Fresnel Magnetic Imaging of Ultrasmall Skyrmion Lattices

TL;DR

Researchers used a special electron microscopy technique to study tiny magnetic skyrmion lattices and found unexpected contrast patterns due to imaging limitations.

Contribution

The study reveals periodic chiral-reversal in ultrasmall skyrmion imaging and identifies its origin in the sinusoidal contrast transfer function.

Findings

Periodic chiral-reversal phenomenon is observed in ultrasmall skyrmion lattice imaging.

Chiral misinterpretations are attributed to sinusoidal modulation of the contrast transfer function.

Findings highlight limitations of conventional Fresnel contrast analysis for nanoscale magnetic imaging.

Abstract

Magnetic skyrmions with ultrasmall nanometric dimensions hold significant promise for next‐generation high‐density spintronic devices. Direct real‐space imaging of these topological spin textures is critical for elucidating their emergent properties at the nanoscale. Here, Lorentz transmission electron microscopy studies of nanometric skyrmion lattices in B20‐structured Mn0.5Fe0.5Ge crystals are presented using the Fresnel mode. According to conventional chiral discrimination methods relying on static bright‐dark contrast, an abnormal periodic chiral‐reversal phenomenon is demonstrated, retrieved through the transport of intensity equation analysis of defocus‐dependent Fresnel images. Through systematic off‐axis electron holography experiments and numerical simulations, these chiral misinterpretations are attributed to the sinusoidal modulation mechanism of the contrast transfer function that correlates with both defocus values and skyrmion dimensions. The findings establish quantitative limitations of conventional Fresnel contrast analysis for ultrasmall skyrmions while revealing fundamental insights into defocus‐mediated phase‐to‐intensity conversion processes in nanoscale magnetic imaging.

This study uncovers periodic contrast reversal in ultrasmall skyrmion lattice imaging through defocus‐modulated Fresnel magnetic microscopy, elucidating its origin in the sinusoidal contrast transfer function mechanism and providing fundamental insights into defocus‐mediated phase‐to‐intensity conversion processes in nanoscale magnetic imaging.

Figures

a,b) Schematic spin configuration of skyrmions when the external magnetic field B is applied downwards. c,d) Schematic illustrations of Lorentz‐TEM used in defocused conditions to observe skyrmions and determine their helicity. The orange lines represent the electron beam.

a) The evolution of magnetic structures under out‐of‐plane magnetic fields at 95 K in a 150 nm‐thick Mn0.5Fe0.5Ge lamella. Defocus, 20 µm. b) Magnetic phase diagram of magnetic skyrmion as a function of temperature and magnetic field. SkL, FM, and PM represent skyrmion lattice, ferromagnetic, and paramagnetic states, respectively. The color bar in the phase diagram indicates the skyrmion density per 104 nm2.

a–e) Fresnel images obtained at defocus values of 20, 60, 100, 140, and 180 µm. f–j) The TIE analysis corresponding to the Fresnel images in (a–e). The inset in the lower left corner of figures (f,h,j) shows magnified views of the regions enclosed by the white dashed boxes, with white arrows indicating the direction of the helicity of the identified skyrmions. The colorwheel represents the in‐plane magnetizations. k) The contrast intensity at the center of the yellow region in figure (a) as a function of defocus value. l) Profiles of the magnetic phase shift extracted from the yellow rectangular regions in figures (a–c) (corresponding to defocus values of 20, 60, and 100 µm, respectively).

a) Reconstructed magnetic phase obtained from electron holography. The inset on the right shows a magnified image of holographic interference fringes with a spacing of 2.97 nm. b) Magnetic induction derived from the analysis of (a). c) Enlargement of the region enclosed by the red dashed line in (b), where white arrows indicate the helicity of the skyrmions. The colorwheel represents the in‐plane magnetizations.

