Unveiling the three-dimensional magnetic texture of skyrmion tubes.

Sample preparation

On the basis of the results of crystal growth by chemical vapour transport in the system Fe/Ge 48 single crystals of FeGe in the B20 structure were grown via chemical transport reaction using iodine as a transport agent. Starting from a homogeneous mixture of the element powders iron (Alfa Aesar 99.995%) and germanium (Alfa Aesar 99.999%) the cubic modification of FeGe crystallized by a chemical transport reaction very slowly in a temperature gradient from 850 K (source) to 810 K (sink) and a transport agent concentration of 0.2 mg cm −3 iodine (Alfa Aesar 99.998%). The chemical vapour transport was made perpendicular to the tube axis over a diffusion distance of 38 mm. Selected crystals were characterized by EDXS, WDXS and especially X-ray single crystal diffraction to verify the present modification. The preparation of the FeGe needle was carried out via FIB technique on a Thermo Fisher Scientific Helios 660 operated at 30 kV. A rough cut of the needle geometry (700 × 700 nm 2 ) was performed with currents of 790 and 430 pA. For further fine shaping (300 × 300 nm 2 ) the current was reduced to 80 and 40 pA. The final polishing was carried out at 24 pA. To remove preparation residue, the needle was finally cleaned in a Fischione Model 1070 NanoClean for 1 min.

Acquisition and reconstruction of the holographic tilt series

Holographic tilt series were recorded at an FEI Titan G2 60-300 HOLO in Lorentz-Mode (conventional objective lens switched off) operated at 300 kV. The voltage of the electrostatic Möllenstedt bisprism was set to 120 V leading to a fringe spacing of 2.3 nm in the electron hologram ( Supplementary Information ). For the acquisition of the latter, a GATAN K2 Summit direct detection camera in counting mode was used yielding a holographic fringe contrast of 40%. The acquisition process was performed semi-automatically with an in-house developed software package 49 to collect three holographic tilt series consisting of object and object-free empty holograms, two at 95 K and one at room temperature. For the first tilt series at 95 K, the angle between the needle and tilt axis amounted to 30°. For the second tilt series, the specimen was manually rotated outside the microscope in-plane by 70° (ideal is 90°) resulting in an angle between the needle and tilt axis of −40° ( Supplementary Information for the details). The tilt range of each tilt series was from −66° to + 65° in 3° steps. To obtain the full phase shift (>2π), the phase images were unwrapped automatically by the Flynn algorithm and manually at regions, where the phase signal was too noisy or undersampled, by using previous knowledge of the phase shift (for example, from adjacent projections) 40 . Potential phase wedges in vacuum caused by the magnetic stray field of the ring were corrected in all three-tilt series. An analysis of these stray-field contributions is presented in the Supplementary Information .

