The most common techniques for imaging the magnetic domains in TEM are Foucault and Fresnel modes, and their corresponding schematics are shown in Fig. 1.^[27] For the purpose of illustration, a simple specimen comprising oppositely magnetized 180 °C domains is assumed. When the parallel electron beam passes through the magnetic domain region, the Lorentz force leads to the deflection of the electrons and the diffraction spot splits into two by following the right-hand rule as shown schematically in Fig. 1(c). One split spot contains information about domains with magnetic moments lying in one direction, and the other spot contains information about the antiparallel domains.^[24,27,28]

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of ray diagram indicating the paths of electrons passing through a magnetic specimen, together with the magnetic domain or domain wall contrast for the Foucault [(a)–(b)] and Fresnel modes [(c)–(e)] of Lorentz TEM.[27]

In the Foucault mode as shown in Figs. 1(a)–1(b), one of the split spots due to the deflection of the electrons in the diffraction pattern is selected by an objective aperture in the focal plane to image the magnetic domains while keeping the imaging lens in-focus. The domains with antiparallel magnetization show bright and dark contrast depending on which split spot is selected. The magnetization direction in the domains can be identified if the relative direction of aperture displacement with respect to the image is known. To obtain high-quality Foucault-mode images, the back-focal plane of the objective lens and the blocking aperture must be as near coplanar as possible. It is inconvenient to carry out dynamic imaging of magnetization in Foucault mode under external fields since both the electron beam and the aperture position relatively shift and the split spots change with external fields.

For imaging the magnetic domains in Fresnel mode as shown in Figs. 1(c)–(e), the deflecting electrons are focused in the final image plane and no magnetic contrast appears in the in-focus conditions. However, when the Lorentz TEM is in the over-focused condition, the electron deflection induces a decreased intensity contrast because the electrons are deflected away from the domain wall. This results in the appearance of dark contrast lines in the domain wall regions. Similarly, bright contrast lines appear in under-focused conditions due to the increased electron density caused by the converged electrons. In this sense, the inversion of the magnetic contrast at the domain walls is observed between the over- and under-focused images. This is the classic Fresnel imaging mode. Different from the Foucault mode, the bright and black contrasts indicate the domain wall position, not the magnetization from inner magnetic domains. It should be noted that the spatial resolution of Fresnel mode is not as high as that of Foucault mode because the images must be recorded at a relatively large objective lens defocus in order to manifest the contrast of domain walls. The advantages of Fresnel imaging mode are its ease of operation and significant contrast. It is also convenient for performing real-space in situ observation, allowing recording of dynamic domain evolution behavior during magnetizing and manipulation under external fields.^[29]

In order to quantitatively determine the direction of magnetization and the spin configuration of magnetic domains using Lorentz TEM images, a method named transport-of-intensity equation (TIE) is generally adopted based on Fresnel mode, which is briefly introduced as follows. When a plane wave of electrons passes through a TEM specimen, the direction of transmitted electrons varies owing to the Lorentz force caused by the interaction between electron wave and specimen, and hence the different image intensity is produced. The image intensity includes both the amplitude and phase information of the transmitted electron wave. TIE can be used to detect the phase information from the intensity distribution as follows:^[30]

Fig. 2.

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	Fig. 2. (color online) Schematic intensity distribution of Lorentz TEM images at under-focus, just-focus, and over-focus conditions for TIE method

Therefore, the lateral magnetization distribution can be quantified using the above TIE method to analyze the electron intensity differences of Lorentz TEM images with different focus values. This knowledge has been developed into a commercial QPt software embedded in widely used Digital Micrograph (DM) TEM interface.

Differential phase contrast (DPC) in Lorentz TEM is a high-resolution magnetic imaging technique for visualizing electromagnetic field distribution via measuring the deflection of an electron beam at each beam-scan point.^[28,31,32] The differential phase contrast (DPC) technique, principally described in Fig. 3(a), requires a field emission gun source to create a very high monochromatic electron beam with converged probe scanning across the specimen similar to the normal scanning transmission electron microscopy (STEM).^[33] When the electron beam scans through the specimen, a cone of illumination beam carrying the differential or gradient of the electron wave phase is projected onto a circular quadrant detector. If the specimen is non-magnetic, the disc will be centered on the detector. If the specimen is magnetic, the Lorentz force deflects the electrons and shifts the disc to a position that is no longer concentric on the detector. Each segment of the detector measures a separate electron signal. The different signals taken from opposite segments of quadrants provide the magnetic induction information as two orthogonal components. The resolution in DPC Lorentz TEM depends on the probe size and the best resolution commonly achieved is in the range of 2 nm–5 nm.^[34,35] In recent past, for aberration-corrected STEM, the best resolution of about 0.9 nm in DPC Lorentz STEM was reported.^[31] High resolution, straightforward interpretation, and easy mapping of magnetic induction are the main advantages of DPC. For example, the DPC STEM images demonstrate the spin configuration of the transition from a single-skyrmion state (Figs. 3(b)–3(c)) to a triple-skyrmion state (Figs. 3(d)–3(e)) in a very narrow range of perpendicular magnetic fields at 295 K in Co₈Zn₈Mn₄.^[33]

