Dynamics of Coupled Vortices in a Pair of Ferromagnetic Disks

We here experimentally demonstrate that gyration modes of coupled vortices can be resonantly excited primarily by the ac current in a pair of ferromagnetic disks with variable separation. The sole gyration mode clearly splits into higher and lower frequency modes via dipolar interaction, where the main mode splitting is due to a chirality sensitive phase difference in gyrations of the coupled vortices, whereas the magnitude of the splitting is determined by their polarity configuration. These experimental results show that the coupled pair of vortices behaves similar to a diatomic molecule with bonding and antibonding states, implying a possibility for designing the magnonic band structure in a chain or an array of magnetic vortex oscillators.

Figure 1

(a) Schematic diagram of the measurement circuit and a SEM image of the sample. Two copper electrodes are attached to one of the disks in the Permalloy disk pair and the enveloped core is excited by a radio frequency current. Dynamics of the cores can be detected as dc voltages through the spin-torque diode effect utilized by a resistance oscillation associated with the core gyration. A lock-in technique is adopted at room temperature. (b) Frequency dependence of the normalized dc voltage $V_{\mathrm{dc}}/I_{\mathrm{ac}}$ measured for an isolated disk (green triangles) and for the paired disks with different polarities; black squares for $p_1{p}_2=1$, and red circles for $p_1{p}_2=-1$. The ac current amplitudes $I_{\mathrm{ac}}$ used for the measurements are $I_{\mathrm{ac}}=3.8\mathrm{mA}$ and 7.1 mA for the single disk, and $I_{\mathrm{ac}}=6.3\mathrm{mA}$ for the paired disks with the edge to edge distance $d=75\mathrm{nm}$. Solid curves in each spectrum represent the best fit to the data points using Eq. (1), thereby the dipolar coupling is evaluated.

Figure 2

(a) Simulated time evolutions of vortex cores at resonance frequencies for $(p_1{p}_2,c_1{c}_2)=(1,1)$ under ac currents (365 and 390 MHz). Blue solid lines show the motions of the current-excited core in the left disk and red lines correspond to those of the indirectly excited core in the right disk. (b) Dispersion relations of amplitude of steady gyrations. Values of $\delta y_{\max }$ show the radii of steady gyrations (50 nsec after beginning of the current flow). Black squares correspond to the parallel polarities $p_1{p}_2=1$ and red circles to antiparallel polarities $p_1{p}_2=-1$ for opposite chiralities $c_1{c}_2=-1$. Results of same chiralities $c_1{c}_2=1$ are plotted by open symbols. Simulation results for a single vortex are also presented by green triangles for comparison.

Figure 3

Schematic diagrams of four different resonance modes. Each mode is identified with the sign of polarities $p_1{p}_2$, chiralities $c_1{c}_2$, and the phase difference of the gyrations.

Figure 4

Eigenfrequencies of the coupled gyration modes as a function of the separation distances. The simulation and experimental results are shown in (a) and (b), respectively. The separation distance is given by a dimensionless value $d_n=d/R$. (c) Estimated values of coupling strength ${\eta }_x$ from both experiment and simulation with different separation distances. A $d^{-6}$ curve from the rigid vortex model is also plotted as a dotted line.