Correlation between spin structure oscillations and domain wall velocities

Magnetic sensing and logic devices based on the motion of magnetic domain walls rely on the precise and deterministic control of the position and the velocity of individual magnetic domain walls in curved nanowires. Varying domain wall velocities have been predicted to result from intrinsic effects such as oscillating domain wall spin structure transformations and extrinsic pinning due to imperfections. Here we use direct dynamic imaging of the nanoscale spin structure that allows us for the first time to directly check these predictions. We find a new regime of oscillating domain wall motion even below the Walker breakdown correlated with periodic spin structure changes. We show that the extrinsic pinning from imperfections in the nanowire only affects slow domain walls and we identify the magnetostatic energy, which scales with the domain wall velocity, as the energy reservoir for the domain wall to overcome the local pinning potential landscape.

For f > f crit , the domain wall cannot track the rotating magnetic field and θ grows continuously. Radial vortex core displacement. The calculated vortex core displacement δr (black line) and the radial vortex core position from micromagnetic simulations (red circles) are plotted as a function of time. By fitting the radial potential stiffness κ r we find very good quantitative agreement between the analytical expression for δr and the vortex core position in the micromagentic simulation.

Supplementary Note 1
Domain wall motion can be analytically described by a one-dimensional (1D) model for the collective coordinates domain wall position q and polar angle ψ [34]. Here we derive the 1D model for domain wall motion in circular nanowires driven by a rotating magnetic field and we numerically find that in contrast to our observation the steady-state domain wall motion below the Walker breakdown occurs at constant velocity and without precession (constant polar angle ψ, corresponding to no changes in the domain wall spin structure).
The driving magnetic field rotates at constant angular velocity and constant amplitude (S1) To reflect the circular geometry, we assume periodic boundary conditions for the domain where R is the radius of the circular nanowire. The driving field is the tangential component where γ 0 is the gyromagnetic ratio, α is the Gilbert damping parameter, M S is the saturation magnetization, K d is the transverse anisotropy and ∆ 0 is the domain wall width in equilibrium.
To numerically solve the model, we assume realistic material parameters and dimensions We consider only the case where B t is smaller than the Walker field µ 0 where the domain wall velocity is proportional to the tangential field amplitude [2] The maximum domain wall velocity at B t = B (θ = π/2) defines a critical field rotation frequency for the domain wall to follow the field For f < f crit , the domain wall lags behind the rotating field (θ > 0) and the tangential field component grows until the domain wall velocity matches the field rotation speed (v dw = 2πf R). The steady-state motion in this case is at constant velocity without precession of the polar angle ψ, as shown in Supplementary Figure S1. If the field rotation frequency exceeds the critical field rotation frequency f > f crit , the domain wall cannot track the the rotating magnetic field.

Supplementary Note 2
Here, we calculate analytically the radial vortex core displacement δr based on the velocities obtained from micromagnetic simulations. We assume a parabolic potential along the radial direction, due to the shape anisotropy: where w is the ring width, V 0 = κ r w 2 /8 is the potential depth and κ r is the radial potential stiffness. Assuming that the restoring force F rf = −ê r ∂V r /∂δr from the parabolic potential counteracts the sum of the gyroforce G × v and F t , we can calculate the radial vortex core displacement δr because the gyroforce and F t are always pointing parallel. The potential stiffness κ r is found by fitting δr to the vortex core displacement from the micromagnetic simulations, the best fit is found for κ r = 8.9 ± 0.1 × 10 −4 kg s −2 , which agrees quantitatively with previous measurements [39]. The resulting time evolution of the vortex core displacement and the radial vortex core position from the micromagnetic simulations are shown in Supplementary   Figure S2.