Abstract
Magnetic skyrmions are key candidates for applications in memory, logic and neuromorphic computing. An essential property is their topological protection that is caused by the swirling spin texture and described by a robust integer winding number. However, this protection is strictly enforced only in the continuum, and so the atomic lattice present in all real materials leaves a loophole for switching the winding number. Hence, understanding the microscopic mechanism of this unwinding is crucial for enhancing the stability of skyrmions. Here we use spinpolarized scanning tunnelling microscopy to locally probe skyrmion annihilation by individual hot electrons. We tune the collapse rate by up to four orders of magnitude by using an inplane magnetic field, and observe distinct transition rate maps that either are radial symmetric or exhibit an excentric hotspot. We compare these maps to atomistic spin simulations based on parameters obtained from firstprinciples calculations and find that the maps are explained by a radial symmetric collapse at zero inplane magnetic field and a transition to the recently predicted chimera collapse at finite inplane magnetic fields. These insights into the transient state of the skyrmion collapse will enable future enhancement of skyrmion stability and designs for intentional skyrmion switches.
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Data availability
Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
All codes used to evaluate the data are available from the corresponding author upon request.
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Acknowledgements
We gratefully acknowledge helpful discussions with S. Lounis, S. Blügel, G. Volovskiy, A. Schlenhoff, M. Liebmann, M. A. Goerzen and T. Sigurjónsdóttir, and financial support by the German Science Foundation (grant no. PR 1098/11), Russian Science Foundation (grant no. 197210138), Icelandic Research Fund (grant no. 184949052) and Alexander von Humboldt Foundation.
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F.M. and B.P. conducted the experiments and C.H., M.P. and M.M. supervised. S.v.M. performed all atomistic spin simulations and P.F.B. and S.H. supervised. F.M. and M.M. provided the original idea of the experiment, and P.F.B. and S.v.M. contributed the idea of a more detailed comparison with DFTbased calculations. In particular, S.v.M., P.F.B. and S.H. introduced the chimera mode as an explanation for the experimental data. F.M. with the help of C.H. evaluated all experimental data, performed comparisons to calculated data and conducted the calculations of heat distribution of a hot electron. M.M. wrote the first version of the manuscript and F.M. and S.v.M. supplied the images. All authors contributed to multiple discussions and the final version of the manuscript.
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Extended data
Extended Data Fig. 1 Telegraph noise in switching maps.
a, Map of skyrmion collapse rate, B_{∥} = 1.5 T, B_{⊥} = 1.5 T, V = 610 mV, I = 60 nA, Δt = 150 s (same as Fig. 3f). Large (small) circle marks skyrmion area (hot spot of collapse rate). Colored rectangles with numbers mark positions, where time traces and histograms are displayed in (b), (c). b, Left: time trace (orange curve) on position marked as 1 in (a). The deduced digital values (blue, dashed curve) using a threshold (black, dashed line) are shown with upper (lower) value marking the ferromagnetic (skyrmion) state. The skyrmion probability P_{SK} and the number of switching events n are marked on top. These numbers are used to calculate the collapse rate rate_{collapse} by the given formula for a time interval Δt of recording. Right: histograms of measured dI/dV values during the time trace. c, Time traces as exemplarily shown in (b) (left) and histograms of measured dI/dV values during the time trace (right) for the positions marked in (a) by the same number and frame color. Deduced rates for collapse and creation, skyrmion probability P_{SK} and number of switching events n are marked on top in each frame. Frames are ordered for pixels within the skyrmion interior (left) and at the rim of the skyrmion (right). While the histograms from the interior are always bimodal, additonal peaks partly appear at the rim. They are likely caused by switching of neighboring skyrmions. (More time traces and detailed discussion: Supplementary Fig. S3).
Extended Data Fig. 2 Switching maps for skyrmion B marked in Fig. 6a.
Images in each column are recorded at the same B_{∥} and B_{⊥} indicated on top. ac, STM topography, V = 610 mV, I = 2 nA. Circles mark the skyrmion area with black dot as center. Defects (bright dots) at the skyrmion rim are visible. The skyrmion slightly shifts between images in contrast to skyrmion A (Supplementary Section S1). df, Simultaneously recorded dI/dV maps. gi, Maps of skyrmion collapse rates as deduced from time traces recorded at each pixel (Extended Data Fig. 1), Δt = 150 s, V = 610 mV, I = 200 nA. The direction of B_{∥} is marked. As in Fig. 3f–h, the hot spot at the rim switches position along B_{∥} by sign change of B_{⊥} or B_{∥} and is always located close to the darkest spot within the corresponding dI/dV images. jl, Maps of creation rates obtained from the same time traces, being rather homogeneous in the skyrmion area. mo, Maps of skyrmion probability P_{SK} obtained from the same time traces. The probability close to one shows that the skyrmion state is more stable than the ferromagnetic state. Circles are at identical positions in each column.
