(1) the assistance with Fourier transfer (FT), the inverse Laplacian x,y could be replaced by [12] (2)

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1 Magnetic hard nanobubble: a possible magnetization structure behind the bi-skyrmion Yuan Yao 1, *, Bei Ding 1, 2, Jie Cui 1, 2, Xi Shen 1, Yanguo Wang 1, Richeng Yu 1, 2 1. Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing , China 2. School of Physical Sciences, University of Chinese Academy of Science Beijing , China *Corresponding author, yaoyuan@iphy.ac.cn Abstract The magnetic hard bubbles in the scale of hundreds of nanometers have been analyzed with simulation in Lorentz transmission electron microscopy (LTEM). The results from image processing with transport of intensity equation (TIE) have revealed the emergence and evolution of bi-spiral features in the recovered magnetic structures. Systematic studies demonstrated that the parameter in TIE could modulate the details of these specious bi-skyrmions and the projection characteristics of LTEM imaging may distort the artificial magnetic configuration heavily when sample is tilted. Skyrmion is a hotspot in the magnetic materials research community recently, which possesses a fix chiral character and is considered as a promising candidate for high density [1, 2] memory and other modern electronic devices. Lorentz transmission electron microscopy (LTEM) is a powerful tool to characterize features in magnetic materials at the nanometer scale. With image post-processing methods, such as transport of intensity equation (TIE), various skyrmions have been disclosed by LTEM. Besides of the skyrmions with unique ±1 chirality, a special bi-skyrmion containing tow side-by-side antispiral contours is also an interesting magnetic configuration. [3, 4] Contrary to those novel ±1 skyrmions, the formation of the bi-skyrmion with the skyrmion number (Ns) of 2 [3] is still unclear. Takagi et al suggested a mechanism including Dzyaloshinskii-Moriya interaction (DMI) to explain the bi-skyrmion arrays in non-centrosymmetric Cr 11 Ge 19. [5] But for bi-skyrmions in some centrosymmetric compounds, such as La 2-2x Sr 1+2x Mn 2 O [3, 6] 7 and NiMnGa alloy [4, 7], the real feature of the bi-skyrmion is waiting for being illustrated. In our previous work, a simple Neel type magnetic nanodisk can show the bi-chiral structure in the recovered magnetization distribution when its symmetric axis deviates from the incident electron direction in LTEM. [8] Such artifact comes from the projection characteristics of imaging in LTEM, which can disappear when tilting the Neel domain along the electron beam rightly. In this report, another common magnetic nanostructure, magnetic hard nanobubble, is studied systematically by LTEM image simulations and image post-processing with TIE method. The results show that the bi-skyrmion may be reproduced in the retrieved image of the magnetic hard nanobubble if an improper parameter is selected in TIE processing, no matter how the specimen is placed in the LTEM.

2 a) b) Fig. 1 a) Schematic three dimensional configuration and b) top view of magnetization in a hard bubble, the arrows denote the magnetization orientation. A magnetic hard bubble (HB) is a mixture of Bloch and Neel type domain walls. It is also defined as type II magnetic bubble because of its zero chiral number, while a pure Bloch domain wall is named simple or soft bubble (SB) with a nonzero chirality. [9] Fig.1 is the schematic picture of an isolated HB where two semi-circle Bloch domain walls are separated by two Neel type gaps. The gaps are denoted as Bloch lines. [9] In a more general situation, there are even dozens of Bloch lines if the bubble size exceeds hundreds of micrometers. It is necessary to point out that the magnetization configuration in Fig. 1 is just a supposed model to portray the main feature of HB. In order to obtain the accurate distribution, the magnetization structures of the hard nanobubbles were numerically calculated in a specimen with a given size by the object oriented micro-magnetic framework (OOMMF) software based on solving the Landau-Lifshitz-Gillbert (LLG) equation. [10] Then the generated magnetization data were input to the home-made program to compute the induced magnetic field in a space containing the specimen and simulate the LTEM images with different placements and optical parameters, such as overfocus and underfocus Fresnel images in different tilting conditions. The recovered exit electron wave phase φ(x,y) and map of magnetization distribution were obtained by image post-processing with a home-made TIE plugin written in Digital Micrograph script. TIE method is a widely used implement to retrieve the phase φ(x,y) of the exit wave in electron microscopy by using several images at different defocuses, usually three images at over-focus, in-focus and under-focus, [11] (,, ) 2 I xyz xy, xy, 2 z (1) ϕ ( xyz,, 0 ) = k xy, xy,? I( xyz,, 0 ) where z is the electron propagation direction and k is the electron wave vector. The partial differential of intensity means the intensity of images variation along z, which can be substituted by numerical differential between the image intensity at different focus I z. Under 2 the assistance with Fourier transfer (FT), the inverse Laplacian x,y could be replaced by [12] ( xy, ) f 2 1 xy, f( xy, ) = 2 qxy (, ) where q(x,y) is the spatial frequency in the plane which is perpendicular to the beam direction. Thus TIE solves the phase problem so fast that it is convenient to deal with the LTEM images to reveal the magnetic structures in specimen, booming the current research in magnetic skyrmions. [13] However, eq. 2 is divergent and the noise is magnified when q(x,y) approaching zero. So a small nonzero constant q 0 must be appended to avoid the divergence and suppress [12, 14] low-frequency noises in the actual realization (2)