Fig. 3.

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	Fig. 3. (color online) (a) Schematic illustration of the electron beam deflection in a specimen using a segmented STEM detector and top view of the four segments detector presenting the results of a not-deflected beam and deflected beam after passing through the specimen. [(b)–(e)] DPC-STEM experimental observation about magnetic skyrmion in Co8Zn8Mn4.[33]

Another magnetic imaging technique in Lorentz TEM is off-axis electron holography, which is based on the interference of electron waves passing through the sample (object wave) and the vacuum (reference wave), as shown schematically in Fig. 4(a).^[36–38] The critical requirement for electron holography is a coherent source of electrons together with a thin and uniform TEM specimen. When the electron wave passes through the sample, the phase shift in the exit wave contains the sample information of both magnetic and electric potentials. An electron biprism inserted between the back focal plane and the image plane superimposes the exit wave and the reference wave resulting in an interference pattern (hologram) in the image plane. From the hologram, the electron wave can be reconstructed in the form of two images, an amplitude image and a phase image. The measured phase shift can be expressed in the form:^[39–41]

Fig. 4.

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	Fig. 4. (color online) (a) Principle of electron holography in Lorentz TEM;[41] [(b)–(e)] The magnetic flux maps of the skyrmion lattice in Fe0.5Co0.5Si.[37]

Magnetic vortex has been studied mostly in permalloy with the pattern constriction to exemplify its potential applications in magnetic random access memory.^[56,57,60] Lorentz TEM observation gives direct and clear images about the topological magnetic vortices and particularly their dynamic behavior in correlation to the application. Figure 6 demonstrates the current-excited magnetization dynamics in a permalloy nanowire by in situ Lorentz TEM and the quantitative distribution of magnetic flux lines by electron holography. The pulsed electric current of 2.0 × 10¹¹ A/m² with 300-ns interval was applied to the uniformly magnetized state of the nanowire from right to left at zero magnetic field as shown in Fig. 6(a). Chained vortices and crosstie walls nucleated from the uniformly magnetized state due to the stimulation of the electric current pulse as shown in Fig. 6(b) and Fig. 6(e). Such nucleated vortices annihilated immediately before the next current pulse arrived and turned into a uniformly magnetized state in probability as high as 90% (Fig. 6(d) and Fig. 6(c)). Interestingly, paired domain walls nucleated from the uniformly magnetized domain state as shown in Fig. 6(d).^[57]

Fig. 6.

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	Fig. 6. (color online) (a)–(c) Lorentz images of the magnetization reversal in the uniformly magnetized domain state at zero magnetic field; (e)–(g) showing the corresponding distribution of magnetic flux lines via holography in panels (b)–(d).[57]

Besides the typical vortex study in patterned permalloy, amorphous Ce₁₄Fe₈₀B₆ ribbon has been recently explored to demonstrate the generation and manipulation of magnetic vortices without pattern constriction.^[61] Figure 7 shows the temperature dependence of vortex–antivortex (V–AV) evolution. Figure 7(b) shows more V/AV in the chains of cross-tie domain walls (DWs) at lower temperatures of 278 K than at 293 K (Fig. 7(b)). Figures 7(c)–7(f) demonstrate the temperature dependence of the magnetic textures in the red-selected region without obvious defects (Fig. 7(b)). Upon cooling from 293 K to 123 K, the number of magnetic vortices is increased because of the pairwise nucleation of vortices and anti-vortices. The inset of Fig. 7(c) shows the in-plane magnetic configuration of the cross-tie wall, reconstructed by TIE analysis, similar to the spin configuration of the schematic vortex and antivortex along the cross-tie domain wall as shown in Figs. 7(g)–7(h).^[10,62]

Fig. 7.

