Quantification of Magnetic Surface and Edge States in an FeGe Nanostripe by Off-Axis Electron Holography

Dongsheng Song, Zi-An Li, Jan Caron, András Kovács, Huanfang Tian, Chiming Jin, Haifeng Du, Mingliang Tian, Jianqi Li, Jing Zhu, and Rafal E. Dunin-Borkowski National Center for Electron Microscopy in Beijing, Key Laboratory of Advanced Materials (MOE) and The State Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, 100084 Beijing, China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, 100190 Beijing, China Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany High Magnetic Field Laboratory, Chinese Academy of Sciences, 230031 Anhui, China

Introduction.-In certain physical systems, peculiar boundary states can arise because of topology and geometry considerations. Such boundary states include edge current modes in quantum Hall systems [1], edge metallic states in topological insulators [2], and spin boundary states in chiral magnets [3]. In chiral magnets, noncollinear spin textures at surfaces and edges surround magnetic Skyrmions: particlelike swirling spin textures [4]. Rohart and Thiaville have theorized [3] that the Dzyaloshinskii-Moriya interaction (DMI) [5,6] in a chiral magnet can lead to specific micromagnetic boundary conditions that twist the magnetization at sample edges, with a feature length that is a fraction of a helix period [7]. Such edge spin configurations preserve the magnetic chirality around the sample edges and play a key role in current-induced Skyrmion motion in nanostripes [8]. Moreover, the enhanced stability of Skyrmions in systems that have reduced dimensions has been attributed theoretically to the presence of peculiar spin textures at chiral boundaries (surfaces and edges) [9,10]. Experimentally, chiral edge states have been inferred indirectly from magnetotransport measurements [11] and using magnetic imaging [12]. Given the importance of chiral boundary states for understanding Skyrmion physics and exploiting Skyrmionbased applications, it is important to experimentally quantify the local magnetic properties of chiral boundary states and their influence on Skyrmions in nanostructures.
Advanced magnetic imaging techniques that have been used to image magnetic Skyrmions include spin-polarized scanning tunneling microscopy (SP-STM) [13], x-ray magnetic circular dichroism in scanning transmission x-ray microscopy (STXM) [14], and magnetic force microscopy (MFM) [15]. However, STXM and MFM are not well suited to resolving fine chiral edge and surface states because of their relatively poor spatial resolution (a few tens of nm). SP-STM has stringent requirements for clean material surfaces and probes only surface spins. Transmission electron microscopy (TEM) based techniques such as Lorentz microscopy (LM) [12,16], off-axis electron holography (EH) [17][18][19], and differential phase contrast (DPC) imaging [20,21] have been used to image magnetic Skyrmions in thin films with sub-10-nm spatial resolution. However, resolving Skyrmionic spin textures in nanostructures is highly challenging because of the required spatial resolution and magnetic sensitivity. In particular, the LM technique is not suitable for studying magnetic nanostructures that have sizes far below 100 nm because the formation of Fresnel fringes arising from changes in specimen thickness at the boundaries of nanostructures complicates interpretation of the magnetic contribution. Off-axis EH provides the phase of the electron wave function that has passed through a sample, allowing the magnetic signal to be separated more easily than using other TEM-based techniques when studying Skyrmions in confined nanostructures [22,23].
In this Letter, we use state-of-the-art off-axis EH to investigate the formation of magnetic edge states in an FeGe nanostripe in the presence of magnetic fields and as a function of temperature. We determine the projected inplane magnetization distributions of chiral edge twists and Skyrmions from recorded phase images using a newly developed magnetization reconstruction technique. We perform quantitative measurements from the reconstructed magnetization distributions, in order to assess the characteristic penetration lengths of edge twists, the sizes of confined Skyrmions, and their saturation magnetizations at different temperatures.
Experimental details.-A thin nanostripe of noncentrosymmetric single crystalline B20 FeGe was prepared using focused ion beam milling, as described elsewhere [12,22]. Structural characterization was performed using an FEI F20 TEM operated at 200 kV and is given in the Supplemental Material [24] (Figs. S1 and S2). For magnetic imaging, we used an image-aberration-corrected FEI Titan 80-300 TEM equipped with a Lorentz lens and two electron biprisms. The use of a liquid-nitrogen-cooled specimen holder (Gatan model 636) allowed the sample temperature to be varied between 95 and 370 K. The objective lens of the microscope was used to apply out-of-plane magnetic fields of between 0 and 1.5 T. A cumulative acquisition approach was used to record 20 electron holograms, with an exposure time of 6 s for each hologram. The holograms were reconstructed by a standard Fourier-transform-based method using customwritten Matlab codes. The final unwrapped phase images were averaged to improve the signal to noise ratio.
