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Dynamically Stabilized Magnetic Skyrmions


Dynamically Stabilized Magnetic Skyrmions

Y Zhou et al. Nat Commun.


Magnetic skyrmions are topologically non-trivial spin textures that manifest themselves as quasiparticles in ferromagnetic thin films or noncentrosymmetric bulk materials. So far attention has focused on skyrmions stabilized either by the Dzyaloshinskii-Moriya interaction (DMI) or by dipolar interaction, where in the latter case the excitations are known as bubble skyrmions. Here we demonstrate the existence of a dynamically stabilized skyrmion, which exists even when dipolar interactions and DMI are absent. We establish how such dynamic skyrmions can be nucleated, sustained and manipulated in an effectively lossless medium under a nanocontact. As quasiparticles, they can be transported between two nanocontacts in a nanowire, even in complete absence of DMI. Conversely, in the presence of DMI, we observe that the dynamical skyrmion experiences strong breathing. All of this points towards a wide range of skyrmion manipulation, which can be studied in a much wider class of materials than considered so far.


Figure 1
Figure 1. Different skyrmion stabilization mechanisms.
Panels (ae) show skyrmions with skyrmion number formula image. (a) A bubble skyrmion stabilized by dipolar interactions which may exist as a left- or right-handed version. Its size typically exceeds that of skyrmions stabilized by DMI. (b,c) DMI stabilized skyrmions: (b) A chiral skyrmion as favoured in B20-type materials such as MnSi. (c) A hedgehog skyrmion as favoured by interfacial DMI. Note that in both cases the domain-wall chirality when passing through the skyrmion centre is fixed by the DMI and not degenerate as in a. (d) Dynamically stabilized magnetic skyrmion which requires neither dipolar interactions nor DMI and which exhibits precession around the (vertical) easy-axis anisotropy. During precession, the skyrmion number is strictly conserved and the skyrmion sweeps across all possible chiral and hedgehog configurations. In the presence of either dipolar interactions, DMI, or an Oersted (Oe) field, the skyrmion diameter varies periodically in time (breathing). (e) For vanishing dipolar interactions (DDI), DMI and Oe fields, and in absence of damping, the skyrmion precesses uniformly and breathing disappears. Note that panel e only shows half a period of the precessional motion to emphasize that both chiral and hedgehog configurations are being covered in the DS states.
Figure 2
Figure 2. Nucleation and tuning of a DS.
Micromagnetic simulation of a NC-spin torque oscillator with radius 40 nm and at J=6 × 108 A cm−2 showing the nucleation of a DS without DMI and dipolar interactions from an initial ferromagnetic (FM) state and the subsequent tuning of the DS by spin transfer torque, α, DMI, and dipolar interaction. (a) Top view of the spin structure at different times of the simulation; the black circle indicates the NC; The blue and red colour of the colour bar represent the downward (mz=−1) and upward (mz=+1) orientations of the magnetizations, respectively. (b) top view of the topological density at the same times as in a; (c) Time trace of the out-of-plane mz component averaged over the simulation area and time trace of the skyrmion number; dashed vertical lines correspond to the snapshots above. For 0<t<1 ns, the system undergoes an initial relaxation from FM state. For 1 ns<t<3 ns, a DS is nucleated under finite spin transfer torque assisted by the corresponding Oe field without DMI. For 3 ns<t<5 ns, the Oe field is artificially turned off and the breathing disappears. For 5 ns<t<7 ns, both current and damping are turned off. For 7 ns<t<9 ns, a moderate DMI is added and the breathing resumes. For 9 ns<t<11 ns, we only turn on dipolar interactions (both Oe field and DMI are off) and minor breathing is again observed. For 11 ns<t<13 ns, the DS will dissipate into a uniform ferromagnetic state if only damping is turned on without spin torque.
Figure 3
Figure 3. Linear scaling law of a DS.
(a) DS radius versus current density for two NC sizes and three different strengths of the perpendicular magnetic anisotropy; (b) DS frequency as function of current density; (c) Linear scaling law between the DS frequency and inverse radius, for currents at threshold and below, in excellent agreement with the analytical prediction. For panels (ac): red square corresponds to NC radius of 40 nm and Ku=0.5 MJ m−3; green square corresponds to NC radius of 40 nm and Ku=0.7 MJ m−3; blue square corresponds to NC radius of 40 nm and Ku=1 MJ m−3; red dot corresponds to NC radius of 30 nm and Ku=0.5 MJ m−3; green dot corresponds to NC radius of 30 nm and Ku=0.7 MJ m−3; blue dot corresponds to NC radius of 30 nm and Ku=1 MJ m−3. (d) The DMI-induced breathing amplitude ν=(RmaxRmin)/(Rmax+Rmin) as a function of D0, in which the simulation results are again in excellent agreement with the theory. The filled square represents the micromagnetic simulation results and the solid line represents the theory. The inset shows coordinates used in the theory: χ is the (real space) polar angle, R is the skyrmion radius, and φ the azimuthal angle of the magnetization m relative to the radial direction. For main panels ac, the Gilbert damping coefficient α=0.3, with no DMI but with Oe field.
Figure 4
Figure 4. Current-induced motion of a DS in a nanowire without DMI.
