Vortex circulation control in large arrays of asymmetric magnetic rings

Arrays of asymmetric Permalloy nanorings with a decentered inner hole are studied by broadband inductive spectroscopy in the gigahertz range. The spectra display a set of characteristic spin-wave eigenmodes. We find that these eigenmodes probe the absolute vortex circulation direction of a ring. The eigenmode spectra and the signal strengths provide evidence that the asymmetric rings in the large array exhibit an ordered vortex circulation state. The degree of ordering is better than 97%. Micromagnetic simulations reveal that the circulation control is achieved by suppression of domain-wall propagation as the switching mechanism. The deterministic vortex circulation is an important prerequisite for the application of rings in magnetoelectronics.

Figure 1

(Color online) Absorption spectra for (a) the symmetric rings and (c) the asymmetric rings (black: high absorption). $H$ was applied along the CPW starting at ${\mu }_{0}H=-89\mathrm{mT}$. Panels (b) and (d) display the extracted resonance frequencies relevant to determine the vortex circulation direction. Vertical lines indicate the switching fields for the transitions from onion to vortex and to reverse onion state. Insets show scanning electron micrographs of a ring from each array and a simulated spin-wave pattern at $-89\mathrm{mT}$ (left, compare color bar in Fig. 2 for amplitude).

Figure 2

(Color online) (a) Experimental data and (b) simulated absorption spectrum for an asymmetric ring at $89\mathrm{mT}$ in the onion state (iv). Colored panels (i)–(v) show the amplitude distribution of spin-wave excitations (compare to color bar for amplitude). Graphs (iv) and (vi) depict the static $\mathit{M}$ (arrows). (c) Calculated absorption spectra at $-15\mathrm{mT}$ (solid, black) and $-20\mathrm{mT}$ (dash dotted, red) simulating the minor-loop configuration of Fig. 4a: arrows indicate the opposite magnetic-field dispersion of modes $f_{w2}^a$ and $f_{w1}^p$ consistent with the measured eigenmodes at $H<0$.

Figure 3

(a) Magnetic-field dispersions of modes $A_{w1}$ and $A_{w2}$ as expected for an array of asymmetric rings obeying a statistical distribution of clockwise and counterclockwise vortex circulation directions. (b) Sketch of the two possible vortex circulation directions. Note the different $(\mathit{M},\mathit{H})$ orientations for the side arms. The cones illustrate local spin excitations.

Figure 4

(Color online) Minor loops in the vortex state: (a) before measuring the eigenmodes (open circles, green: $A_{w1}$; open squares, blue: $A_{w2}$), fields ${\mu }_0{H}_{\mathrm{his}}$ of $-89$ and $+25\mathrm{mT}$ were applied perpendicular to the symmetry axis. The array appeared to be in an ordered vortex configuration. (b) Prior to acquisition of data (open circles), fields ${\mu }_0{H}_{\mathrm{his}}$ of $+89$ and $-15\mathrm{mT}$ were applied parallel to the symmetry axis. Four modes were found, indicating a statistical distribution of vortex circulation directions. $H$ was applied along the CPW. Light gray symbols: dispersion coming from ${\mu }_{0}H=-89\mathrm{mT}$ (gray arrows).

Figure 5

(a) Magnetization $\mathit{M}$ (small arrows) for $\mathit{H}$ (large arrows) applied perpendicular to the symmetry axis of a ring. The buckling (middle panel) is found in the micromagnetic simulation for $25\mathrm{nm}<t⩽33\mathrm{nm}$. (b) $\mathit{M}$ for $\mathit{H}$ applied along the symmetry axis. Both ring arms are energetically equivalent and the vortex walls can propagate to left or right with equal probability (curved arrows). The vortex circulation direction is not deterministic.