Supplementary Figures

Transcription

1 Supplementary Figures Supplementary Figure 1: Bloch point formation during skyrmion annihilation. Skyrmion number in layers with different z-coordinate during the annihilation of a skyrmion. As the skyrmion number has to change by one during this process, a Bloch point like singularity has to mediate this annihilation process. Initially, both the top and bottom layers have the same skyrmion number. At time t=1.7155ns, the numerical simulation shows different skyrmion numbers at top and bottom of the layer, demonstrating that a 3D Bloch point moves vertically through the film (cf. inset). The skyrmion number in the top film differs slightly from one due to the discreteness of the numerical grid as opposed to the continuous spin distribution assumed in micromagnetics. During the annihilation process of a skyrmion, the continuity of the magnetization field has to be violated. This occurs via surface nucleation and motion of a Bloch point which corresponds to a point singularity in a continuum micromagnetic description. In the finite grid of a numerical simulation the Bloch point will in general sit between lattice sites but can be detected by a nonvanishing winding number of the magnetization field on a compact manifold surrounding the hedgehog-type defect. It is shown that the winding numbers on top and bottom layer of the film are different during skyrmion annihilation which indicates the presence of a localized Bloch point in the bulk of the ultrathin layer rather than a nodal line vanishing magnetization (in a continuum description) across the thickness of the film. The Bloch point is also visible in the inset which shows a vertical cut through the film.

2 Supplementary Figure 2: Theoretically calculated periodic breathing. The breathing is accompanied by nonuniform precession due to finite DMI.

3 Supplementary Figure 3: Effect of various material parameters on the nucleation current density of dynamical skyrmions. The nucleation current density J c as a function of a perpendicular magnetic anisotropy strength K u ; b NC radius; c exchange constant A; d damping constant α.

4 t1 = 0.2 t2 = 1.44 t3 = 1.48 t4 = 1.52 t5 = 2 t6 = 2.22 t7 = 3.58 t8 = 6.38 t9 = 7.4 t10 = 8.48 t = t12 = 12 1 t13 = 12.6 S m z Time (ns) Supplementary Figure 4: Dynamical skyrmion nucleation at reduced nucleation current with small damping. The dynamical skyrmion is nucleated at reduced current density of A cm-2 with small damping of α=0.05. Supplementary Figure 5: Skyrmion transport between two nanocontacts. Snapshots at times a t=0 ns, b 0.15 ns, c 0.35 ns, d 0.5 ns, e 1 ns for the motion of a dynamical skyrmion between two nanocontacts with diameter of 25 nm spaced at 100nm. The white circles indicate the nanocontacts. The dynamical skyrmion recovers its original size at the second nanocontact. In general an increase in damping will shorten the distance across which the skyrmion can be transported. Here we show results for larger damping =0.08 and a current density of A m-2, where transport across a nanocontact distance of 100 m is possible.

5 Supplementary Figure 6: Skyrmion transport through an array of nanocontacts. The white circles indicate the nanocontacts. Snapshots in intervals of 0.5ns from a to g for the motion of a dynamical skyrmion between four nanocontacts with diameter of 25 nm spaced at 100nm. Note that the dynamical skyrmion recovers its original size at each nanocontact. Here the damping constant is chosen as =0.08 and a current density of A m -2 is assumed. In general, equidistantly placed nanocontacts may allow for transport across arbitrarily large distances. Supplementary Figure 7: Nucleation mechanism of dynamical skyrmion in presence of large DMI. a the magnetisation distribution at seven selected times during the transition from non-topological droplet to dynamical skyrmion. b the corresponding DMI energy density distribution E DMI. The color scale for the energy density is normalized by a factor of 10 6.

6 Supplementary Figure 8: Example of nucleation in the presence of defect. In this case, a reduced anisotropy in the black box. a the magnetisation distribution at seven selected times during the transition from non-topological droplet to dynamical skyrmion. b the corresponding DMI energy density distribution E DMI. Supplementary Figure 9: The change of the radius of the static skyrmion as a function of current.