Numerical simulations of Fresnel imaging for ultrasmall skyrmions. a) simulated magnetic phase. b) Magnetic induction derived from the analysis of (a). c) The dependence of the Δf on skyrmion size (λ sk). d–h), Simulated Fresnel images obtained at defocus values of 10, 67, 100, 130, and 160 µm. i–m) The TIE analysis corresponds to the Fresnel images in (d–h), with white arrows indicating the direction of the helicity of the identified skyrmions. n) The contrast intensity at the center of the yellow region in figure (d) as a function of defocus value. o) Profiles of the magnetic phase shift extracted from the yellow rectangular regions in figures (d–f) (corresponding to defocus values of 10, 67, and 100 µm, respectively). The colorwheel represents the in‐plane magnetizations. The scale bar, 20 nm.

Full text

Introduction

1

Magnetic skyrmions, characterized by their small size,^[^ 1, 2, 3 ^]^ low energy consumption,^[^ 4, 5 ^]^ rapid dynamic response,^[^ 6, 7 ^]^ and robust stability,^[^ 8, 9 ^]^ are pivotal for representing data bits and could revolutionize information processing and storage.^[^ 10, 11, 12, 13, 14 ^]^ Recent research has predominantly focused on larger‐scale structures (>50 nm),^[^ 15, 16, 17 ^]^ while studies addressing skyrmions at the 10‐nm level remain relatively limited. Reducing the size of skyrmions allows for more data bits to be accommodated within the same physical space, thereby offering the possibility of achieving higher‐density information storage and promoting the development of more compact and efficient storage devices.^[^ 18, 19, 20 ^]^

The evolving paradigm of magnetic characterization methodologies exhibits complementary strengths and intrinsic limitations: Spin‐polarized microscopy^[^ 21 ^]^ achieves sub‐nanoscale magnetic resolution but demands atomically flat surfaces and extremely low temperatures. Neutron diffraction probes reciprocal‐space representations of macroscopic magnetic ordering, yet lacks spatial specificity for resolving isolated nanoscale spin textures.^[^ 1 ^]^ In contrast, Lorentz transmission electron microscopy (Lorentz‐TEM) operated in Fresnel mode leverages electron wave phase modulation to achieve direct real‐space mapping of magnetic flux distributions. Its advantages are particularly manifested in three key dimensions: 1) Sub‐nanometer resolution for resolving helical/vortex‐type spin topologies^[^ 22, 23 ^]^; 2) Chirality visualization for conventionally‐sized skyrmions can be achieved by leveraging the contrast differences between bright and dark regions produced via objective lens defocusing, enabling direct differentiation between clockwise and counterclockwise magnetic moment distributions^[^ 24, 25 ^]^; and 3) compatibility with in situ dynamic interrogation of magnetic evolution processes.^[^ 26, 27, 28, 29, 30 ^]^ Synergistic integration with off‐axis electron holography further minimizes delocalization artifacts, establishing this multimodal approach as a powerful toolkit for nanoscale magnetic structure elucidation.^[^ 31, 32, 33 ^]^ However, Fresnel magnetic imaging has some limitations: it cannot distinguish Néel‐type spin textures from uniform magnetizations, and the arbitrary filter parameters used in Transport of Intensity Equation (TIE) analysis risk introducing artificial spin textures.^[^ 34, 35, 36, 37, 38 ^]^ A critical question warranting in‐depth investigation is whether the established criteria for identifying conventionally‐sized skyrmions−such as the direct chirality determination method based on Fresnel contrast−remain valid when applied to their ultrasmall counterparts. This fundamental issue of size‐dependent validity directly impacts our mechanistic understanding of magnetic imaging principles. Comprehensive elucidation of the physical mechanisms governing Fresnel magnetic imaging, particularly its inherent scale correlation with spin textures, proves essential for unlocking the full potential of ultrasmall skyrmions in advanced technological applications.

In this study, we implement an integrated methodology combining Fresnel mode, off‐axis electron holography, and micromagnetic simulations to investigate 10‐nm‐scale skyrmion lattices in Mn_0.5_Fe_0.5_Ge thin lamella.^[^ 23, 39, 40 ^]^ Previous studies have shown the experimental imaging of 10‐nm skyrmions in MnGe and Mn_1‐_ * x Fe x *Ge systems.^[^ 23, 41 ^]^ In contrast, our systematic experiments uncover a periodic contrast inversion phenomenon under varying defocus conditions, with inversion periodicity exhibiting a strong correlation with skyrmion dimensions. Our work clarifies that relying solely on black‐and‐white contrast for determining the chirality of ultra‐small skyrmion lattice in Fresnel‐mode imaging can lead to misinterpretation. Furthermore, we explored this characterization method to elucidate the origins of the periodic contrast variations observed in the nanometric skyrmion lattice as a function of defocus amount in the Fresnel mode.