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	Fig. 7. (color online) (a)–(f) Lorentz TEM images for temperature dependence of cross-tie domain wall evolution in amorphous Ce14Fe80B6 ribbon;[61] (c) the enlarged area of red rectangle region of panel (b). The inset of panel (c) shows the corresponding TIE analyzed spin texture of cross-tie wall. Panels (g)–(h) show the schematic spin configuration.[10,62]

The V/AV propagation and nucleation/annihilation under electric current excitation observed via Lorentz TEM are shown in Fig. 8, with the schematic current-driven microdevice in Fig. 8(i). An individual vortex (marked by a yellow rectangle in Fig. 8(a)) moves towards the edge and disappears (Fig. 8(b)) with increasing electric current. The contrast of the magnetization ripple changes significantly and almost all the vortices completely annihilate at 432 mA (Fig. 8(c)), indicating the magnetization transition process. The vortices and anti-vortices nucleate again with further increasing the current to 622 mA (Fig. 8(d)) and become a stabilized cross-tie DW at 676 mA (Fig. 8(e)). The vortices and anti-vortices annihilate under a reversed electric current (Figs. 8(f)–7(g)) and then re-nucleate (Fig. 8(h)) at 969 mA, which remains stabilized even after the electric current is switched off. The summarized curves (Fig. 8(j)) indicate the numbers of vortex cores versus current. The in-plane magnetization change is presented via the changing ripples and vague vortex cores due to the strong interactions between electric current and magnetic moments while increasing electric current. This real-space observation of the nucleation and annihilation behaviors under electric current provides a fruitful playground for fundamental physics and the exploration of the novel application of accumulated rare-earth resources.^[61]

Fig. 8.

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	Fig. 8. (color online) (a)–(h) In situ Lorentz TEM observation for electric current dependence of cross-tie domain wall evolution. (i) Schematic electric-current manipulation device. (j) The summarized number of vortex cores versus current.[61]

Magnetic skyrmions were originally introduced by the British physicist Tony Skyrme to describe localized, particle-like configurations in the field of pion particles.^[63] In recent years, a skyrmion crystal (SkX) with a non-centrosymmetric B20 cubic structure was experimentally identified for the first time in a helimagnet MnSi by small-angle neutron scattering (SANS) measurements. In the ground state, the magnetic structure shows a proper screw structure with a periodicity of ∼ 19 nm, which is induced by Dzyaloshinskii–Moriya interaction (DMI) originating from the symmetry-broken structure. When an external magnetic field is applied, SkX with six-fold symmetry emerges perpendicular to the external magnetic field as shown in Fig. 9(a).^[11] However, the magnetization configuration in real space is not feasible using SANS. Lorentz TEM imaging together with TIE method clearly shows the spin configuration of skyrmions and prompts the overwhelming study since the first demonstration in thin plates of helimagnets Fe_0.5Co_0.5Si (Fig. 9(b)).^[19] Later, skyrmions were discovered in several magnetic alloys with the same asymmetric B20 structure.^[12,64,65] Among them, FeGe (Fig. 9(c)) has the highest Curie temperature of about 278 K near room temperature.^[12,66] Recent studies have revealed different manipulations to stabilize skyrmions in terms of temperature and magnetic field range for prompting its application. When the plate thickness (15 nm) is smaller than the helical period, i.e. a magnetically two-dimensional (2D) state, the T–B skyrmion phase window is broadened as shown in Fig. 9(c) in comparison to the original limited temperature range of B20 compounds. The realization of room-temperature skyrmions and the current-driven behavior is critical for its practical application in information storage and logic technologies, therefore, the exploration for new material and external-field manipulations have always drawn significant attention.^{[6,58,67–69]}

Fig. 9.

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	Fig. 9. (color online) (a) The skyrmion crystal first identified by SANS measurements in MnSi.[11] (b) and (c) Lorentz TEM observation of skyrmion phase in Fe0.5Co0.5Si[19] and FeGe.[12,66]