7 Supplementary Figure 10: Example of dynamical skyrmion nucleation in the presence of anisotropy disorder. The disorder is introduced in the simulations by normally distributing the anisotropy with mean 0.7 MJ m -3 and 45% variation throughout grains generated by Voronoi tesselation, as illustrated in panel a. The corresponding magnetisation snapshots and the evolution of different components of the average magnetisation are shown in b and c, respectively. Supplementary Figure 11: Example of dynamical skyrmion nucleation at T=300 k. a and b are the top-views of the spin structure and normalized topological density at four different times; c time-trace of the three magnetisation components averaged over the simulation area with dashed vertical lines corresponding to the snapshots above; d time-trace of the skyrmion number.

8 Supplementary Figure 12: Different boundary conditions. Panel a shows three different type of boundary conditions: Black curve: uniform α across the sample; Red curve: linearly increasing of α near the edge; Green curve: exponentially increasing in α close to the disk boundary. Panel b illustrates the boundary conditions have negligible effects in our case where the disk size is sufficiently larger than the typical skyrmion size. In the main text of the paper, the simulations are performed with open boundary conditions, i.e., damping is set to be uniform across the whole disk including the boundary. In order to study the possible influence of spin wave reflections at the sample edge we have compared the following boundary conditions: constant, linear, and exponential increase in damping as the edge of the sample is approached. Note that the exponentially increasing α ensures that there are no abrupt changes in damping thus eliminating spurious artifacts in the simulations. Finally, we have compared with simulations utilizing a very large disk with a diameter of 1.6 m with a constant damping. Although only the skyrmion number as a function of time is shown panel b, the simulations show almost identical results in terms of average magnetisation components and the total energy. The black (uniform), red (linear), and green (exponential) curves nearly overlap. The slight difference in skyrmion number for the 1.6 m disk (blue curve) is simply due to the contribution of the spins at the edges. These results clearly demonstrate that the boundary conditions have negligible effects when the disk size is sufficiently larger than the typical skyrmion size, as expected.

9 Supplementary Figure 13: Benchmarking the simulations with the Object Oriented MicroMagnetic Framework (OOMMF) software. Panels a-c: Nucleation and field-toggling of a dynamical skyrmion; Panels d-g: Nucleation and current-toggling of a dynamical skyrmion. All simulations presented in the main paper are obtained by Mumax3. Here, we compare our results with the OOMMF code. Our results, agree with those obtained by OOMMF, and thus serve as a further validation of our Mumax3 simulations.

10 Supplementary Notes Supplementary Note 1: Theory Consider magnetisation configurations parametrized by polar coordinates with m=(sinθcosϕ, sinθsinϕ, cosθ) the magnetisation unit vector. In section Dynamical skyrmions as a generic solutions of the main text we are interested in skyrmions with a radius considerably larger than the domain wall width as sustained by the nanocontact. Such skyrmions can be described by the following variational ansatz ( ( )) (1) where r is the radial coordinate, and both r and the time dependent skyrmion radius R(t) are measured in units of. The skyrmion radius R=R(t) 1 is defined as that of the circle where m z =0, and we assume the skyrmion-core to point downward, i.e. θ(0) = π and θ 0 for r R. The azimuthal angle of the magnetisation is assumed to be time dependent and given by ( ) ( ), where is the polar angle in the plane, i.e. ( ) (cf. Fig. 2e inset of the main text). An angle describes a hedgehog-type skyrmion with a radially outward pointing magnetization, whereas describes a vortex-type skyrmion with a tangential magnetization. A dynamically varying ( ) will interpolate between hedgehog and vortex-type skyrmion configurations. Inserting this variational ansatz into the Landau-Lifshitz-Gilbert-Slonczewski equation we obtain after a 2D integration for R>1, ( ) ( ) ( ),. (2) Here a c is the radius of the nano-contact (NC) measured in units of. The dot indicates a derivative with respect to time which is measured in units of the inverse anisotropy frequency M s /2γK u, and α is the damping constant. The last term in the Supplementary Equation 2 is the spin torque term with ( ) the smoothed Heaviside theta function and the (dimensionless) spin torque amplitude ( ) (3) where ( ). The function ( ) is responsible for the radial skyrmion dynamics and it contains contributions from Dzyaloshinski-Moriya interaction (DMI), Oersted-field, and dipole-dipole interaction (DDI), respectively, ( ) where for the dipolar interaction proportional to we assumed the film to be thick compared to, while its amplitude will be reduced for a thinner film. Here we also defined natural dimensionless versions of the DM constant, the Oersted-field, and the inverse quality factor of a perpendicularly magnetized film,,, (5) where the latter two are <<1 in realistic systems. We also assumed for simplicity that D 0 <<1 in line with the small DMI case studied in the Section Dynamical skyrmions as generic solutions of the main text. Note that ( ) for h Oe =0. In this case, the equations of motion for reduce to ( ) which can be readily integrated. Supplementary Figure 2 shows for D 0 =0.25 both ( ) and ( ). Periodic breathing accompanied by nonuniform precession can clearly be identified. (4)