Extended Data Fig. 3 Adapting dI/dV contrast from atomistic spin simulations to experimental dI/dV maps.
ad, Measured dI/dV maps of skyrmion A (Fig. 6a) at the B fields marked on top (same as Fig. 3ad). Large (small) circles enclose the skyrmion area (hot spot of collapse rate) with dot as circle center. Arrows mark directions of profile lines in (m)(p). eh, Calculated dI/dV maps consisting of overlapping contributions of NCMR and tunneling magnetoresistance using the spin configurations of (i)(l) (methods). Circles and arrows are identical to the images on top. il, Atomic magnetization vectors m_{i} (arrows) with color marking \({m}_{i}^{z}\) as calculated by the atomistic spin model. Circles are identical to the larger circles of the upper images in the same column. (e)(l) B_{⊥}, B_{∥} as marked above (e)(h). mp, Profile lines through measured (black) and simulated (dotted blue) dI/dV images at the positions marked by arrows in (a)(h) (same curves are shown in Fig. 3il).
Extended Data Fig. 4 Calculated minimum energy paths: different energetic contributions.
a, The energy contributions along the minimum energy path given relative to the initial relaxed skyrmion state for the radial symmetric collapse of an isolated skyrmion, B_{⊥} = 3.2 T, B_{∥} = 0.0 T. b, Same as (a) for the chimera collapse, B_{⊥} = 3.2 T, B_{∥} = 1.5 T. The legend applies to both images. Black: total energy along the deduced minimum energy path (methods), red: contribution of exchange energy, blue: contribution of DzyaloshinskiiMoriya interaction (DMI), pink: contribution of magnetocrystalline anisotropy energy, green: contribution of Zeeman energy caused by the applied B. The saddle point (SP, maximum of black curve) is dominated by exchange energy in both cases. For (a), the DMI contribution raises from the start, while the anisotropy and Zeeman energy shrink simultaneously. For (b), the DMI contribution starts rising only shortly prior to the SP, hence contributing much less energy penalty to the SP, while Zeeman and anisotropy contributions barely lower the SP energy and shrink only after the SP has been overcome (detailed analysis in Supplementary Sections S8/S9).
Extended Data Fig. 5 Energy density maps of different contributions at the saddle points.
Energy densities at the saddle point relative to the relaxed ferromagnetic state (FM) for radial symmetric (left column) and chimera (right column) collapse of an isolated skyrmion at B as marked on top. The hot spot is dominated by exchange energy in both cases with minor contributions of DzyaloshinskiiMoriya interaction (DMI) for the chimera collapse. The total energy lowering by DMI with respect to FM is, however, more extensive for the chimera collapse due to the larger area of adequately winding spins (detailed analysis in Supplementary Sections S8/S9).
Extended Data Fig. 6 Comparison of hot spots at saddle point for the same B.
a, Magnetization directions of the atoms (arrows) of the skyrmion, relaxed by spin dynamics and the velocity projection optimization method (methods), and colored according to the m_{z} contribution. The white circle marks the skyrmion radius deduced from calculated dI/dV image (b) (methods). b, Simulated dI/dV map of the relaxed skyrmion using the NCMR contrast only^{23} (methods). Arrows mark directions and interpolation widths of profile lines in (g,h). c,d, Energy density map for the radial symmetric (c) and chimera collapse (d) at the saddle point as calculated by GNEB. The energy is relative to the relaxed ferromagnetic state (FM). e,f, Magnetization structure at the saddle point (SP) for radial symmetric (e) and chimera (f) collapse. g,h, Profile lines along arrows in (c), (d) for (g), (h), respectively, (green) and along arrows in (b) (black). Dashed lines mark maxima (minima) of the green (black) curves. The maximum of the energy density in (h) is slightly shifted away from the center with respect to the simulated NCMR contrast minimum. While the maximum energy density of the radial collapse remains close to the skyrmion center, it is offset by 1 nm for the chimera collapse (detailed discussion in Supplementary Section S10).
Extended Data Fig. 7 Calculated energy barriers and prefactors for skyrmion creation.
ac, Energy barriers (black) and prefactors (blue) for the radial symmetric (open symbols) and chimera type creation (full symbols) of an isolated skyrmion at varying B_{∥} according to atomistic spin simulations via GNEB and transition state theory based on parameters from DFT (methods). (a) B_{⊥} = 3.2 T, (b) B_{⊥} = 4.0 T, (c) B_{⊥} = 4.3 T. d, Thermal creation rates at B_{⊥} = 4.3 T resulting from (c) using eq. (1) for three different temperatures with filled (open) circles employing the chimera (radial symmetric) creation process. Trends are similar as for the collapse mechanisms (Fig. 5), but the creation rates (d) depend much less on B_{∥} than the collapse rates (detailed discussion in Supplementary Section S11).
Extended Data Fig. 8 Comparison of calculated and experimental skyrmion creation rates.
