Abstract text (incl. figure legends and references)
Lorentz transmission electron microscopy (LTEM) and off-axis electron holography (EH) are the methods of choice to image magnetism at the nanometer scale. Both techniques exploit the electro-magnetic Aharonov-Bohm effect imprinting a phase shift on the wave function of highly-energetic electrons after transmitting a thin sample. However, for investigation of three-dimensional (3D) nanomagnetism [1], which involves complex 3D magnetic configurations such as exotic domain walls (e.g. of Bloch point type) or magnetic textures (e.g. (braided) Skyrmion tubes, chiral bobbers, Hopfions), these 2D techniques are insufficient for a unique mapping of the underlying magnetic structure. To overcome this problem EH is combined with electron tomography to holographic vector-field electron tomography (VFET). After recording and reconstructing holographic tilt series, the resulting phase image tilt series are input in tomographic reconstruction algorithms to obtain both the mean inner potential (MIP) and the magnetic induction B in 3D. Here, at least three tilt series are required to reconstruct two linearly-independent B-field components, and the MIP contribution to the phase. The latter is sensitive to the 3D morphology and chemical composition of the sample. The third B-field component can be computed by solving div.B=0.
We discuss state-of-the-art VFET including comparison to x-ray-based magnetic vector-field reconstruction and methodological challenges, e.g., imperfect and incomplete projection data, alignment errors, as well as regularization schemes in the tomographic reconstruction algorithms. Moreover, we present examples how VFET has been applied to obtain 3D magnetic vector fields with a few nanometer resolution: (I) We reconstructed in 3D two remanent magnetic states including their peculiar domain walls in CoNi nanowires with large transversal magnetocrystalline anisotropy [2]. (II) We revealed with holographic VFET Bloch-type Skyrmion tubes (SkTs) in an FeGe needle-shaped specimen [3]. This experiment is challenging because it has to be performed under cryogenic conditions (provided by an LN-cooled sample holder) in the presence of an out-of-plane applied B-field (provided by a ring magnet of Sm2Co17 underneath the specimen), stabilizing the Skyrmion lattice in the FeGe needle. Fig. 1 shows a 3D reconstruction obtained from three holographic tilt series recorded over a tilt range of ±66° in 3° steps. The 3D MIP distribution was determined from a tilt series recorded at room temperature, at which FeGe is paramagnetic. The resulting tomogram was then subtracted from tomograms reconstructed from two tilt series recorded at 95 K to determine the 3D B-field of the FeGe sample in its Skyrmion phase. These results give direct insights into 3D Skyrmion spin texture. An analysis of, e.g., planar energy density maps across the SkTs reflecting the interplay between symmetric exchange and antisymmetric Dzyaloshinskii–Moriya interaction, as well as a comparison with micromagnetic simulations is presented.
Figure 1. 3D magnetic induction of the skyrmions in an FeGe sample. Left: Color-coded components Bx and By with arrow plots show the direction and magnitude of the magnetic field. Right: Extracted single SkT with arrow plots at a top, middle and bottom plane.
[1] A. Fernandez-Pacheco et al., Nat Commun 8, 15756 (2017)
[2] I. M. Andersen et al., Phys. Rev. Res. 3, 033085 (2021)
[3] D. Wolf et al., Nat. Nanotechnol. 17, 250 (2022)