In TEM, a high-energy (100 keV–1000 keV) electron beam is transmitted through a thin specimen. When the beam passes through a region of magnetic induction in the sample, the electrons are deflected by the Lorentz force, as shown in Fig. 2. The Lorentz force is expressed as follows:

Fig. 2.

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	Fig. 2. Scheme showing the deflection of fast electrons by Lorentz force after passing through the magnetic sample.

The Lorentz deflection angle along the x direction is given by^[30]

Lorentz-TEM allows to dynamically record the changes in magnetic microstructures in either the absence or presence of an applied magnetic field. The image contrast results from the in-plane component of magnetization in the specimen, yielding bright or dark intensity bands at the position of magnetic domain walls. The Fresnel and Foucault imaging modes are the most commonly used. Figure 3 illustrates the mechanism of magnetic contrast for a simple specimen containing domains in the plane of the film separated by 180° domain walls.^[31] The Lorentz force deflects the electrons passing through the specimen and splits each of the diffraction spots into two in the back focal plane of the objective lens. Each split spot contains information about the magnetic induction from the domains in which magnetization is found.

Fig. 3.

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	Fig. 3. Scheme representing the origin of the Lorentz deflection in Lorentz-TEM for a magnetic film containing 180° domains magnetized in two directions. Fresnel images were obtained by over- or under-focusing the objective lens. Foucault images were obtained by displacing the objective aperture to choose the split diffraction spots by domains with different magnetization direction.[31]

2.2.2. Foucault mode

In the Foucault mode, the Lorentz lens is in focus, and split spots appear in the diffraction pattern. By displacing the aperture in the back focal plane to select the specific spots, the contrast resulting from the corresponding magnetic domain is formed, similar to the diffraction contrast. The domains in which electrons are deflected through the aperture are bright, whereas domains with magnetization antiparallel to that of the bright domains are dark (Fig. 3). When there are several domains in the illuminated area, as seen in Fig. 4(b), the electron beam deviates in different ways according to the different domains.^[32] This results in a split diffraction spot, Fig. 4(a).^[32] From this experiment, it can be immediately determined whether the direction of magnetization is normal to the direction of spot splitting.

Fig. 4.

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	Fig. 4. (a) Splitting of 200 spots from a Ni3Mn sample caused by the presence of magnetic domains. (b) Image from all of the split spots in (a) showing the four domains that scattered electrons into various spots.[32]

The contrast in Foucault images is sensitive to the position of the aperture. Therefore, the technique is not quantitative if only a single pair of Foucault images is recorded along each aperture. In addition, as mentioned above, the deflection angle is very small compared to Bragg scattering. The position of the aperture has to be carefully set in the experiments to distinguish each split spot.

Images of magnetic domain walls can be obtained via the Fresnel mode of Lorentz-TEM. More recently, the TIE method^[33] was developed to allow Fresnel images to quantitatively reconstruct the phase associated with magnetic thin films,^[34,35] thus obtaining further magnetic information.

Mathematically, the TIE corresponds exactly to the Schrödinger equation for high-energy electrons in vacuum under the small-angle approximation. The TIE formula can be written as follows:^[36]

This equation relates the phase distribution with the intensity distribution and its derivative with respect to the direction of wave propagation. Note that TIE does not assume any scattering model, and thus can be used TIE even for strong phase objects. Since the small-angle approximation for the Schrödinger equation is the basis for microscopy techniques that use high-energy electrons, TIE may be universally applied in electron microscopy.

Because TIE is a partial differential equation, a practical way to solve it was devised.^[36] The final expression for the numerical solution is

One of the difficulties of using TIE is to estimate the intensity derivative with respect to z as accurately as possible. The intensity derivative is a mathematical concept, and in practice, a finite difference is calculated to approximate it. Figure 5 schematically shows an experimental situation for the application of the TIE, in which the image intensities were measured at a few consecutive planes. Image processing was performed in the QPT software for Digital Micrograph.^[37]

Fig. 5.

