Small angle electron diffraction and deflection

Small angle scattering of neutrons and x-rays, usually referred to as SANS and SAXS respectively, are widely used to analyze internal structures of materials such as micrometric domain structures, fine textures, long-range periodic order, and particle structures from nano to micro scales.^1,2 Neutrons also have sensitivity to magnetic moments localized at atoms, which enables SANS analysis of magnetic textures in the reciprocal space.³ 

In the case of electrons, transmission electron microscopy provides both real-space images and reciprocal space (i.e., diffraction) data of materials at atomic scales.⁴ Bragg diffraction angle of the crystal is typically of the order of 10^-2 radian since the crystal has a lattice with a few hundred pico meters periodicity and the wavelength of electrons is about several pico meters in a conventional transmission electron microscope (TEM). When materials have long-range periodic order at nano or micro scales, additional Bragg diffraction will appear in the reciprocal space at the angle several orders of the magnitude smaller than that due to the long periodicity. Thus, a detection of electrons scattered at small angle, which is referred to as small angle electrons scattering (SAES), should be required to examine internal structures of materials in the reciprocal space.⁵ 

Electrons are also deflected at small angles by Lorentz force in magnetic materials. In magnetic elements in functional materials⁶ and spintronic devices,⁷ observation of Lorentz deflection of electrons at the small angle turns to be of significance because Lorentz deflection angle of electrons becomes smaller in modern miniaturized magnetic devices composed of thinner magnetic films. In 1960's, there were a few pioneering works on small angle Bragg diffraction and Lorentz deflection of electrons (SAEDs) in magnetic materials.^8,9

The Lorentz deflection angle β due to the magnetic moments in magnetic materials is given as a function of their saturation induction B₀ and specimen thickness t by the equation of

where e is the electric charge, h is Planck's constant, and λ is the wavelength of electron.¹⁰ When a TEM machine with 200 kV acceleration voltage (λ = 2.5 pm) is used in observing magnetic materials, β is 1.3 × 10^-5 and 3.7 × 10^-6 radians in the case of iron (B₀ = 2.15 T) and nickel (B₀ = 0.61 T) with t = 10 nm, respectively. These angles are three or four orders of magnitude smaller than the Bragg diffraction angle of crystals. Therefore, the camera length, which correspond to a magnification of the diffraction data, should be over several hundred meters for detecting such small angle of Lorentz deflection. In the present work, we have successfully observed small angle diffraction and/or Lorentz deflection pattern down to at the order of 10^-6 radian. For this purpose, we have optimized an electron optical system and prepared the long-distance camera length up to 3000 m.

Figure 1 shows an optical system for obtaining long-distance camera lengths. In order to form a small-sized crossover above the specimen position, condenser lenses are strongly excited. As a result, small divergence angle of incident electrons are appropriately guaranteed in the experiments. In this optical system, diffraction pattern that is virtually formed at the crossover plane is imaged as a real pattern on an object plane of the image-forming lens by using long focal objective lens. Then, the diffraction pattern is enlarged by the image-forming lens system. Consequently, the diffraction pattern around the zero-order spot is largely magnified and thus small angle diffraction and/or Lorentz deflection of electrons can be detected.

The objective aperture originally installed in a TEM (given black horizontal lines in Fig. 1) is used as a selected-area aperture although it is placed in the out-of-focus condition of a few millimeters below the specimen. Instead, the selected-area aperture originally installed, as described by gray horizontal lines in Fig. 1, works as an objective aperture (i.e., an aperture in the reciprocal space) when the diffraction pattern is formed on the appropriate position. Foucault image of Lorentz microscopy¹⁰ is obtainable by using the selected-area aperture when the specimen image is formed by using the image-forming lenses at the magnifying system. The camera length can be controlled by the image-forming lenses.

The optical system shown in Fig. 1 is successfully realized in a LaB₆ thermal emission-type electron microscope JEM-2010 (acceleration voltage is 200 kV). JEM-2010 has three image-forming lenses and one projection lens. Specimen is located at the normal position inside the objective lens. The long focal objective lens is focused on a crossover above the specimen to perform small angle electron diffraction and/or Lorentz deflection observation. When largely magnified specimen images are required, the conventional optical system is reconstructed by normally using the objective lens. All diffraction and/or Lorentz deflection patterns are recorded on ordinary electron microscope films Fuji-FG.