A DS can be transported coherently between NCs along a nanowire without DMI, illustrating its quasiparticle nature. Initially, the skyrmion is nucleated and pinned at the left NC with a diameter of 25 nm. After switching off the spin transfer torque, the DS is released, continues to precess, and is transported coherently by an in-plane current to a second NC at a distance of 150 nm where it is trapped again and restored to its original size. Here the in-plane current density is 1.8 × 108 A cm−2 and the damping constant is set to α=0.05 with sample dimensions of 250 nm × 75 nm × 1 nm. Here β=α is chosen but very similar results are obtained for βα similar to the case of DMI stabilized skyrmions. The white circles indicate the NCs. The red and blue colour indicate up and down magnetization, respectively.
Figure 5
Figure 5. Nucleation and field toggling of a DS in presence of DMI.
Panels (ac) show, respectively: the top view of the spin structure at selected simulation times where the white circle indicates the NC of 15 nm radius; the topological density at the same times where the colour-scale is normalized; the time trace of the out-of-plane magnetization component mz averaged over the simulation area; and time trace of the skyrmion number (green). Panels (ac) show droplet nucleation at J=8.3 × 108 A cm−2 and μ0Ha=0.3 T, which remains stable for several periods until about t=0.2 ns when it becomes increasingly susceptible to anti-skyrmion perturbation formula image. These perturbations eventually (t=0.3 ns) give way to the formation of a DS with formula image. When the applied field is turned off at t=0.7 ns, the DS rapidly dissipatively shrinks into a static DMI stabilized skyrmion. If the field is again turned on, the skyrmion can be transformed into a DS in a reversible manner. Finally, if both the field and the current are turned off, the static skyrmion contracts to its equilibrium size given by the material parameters.
Figure 6
Figure 6. Nucleation and current toggling of a DS in presence of DMI.
Panels (ac) show a DS nucleation as a function of current and fixed field, where the white circle indicates the NC of 15 nm radius. For 0 ns<t<1 ns, a magnetic droplet is nucleated when J=7.2 × 108 A cm−2. For 1 ns<t<2 ns, a DS forms when current density is increased to 8.3 × 108 A cm−2. For 2 ns<t<3 ns, a DS remains stable even when the current density is decreased to 6.7 × 108 A cm−2, well below the threshold nucleation current density. For 3 ns<t<4 ns, the DS collapses into a uniform ferromagnetic state when the current density is further decreased to 6.3 × 108 A cm−2. The current density is varied as follows: J=7.2 × 108 A cm−2 for 0<t<1 ns, 8.3 × 108 A cm−2 for 1 ns<t<2 ns, 6.7 × 108 A cm−2 for 2 ns<t<3 ns, and 6.3 × 108 A cm−2 for 3 ns<t<4 ns. Panel (d) schematically shows the current pulses applied during the simulation.
Figure 7
Figure 7. Frequency, size and stability of DS.
The frequency of the DS is shown as solid lines and filled circles for: (a) μ0Ha=0.3 T while the current density is varied; (b) J=8.3 × 108 A cm−2 while the applied field is varied. The corresponding droplet frequency (DMI=0) is shown as dotted lines and hollow circles. The Zeeman and FMR frequencies are shown as dashed lines. The radius of the droplet (hollow circles) and the time averaged radius of the DS (filled circles) measured in units of NC radius where the error bars indicate the total range of radii values as a function of: (c) current density; (d) applied perpendicular magnetic field strength. The DS frequency decreases rapidly with increasing radius (increasing current) in a. As the field increases in b, the DS becomes stiffer, reducing the breathing and making the dynamics resemble that of the droplet. (e) Nucleation of a droplet (hollow circle), DS (filled rainbow circle) and static skyrmion (green filled circle) at different fields and currents. (f) Sustainability of the DS over a very wide range of current and field. Note that the field axis is non-linear to reach the final collapse of the DS at very high fields. The DS was nucleated using the conditions highlighted by the pink square. (g) The schematic representation of different states. A droplet is represented by hollow circle; a DS is represented by filled rainbow circle; a static skyrmion is represented by green filled circle. The red and blue squares represent the upward and downward orientations of all the spins, respectively.
Figure 8
Figure 8. Droplet and DS injection locking.
Injection locking diagram of (a) a droplet and (b) a DS as a function of the frequency of the injected microwave current (white dashed lines). The injected signal is a pure tone with amplitude 0.2 times the bias current density. The full current-induced Oe field is also included in these simulations. While the droplet is not visibly injection locked for the considered parameters, the DS exhibits a very large phase locking bandwidth of ∼3 GHz. Outside of the locked region the DS exhibits strong intermodulation products further demonstrating its strong interaction with the injected current.
Figure 9
Figure 9. Droplet nucleation and skyrmion read-out.
(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-dependent frequency of the precessing magnetization. A static skyrmion is initialized as the micromagnetic ground state at t1. By applying a current density J=7.2 × 108 A cm−2, a droplet is nucleated (t2). The droplet acts as an attractive source for the static skyrmion until it merges at t3. The droplet absorbs the skyrmion topology as evidenced by the reduction of the in-plane magnetisation and the frequency drop as the DS forms at (t4).

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