Tomographic reconstruction

All three phase image tilt series were aligned, that is, corrected for image displacements with respect to their common tilt axis by cross-correlation, centre-of-mass method and common-line approach 33 . The thereby obtained aligned datasets correspond to the following linear projection laws (Radon transformations): 1 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${varphi }_{{mathrm{e}}}left(p,theta ,zright)={C}_{{mathrm{E}}}{iint }_{{{{{{mathbf{e}}}}}}cdot {{{{{mathbf{r}}}}}}}{{varPhi }}left(x,y,zright){{{rm{d}}}}x{{{rm{d}}}}y$$end{document} φ e p , θ , z = C E ∬ e ⋅ r Φ x , y , z d x d y and 2 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{partial {varphi }_{{mathrm{m}}}(p,theta ,z)}{partial z}=frac{e}{hslash }{iint }_{{{{{{mathbf{e}}}}}}cdot {{{{{mathbf{r}}}}}}}{B}_{p = y,x}left(x,y,zright){{{rm{d}}}}x{{{rm{d}}}}y.$$end{document} ∂ φ m ( p , θ , z ) ∂ z = e ℏ ∬ e ⋅ r B p = y , x x , y , z d x d y . Here, C E is a kinetic constant depending solely on the acceleration voltage, p and z are the 2D detector coordinates, θ the tilt angle, r = ( x , y ) T and documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{{{{mathbf{e}}}}}}={(cos theta ,sin theta )}^{T}$$end{document} e = ( cos θ , sin θ ) T . The index to the integral indicates a collapse of the 2D integral to the projection line defined by e ⋅ r . The subsequent tomographic 3D reconstruction of the aligned phase tilt series (that is, the inverse Radon transformation) was numerically carried out using weighted simultaneous iterative reconstruction technique (W-SIRT) 50 . The three resulting tomograms represent the incremental 3D phase shift per voxel that we refer to as 3D phase maps. The two 3D phase maps obtained at 95 K were released from their electrostatic contribution by superposition and subtraction of the 3D phase map obtained at room temperature. Then, the derivation of each of the two resulting magnetic 3D phase maps in directions perpendicular to both the experimental tilt axis and tilt directions using an appropriate Fourier filter (Fourier-slice theorem) as well as multiplication with the factor ℏ / e leads to one component of the magnetic induction in the respective direction. Since the specimen was rotated only by 70° in the underlying tomographic experiment for the reconstruction of these two B field components, one of them was projected on the orthogonal direction of the other to receive finally the 3D B x and B y components. 3D visualization was performed using the Avizo software package (ThermoFisher Company) and the Mayavi Python package. A verification of the experimental workflow repeated on simulated data is provided in Supplementary Information .

Calculation of the third magnetic B field component

The third B field component B z is obtained by solving Gauss’s law for magnetism documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{{rm{div}}}} {{{{{mathbf{B}}}}}}=0$$end{document} div B = 0 with appropriate boundary conditions on the surface of the reconstruction volume. Here, we used periodic boundary conditions for solving this differential equation in Fourier space endowed with coordinates k , that is, 3 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${B}_{z}left({{{bf{k}}}}right)=-frac{{k}_{x}{B}_{x}left({{{bf{k}}}}right)+{k}_{y}{B}_{y}left({{{bf{k}}}}right)}{{k}_{z}}$$end{document} B z k = − k x B x k + k y B y k k z The zero frequency component (integration constant) was fixed by setting the average of B z to zero on the boundary of the reconstruction volume. To suppress noise amplification by this procedure, a Butterworth-type low-pass filter was applied. Magnetic energy densities Following ref. 33 the exchange energy density may be split into contributions from magnetic charges, currents and surface terms documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$frac{A}{{M}_{rm{s}}^{2}}left({left(nabla cdot {{{{{mathbf{M}}}}}}right)}^{2}+{left|nabla times {{{{{mathbf{M}}}}}}right|}^{2}right)-{{w}}_{{{{rm{surf}}}}}.$$end{document} A M s 2 ∇ ⋅ M 2 + ∇ × M 2 − w surf . In the magnetostatic limit considered here, the magnetization in the second term may be replaced by B / μ 0 and can be reconstructed from the tomographic data. In the case of the DM interaction, we have the following identities 4 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{array}{ll}{E}_{{{mbox{DM}}}}left[{{{{{mathbf{M}}}}}}right]&=frac{D}{{M}_{rm{s}}^{2}}int {{{{{mathbf{M}}}}}}cdot left(nabla times {{{{{mathbf{M}}}}}}right){{{rm{d}}}}V\ &=frac{D}{{mu }_{0}^{2}{M}_{rm{s}}^{2}}int {{{{{mathbf{B}}}}}}cdot left(nabla times {{{{{mathbf{B}}}}}}right){{{rm{d}}}}V+frac{D}{{M}_{rm{s}}^{2}}int nabla cdot left({{varPhi }}{{{{{{mathbf{j}}}}}}}_{{mathrm {b}}}right){{{rm{d}}}}V\ &=frac{D}{{mu }_{0}^{2}{M}_{rm{s}}^{2}}int {{{{{mathbf{B}}}}}}cdot left(nabla times {{{{{mathbf{B}}}}}}right){{{rm{d}}}}V+frac{D}{{M}_{rm{s}}^{2}}oint!!!int {{{{{{mathbf{w}}}}}}}_{{{{rm{surf}}}}}cdot {{{rm{d}}}}{{{{{mathbf{S}}}}}}end{array}$$end{document} E DM M = D M s 2 ∫ M ⋅ ∇ × M d V = D μ 0 2 M s 2 ∫ B ⋅ ∇ × B d V + D M s 2 ∫ ∇ ⋅ Φ j b d V = D μ 0 2 M s 2 ∫ B ⋅ ∇ × B d V + D M s 2 ∮ ∫ w surf ⋅ d S Here j b denotes the bound current and Φ the scalar magnetic potential. The last line identifies that part of the DM energy density, which can be derived solely from the B field, and may be identified as a volume contribution, which can be reconstructed from tomographic data. The remainder can be collapsed to a surface term.