Results and discussion.-Spin configurations at surfaces and edges.- Figure 1(a) shows a schematic representation of edge twists and Skyrmions in a chiral nanostripe, whose width spans two edges and one Skyrmion. The colors denote in-plane (x, y) magnetization components. This simplified model illustrates Bloch-type edge twists and a Skyrmion, in which the spins are perpendicular to radii that point outward from the center of the Skyrmion to its edge [see the magnified part of Fig. 1(a)]. Figure 1(b) shows a three-dimensional (3D) Skyrmion with chiral surface twists, which are characterized by an in-plane rotation angle θ. Such twisted spin textures at the surface have been predicted to lower Skyrmion energetics and to provide thermodynamical stability of a Skyrmion lattice against the formation of conical states in a thin film [9,10].
It should be noted that the formation of chiral surface and edge states results from the volume energy in the DMI term, without including modified magnetic parameters at the surface [3,9,10]. When considering magnetic properties that are specific to the surface, the edge spin configurations can be complex, and various forms of spin texture at specimen edges have been discussed theoretically in achiral and frustrated-type magnets [32].
Reconstruction of in-plane magnetization distributions from magnetic phase images.-We first used LM to visualize the magnetic-field-driven magnetization transitions in an FeGe nanostripe, as seen in the Supplemental Material [24] (Fig. S3). Figure 2(a) shows a representative LM image recorded at 95 K in a 300 mT applied field, with the circular contrast features corresponding to individual magnetic Skyrmions. Strong intensity oscillations [a line profile in Fig. 2(b)], which arise from the abrupt sample thickness change and the defocused imaging conditions, complicate the interpretation of the weak magnetic contribution to the contrast at the edges. In Fig. 2(b), Skyrmion contrast features (marked "1" and "2") are much weaker than the oscillatory contrast (marked "A" and "B"). Retrieval of the magnetic structure at the specimen edges from such LM images is practically unfeasible [12].
To quantify the magnetic structure, we separated the magnetic contribution to the phase from the unwanted electrostatic (mean inner potential) contribution using off-axis EH, as described in Figs. 2(c)-2(e).   When the FeGe nanostripe was zero-field cooled from room temperature to 240 K, it was in a magnetic ground state of spin helices [ Fig. 3(a)]. When a magnetic field of 100 mT was applied normal to the plane of the nanostripe, the spin helices evolved into a Skyrmion lattice [ Fig. 3(b)]. The internal spin structures of individual Skyrmions adopted hexagonal rather than circular shapes, as a result of their proximity to each other in a lattice state [18,21,33]. Interestingly, the hexagonal Skyrmions are elongated in the vertical direction in the nanostripe. Quantitative measurements of the Skyrmion elongation are presented in Fig. 4(b). The elongation is thought to result from the geometrical confinement effect, with the Skyrmions flexibly changing their shape in response to the specimen geometry [19,20,22]. The elongated distortions became less pronounced with increasing applied magnetic field: the Skyrmions adopted more circular shapes and decreased in size in a 200 mT field [ Fig. 3(c)], and collapsed in a 300 mT field [ Fig. 3(d)].
The Skyrmions are always accompanied by chiral edge twists, as marked by white arrows in Figs. 3(b)-3(c) The chiral edge twists persist up to large values of applied magnetic field, when the Skyrmions collapse completely [ Fig. 3(d)]. Such chiral edge states [see also the schematic diagram in Fig. 1(a) increasing field, suggesting that the interaction between the edge and the Skyrmions is repulsive [8,34], (iii) the edge twists exhibit curved shapes at lower values of applied field [100 mT in Fig. 3(b)] and become less curved in higher fields [ Fig. 3(c)], as denoted by white lines in Figs. 3(b), 3(c), and (iv) the edge states persist up to very high applied magnetic fields, at which the Skyrmions are transformed to a conical or a field-polarized ferromagnetic state [ Fig. 3(h)] [22].
