Rotating edge-field driven processing of chiral spin textures in racetrack devices

Topologically distinct magnetic structures like skyrmions, domain walls, and the uniformly magnetized state have multiple applications in logic devices, sensors, and as bits of information. One of the most promising concepts for applying these bits is the racetrack architecture controlled by electric currents or magnetic driving fields. In state-of-the-art racetracks, these fields or currents are applied to the whole circuit. Here, we employ micromagnetic and atomistic simulations to establish a concept for racetrack memories free of global driving forces. Surprisingly, we realize that mixed sequences of topologically distinct objects can be created and propagated over far distances exclusively by local rotation of magnetization at the sample boundaries. We reveal the dependence between chirality of the rotation and the direction of propagation and define the phase space where the proposed procedure can be realized. The advantages of this approach are the exclusion of high current and field densities as well as its compatibility with an energy-efficient three-dimensional design.

For the systems, discussed in the main text, we have used a discretization into cubic simulation cells. However, ultrathin films such as the Pd/Fe bilayer on Ir(111) typically grow in hexagonal close-packed (hcp) stacking. Here, we strictly use the hcp structure of magnetic adatoms and show the applicability of the presented writing and deleting mechanism also to these cases. Additionally, the skyrmions are stabilized by a combination of the DMI and the geometric confinement entering as fixed boundary conditions. This represents another example of nanometer sized skyrmions in the absence of global magnetic fields.
For the atomistic simulations, classical Heisenberg magnetic moments S i = (S i x , S i y , S i z ) of unit length µ i /µ s are placed at each lattice point. The energy of the system is given by the where J > 0 is the ferromagnetic exchange coupling between nearest neighbors, D ij is the DMI vector, B z is a global magnetic field, K z is the perpendicular magnetic anisotropy and B i (t) is a local space-and time-dependent magnetic field. The material parameters were set to K z = 0.05 J, D = 0.07 J and the magnetic moments at the edges are fixed to S =ê z to stabilize skyrmions in the absence of an additional external field. The spins at x = 0 were rotated parametrically. The time scale in these calculations can be expressed by dt = τ µ s /γJ with the reduced time step τ and the gyromagnetic ratio γ. For µ s = 2µ B , J = 5.72 meV and τ = 0.01 a simulation step corresponds to the time span of ≈ 10 −14 s.
The frequency of rotation at the edge of the stripe was varied between 1 and 10 GHz. The propagation velocities were similar to those found in micromagnetic simulations.
The dynamics of the free spins is determined by the LLG equatioṅ with the effective magnetic field Fig. S1 shows a sequence of snapshots of the creation and deletion of magnetic skyrmions using atomistic LLG simulations on a triangular lattice with 8 × 100 spins. The excited 2 line of magnetic moments is visualized by the gray screw, also indicating the rotational sense. As shown in the micromagnetic simulations presented in the main text, skyrmions can be injected into the sample. In Fig. S2 the spins along the screw are rotated in opposite direction. Particularly interesting is that the rotation against the DMI chirality destroys skyrmions only on one side of the excited line as can be seen in time steps t 3 to t 6 of Fig. S2.
In conclusion, this shows that skyrmions stabilized by DMI and fixed boundary conditions can be created and annihilated similar to the open systems discussed in the main text.
Furthermore, the hexagonal lattice has no fundamental impact on the proposed control mechanism.  The creation processes of either domain walls or skyrmions discussed in the main paper have been modeled by a simplified rotation scheme, namely a uniform rotation of a distinct number of involved edge magnetic moments due to a rotating magnetic field. While for domain walls this intuitively is the optimal scheme, a rotation respecting the intrinsic circular form of skyrmions is a more adequate skyrmion creation process. As we will show in the following, such a tailored rotation allows for higher creation rates, which in the context of information technologies is a desirable goal. We Here σ is a measure of the width of the Gauss distribution, y refers to the position of the magnetic moment and y 0 to the center of the skyrmion, coinciding in the following with the center of the edge. Fig. S3(a) illustrates the scheme of the Gauss rotation. The moments (black arrows) rotate around the spatially dependent rotation axis (red arrow) and are depicted in their initial state (dashed line) and after half a rotation (continuous line). The angle ϕ between m(t = 0) and m(t = T 2 ), T being the period of the rotation, depends on the Gauss function in equation 5, namely cos(ϕ(y)) = −2 exp − y − y 0 σ 2 + 1.
Using this rotation, we find the creation diagram of the Gauss rotation, see Fig. S3(c). The Gauss rotation yields the possibility of skyrmion creation at significantly higher frequencies up to 20.8 GHz, i.e., a creation of a skyrmion within 50 ps, for an optimal Gauss width σ = 20. This underlines the increased efficiency of the rotation, if it adequately forms the skyrmion shape. Starting from a field-polarized state, two writing operations and one deletion with reversed rotational sense of the external field are performed. System size: 60×30×1c 3 , c = 0.233 nm, with material parameters for Pd/Fe/Ir(111); ν = 2 GHz, B = 2.5 T