Online content Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41565-021-01031-x.

Supplementary information Supplementary Information Supplementary Figs. 1–14 and Discussion. Supplementary Video 1 Circulating magnetic induction B of a skyrmion lattice in a FeGe needle. On the left side, the animation shows the volume rendering of the reconstructed out-of-plane component B z with the colour coding red (+0.4 T), white (0 T) and blue (−0.4 T). In the right panel, the superimposed volume renderings of the reconstructed in-plane components B x (horizontal) with colour coding red (+0.3 T), white (0 T), green (−0.3 T) and B y (vertical) with colour coding blue (+0.3 T), white (0 T), yellow (−0.3 T) is displayed. The size of the specimen is approximately 240 nm in the thickest part (top). The animation sequence is as follows: time action, 0–8 s, rotation of both volumes around the vertical axis by 360°. 8–10 s, rotation of the B z component (left) around the vertical axis by −80° circle side view of B z . 10–12 s, cutting of the volumes from the side (position indicated by a red slice). 12–14 s, reverse cutting with rotation of B z around the vertical axis by 80°. 14–16 s, cutting of both volumes from the front. 16–18 s, rotation of both volumes around the vertical axis by −90° circle cut side view. 18–20 s, reverse slicing from right to left and rotation (back) around the vertical axis by a 90° circle. The initial orientation of the two volumes is reached. Supplementary Video 2 Z slicing of the magnetic induction B of a skyrmion lattice in a FeGe needle. In the upper panel, the animation shows the volume rendering of the B x component from the side ( y z view). In this representation, the SkTs are visualized as green–red pairs. The lower panel displays z slices from the top ( x y view) of the reconstructed in-plane components as arrow plots, as well as B x (vertical) with colour coding red (+0.2 T), white (0 T), green (−0.2 T) and B y (horizontal) with colour coding blue (+0.2 T), white (0 T), yellow (−0.2 T). The z slices are simultaneously shown in the upper panel to illustrate their position. Supplementary Video 3 Slicing of the magnetic induction B of a skyrmion lattice in a FeGe needle. On the left side, the animations show arrow plots taken at slices parallel to the nearest neighbour direction q 1 (see main text). The positions of the slices are indicated as black rectangles on the right, where the superimposed volume renderings of the reconstructed in-plane components B x (horizontal) with colour coding red (+0.3 T), white (0 T), green (−0.3 T) and B y (vertical) with colour coding blue (+0.3 T), white (0 T), yellow (−0.3 T) are visualized. The colour coding of the arrows is consistent with the colours in the volume rendering (colour wheel). Supplementary Video 4 Slicing of the magnetic induction B of a skyrmion lattice in a FeGe needle. On the left side, the animations show arrow plots taken at slices parallel to the nearest neighbour direction q 2 (see main text). The positions of the slices are indicated as black rectangles on the right, where the superimposed volume renderings of the reconstructed in-plane components B x (horizontal) with colour coding red (+0.3 T), white (0 T), green (−0.3 T) and B y (vertical) with colour coding blue (+0.3 T), white (0 T), yellow (−0.3 T) are visualized. The colour coding of the arrows is consistent with the colours in the volume rendering (colour wheel).

Supplementary information The online version contains supplementary material available at 10.1038/s41565-021-01031-x.