Coupled oscillations of vortex cores confined in a ferromagnetic elliptical disk

By solving the Thiele equation with simultaneous application of a radio-frequency (rf) magnetic field $\left({\mathit{h}}_{\mathrm{rf}}\right)$ and an rf spin current $\left({\mathit{j}}_{\mathrm{sp}}\right)$, the dynamic susceptibility of exchange-coupled vortices in response to ${\mathit{h}}_{\mathrm{rf}}$ and ${\mathit{j}}_{\mathrm{sp}}$ was obtained. It was found that the four eigenmodes expected for two vortices trapped in a magnetic elliptical disk were coupled to different components of ${\mathit{h}}_{\mathrm{rf}}$ and ${\mathit{j}}_{\mathrm{sp}}$. As a consequence, orthogonal ${\mathit{h}}_{\mathrm{rf}}$ and ${\mathit{j}}_{\mathrm{sp}}$ (which are simultaneously generated by the application of an rf current to an elliptical disk) can excite two modes with different eigenfrequencies. This result suggests that a fieldlike nonadiabatic torque caused by an rf spin current can be spectroscopically distinguished from the one caused by the rf magnetic field.

Figure 1

A pair of vortices confined in an elliptical disk. (a) Micromagnetic simulation of the magnetization distribution. Left: Colors represent the in-plane magnetization direction. Right: Black and white represent downward and upward out-of-plane magnetization, respectively. Gray denotes regions with no out-of-plane magnetization component (in-plane magnetization component only). (b) The coordinate system used in this study. The origin of the coordinate system is placed at the center of the elliptical disk.

Figure 2

(a) Schematic of magnetic moment distribution. A diamond-shaped magnetic domain (i.e., quadrilateral ABCD) appears at the center of the disk. Line segments AB, BC, CD, and DA correspond to ${90}^{\circ }$ domain walls. (b) The coordinate system for the magnetic field calculation and the coordinate model for the magnetostatic energy in the line segment AB. (c) Numerically calculated values of the magnetic field $H(x^{\prime },0)$ as a function of $x^{\prime }$ at $\theta =44.7^{\circ }$. (d) Integration area to calculate the magnetostatic energy from the magnetic-field distribution. The integration area shown in (d) is simplified to (e).

Figure 3

Schematic geometry of domain walls in the elliptical disk when two cores move individually along the $+y$ and $-y$ directions.

Figure 4

Micromagnetic simulation calculating the magnetostatic energy as a function of (a) $r_x$ and (b) $r_y$. Solid circles and lines indicate values estimated by the simulation results and the fitting curves, respectively.

Figure 5

Depictions of the core motions and eigenfrequencies derived by analytical calculation. First row: Schematics of the excitation source and core motions. The open and closed circles represent vortex cores with positive and negative polarity, respectively. “$\times$” marks the center of gyration. Second and third rows: Core displacement along the $x$ and $y$ axes as a function of time. The red dashed line and the blue solid line correspond to the left ($j=1$) and right ($j=2$) cores, respectively. Fourth row: Formulas of the eigenfrequencies.

Figure 6

Depictions of the core motions and eigenfrequencies derived from micromagnetic simulation. First row: Schematics of the excitation source and core motions. The open and closed circles represent vortex cores with positive and negative polarity, respectively. “$\times$” marks the center of gyration. Second and third rows: Core displacement along the $x$ and $y$ axes as a function of time. The red dashed line and the blue solid line correspond to the left ($j=1$) and right ($j=2$) cores, respectively. Fourth row: Observed eigenfrequencies.

Figure 7

(a) Gyration radius of the cores as a function of the frequency when the rf spin current and the rf field are orthogonally applied at the same time. (b) Gyration radius of the left core ($j=1$) as a function of the frequency with various $\beta$, when the polarity is $p_1=+p_2$ and $u_x+h_y$ is applied. (c) $\beta$ dependence of the gyration radius at the eigenfrequency of the rf spin current excitation (red solid line), and that of the rf field excitation (blue dashed line).