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	Fig. 5. Estimation of the intensity derivative along the direction of wave propagation.[36]

Conversely, according to the Maxwell–Ampére equations, in a magnetic field, ϕ (xyz) and the magnetization have the following relationship^[24]

Off-axis EH examines electron-transparent samples using defocused illumination from a highly coherent field-emission gun electron source. The region of interest should be positioned as shown in Fig. 6, so that approximately half the field of view is covered. Applying a voltage to an electron biprism results in the overlap of a reference electron wave that passes through the vacuum with one that passes through the sample (Fig. 6). The amplitude and phase shift of the wave are respectively recorded in the intensity and position of the hologram. Changes in the position or spacing of the interference fringes corresponds the relative phase shift of the electron wave that passes through different sections of the sample. When studying magnetic materials, a Lorentz lens allows to operate the microscope at high magnification with the objective lens switched off and the sample in magnetic-field-free conditions, Fig. 6(b).

Fig. 6.

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	Fig. 6. Scheme of microscope geometry for off-axis electron holography: (a) electrostatic phase shifts corresponds the projected potential times the sample thickness, (b) slope of phase shift corresponds to the sample magnetization.[38]

An expression for the intensity distribution in an off-axis electron hologram can be obtained by considering the addition of a plane reference wave to a complex specimen wave as follows:^[39]

Three separate contributions are considered for the intensity distribution in the hologram: both the reference and specimen image intensity, and a set of cosinusoidal fringes. The local phase shifts and amplitudes are equivalent to the phase and amplitude of the electron wavefunction in the image plane. To extract phase and amplitude information, a Fourier transform is performed for the hologram as follows:^[39]

Equation (9) describes a peak centered at the origin corresponding to the Fourier transform of a conventional bright-field TEM image of the sample, and two sidebands centered at and corresponding to the Fourier transform of the desired image wavefunction. To recover the complex electron wavefunction of the object, one of the two sidebands in the Fourier transform is selected, and then the amplitude and phase are extracted through the inverse Fourier transform.

Figure 7 illustrates how a phase shift is extracted using off-axis EH. The original recorded hologram is presented in Fig. 7(a), an enlargement showing the bending of the holographic interference fringes at the edge of the sample can be seen in Fig. 7(b). Next, the Fast-Fourier transform (FFT) of the hologram is calculated, Fig. 7(c), yielding a central peak and two sidebands that are complex conjugates of each other. These sidebands contain the phase and amplitude information about the sample that cannot be recovered from the central autocorrelation peak. Then, the inverse FFT is calculated selecting only one of the sidebands, in Fig. 7(c). Finally, the reconstructed complex image is obtained, Fig. 7(d). A color-coded phase image showing details of the magnetic induction is presented in Fig. 7(e).

Fig. 7.

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	Fig. 7. Overview of the steps for quantitative phase imaging using off-axis electron holography: (a) original hologram, (b) enlargement showing bending of interference fringes, (c) FFT of hologram, (d) reconstructed complex image calculated by inverse Fourier transform from FFT sideband, (e) color-coded phase image showing magnetic induction.[38]

The recorded phase can be used to measure the electrostatic potential and the in-plane component of the magnetic induction in the sample. Neglecting the effects of dynamical diffraction, the phase can be expressed as follows:^[40]

Nanostructures from bulk chiral magnets have been fabricated using focus ion beam (FIB) lithography. The surface atoms of the materials acquire high energy from the irradiated heavy metal ions (e.g., Ga³⁺ and Xe³⁺) and can thus escape from the mother material. Because the ion beam was focused, the minimum linewidth irradiated could in the nm range, enabling us to fabricate high-precision nanostructures. Furthermore, when irradiating the materials in a metal organic atmosphere, metal organics could be deposited in the irradiation zone after decomposition. Using the SEM-FIB dual-beam system, we were able to fabricate nanostructures with various shapes, Fig. 8. The nanostructures were then transferred into copper chipsusing a micromanipulator. They were then thinned onto plates less than 150-nm thick for the TEM measurements.^[11,22,42]

Fig. 8.