Results and Discussion

2

Field‐Driven Magnetic Evolution and Magnetic Phase Diagram in Mn0.5Fe0.5Ge

2.1

The Mn_0.5_Fe_0.5_Ge crystal investigated in this study exhibits a B20 structure,^[^ 23 ^]^ belonging to the non‐centrosymmetric space group P2_1_3, with a Curie temperature T C of ≈170 K (see Figure S1, Supporting Information). In this chiral lattice helimagnet, the spin system is modeled by the following effective Hamiltonian^[^ 42 ^]^:

$$[eqn]$$

Where M denotes the spatially varying magnetization, J represents the ferromagnetic exchange interaction, α is the DMI constant, and r is the 3D position vector. In the ground state, an appropriate helical magnetic structure is stabilized. Within this model, the magnitude of the wavevector q is proportional to α/J. Due to the asymmetric DMI, the chirality of the spin helical structure (hereafter referred to as magnetic helicity) depends on the sign of α. It is known that the sign of α is determined by both the crystal chirality and the sign of the spin‐orbit coupling. When the propagation vector q is parallel (or antiparallel) to M 1×M 2 (where M 1 and M 2 are the magnetic moments aligned along the q direction), we define it as a right‐handed (or left‐handed) helix.

Upon applying an external magnetic field B perpendicular to the lamella, skyrmions emerge in the helimagnet. Under these conditions, two configurations of skyrmions can be stabilized: the magnetic moments rotate clockwise (CW) or anticlockwise (ACW) in the plane, corresponding to left‐ and right‐handedness, as shown in Figure 1a,b. The chirality of skyrmions is not only a cornerstone of their topological stability but also a critical parameter in regulating their physical properties, such as mobility and response to external fields,^[^ 43 ^]^ as well as in enabling functional applications like data storage and logic devices.^[^ 44 ^]^ A profound understanding of chirality will drive advancements in topological spintronics, particularly in achieving high‐density information storage technologies. Lorentz‐TEM is a powerful tool for visualizing the real‐space magnetization distribution of topological spin textures. The incident electron beam is deflected by the Lorentz force due to the local in‐plane magnetic induction within the sample, and the spatial variation of the in‐plane magnetization results in convergence (bright contrast) or divergence (dark contrast) on the defocused image planes. Using this method, helical magnetic structures appear as stripes. When the objective‐lens current of the TEM generates an applied magnetic field B, Lorentz‐TEM images contain sufficient information to reflect the helicity of the skyrmions. As illustrated in Figure 1c,d, an in‐plane CW (ACW) magnetic moment configuration acts as a convex (concave) lens, forming bright (dark) spots on the defocused image planes.^[^ 45, 46, 47 ^]^ Consequently, in defocused images, the position and helicity of conventionally‐sized skyrmions are simultaneously visualized as spotty images and their contrast.

a,b) Schematic spin configuration of skyrmions when the external magnetic field B is applied downwards. c,d) Schematic illustrations of Lorentz‐TEM used in defocused conditions to observe skyrmions and determine their helicity. The orange lines represent the electron beam.