11 Supplementary Note 2: Dependence of the nucleation current of a dynamical skyrmion on different material parameters We have carried out a series of simulations where we vary the material parameters including: damping α, perpendicular magnetic anisotropy (PMA) K u, NC radius, and exchange constant A. The spin polarization ratio is set to be P=0.3, a typical value for polarizer layers such as Co and CoFe. It is found that the damping coefficient α has the most significant effect on reducing the nucleation current, J c, of dynamical skyrmion, as shown in Supplementary Figure 3. With increasing K u, while keeping the other parameters unchanged, i.e. α=0.1, NC radius of 50 nm, A =15 pj/m, the critical current J c increases quasi-linearly with K u, as shown in Supplementary Figure 3a. Supplementary Figure 3a also reflects that the dynamical skyrmion exists for a wide range of K u. As a function of NC radius, Supplementary Figure 3b shows that the current density exhibits the expected behaviour for a constant current. On the other hand, J c increases linearly with exchange constant A, as shown in Supplementary Figure 3c while fixing the other material parameters, i.e. α=0.1, NC radius of 50 nm, K u =0.7 MJ m -3. Most importantly, the current density J c for the dynamical skyrmion nucleation can be significantly reduced by using a smaller damping coefficient, as shown in Supplementary Figure 3d. For materials with α = 0.05, the critical current density will be reduced to A cm -2 for P=0.3. However, if we set α << 0.05, as is usually associated with soft materials such as CoFeB (α ~0.015), multi-domain structures are favored instead of a single skyrmion 1. For a combination of smaller damping and reduced PMA (α=0.05, K u =0.3 MJ m -3 ), J c can be further reduced to A cm -2 with a larger NC radius of 80 nm. Therefore it should be possible to nucleate and observe the dynamical skyrmion for practical experimental conditions with realistic material parameters. The Dzyaloshinskii-Moriya interaction (DMI) strength D is set to zero in the above simulations. We also found that the nucleation current slightly decreases for 0 < D < 4 mj m -2. Outside of this range the skyrmion cannot be nucleated and it will evolve into more complex helical states (e.g. multi-domains). In summary, Supplementary Figure 3 demonstrates that the critical current density can be dramatically reduced by more than one order of magnitude if we consider a soft material with smaller damping with a combination lower PMA and larger NC size, consistent with the case of static skyrmion by Sampaio et al. 1. Furthermore, Supplementary Figure 4 highlights that the properties of the dynamical skyrmion for a smaller damping of α=0.05 remain qualitatively the same as for α=0.3 used in the paper. The current density is reduced to ~1/4 of the current density in Fig. 2 of the main text due to the reduced damping, consistent with the case of static skyrmion by Sampaio et al. 1. These results clearly demonstrate that the primary conclusions drawn from our work remain qualitatively unchanged for smaller damping.