3 ( xy, ) f 2 1 xy, f' ( xy, ) 2 2 qxy (, ) + q0 = The magnetization in the image plane could be deduced from the partial differential of the obtained phase φ(x,y) [15] ' ' ϕ( xy, ) ' ' ϕ( xy, ) Mx Bx, M y By y x It should be emphasized that M x and M y (or B x and B y ) should not be the corresponding actual magnetic field because the phase is the integral of the vector potential A (x, y, z) along the electron path, φ(x, y) = e ħ + + (2 ) A (x, y, z) dz = e A ħ z(x, y) dd, if electrons travel along z direction. The initial configuration of a hard nanobubble in Fig. 1 relaxes and reaches a stable state after the calculation in OOMMF. Fig. 2a indicates that the magnetization pattern in the middle x-y plane of the nanobubble is similar to the initial setup but the magnetization distributions on two surfaces change remarkably. It is reasonable because the magnetostatic energy should be minimal if the magnetizations parallel to the surface. Actually, if observing the x-z plane, M changes greatly beyond the expectation (Fig. 2b). (3) a) b) Fig. 2 The magnetization distribution a) in the different x-y planes of a relaxed hard nanobubble and b) in the x-z plane drawn by white dash lines in a). The arrows denote the size and direction of in-plane components while the color visualizes the magnitude of the out-plane components. The image sizes are 278 nm 278 nm in a) and 278 nm 115 nm in b), respectively. The simulated over-focus and under-focus LTEM images of the nanobubble without tilting are shown in Fig. 3. Two arches can be distinguished clearly. Each arch consists of one white and one black stripes in same sequence if scanning from left to right, implying same magnetization orientation in the two parts of the domain wall. Indeed, the contrast in the LTEM images could directly reflect some properties of the domain walls without any image processing, such as the shape, size and parallel or anti-parallel features. After TIE processing, the integral projections of M x and M y were generated and are displayed in the first row of Fig. 4a and 4b, together with different q 0 needed in eq. 2. To verify the influence of specimen tilting, the results from different tilted conditions are also presented in Fig. 4 and Fig. 5, corresponding two symmetric x and y axes of the hard nanobubble, respectively. The retrieved M x and M y are similar to the real M x and M y when q 0 is very small. As q 0 increases, the patterns of M x and M y vary and the

4 magnetizations opposite to the initial two arches appear in the core of nanobubble. The typical bi-spirals or bi-skyrmions emerge obviously. For a large q 0, the magnetizations in the core can separate and new arches are formed around the nanobubble. Specimen tilting also causes the distortion in final features. Similar behavior can be found in the experimental survey for NiMnGa alloys, as shown in Fig. 5 where just the effect of q 0 is investigated. These results illustrate that the processing parameter q 0 and specimen placement in LTEM can complicate the recovered magnetic configurations and bring the confusion in interpreting the actual magnetic structures in the samples. Fig. 3 The simulated LTEM images of a hard nanobubble at different defocuses. (Accelerate voltage: 200 kev, Cs: 5 m, Cc: 100 Å) a) b) Fig. 4 The influence of q 0 and the sample tilting around different symmetric axes, retrieved with TIE processing (1 pixel = nm -1 ). a) b) Fig. 5 a) The LTEM images of NiMnGa and b) the retrieved magnetic configuration with different q 0 (1 pixel = nm -1 ). The reproduce of the bi-skyrmions in a traditional magnetic hard nanobubble may provide a