We first examined the magnetic structures within a 150 nm‐thick Mn_0.5_Fe_0.5_Ge lamella using Lorentz‐TEM. As shown in Figure 2a, at 95 K and in the absence of an external magnetic field, we observed a typical ground‐state helical magnetic domain structure. As the perpendicular magnetic field was gradually increased to 300 mT, the helical domains rapidly transformed into skyrmions, which further evolved into a compactly packed skyrmion lattice configuration. The size of the skyrmions (λ sk) was ≈15 nm. Upon further increasing the magnetic field to 575 mT, the skyrmions gradually annihilated and eventually transitioned into a ferromagnetic (FM) state. By systematically varying the magnetic field at different temperatures, we constructed a magnetic phase diagram that describes the evolution of magnetic domains (see Figure 2b). Below the T C, small‐sized skyrmions in Mn_0.5_Fe_0.5_Ge persist primarily in a lattice form over a wide range of magnetic fields.

a) The evolution of magnetic structures under out‐of‐plane magnetic fields at 95 K in a 150 nm‐thick Mn0.5Fe0.5Ge lamella. Defocus, 20 µm. b) Magnetic phase diagram of magnetic skyrmion as a function of temperature and magnetic field. SkL, FM, and PM represent skyrmion lattice, ferromagnetic, and paramagnetic states, respectively. The color bar in the phase diagram indicates the skyrmion density per 104 nm2.

Fresnel Imaging of the Skyrmion Lattice in Mn0.5Fe0.5Ge

2.2

During the characterization of a 15‐nm skyrmion lattice using Fresnel imaging at various defocus values, we observed a significant reversal in magnetic contrast as a function of defocus. The TIE solution process requires the introduction of a non‐zero constant, q 0, to prevent numerical divergence^[^ 48 ^]^; however, its value necessitates careful determination. In this study, a value of q 0 = 9 × 10^−^⁶ was adopted for the TIE reconstruction. Systematic comparative analysis confirmed this value to be optimal (see Figure S2, Supporting Information), and it does not compromise the reliability of the conclusions drawn. As shown in Figure 3a, at a defocus value of 20 µm, the skyrmion exhibits bright contrast (white spots), and the corresponding TIE analysis (Figure 3f) reveals a CW helicity. When the defocus value is increased to 60 µm (Figure 3b), the bright contrast becomes very faint, which is also reflected in the TIE analysis (Figure 3g). Further increasing the defocus to 100 µm (Figure 3c) results in a complete transformation from bright to dark contrast (black spots), with the TIE analysis (Figure 3h) indicating an ACW helicity. We define this transition from bright to dark (or dark to bright) contrast as contrast reversal, and the corresponding difference in defocus values is denoted as Δf, where Δf = 80 µm in this case. Upon further increasing the defocus to 180 µm (Figure 3e), the contrast of the skyrmion reverts from dark to bright, as evidenced by the TIE analysis (Figure 3j), showing that the helicity returns to a CW direction. The intermediate states during this transition, depicted in Figure 3d,i, exhibit almost no discernible magnetic contrast.

a–e) Fresnel images obtained at defocus values of 20, 60, 100, 140, and 180 µm. f–j) The TIE analysis corresponding to the Fresnel images in (a–e). The inset in the lower left corner of figures (f,h,j) shows magnified views of the regions enclosed by the white dashed boxes, with white arrows indicating the direction of the helicity of the identified skyrmions. The colorwheel represents the in‐plane magnetizations. k) The contrast intensity at the center of the yellow region in figure (a) as a function of defocus value. l) Profiles of the magnetic phase shift extracted from the yellow rectangular regions in figures (a–c) (corresponding to defocus values of 20, 60, and 100 µm, respectively).

To quantitatively evaluate the variation in magnetic contrast intensity of a 15‐nm skyrmion lattice under different defocus conditions, we have plotted Figure 3k, which illustrates the trend of contrast intensity at the skyrmion center as a function of defocus value. The figure delineates the periodic transformation process experienced by the contrast intensity at the skyrmion center, transitioning from bright to dark, then bright again, and finally back to dark. Figure 3l presents the magnetic phase shift profiles extracted from the yellow rectangular regions in Figure 3a–c, depicting how the magnetic phase shift within an entire skyrmion area varies with position under varying degrees of defocus (20, 60, and 100 µm).

