Two-dimensional characterization of three-dimensional nanostructures of magnetic bubbles in Fe 3 Sn 2

We report differential phase contrast scanning transmission electron microscopy (TEM) of nanoscale magnetic objects in Kagome ferromagnet Fe 3 Sn 2 nanostructures. This technique can directly detect the deflection angle of a focused electron beam, thus allowing clear identification of the real magnetic structures of two magnetic objects including three-ring and complex arch-shaped vortices in Fe 3 Sn 2 by Lorentz transmission electron microscopy imaging. Numerical calculations based on real material-specific parameters well reproduced the experimental results, showing that the magnetic objects can be attributed to integral magnetizations of two types of complex three-dimensional (3D) magnetic bubbles with depth-modulated spin twisting. Magnetic configurations obtained using the high-resolution TEM are generally considered as two-dimensional (2D) magnetic objects previously. Our results imply the importance of the integral magnetizations of underestimated 3D magnetic structures in 2D TEM magnetic characterizations.

They were first found in non-centrosymmetric magnetic compounds, where chiral Dzyaloshinskii-Moriya interactions (DMIs) bend the magnetic moments [20][21][22][23]. The unique feature of magnetic skyrmions is their non-trivial topology defined by unit topological charge [24]. Unlike the chiral DMI-induced skyrmions, magnetic bubbles originate from the interplay of four types of interactions including ferromagnetic exchange coupling, dipolar-dipolar interaction (DDI), uniaxial anisotropy, and Zeeman energy. Competition among the first three interactions leads to stripe domains, which may change into a magnetic bubble when applying an external field. There are two types of magnetic bubbles according to the rotation sense of the cylinder domain wall ( Figure S1). One is a type-I magnetic bubble stabilized by a perpendicular magnetic field with a clockwise or anticlockwise closure cylinder domain wall contributing to a similar integer topological winding number as a skyrmion; type-I magnetic bubbles are thus renamed skyrmion bubbles [25][26][27][28]. The other one is a type-II magnetic bubble stabilized by a tilted magnetic field with magnetization in the partially reversed cylinder domain wall, with all domain wall magnetizations pointing toward the in-plane field component. However, such a domain wall in a type-II magnetic bubble contributes to a zero winding number and is topologically trivial [27].
The first wave of interest in magnetic bubbles occurred in the 1970s-1980s, motivated by experimental and theoretical studies of potential bubble memory [29,30].
Although these two types of bubbles are well understood within the theoretical framework describing uniaxial ferromagnets, a recent study on a typical uniaxial ferromagnet Fe 3 Sn 2 found new exotic spin whirls beyond conventional magnetic bubbles by Lorentz transmission electron microscopy (Lorentz-TEM) [25,28]. Two typical examples of such new spin whirls are three-ring and complex arch-shaped vortices characterized by a series of concentric circular stripe domains and one or multiple bound pairs of rotating magnetic whirls, respectively. Such magnetic structures were also observed in other uniaxial ferromagnets [26,31]. These objects are nanoscale size, which implies that they can be applied as information carriers in spintronic devices [17]. However, they are neither detected by other magnetic imaging methods nor in simulations conducted under realistic conditions. Moreover, a recent study demonstrated that the improper filter parameter in the transport of intensity equation (TIE) analysis of Lorentz-TEM imaging of type-II bubbles can lead to artificial biskyrmion structures [33].
Three-dimensional (3D) magnetic structures have become an active research topic because they are important in understanding novel experimental phenomena and potential applications [4,23,[36][37][38][39][40]. It has been suggested that the chiral exchange interactions play important roles in tailoring 3D magnetic structures in synthetic antiferromagnets for potential 3D spintronic systems [39,40]. 3D magnetic skyrmions in B 20 magnets induced by DMI have been proposed to understand the stability of zero-field target skyrmions and attractive interactions between skyrmions [4,23].
Magnetic skyrmion bubbles have also been predicted with depth modulated spin twisting induced by DDI [41]. One typical characteristic of 3D magnetic skyrmion bubbles is that skyrmions near two surfaces have nearly contrary Néel-twisting. This characteristic has been observed in magnetic multilayers by some surface-sensitive magnetic detection methods [36][37][38]. TEM is a real-space imaging of integral magnetic field over depth with ultrahigh spatial resolution. Magnetic configurations in thin nanostructures have been typically considered as quasi two-dimensional (2D) magnetic objects using TEM [19,25,26,28,31]. However, one may clarify real 3D magnetic structures from the difference in integral magnetization over depth. This rule has been used to identify 3D chiral bobbers from integral phase shifts weaker than skyrmion tubes using TEM [3]. The depth-modulated 3D magnetic bubbles are also expected to show more complex integral magnetizations over the depth and are detected using 2D TEM magnetic imaging. The underestimated complex integral magnetizations of 3D magnetic bubbles may clarify the physics behind the complex three-ring and arch-shaped vortices in Fe 3 Sn 2 through TEM, which is more readily considered as 2D magnetic configurations in thin nanostructures [25].
Here, we investigate the magnetic objects in an Fe 3 Sn 2 nanodisc using