12 Supplementary Note 3: Nucleation mechanism of the dynamical skyrmion in the presence of large DMI We show the temporal evolution of spin textures and DMI energy density in Supplementary Figure 7. The parameters used in the simulation are: D=3 mj m -2, a current density of J= A cm -2, and a damping coefficient of α=0.1. The initial magnetisation configuration corresponds to a uniform state with most of the spins oriented along +z. After a relaxation of 0.2 ns (not shown), a current density of J= A cm - 2 is turned on and applied for 10 ns. At t 1 =0.312 ns, finite in-plane components of the spin appear due to the application of the current pulse, which exerts spin transfer torque on the central spins inside the NC region. At t 2 =0.334 ns, a magnetic droplet develops in which all the spins in the vicinity of the NC are dynamic and precess at a single frequency. During ns < t < 3.25 ns, the skyrmion number remains approximately zero, being topologically equivalent to the initial FM state. The blue and red regions of the DMI energy distribution correspond to the spin distribution with different spin chiralities. For the reversed part surrounded by the mixed-chirality pseudo-wall, the DMI favored chirality region (blue colored region) will expand continuously and compresses the red part into a small regime with large positive DMI energy density (See t=3.258 and ns). At t 6 =3.268 ns, the small defect region eventually disappears and a dynamical skyrmion with skyrmion number S 1 nucleates. At t 7 =3.27 ns, the dynamical skyrmion has an almost circular shape. This transformation process is mediated by a Bloch point pair (t=3.258) and a single Bloch point on the surface (t=3.266 ns). Supplementary Note 4: Robustness of the dynamical skyrmion against disorder and finite temperature Supplementary Figure 3 shows that the dynamical skyrmion is robust against a wide range of material parameter variations. Here we present our micromagnetic simulations of the dynamical skyrmion nucleation in the presence of disorder and finite temperature. An example of the effect of a defect on the nucleation of a dynamical skyrmion is presented in Supplementary Figure 8. In contrast with what is presented in Supplementary Figure 7 where there is a uniform, K u =0.7 MJ m -3, anisotropy throughout the sample, we set K u =0.5 MJ m -3 in the region inside the 3 nm black square box. Interestingly, the nucleation of the Bloch point pair is facilitated by the presence of the defect, as evidenced by the appearance of the Bloch point in the vicinity of the defect region. The transition time from the initial FM state is also reduced from ~2.9 ns to ~0.57 ns, as can be seen by comparing Supplementary Figure 7 and Supplementary Figure 8. This example clearly demonstrates that the distribution of the disorder in realistic samples might facilitate the nucleation of a dynamic skyrmion. The topological protection of static skyrmions can drastically reduce the influence of material variations and defects. In a similar way, the dynamical skyrmion is robust against material variations and disorder. We define grain-like regions using Voronoi tessellation in the simulated material 2, with each grain having a random K u subject to a normal distribution with mean value 0.7 MJ m -3 and 45% variation, as illustrated in Supplementary Figure 10a. The corresponding magnetisation snapshots and the evolution of the different components of the average magnetisation are shown in Supplementary Figure 10 b-c, respectively. The dynamical skyrmion is well-preserved, as shown in Supplementary Figure 10 b-c. Furthermore, we have also carried out finite temperature simulations, as shown in Supplementary Figure 11. Although there are strong local distributions of the spin dynamics due to the thermal effects (t 2 -t 4 in

13 Supplementary Figure 11a), the dynamical skyrmion can be nucleated and sustained at room temperature (t 5 -t 13 ). The strong breathing of the dynamical skyrmion is also robust against thermal fluctuations, as indicated by the temporal stability of the integrated skyrmion density and strong oscillations of the m z component. In summary, we have confirmed that the dynamical skyrmion is robust against perturbations and disorder at room temperature. In addition, we show that the dynamical skyrmion properties do not depend on the boundary conditions implemented in the simulations, as discussed in Supplementary Figure 12. Finally, we have also compared our results with the OOMMF code 3 in Supplementary Figure 13. Our results, agree with those obtained by OOMMF, and thus serve as a further validation of our Mumax3 simulations. Supplementary References 1. Sampaio, J., Cros, V., Rohart, S., Thiaville, A. & Fert, A. Nucleation, stability and currentinduced motion of isolated magnetic skyrmions in nanostructures. Nature Nanotechnology 8, (2013). 2. Vansteenkiste, A. & de Wiele, B. V. Mumax: A new high-performance micromagnetic simulation tool. Journal of Magnetism and Magnetic Materials 323, 2585 (2011). 3. Donahue, M. J. & Porter, D. G. OOMMF User's Guide, Version 1.0, Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (1999).