Thermally induced skyrmion creation rates at different T deduced with the help of eq. (1) using the theoretical parameters from atomistic spin simulations of Extended data Fig. 7a (colored lines, B_{⊥} = 3.2 T) and experimental skyrmion creation probabilities per single electron (Fig. 6g) extrapolated to creation rates for an assumed I = e/2 fs = 75 μA (symbols, B_{⊥} = 1.5 T). This large I would establish a quasiconstant temperature of 45 K within the hot spot area of increased energy density at the SP according to our finite element simulation as displayed in Fig. 6h (detailed discussion in Supplementary Section S11). Experimental data of two skyrmions are displayed exhibiting barely any difference. Error bars result from the uncertainty of the adaption of I to a continuous T.
Supplementary information
Supplementary Information
Supplementary Discussions 1–12 and Figs. 1–17.
Supplementary Video 1
Evolution of total energy, total topological charge, spin structure, energy density map and map of topological charge per triangle along the minimum energy path for radial symmetric skyrmion collapse, B_{⊥} = 3.2 T, B_{∥} = 0.0 T.
Supplementary Video 2
Evolution of total energy, total topological charge, spin structure, energy density map and map of topological charge per triangle along the minimum energy path for chimera collapse, B_{⊥} = 3.2 T, B_{∥} = 1.5 T.
Supplementary Video 3
Evolution of total energy, total topological charge, spin structure, energy density map and map of topological charge per triangle along the minimum energy path for radial symmetric skyrmion collapse, B_{⊥} = 4.3 T, B_{∥} = 1.5 T
Supplementary Video 4
Evolution of total energy, total topological charge, spin structure, energy density map and map of topological charge per triangle along the minimum energy path for chimera collapse, B_{⊥} = 4.3 T, B_{∥} = 1.5 T.
Source data
Source Data Fig. 1
Calculated minimum energy paths of radial (B_{∥} = 0 T) and chimera (B_{∥} = 1.5 T) collapse, both at B_{⊥} = 3.2 T in Fig. 1b.
Source Data Fig. 2
Experimental telegraph noise in Fig. 2e and histogram data in Fig. 2f, both at I = 10 nA, V = 610 mV, B_{⊥} = 1.5 T and T = 6 K.
Source Data Fig. 3
Experimental profile lines of collapse rate and measured dI/dV signal compared with calculated dI/dV signal in Fig, 3i–l.
Source Data Fig. 4
Profile lines of calculated energy density for the saddle point GNEB image and calculated NCMR contrast for the initial GNEB image of the radial collapse in Fig. 4i and the chimera collapse in Fig. 4j using minimum energy path simulations based on an atomistic spin model parametrized from DFT calculations.
Source Data Fig. 5
Calculated energy barriers and prefactors divided by temperature for the radial and chimera collapse at different out of plane magnetic fields in Fig. 5a–c and thermal collapse rates at B_{⊥}=4.3 T in Fig. 5d, using atomistic spin simulations via GNEB and transition state theory based on parameters from DFT.
Source Data Fig. 6
Measured collapse and creation rate versus tunneling current at different in plane magnetic fields at B_{⊥} = 1.5 T in Fig. 6d,e. Collapse probability per electron versus inplane magnetic field in Fig. 6f,g as deduced from linear fits to the raw data in Fig. 6d,e. Time development of simulated mean temperature in surface area within radius r in Fig. 6h, obtained by a finite element method. Calculated thermal collapse rates compared with experimental collapse rates extrapolated from the single electron probabilities in f and assuming I = 75 μA in Fig. 6i.
Source Data Extended Data Fig. 3
Measured telegraph noise and deduced histograms with 100 bins at I = 60 nA, V = 610 mV, B_{⊥} = 1.5 T, B_{∥} = 1.5 T, T = 6 K.
Source Data Extended Data Fig. 4
Energy contributions over number of minimum energy path image for the radial (B_{⊥} = 3.2 T, B_{∥} = 0 T) and chimera (B_{⊥} = 3.2 T, B_{∥} =1.5 T) collapse.
Source Data Extended Data Fig. 6
Profile lines of calculated energy density for the saddle point GNEB image and calculated NCMR contrast for the initial GNEB image for radial and chimera skyrmion collapse, (both: B_{⊥} = 4.3 T, B_{∥} = 1.5 T).
Source Data Extended Data Fig. 7
Calculated energy barriers and prefactors of the skyrmion creation via the radial and chimera mechanism over inplane magnetic field at B_{⊥} = 3.2 T, B_{⊥} = 4.0 T and B_{⊥} = 4.3 T in Extended Data Fig. 7a–c and thermal creation rates at B_{⊥} = 4.3 T in Extended Data Fig. 7d, using atomistic spin simulations via GNEB and transition state theory based on parameters from DFT.
Source Data Extended Data Fig. 8
Experimental (B_{⊥} = 1.5 T) and calculated (B_{⊥} = 3.2 T) creation rates versus inplane magnetic fields B_{∥}.
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Muckel, F., von Malottki, S., Holl, C. et al. Experimental identification of two distinct skyrmion collapse mechanisms. Nat. Phys. 17, 395–402 (2021). https://doi.org/10.1038/s41567020011012
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