RESULTS AND DISCUSSIONS
Identification of a multi-ring vortex We first focus on the three-ring vortex in Fe 3 Sn 2 uniaxial ferromagnet. An Fe 3 Sn 2 nanodisc (diameter, ca. 1550 nm; thickness, ca. 140 nm) with (001)-oriented out-of-plane direction is chosen for DPC-STEM measurements ( Figure 1f and Figure   S3) and micromagnetic simulations (see supplementary simulation method) [42].
Lorentz-TEM is also performed for comparison. TEM magnetic imaging is discussed in detail in supporting information [11][12][13]15,[43][44][45][46][47][48][49]. Stripe domains are observed at zero field, which transfer into circular domains when a magnetic field is applied out-of-plane (Figures 1a-1c). However, once the circular domains are formed, they may persist as the field decreases ( Figure 1d). In such a case, the Lorentz-TEM image gives rise to a three-ring vortex at low field ( Figure 1e) that transfers into a normal bubble skyrmion when the field is increased. In Figure 2a, a field-driven process of one bubble by Lorentz-TEM is shown as an example. At a low field, a black dot in the center is surrounded by outer rings, which is different from a conventional skyrmion image [7,13,19]. The Lorentz contrast of a normal skyrmion is composed of only a black or white circle [5,6,19]. Such distinctness implies complexity in the magnetic objects. When using the TIE method, the reconstructed magnetic configuration is characterized by a series of concentric stripe domains with opposite rotation sense between neighboring magnetic rings (Figures 2 b1-b3), forming a three-ring vortex.
At a high field, a normal skyrmion-like image is observed (Figures 2 b4-b5).
Assuming these nanoscale magnetic objects are arranged in thin nanostructures of uniform magnetization, such complex vortices with multiple rings and field-driven transition cannot be well reproduced in 2D uniaxial ferromagnets. However, we noted that the TEM method can only detect the integral in-plane magnetizations over the depth [5,6,19,45,46]. We noted the Q factor of Fe 3 Sn 2 determined by the ratio of uniaxial magnetic anisotropy (~54.5 kJ/m 3 ) to shape anisotropy (~244 kJ/m 3 ) is smaller than 1. In this case, DDI interaction could lead to the closure of cross-sectional bubble domains, which reveals Néel-twisting at the surface and Bloch-twisting in the middle [41]. Such Néel-twisting at two surfaces of 3D magnetic skyrmion bubbles with contrary chirality has been identified in reciprocal momentum space by a surface-sensitive resonant elastic X-ray scattering in magnetic multilayer films [8,36,37]. Using a nitrogen-vacancy magnetometer, a skyrmion in the surface layer has contrary chirality to intrinsic chiral interaction, which also implies the validity of the proposed 3D magnetic skyrmion bubbles [38]. Furthermore, more complex integral in-plane magnetizations over depth of 3D skyrmion bubbles that are measured using 2D TEM magnetic imaging will be expected and may explain the complex three-ring vortex (Figure 2b). We thus performed 3D numerical simulations of the Fe 3 Sn 2 nanodisc, which showed field-driven evolutions of magnetic structures   The aforementioned consistency further enables us to analyze the origin of the two-ring vortex. The simulated 3D cross-section spin configuration of a two-ring vortex at a typical field is shown in Figure 3a. A rugby ball-like 3D structure is obtained, in which hybrid skyrmions along the sample thickness ranged from Néel-to Bloch-type with increasing depth below the surface, which is attributed to the DDI-induced vortex-like cross-sectional configurations. The surface layers host mainly Néel-type skyrmions with radially inward and outward pointing spins in the upper and bottom layers, respectively (Figures 3b and 3d). The Lorentz-TEM and DPC-STEM only detect the averaged in-plane magnetization, but much of the averaged in-plane magnetization cancels itself out, thus leading to a weak vortex core in the center (Figure 3e). From the 3D structure, it is readily understood that the outside ring originates from the Bloch-type skyrmions in the middle layers (Figure 3c), indicating that the two-ring vortex is intrinsically a type-I skyrmion bubble with depth-modulated spin configurations. Interestingly, when the field increased, the size of the outer ring, which comprises contributions from Bloch-type skyrmions in the middle layers, decreases from ca. 216 nm at B ≈ 90 mT to ca. 128 nm at B ≈ 450 mT. However, the size of the internal vortex-like core maintains constant (ca. 120 nm).