3.2.1. Magnetic skyrmions in bulk magnets

Up to now, numerous magnetic materials hosting skyrmions,^[14,66,70] which could be generated via different mechanisms including the dominated symmetry-broken DMI and dipolar interactions, have been discovered. Using Lorentz TEM, biskyrmion spin configuration is identified over a wide temperature range of 16 K–340 K in hexagonal centrosymmetric (Mn_1−xNi_x)₆₅Ga₃₅ (x = 0.5) (MnNiGa) alloy. The biskyrmion evolution from ground stripe domains is clearly demonstrated in Fig. 10.^[71] The overall skyrmion distribution in terms of temperature and magnetic field (Fig. 10(i)) exposes the indispensable requirement of magnetic field for equilibrium skyrmion phase and less biskyrmion density away from T_C. Field-cooling (FC) manipulation has shown effective in extending skyrmions to a wider temperature and magnetic field range. Metastable SkX state is extended below the equilibrium SkX temperature range via FC manipulation evidenced by neutron diffraction and Hall resistivity measurements in β-Mn-type Co₈Zn₈Mn₄ ^[64] and MnSi,^[72] respectively. However, the nontrivial zero-field skyrmions cannot be obtained in the equilibrium range via FC in the above skyrmion materials and the indirect detection methods do not depict the dynamic behavior. The in situ Lorentz TEM observation gives clear images depicting the generation and sustainability of the robust field-free biskyrmion lattice via appropriate FC manipulation from above T_C in centrosymmetric MnNiGa alloy as shown in Fig. 11. The nucleation of biskyrmion phase is energetically favored under appropriate magnetic field while cooling across T_C; the topological protection and the increased energy barrier help to stabilize biskyrmion phase without transforming into stripe domains. The detailed magnetic evolution under different FC procedures are shown in Fig. 12, which demonstrates that the magnetic field strength during FC plays a critical role, and the optimum magnetic field strength to generate highest-density biskyrmions is obtained by comparing the biskyrmion densities in presence of too strong and weak fields.^[18] The high-density zero-field biskyrmion lattice sustains over the whole temperature range of 16 K–338 K (Fig. 11(b)) after the optimized FC manipulation, promoting the potential applications in nonvolatile memory devices.

Fig. 10.

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	Fig. 10. (color online) (a)–(h) Magnetic field dependence of biskyrmions in real-space Lorentz TEM images at room temperature and the corresponding spin texture obtained by TIE analysis.[71] (i) The contour mapping of the external magnetic field and temperature for biskyrmion distribution in MnNiGa extracted from Lorentz TEM data.[18]

Fig. 11.

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	Fig. 11. (color online) (a) Lorentz TEM image of zero-field biskyrmion lattice at room temperature after FC manipulation. (b) Phase diagram obtained by contour mapping biskyrmion density, as a function of external magnetic field and temperature, based on the Lorentz TEM data after 50-mT FC.[18]

Fig. 12.

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	Fig. 12. (color online) (a)–(d) Magnetic domain dependence on temperature at 50 mT (e)–(h) Biskyrmion density dependence on the different magnetic fields while FC process. (i)–(l) Biskyrmion distribution after increasing the magnetic field to complete skyrmion state based on panels (e)–(h). The experimental procedures are shown on top of the column. The scale bars are 200 nm.[18]

3.2.2. Magnetic skyrmions in multilayers

Tremendous progress about exploring new skyrmion materials has been made in the multilayer thin film since the discovery of the blowing skyrmion behavior via electric currents in a sandwiched Ta/CoFeB/TaO multilayer with interfacial DMI.^[73] Room temperature skyrmions, with their electric field-driven behavior and compatibility with current spintronic fabrication techniques in thin films can be readily integrated into the spintronic devices. The skyrmion characterization and dynamic behavior can be studied using various instruments. For example, writing and deleting a single skyrmion is studied using spin-polarized tunneling currents at low temperatures using spin-polarized scanning tunneling microscopy (SP-STM).^[74] Blowing out room-temperature skyrmions due to electrical currents are observed using skyrmion Hall effect by magneto-optical Kerr effect (MOKE).^[73,75] Artificial skyrmions are detected due to the interlayer coupling in thin films by scanning electron microscopy with polarization analysis (SEMPA).^[76]