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	Fig. 8. (a) Schematic diagram of dual-beam microscopy combined with electron and ion optics. (b) In a typical sample fabrication process, a cylinder is obtained by rotating and tilting the sample stage. This system allows for sub-nanometer SEM imaging and also to produce nanostructures with specific geometries with precise prototyping capabilities.

Lorentz-TEM is an important technique to analyze magnetic structures with a resolution of ∼2 nm. Skyrmions have been widely investigated using Lorentz-TEM, revealing several unique features.^[7,10,24] A typical procedure to obtain the magnetic configurations is shown in Fig. 9. In particular, we stabilized the skyrmion phase in an extended temperature-field region in thin plates that was larger than the corresponding bulk material, in bulk material the region hosting skyrmion phase was only in a tiny temperature-field region close to the critical temperature. The magnetic anisotropy or spatial confinement effect is suggested to be responsible to the stabilization of skyrmions.^[43–45]

Fig. 9.

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	Fig. 9. Schematic diagram of the in-plane magnetic configurations obtained for the skyrmion phase. Using Lorentz-TEM, three images were acquired in the same region under three different defocus conditions. The in-plane magnetization distribution map was generated from the phase image by applying Eqs. (3)–(5) (Subsection 2.3). The color wheel represents the magnetization direction at every point.

Observing the skyrmion phase in a small sample in the scale of a single skyrmion size is very attractive because it is directly related to the size of a real device. The Fresnel fringes obtained at under- or over-focus conditions, however, hampered the determination of magnetic structures at the edge of the nanostructure.^[11,22] A large Fresnel fringes region was obtained for a larger defocus distance. It was essential to increase the defocus distance to observe the magnetic structure with weak in-plane moments, which in turn enlarged the Fresnel fringes. Note that the Fresnel fringes in the nanostructures with metallic organic Pt–C interfaces could be greatly reduced compared to those with a vacuum interface, enabling the magnetic imaging of a nanostructure with a size down to ∼150 nm for a typical skyrmion host material (i.e., FeGe).^[10,21] For a more accurate determination of the magnetic structure in samples under 150 nm, EH is preferable because it performed under in-focus conditions.

Figure 10 shows the skyrmion arrangements in the helimagnetic FeGe nanostructures.^[11,22] Closely packed skyrmions could be observed at high temperature. At low temperatures, one helical stripe was transferred into the skyrmion. The following size effect was demonstrated: for the nanostrips, the number of skyrmion chains decreased as the nanostrip width decreased. It is worth noting that a single chain of skyrmions was observed in the ∼130-nm wide FeGe nanostrip, indicating that skyrmion-based racetrack memory could be achieved. Similarly, the size of the nanodisks influenced the whole skyrmion number for the cluster states. In addition to the size effect, we also investigated the temperature and field effects using Lorentz-TEM. For example, skyrmion clusters in nanodisks with different geometrical configurations and field-driven quantum annihilation of skyrmion numbers have been observed.^[11,22] In a FeGe nanodisk at an elevated temperature, close to its Curie temperature, thermal fluctuations enabled the quantum jump between two metastable skyrmion phases with different skyrmion numbers. In a 330-nm FeGe nanodisk, four skyrmions appeared to change their positions with a clockwise or anticlockwise rotation as a result of thermal fluctuations.^[11]

Fig. 10.

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Fig. 10. Size dependence of the skyrmions arrangements of the FeGe nanostrip in panel (a) and nanodisk in panel (b). The measured temperature was 220 K and a magnetic field was applied to stabilize the skyrmion state. W and R represent the width of the nanostrip and the diameter of nanodisk, respectively. The red numbers in panel (a) and panel (b) stand for the number of skyrmion chain and number of isolated skyrmions, respectively.[11,22]