Accordingly, at high field, the internal core and outer ring mix, leading to only one vortex ( Figure S6), which may be responsible for traditional small-size one-ring skyrmion bubbles observed in Fe/Gd films with a comparable Q factor as Fe 3 Sn 2 [41].
Such agreement between the experimental and simulation results verifies the complex 3D structure of the type-I bubble skyrmion, which may give a general understanding of bubble skyrmions in uniaxial ferromagnets with a relatively small Q factor [27,41]. We Noted that the presented two-ring vortices are distinct from the previously proposed two-ring bubbles in BaFeScMgO [31], which are typically target Following the procedure outlined previously to investigate the type-I bubble, here we discuss the type-II bubble to clarify the complex arch-shaped vortex [25,31].
According to our experiments, such a vortex can be easily obtained by slightly tilting the external field ( Figure S7). The Lorentz contrast of such a vortex shows Φ-shaped ring with two strong contrasts on the top and bottom (Figure 4a). A weak line contrast in the center linking the two strong ones is also observed. Using the TIE method, the reconstructed magnetic configuration is characterized by multiple bound pairs of rotating magnetic whirls (Figure 4b).
The simulated averaged in-plane magnetic configuration (Figure 4c) shows a Φ-shaped spin whirl with the onion-like characteristic of a type-II bubble [27], which is confirmed using DPC-STEM images (Figure 4d) However, the magnetic configuration reconstructed by TIE (Figure 4f) is entirely different from simulations and DPC-STEM images (Figures 4c and 4d). Therefore, we believe this magnetic object in Figures 4b and 4f is an artificial magnetic configuration created by TIE analysis.
We compared the actual magnetic configuration and artificial magnetic configuration to obtain more insight into this issue. Interestingly, a nearly uniform ferromagnetic background is obtained if we subtract the magnetic configuration in We further show that the Φ-shaped spin whirl originates from a rugby ball-like 3D structure ranging from Néel-to Bloch-type with increasing depth below the surface ( Figure 4h). The outside ring originates from the Bloch-type type-II bubble in the middle layers, and the central line comes from the averaged in-plane magnetization over the upper and bottom surfaces ( Figure S8).

CONCLUSIONS
In summary, using DPC-STEM magnetic imaging, we showed that 2D integral magnetizations of 3D type-I and II magnetic bubbles can well explain the multi-ring and arch-shaped vortices, respectively. The experimental observations are well reproduced by numerical calculations of real 3D magnetic nanostructures. We further analyzed the intrinsic origin of artifacts of magnetic contrast from Lorentz-TEM. Our results also imply that other unexplained magnetic configurations by TIE should be re-examined using other 2D TEM methods to consider their real 3D magnetic nanostructures [26,31]. In comparison to surficial magnetic configurations of the next step is to study the dynamics induced by current to build a purely bubble-based spintronic device [28].

METHODS
We prepared bulk Fe 3 Sn 2 crystals by chemical vapor transport and fabricated the Fe 3 Sn 2 nanodisc using a focused ion beam and scanning electron microscopy dual beam system (Helios NanoLab 600i, FEI). The magnetic imaging of the Fe 3 Sn 2 nanodisc was performed on a TEM (Talos F200X, FEI) operated at 200 kV. Micromagnetic simulations were performed using a GPU-accelerated program: Mumax3. For details about the methods, refer to the supplemental information.

Supporting Information
Supplementary data are available at NSR online.