In the experimental scheme mentioned above, we have confirmed that the camera length increases up to 3000 m by changing a third image-forming lens as shown in Fig. 2(a). Camera lengths obtained here are three or four orders of magnitude larger than those used in TEM crystalline analyses. Typically, the optical system of commercial TEMs offers the camera length of sub meters to a few meters.

Figure 2(b) is an image of a carbon replica grating of 2000 lines/mm, corresponding to a lattice spacing of 500 nm (5 × 10^-6 radian for 2.5 pm electron wavelength). The diffraction pattern of the replica grating is shown in Fig. 2(c), which is recorded at the camera length of 700 m on the film. First and higher order Bragg diffraction spots of the periodic grating are clearly observed with a spacing of 5 × 10^-6 rad. The divergence angle of incident electrons is estimated to be about 1 × 10^-6 radian from the size of diffracted spots.

Small angle Lorentz deflection of electrons in magnetic materials is successfully observed in the present optical system. Figure 3(a) is a Lorentz (Fresnel) image of Permalloy squares array fabricated on a Si₃N₄ membrane of 30 nm in thickness.¹¹ The size of the square is 1 μm in side widths and 30 nm in thickness. The contrast pattern inside each square in Fig. 3(a) depends on the direction of rotation, i.e., chirality of the in-plane magnetic induction (see white arrows). One of the squares is selected by 2 μm hole aperture indicated by a broken circle in Fig. 3(a) when observing the Lorentz deflection pattern. Four Lorentz deflection spots with a central spot appear in the corresponding selected-area pattern as shown in Fig. 3(b). This is because electrons are magnetically deflected in the four-fold azimuth directions by four magnetic domains formed in the Permalloy square and not deflected in a region of non-magnetic Si₃N₄ membrane outside the square. β is estimated to be 2 × 10^-5 radian, which is in good agreement with the value of 1.82 × 10^-5 radian calculated by Eq. (1) for 30 nm thick Permalloy (B₀ = 1.0 T).

Figures 3(c) and 3(d) are a Fresnel image and small angle Lorentz deflection pattern of a Permalloy disk. Since electrons passing through the disk are deflected to the whole azimuth directions reflecting a circular distribution of magnetic induction in the disk, small-angle Lorentz deflection of the disk in the selected area exhibits a ring pattern together with an undeflected spot at the center. Importantly, the Lorentz deflection angle does not depend on the shape of Permalloy elements. Thus, the ring radius is consistent with the distance between the Lorentz deflection and center spots in Fig. 3(b).

Figure 4(a) is a Fresnel micrograph of 180 degree striped magnetic domains formed in the orthorhombic phase of La_0.825Sr_0.175MnO₃ (LSMO) at 110 K.¹² A thin LSMO film with a thickness of about 200 nm is prepared by an ion-thinning technique. The Curie temperature T_C is 270 K and the period of striped magnetic domains is 550 nm. Figure 4(b) is a diffraction pattern of the 180 degree striped magnetic domains taken at the camera length of 100 m. Two bright spots appear due to the Lorentz deflection by the 180 degree magnetic domains, while the streak between the spots indicates the existence of Bloch-type domain walls. A series of tiny spots between the two main spots, indicated by arrows, correspond to the Bragg diffraction spots by the periodicity of the striped magnetic domain structure, which is working as a phase grating for electrons.^9,10 The Lorentz deflection and Bragg diffraction are obtained at β = 9.0 × 10^-5 rad and 4.3 × 10^-6 rad, respectively. Those give 190 nm thickness for B₀ = 0.78 T⁶ and the stripe period of 580 nm, which are consistent with values described above, respectively. Thus, the Lorentz deflection and Bragg diffraction patterns in Fig. 4(b) indicate clearly that quantitative information on magnetic properties of the magnetic domains is simultaneously recorded in the reciprocal space.

In the present work, we have confirmed that small angle electron diffraction and Lorentz deflection data is successfully observed at the order of down to 10^-6 radian in the conventional TEM machine with LaB₆ thermal emission type. The long-distance camera length up to 3000 m is obtained in the electron optical system constructed in this study. Small angle Bragg diffraction and Lorentz deflection analyses provide a quantitative way to analyze structural and magnetic properties in the reciprocal space. We believe that the present methods should be widely and complementary utilized together with various real-space TEM imaging techniques such as internal structure observations⁵ and magnetic visualizations including Lorentz microscopy^13,14 and electron holography.¹⁵ Furthermore, a phase reconstruction¹⁶ using small angle electron diffraction and Lorentz deflection data¹⁷ will be a novel and interesting method of obtaining microscopic images of internal structures of self-assemble superlattice and magnetic materials.