However, in practice, simply adjusting the defocus value in Fresnel mode does not achieve periodic reversals of the skyrmion chirality. To more accurately characterize the genuine skyrmions in Mn_0.5_Fe_0.5_Ge, we employed off‐axis electron holography (details in Experimental Section). This technique operates under in‐focus conditions to avoid image delocalization that can occur in Lorentz mode. Figure 4a shows the reconstructed magnetic phase image obtained through off‐axis electron holography, and further analysis of this image yielded the magnetic induction distribution (see Figure 4b). As illustrated in Figure 4c, the results demonstrate that the skyrmion helicity is CW.

a) Reconstructed magnetic phase obtained from electron holography. The inset on the right shows a magnified image of holographic interference fringes with a spacing of 2.97 nm. b) Magnetic induction derived from the analysis of (a). c) Enlargement of the region enclosed by the red dashed line in (b), where white arrows indicate the helicity of the skyrmions. The colorwheel represents the in‐plane magnetizations.

The contrast inversion phenomenon observed in Fresnel imaging of a 15‐nm skyrmion lattice has been perfectly reproduced in numerical simulations, as illustrated in Figure 5. Figure 5a shows the simulated magnetic phase image, while Figure 5b presents the magnetic induction derived from further analysis. In the simulated Fresnel mode, the skyrmion contrast exhibits periodic reversals as a function of defocus, as shown in Figure 5d–m. Similarly, we have quantitatively evaluated the variations in magnetic contrast intensity under different defocus conditions, with the results plotted in Figure 5n,o. These simulation results are in excellent agreement with experimental observations. Notably, for non‐periodic individual skyrmions, changes in defocus only affect the overall brightness and sharpness without inducing contrast reversal, as shown in Figure S3 (Supporting Information).

Numerical simulations of Fresnel imaging for ultrasmall skyrmions. a) simulated magnetic phase. b) Magnetic induction derived from the analysis of (a). c) The dependence of the Δf on skyrmion size (λ sk). d–h), Simulated Fresnel images obtained at defocus values of 10, 67, 100, 130, and 160 µm. i–m) The TIE analysis corresponds to the Fresnel images in (d–h), with white arrows indicating the direction of the helicity of the identified skyrmions. n) The contrast intensity at the center of the yellow region in figure (d) as a function of defocus value. o) Profiles of the magnetic phase shift extracted from the yellow rectangular regions in figures (d–f) (corresponding to defocus values of 10, 67, and 100 µm, respectively). The colorwheel represents the in‐plane magnetizations. The scale bar, 20 nm.

To further investigate the impact of skyrmion size on Δf in Fresnel imaging, we conducted a systematic study. Initially, as illustrated in Figures S4–S6 (Supporting Information), the results indicate that neither sample thickness nor magnetic field strength significantly affects Δf. Furthermore, when the sign of the DMI is reversed, although the chirality of the corresponding skyrmion lattice is altered, the critical defocus distance Δf associated with contrast reversal remains unchanged (Figure S7, Supporting Information). Therefore, in our numerical simulations, we varied only the skyrmion size (λ sk) parameter. The simulation results show a rapid increase in Δf as the λ sk increases (Figure 5c), the discrepancy between the simulated and approximatively calculated curves primarily stems from the simplified model's approximation of the skyrmion as a single‐frequency structure (q≈1λsk), whereas the actual imaging contrast in periodic skyrmion lattices arises from multi‐frequency interference,^[^ 49 ^]^ leading to deviations in the predicted Δf values. Despite these deviations, the simplified model still captures the overall trend of increasing Δf with skyrmion size (λ sk). Specifically, when the skyrmion size λ sk is 50 nm, the corresponding Δf reaches 1450 µm (Figure S8, Supporting Information). Moreover, by leveraging temperature to modulate the skyrmion size (Figure S9, Supporting Information), we systematically acquired contrast reversal data across lattices of different dimensions. A clear trend of increasing Δf with λ sk was observed (Figure 5c), consistent with our simulation results, thereby validating the predictive reliability of our method for skyrmions of varying sizes. Notably, the skyrmion lattice in FeGe (λ sk ≈70 nm) also exhibits contrast inversion, with the corresponding Δf value reaching up to ≈3000 µm, as shown in Figure S10 (Supporting Information). Based on these findings, as shown in Figure S11 (Supporting Information), we established two regions on the same sample, with skyrmions sized at 10 and 15 nm, respectively. Due to the differences in Δf between these two regions, within a specific defocus range, it is possible to observe skyrmions with contrasting black‐and‐white contrast on the same sample simultaneously. TIE analysis suggests that the helicity of the skyrmions in these two regions appears opposite; however, in reality, the helicity of both types of skyrmions is consistent.