Lorentz TEM has advantages in demonstrating the spin configuration and dynamic behavior of nanometric magnetic domains under different external fields. The spin configuration of typical Néel-type skyrmions in multilayers can be identified by tilting the specimen based on the magnetic field-induced beam deflection and the contrast mechanism of Lorentz TEM, which can be summarized as the disappearance of contrast at a position and reversal of contrast at relatively opposite angles.^[68,77] Moreover, the multilayers sputtered on the Si₃N₄ membrane windows for direct TEM observation avoid the extrinsic features introduced by additional TEM preparation. Néel-type magnetic skyrmions were successfully identified in Ta(4)/[Pt(3)/Co(1.85)/Ta(3)]₆ multilayers based on their response to opposite tilting angle of 18° as shown in Figs. 13(b)–13(d).^[78] By increasing Co layer thickness to alter the anisotropy, only a few Néel-type skyrmions could be identified, as marked by yellow arrows in Figs. 13(e)–13(g), using the same tilting procedure.^[78] The consistent contrast regardless of any tilting angle indicates non-ideal Néel-type skyrmions as shown in Figs. 13(e)–13(j). Figure 14 shows the sub-50-nm skyrmion distribution with and without electromagnetical manipulation in Ta(4)/[Pt(3)/Co(1.85)/Ta(3)]₆ multilayers,^[79] where the skyrmion density is tunable and can be significantly enhanced by electromagnetical manipulation as compared to the skyrmions density directly induced via magnetic field. Remarkably, these high-density skyrmions sustain at zero field due to the optimized manipulation after both the in-plane current and perpendicular magnetic fields are switched off.

Fig. 13.

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Fig. 13. (color online) Identification of skyrmion spin configuration by tilting samples in real-space Lorentz TEM at 300 K. (a) Schematic diagram of titling process, α is the tilt angle around x-axis. The angle α for Pt/Co(1.85)/Ta is (b) −18°, (c) 0°, and (d) 18°, respectively. The enlarged views of a single skyrmion in the insets (b) and (d) with bright and dark reversed magnetic contrast for the opposite tilting angle are identified as Néel-type skyrmion. The tilting angle α are (e) −18°, (f) 0°, and (g) 18° for Pt/Co(1.95)/Ta, respectively. The tilting angle α are (h) −20°, (i) 0°, and (j) 20° for Pt/Co(2.1)/Ta, respectively. Scale bars in panels (b)–(j) correspond to 50 nm.[78]

Fig. 14.

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	Fig. 14. (color online) The Lorentz TEM image of the Néel-type skyrmions in Ta(4)/[Pt(3)/Co(1.85)/Ta(3)]6 multilayers, demonstrating the comparison between the nonvolatile field-free skyrmions after electromagnetical manipulation and the randomly distributed biskyrmions induced purely by the magnetic field.[79]

Figure 15 demonstrates the different magnetic domain distributions owing to the different electromagnetic effects under the same magnetic field of 270 Oe but different electric current densities. The real-space observation of zero-field magnetic skyrmions in the multilayers at room temperature promotes skyrmions in practical applications.

Fig. 15.

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	Fig. 15. (color online) Magnetic skyrmion evolution by tuning the electric current density, at a fixed magnetic field of 270 Oe, at values: (a) 0, (b) 5.01, (c) 6.81, (d) 7.36 (× 108 A⋅m−2). (e) High-density skyrmions at J = 0. (f) Zero-field and high-density skyrmions at J = 0 and H = 0. The scale bar in panel (a) is 200 nm.[79]

3.2.3. Magnetic skyrmions driven by electric currents

Electron spins play a critical role and contribute differently in functional magnetic materials. In 2008, Parkin et al. proposed that the polarization current drives the domain wall movement via spin transfer torque effect, thereby realizing the information reading and writing in the racetrack memory.^[80] From the aspect of device application, dynamic current-driven motion of magnetic skyrmion is superior to the normal domain wall due to the lower current density requirements. The interactions between the traversed electrons and the topological skyrmions give rise to the topological Hall effect. As a counteraction of the topological Hall effect related to the electron Hall motion, the skyrmion or SkX itself exhibits a Hall motion.^[6] Similar to the FM domain wall motion driven by spin-polarized electric current, the flowing electric current drives the skyrmions not only along the electron flow direction but also along the transverse (Hall) motion as shown in Figs. 16(a).^[6] Figures 16(b) and 16(c) show current-flow dynamics captured from in situ Lorentz TEM movie based on the perpendicular magnetic field of 150 mT applied to generate skyrmion crystal.^[12] Skyrmions in region A start to move at an applied current of 0.41 mA, then the skyrmions in region B move at a current of 0.50 mA. As the current further increases, up to 0.61 mA, skyrmions are no longer static over the whole region. An electric current of 1 mA corresponds to a current density of ∼ 65 A⋅cm⁻². Skyrmion-based devices with such low-current controllability could be further explored for next-generation spintronics.

Fig. 16.