5 hint to explain the observed similar structures in some center-symmetric magnetic materials where DMI does not exist. In reality, LTEM images exhibit the projection characteristics of the magnetic induction B in the space where electrons transmit, not M distribution in the specimen, though B is stronger in the specimen than in free space. So the retrieved phase image masks the real magnetization configuration, especially for the thin specimen which owns a fairly considerable portion of the magnetizations parallel to both surfaces. When the sample changes its orientation relative to the electron beam, the situation is more complicated. Moreover, the magnetic structure recovered with FT assistant TIE method is sensitive to the parameter q 0. In eq. 2, q 0 must be applied to avoid the divergence when q moves toward zero. It is also an adjustment to restrain the low frequency noise. But an inevitable byproduct is that it is like a high pass filter which depresses the low frequency information. In principle, the smaller the q 0 is, the better the fidelity is. This trend is obvious in Fig. 4. However, a very small q 0 may be unrealistic in the experimental characterization because it is difficult to acquire a flat contrast background in the real images where the low frequency signals containing diffraction contrast can overwhelm the magnetic characteristics in the recovered images. On the other hand, much higher q 0 may highlight the high frequency noise. Fig. 6 demonstrates those influences on the experimental images. Therefore, in practical operation, a modest q 0 will attract more attention, when it leads to the clear contours in the resulted images but it brings a misleading risk, particularly tangled with the situation in which the magnetic configuration orientation relative to electron beam is unknown when the sample is inserted into TEM. a) b) Fig. 6 a) The LTEM images of NiMnGa and b) the retrieved magnetic configuration with different q 0 to demonstrate the influence of low frequency contrast and high frequency noise (1 pixel = nm -1 ). The contrast-performance in LTEM images of hard nanobubbles is different from a pure Neel spiral domain wall. Bi-skyrmion contrasts could emerge in a pure Neel spiral domain, if its vertical axis is tilted away from the electron beam in LTEM. [8] But that feature will vanish, when its axis returns parallel to the electron beams, because there is no contrast in the image of the pure Neel domain at this condition. However, whatever the sample direction is, a hard nanobubble always contributes image contrast in LTEM, as shown in Fig. 3. Thus, it may be an effective method to identify what is the actual magnetic structure behind the bi-skyrmions features by tilting the sample in a wide angle range.

6 In summary, the magnetization and LTEM images of a magnetic hard nanobubble have been simulated to investigate the origin of the bi-skyrmions in centrosymmetric magnetic materials. Systematic studies indicate that the retrieved magnetic configures are sensitive to the parameter q 0 of TIE approach and q 0 can modify the obtained magnetic appearance easily. Sample tilting can deteriorate the distortion. The reproduce of bi-skyrmion feature in a hard nanobubble by adjusting q 0 hints that a simple magnetic structure can lead to a complex phenomenon if improper TIE method is applied or the sample is placed inappropriately in LTEM. Acknowledgment This work was supported by the Ministry of Science and Technology of the People s Republic of China (Grant Nos. 2016YFA and 2017YFA ) and National Natural Science Foundation of China (Grant ). References [1] S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials, Springer, Springer International Publishing Switzerland, 2016 [2] J. Seidel, Topological Structures in Ferroic Materials: Domain Walls, Vortices and Skyrmions, Springer, Springer International Publishing Switzerland, 2016 [3] X. Z. Yu, Y. Tokunaga et al, Nat. Commun. 5, 3198 (2013) [4] W. Wang, Y. Zhang et al, Adv. Mater. 28, 6887 (2016) [5] R. Takagi, X. Z. Yu et al, Phys. Rev. Lett. 120, (2018) [6] X. Z. Yu, Y. Tokunaga et al, Adv. Mater. 29, (2016) [7] L. Peng, Y. Zhang et al, Nano Lett. 17, 7075 (2017) [8] J. Cui, Y. Yao et al, J. Mag. Mag. Mater. 454, 304 (2018) [9] A. H. Eschenfelder, Magnetic Bubble Technology, 2 nd ed. Springer-Verlag Berlin Heidelberg New York 1981 [10] M. J. Donahue and D. G. Porter, OOMMF User's Guide, Version 1.0,'' NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (Sept 1999) [11] M. Teague, J. Opt. Soc. Am. 73, 1434 (1983) [12] K. Ishizuka, B. Allman, J. Electron Microsc. 54, 191 (2005) [13] X. Z. Yu, Y. Onose et al, Nature 465, 901 (2010) [14] M. Mitome, K. Ishizuka et al, J. Electron Microsc. 59, 33 (2010) [15] M. Beleggia, T. Kasama et al, Ultramicroscopy 110, 425 (2010)