The Physical Mechanism of Contrast Reversal in Imaging Ultrasmall‐Sized Skyrmion Lattice in Lorentz‐TEM Fresnel Mode

2.3

In TEM, the formation of images is influenced by the contrast transfer function (CTF),^[^ 50, 51, 52, 53 ^]^ which describes how phase information from the sample is translated into contrast variations in the final image. For Lorentz‐TEM, the CTF can be expressed as follows:

$$[eqn]$$

Here, q represents the spatial frequency, f denotes the defocus, λ is the electron wavelength, and ϕ 0 is the initial phase angle. This equation elucidates the defocus‐dependent modulation of phase components across spatial frequencies, while incorporating the constraining effects of spatial coherence on electron wavefront phase distributions.

When characterizing periodic skyrmion lattices, the repetitive magnetic structure units within the lattice produce a series of diffraction spots in Fourier space,^[^ 54 ^]^ each corresponding to a specific spatial frequency q. Variations in defocus f induce electron disturbance‐mediated phase retardation, causing specific diffraction spots to traverse zero‐crossings or extrema of the sinusoidal CTF. This results in periodic intensity fluctuations in real space, termed “contrast reversal.” Such phenomena arise from defocus‐altered interference conditions between diffraction orders.

To approximate the local magnetic structure of a skyrmion as a circular structure, the spatial frequency q can be approximated as the inverse of the characteristic dimension λ sk:

$$[eqn]$$

For a given spatial frequency q, to observe contrast reversal, where the CTF (q, f) changes sign from positive to negative or vice versa, specific defocus values f must be satisfied. According to the CTF expression, contrast reversals occur when πλ q ^2^ f equals π2, 3π2, 5π2, etc. Therefore, the relationship between defocus f and spatial frequency q can be expressed as:

$$[eqn]$$

where n is an integer (0, 1, 2, …). Simplifying this equation yields:

$$[eqn]$$

For larger skyrmions, achieving equivalent phase shifts necessitates greater defocus adjustments due to cumulative electron disturbance under spatial coherence constraints.

Individual skyrmions lack periodic arrangements, precluding discrete diffraction patterns. In Lorentz‐TEM, their imaging relies on intrinsic topological magnetic structures and local field characteristics rather than defocus‐modulated interference. Although defocus influences overall brightness, spatial coherence‐preserved phase stability prevents complex contrast reversal. Single‐skyrmion imaging thus depends on direct phase contrast rather than electron disturbance‐driven interference effects.

Consequently, ultra‐small skyrmion lattices, due to their periodicity, exhibit highly sensitive contrast changes in response to defocus variations in Lorentz‐TEM, manifesting as periodic contrast reversals. Conversely, individual skyrmions, lacking such periodicity, do not display similar contrast reversal behavior. Moreover, as skyrmion size increases and spatial frequency decreases, achieving equivalent contrast reversal requires a greater defocus. It is noteworthy that the TIE is generally recommended for use under small‐defocus conditions.^[^ 55 ^]^ However, for ultrasmall skyrmions with λ sk ≈15 nm, the applicable upper limit of “small defocus” (≈15 µm, Figure S12, Supporting Information) is significantly lower than the defocus values commonly used to characterize larger skyrmions.^[^ 6, 16 ^]^ As a result, the conventionally understood “small defocus” condition may exceed the appropriate range for such dimensions, thereby inducing contrast reversal. Relying solely on black‐white contrast to determine chirality can thus lead to misleading conclusions.

