattice in FeGe. The strengths of external magnetizations are (a) 0 T, (b) 60 mT and (c) 88 mT at 260 K, respectively. The Fresnel images are acquired under overfocus conditions. An enlarged view of the area surrounded by the red dotted line in (b) is shown at (d) 0 T, (e) 50 mT, (f) 55 mT, (g) 60 mT and (h) 88 mT. (i) An enlarged view of the area surrounded by the blue dotted in (b). A, B, C and D correspond to regions where small-angle electron diffraction patterns were acquired as shown in Fig. 8(a)–(d). Figure reproduced with permission from Ref [48]. Fig. 7. Open in new tabDownload slide Formation process of skyrmion lattice in FeGe. The strengths of external magnetizations are (a) 0 T, (b) 60 mT and (c) 88 mT at 260 K, respectively. The Fresnel images are acquired under overfocus conditions. An enlarged view of the area surrounded by the red dotted line in (b) is shown at (d) 0 T, (e) 50 mT, (f) 55 mT, (g) 60 mT and (h) 88 mT. (i) An enlarged view of the area surrounded by the blue dotted in (b). A, B, C and D correspond to regions where small-angle electron diffraction patterns were acquired as shown in Fig. 8(a)–(d). Figure reproduced with permission from Ref [48]. When a magnetic field of 60 mT is applied, two different skyrmion lattices are formed in the regions B and C corresponding to each of the two grain boundary edges as shown in Fig. 7b. This is because the specimen is thin in these areas. SmAED patterns obtained from the regions B and C in Fig. 7b are shown in Fig. 8b and c. The azimuthal angle between the diffraction spots is 120°, and three propagation vectors exist because of the skyrmion lattice. The orientations of the skyrmion lattices in the B and C regions differ by approximately 12°. The position of the magnetic diffraction spots in Fig. 8b and c is 3.6 × 10−5 rad, which corresponds to (70 nm)−1. This distance corresponds to the skyrmion lattice spacing d = (√3/2)ask ∼ 70 nm, as shown in Fig. 7i. Here, ask ∼ 80 nm is the distance between two skyrmions. We note that in Fig. 7f and g, the contrast decreases to zero in the region between the two skyrmion lattices. We attribute this to a conical structure realized in this region with its axis oriented to the external magnetic field direction. This is explained as follows: A spiral magnetic domain was confirmed before magnetic-field application. The contrast disappeared as the magnetic field further increased. Eventually, this area shows a skyrmion lattice above 88 mT in Fig. 7c and h. Therefore, a magnetic structure without contrast appeared between the helical and skyrmion structures. Recent theoretical studies show that a conical structure can be realized between the helical and skyrmion phases [49]. Furthermore, a conical structure whose axis is oriented to the magnetic field (observation direction) does not produce Fresnel contrast because magnetic deflection does not occur, which explains the contrast disappearance in the region. Figure 9 is a Fresnel image of the skyrmion lattice at 240 K after the temperature was decreased from the state of Figs. 7 and 8 at 260 K. The applied magnetic field was maintained at 88 mT during the cooling process. It can be seen that the skyrmion exists over 6 µm and the skyrmion phase has a long-range order. Fig. 9b shows the corresponding SmAED pattern. In addition to the six magnetic diffraction spots observed in Fig. 8, higher-order reflections are observed. The higher-order reflections are evidence of increasing magnetization at a low temperature. In an electron holography experiment [32], magnetization comprising skyrmions increased with decreasing temperature. The electron diffraction intensity is proportional to the magnitude of magnetism [10, 50]. Therefore, this SmAED result also shows the magnetization of skyrmions increased at lower temperatures. Fig. 9. Open in new tabDownload slide (a) Fresnel image and (b) small-angle electron diffraction pattern at 240 K and 88 mT. The camera length is approximately 95 m. Fig. 9. Open in new tabDownload slide (a) Fresnel image and (b) small-angle electron diffraction pattern at 240 K and 88 mT. The camera length is approximately 95 m. Summary In this review, we introduced an optical system for SmAED and observation results of static and dynamic behavior of magnetic domains in functional materials. The optical system and methods described can be used in a conventional transmission electron microscope by simply adjusting lens currents, enabling the analysis of magnetic domains in reciprocal space. It is anticipated that combined SmAED and Lorentz microscopy will be applied to the research of magnetic domain structures and the magnetic-field response of magnetic materials. References 1. Guinier A , Fournet G, and Yudowitch K L ( 1955 ) Small-angle Scattering of X-rays ( Wiley, New York ). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 2. Kostorz G ( 1979 ) Small-angle scattering and its applications to materials science . Treatise Mater. Sci. Technol. (Elsevier) 15 : 227 – 289 . Google Scholar Crossref Search ADS WorldCat 3. 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