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	Fig. 16. (color online) (a) Schematic interaction between skyrmion and electrons demonstrating the topological Hall effect.[6] (b) Schematic diagram of a microdevice in FeGe plate. (c) Lorentz TEM images showing current-driven dynamics of skyrmions at 250 K in FeGe. Scale bar is 100 nm.[12]

Traditional magnetic bubble domains were studied as early as 1970 with the corresponding information storage device commercialized at that time.^[81,82] With the development of high-resolution techniques, the detailed spin configuration inside the domain wall and the discovery of nanometer bubbles led to a new surge of research interests. The topological states of traditional magnetic bubbles are usually various and random, as shown in Fig. 17, depending on the defects and orientations of the BLs inside the domain wall. Hence, the topological number can be 2, 1, or 0 for the magnetic bubbles with different internal structures or the configuration of BLs. The BL inside the bubble domain wall is very sensitive to the excitations from magnetic field. By using in situ Lorentz TEM imaging, the magnetization dynamics through the BL motion were clearly depicted in La_1−xSr_xMnO₃ (LSMO) (x = 0.15–0.2) as shown in Fig. 17. Figure 17(a)–17(d) show the motion of the BLs along the bubble domain wall, accompanied by the deformation of the bubble under the in-plane magnetic field along the a axis. Under the influence of the applied magnetic field, the initial state of magnetic bubbles changes withholding different topological numbers as shown in Figs. 17(a)–17(d), Lorentz TEM images in Figs. 17(i)–17(l) demonstrate the temporal evolution of magnetization accompanied by annihilation of BLs for a selected magnetic bubble under a fixed in-plane magnetic field. A successive evolution within 2.0 s is clearly recorded including the initial generation of a pair of BLs inside the bubble domain wall, the movement of the bubble wall, and the final annihilation. Therefore, in situ Lorentz TEM has good capability of analyzing the spin configuration and providing evidence for the topological transitions driven by external fields and time.^[14]

Fig. 17.

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	Fig. 17. (color online) (a)–(d) Magnetization dynamics through Bloch line (BL) motion induced by the in-plane magnetic fields. (e)–(h) Analyzed spin configurations for the bubbles shown in panels (a)–(d), respectively. (g)–(i) Temporal evolution of magnetization accompanied by annihilation of BLs.[14]

Despite the complex spin configuration in the above-mentioned narrow domain wall region, the domain morphology favors the inside magnetization direction along the easy-axis, which usually requires the quality factor Q (Q = K_u/K_d = μ₀H_k/μ₀M_s) larger than 1. The enhancement of quality factor Q caused by the substitution of rare-earth element Y in MnNiGa contributes to the transition from biskyrmions to bubbles as shown in Fig. 18. The biskyrmion spin configuration composed of two single skyrmions with opposite magnetic helicities in MnNiGa is demonstrated in Fig. 10 and discussed in Subsection 3.2.^[71] Lorentz TEM images identify the magnetic bubbles with random distribution of different spin configurations (Fig. 18(b)) in clear comparison with uniform biskyrmions (Fig. 18(a)) although other magnetic properties like T_C do not change much due to the slightly substituted rare earth Y. The magnified in-plane magnetization textures via TIE analysis demonstrate bubbles “1, 2” with clockwise and counterclockwise winding spins in three concentric rings and bubble “3” with a pair of Bloch lines. To explore the intrinsic nature of the topological transition, the effective uniaxial anisotropy field H_k and the saturated magnetization M_s for MnNiGa and MnNiGaY are measured as shown in Fig. 18(c). Substitution of Y leads to a drop in the calculated stray field energy K_d from 119.1 kJ⋅m⁻³ to 74.8 kJ⋅m⁻³ and uniaxial anisotropy K_u from 82.70 to 74.20 kJ⋅m⁻³. Therefore, the overall quality factor Q increases from 0.69 for MnNiGa to 0.99 for MnNiGaY, which explains the origin of the spin configuration transition from biskyrmions to bubbles well.^[83] These novel magnetic textures directly observed by real-space Lorentz TEM may encourage further exploration of new materials with skyrmionic bubbles in accordance with the magnetic parameters.

Fig. 18.

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Fig. 18. (color online) Topological transition from (a) biskyrmion configuration in MnNiGa[71] to (b) magnetic bubbles with various topological states in MnNiGaY by substitution of rare-earth element Y. (c) M–H curve, for the oriented and isotropic bulk samples, measured parallel and perpendicular to the direction of easy magnetization, and the inset showing the second derivative (d2M/dH2) curves of perpendicular magnetization and the M–1/H curve.[83]