Conclusion

3

This study systematically investigates the ultrasmall 10‐nm skyrmion lattice in B20‐structured Mn_0.5_Fe_0.5_Ge using Fresnel imaging in Lorentz‐TEM. We observe periodic contrast reversals in magnetic images as a function of defocus, which are strongly correlated with the size of the skyrmions. Relying solely on black‐and‐white contrast for chirality determination may lead to misinterpretation. Through off‐axis electron holography and numerical simulations, we elucidate how defocus‐induced contrast changes impact chirality assessment. The simulations successfully reproduce the experimental contrast inversion, validating our findings. Furthermore, we demonstrate that the periodic reversal of magnetic contrast modulated by defocus arises from the sinusoidal modulation mechanism of the CTF in Lorentz‐TEM. This highlights not only the influence of skyrmion size on imaging but also underscores the importance of understanding the role of defocus in Lorentz‐TEM. The integration of multi‐scale imaging techniques emerges as a prerequisite for fully unlocking the potential of ultra‐small skyrmions in ultrahigh‐density spintronic applications.

Experimental Section

4

Sample Preparation

The initial step involved synthesizing polycrystalline Mn_0.5_Fe_0.5_Ge by arc melting Mn granules (99.9% purity), Fe granules (99.9% purity), and Ge granules (99.999% purity) in their stoichiometric proportions. Subsequently, the material underwent high‐pressure synthesis at 8 GPa and 1000 °C for 2.5 h, followed by a slow cooling process to room temperature.

Preparation of Mn0.5Fe0.5Ge Lamellas

The Mn_0.5_Fe_0.5_Ge nanostructured lamellas were fabricated from a bulk single crystal using a standard lift‐out method with an SEM‐FIB dual‐beam system (Helios Nanolab 600i, FEI).

Lorentz‐TEM Measurements

In situ Fresnel imaging was performed in a Lorentz‐TEM (Talos F200X, FEI) with an acceleration voltage of 200 kV. The TEM holder (model 636.6, Gatan) supported variable temperature measurements.

Experimental Details of Off‐Axis Electron Holography

Magnetic induction maps were acquired by using in situ off‐axis electron holography under in‐focus conditions to avoid image delocalization in Lorentz mode. Electron holography was performed using a single electrostatic biprism and a direct electron counting camera with an interference fringe spacing of 2.97 nm. Each experiment involved the acquisition of 30 object holograms, followed by 30 vacuum reference holograms to remove image distortions associated with the imaging and recording systems of the microscope. Averaging of the holograms, which were each acquired using an exposure time of 2 s, was used to improve the signal‐to‐noise ratio.

Micromagnetic Simulations

The zero‐temperature micromagnetic simulations for magnetic domains in 150‐nm‐thick lamellas were performed using MuMax3.^[^ 56 ^]^ The simulated ratio A/D was chosen to be consistent with the skyrmion size varying from 10 to 70 nm according to the relation λ = 4πA/D. The cell size adopted for the simulations was 0.5 nm × 0.5 nm × 5 nm.

Simulation of Lorentz‐TEM Images and Magnetic Phase Shift

Lorentz‐TEM images and magnetic phase images were simulated using the open‐source software Pylorentz.^[^ 57 ^]^ The spin configurations for these simulations were derived from micromagnetic simulation outputs. The simulation parameters included an accelerating voltage of 200 kV and defocus values. To evaluate the potential effects of microscope aberrations on Fresnel image simulations, supplementary simulations incorporating non‐zero aberrations—specifically spherical aberration (C s) and chromatic aberration (C c)—were performed (Figure S13, Supporting Information). Given that the experimental data were obtained using a Talos F200X TEM, which inherently exhibits non‐zero C s, the aberration parameters in the simulations were accordingly set to the default values in the Pylorentz software: C s = 2 × 10⁵ nm and C c = 5 × 10⁶ nm.

Conflict of Interest

The authors declare no conflict of interest.

Author Contributions

Y.Z. and W.L. contributed equally to this work. S.W., H.D., M.T., and J.T. supervised the project. J.T. and Y.Z. conceived the idea and designed the experiments. W.L. synthesized the bulk crystals. Y.Z., H.Z., and Y.W. fabricated the Mn_0.5_Fe_0.5_Ge lamella and performed the TEM measurements with the help of J.J. M.S., and S.Q. performed the simulations. Y.Z., Y.W., and J.T. wrote the manuscript with input from all authors. All authors discussed the results and contributed to the manuscript.

Supporting information

Supporting Information