Characteristic dimensions of edge twist and confined Skyrmions.-We now quantify the characteristic length of the chiral edge twist and the sizes of confined Skyrmions in the FeGe nanostripe. Figure 4(a) shows reconstructed in-plane magnetization distributions under different temperature-field conditions, with dashed lines showing the positions of corresponding line profiles in Figs. 4(b)-4(c). The penetration length of the edge twist was measured between the nanostripe edge (marked "1") and the contrast dip (marked "2"), while the diameter of the Skyrmions was measured between the dips marked 2-3 and 4-5 in the line profiles. Figures 4(d)-4(e) summarize the measurements, which reveal that the edge twist expands slightly towards the nanostripe interior with increasing applied field [see the dashed lines at 95 and 240 K in Fig. 4(d)]. The measured values (14.2, 24.4, 22.5, and 31.6 nm) are a fraction of the helix period L D ¼ 70 nm for FeGe [35], and are in good agreement with predicted penetration lengths [7,8,11]. For the confined Skyrmions, (i) in lower applied magnetic fields, the Skyrmions exhibit a strong elongation [see the horizontal and vertical measurements in Fig. 4(e)], (ii) in higher applied magnetic fields, the Skyrmions become round, with diameters that are larger than the 80 nm characteristic diameter of Skyrmions in an extended thin film [35]. The formation of elliptically deformed Skyrmions with large diameters can be ascribed to a strong confinement effect, as discussed recently for a single Skyrmion chain in an FeGe nanostripe [22]. In the present study, the width of the nanostripe (∼330 nm) spans a pair of edge twists and three rows of Skyrmions. In a nanostripe whose width is larger or smaller, Skyrmions are anticipated to flexibly change their sizes and shapes to accommodate the sample geometry, in accordance with the topological invariance of Skyrmionic spin textures in confined nanostructures.
Quantitative analysis of saturation magnetization.-We further analyzed the reconstructured magnetization distributions to determine the saturation magnetization in the FeGe nanostripe. According to analytical two-dimensional spin models for a magnetic helix and a Skyrmion [17,36], the saturation magnetization M S can be extracted from the maximal in-plane magnetization. We therefore measured M S in selected regions (dashed lines) for the helix and Skyrmion in Figs. 5(a), 5(b) respectively, in order to determine the maximal in-plane magnetization. The procedure is described in Fig. S5 in the Supplemental Material [24]. the expected values. For example, M S;95K ¼ 256 kA m −1 measured from the helices is approximately 80% of the expected value of 318 kA m −1 . This discrepancy is likely to result from a nonmagnetic surface layer on the TEM specimen, as well as from possible inaccuracy in the determination of the specimen thickness [33].
Significantly, M S at 95 K (240 K) measured for the Skyrmions is ∼7% (∼9%) lower than that measured from the helices. This difference can be ascribed to the 3D nature of the Skyrmion model [ Fig. 5(b)], which contains chiral surface twists [9]. A decrease in magnetic phase shift recorded using off-axis EH is therefore expected, resulting in an underestimate of the saturation magnetization M S from a measurement of the maximum in-plane magnetization of Skyrmions. Our comparative analysis of M S for helices and Skyrmions supports a theoretically predicted 3D Skyrmion model with chiral surface twists [9].
Conclusions.-In summary, we have investigated chiral boundary (surface and edge) states in an FeGe nanostripe using off-axis electron holography, with a focus on the quantitative understanding of the in-field evolutions and characteristic penetration lengths of edge twists. The analysis of magnetization distributions reconstructed from electron holographic phase images for edge twists and Skyrmions allows precise measurements of the characteristic sizes of edge states and confined Skyrmions in the FeGe nanostripe. We quantified the saturation magnetization at 95 and 240 K from the maximal values of the reconstructed inplane magnetization for the helices and Skyrmions. An observed difference in measured saturation magnetization between helices and Skyrmions was attributed to the 3D nature of the Skyrmion model, which involved chiral surface twists. Such magnetic boundary states are important for the exploitation of Skyrmion-based devices, as they affect the particlelike properties of topologically protected spin textures and are likely to have a strong effect on magnetotransport properties [8,34,38] and spin wave excitations [39] in chiral magnet nanostructures. From a methodological perspective, the experimental approach that we use opens new avenues for exploring quantum confinement in other complex noncollinear spin systems at the nanoscale.
We thank Nikolai S. Kiselev and Filipp N. Rybakov for helpful discussions. This work was supported financially by the Chinese National Natural Science Foundation