The EH technique has been widely used to investigate the magnetic flux of magnetic materials. For magnetic skyrmions, the first experiment was performed by Park et al..^[30] They used ultrahigh microscopy (1000 kV) with the Lorentz mode, meaning that an external magnetic field could be applied to the sample by exciting the conventional microscope objective lens to a pre-calibrated value. EH enabled the reliable quantification of magnetic states with nanometer spatial resolution and high phase sensitivity. By analyzing the dependence of phase shift on the thickness of thin Fe_0.5Co_0.5Si plates, Park et al. inferred that skyrmions in plates with a certain thickness were in fact homogenous. In the Lorentz mode, there is a strong Fresnel fringe contrast at the sampleʼs edge, contrastingly, EH uses in-focus conditions, which are ideal for imaging skyrmions in magnetic nanostructures. In this case, the contribution of phase change arose from (i) an electrostatic phase that originated from local variations in the sampleʼs thickness and composition, and (ii) a magnetic phase that originated from the magnetic vector potential associated with the sampleʼs magnetism. To obtain the magnetic phase information, the two phases must be separated, in particular, the area close to the edge of the sample where thickness and composition remarkably change.

We developed an experimental procedure to separate the magnetic contribution to the total phases in a magnetic nanostructure using data from Zheng,^[44] Fig. 11. The total recorded phase of a FeGe nanodisk at K under an applied magnetic field is shown in Fig. 11(c). The spotty contrast in the center of the nanodisk arose from the formation of individual skyrmions. The dominant contrast arising from the electrostatic contribution to the phase, however, hindered the interpretation of the detailed skyrmion structure. To separate the two phases, we recorded the electrostatic phase at T ∼300 K, which is above the Curie temperature (T_C∼280 K) of FeGe. We generated a magnetic induction map from the resulting magnetic contribution to the phase using both phase contour lines of spacing and colors to represent the direction and magnitude of the measured in-plane magnetic induction.

Fig. 11.

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Fig. 11. Experimental procedure to obtain the magnetic flux for a FeGe nanodisk: (a) the sample was analyzed at room temperature to acquire the electrostatic (mean inner potential) contribution because the Curie temperature of FeGe was only 278 K, (b) total phase image (containing both the mean inner potential and magnetic contribution) recorded at T ∼95 K, (c), magnetic contribution to the phase obtained by evaluating the difference between the phase images recorded at room temperature and at 95 K, and (d) magnetic induction map of the FeGe nanostrip with a diameter of 160 nm. Data for this procedure are from Zheng.[26]

Using the EH technique, we achieved an important result, that is, the discovery of a magnetic bobber. Generally, skyrmions are believed to be the only chiral nanoscale spin texture in chiral magnets,^[46,47] therefore, pinning a chain of individual sparsely distributed skyrmions to the desired position to achieve data encoding in skyrmion-based memory devices requires a significant effort. In this case, the distance between skyrmions should be quantized to represent the correct number of “0” between “1”. Thus far, however, the approach to maintain such distance between skyrmions was suggested to carve a notch around the edge of the sample. Theoretical work demonstrated that chiral magnets with a certain thickness could stabilize another type of localized excitations, i.e., chiral bobbers, which may coexist along with skyrmions in a wide range of parameters. As seen in Fig. 12, we accomplished the first direct observation of chiral bobbers using EH. Because the magnetic bobber occupied a significantly smaller volume than that of a skyrmion tube, these two states could be easily distinguished by their net magnetization. Each state would yield a different magnetic contrast in the off-axis EH technique because EH provided a high-spatial resolution and quantitative measurements of the phase shift, which originated from interactions between electrons with an in-plane component of magnetization. The existence of two types of localized and movable objects results in a new magnetic data storage concept, in which the two objects act as the binary state of a data bit.

Fig. 12.

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Fig. 12. Discovery of another localized nanoscale spin texture (i.e., magnetic bobber) in a chiral magnet FeGe nanostrip. (a) Magnetic flux measured in an FeGe nanostrip with a thickness of 175 nm. The circular dots with strong contrast on the right side stand for the skyrmion tubes. The weak dot on the left side corresponds the chiral magnetic bobber. (b) The fine spin configurations for bobber and skyrmions.[46]