Complete mapping of the spin-wave spectrum in a vortex-state nanodisk

We report a study on the complete spin-wave spectrum inside a vortex-state nanodisk. Transformation of this spectrum is continuously monitored as the nanodisk becomes gradually magnetized by a perpendicular magnetic field and encounters a second-order phase transition to the uniformly magnetized state. This reveals the bijective relationship that exists between the eigenmodes in the vortex state and the ones in the saturated state. It is found that the gyrotropic mode can be continuously viewed as a uniform phase precession, which uniquely softens (its frequency vanishes) at the saturation field to transform above into the Kittel mode. By contrast, the other spin-wave modes remain finite as a function of the applied field, while their character is altered by level anticrossing.

Figure 1

(a) Micromagnetic simulations performed on a vortex-state FeV nanodisk with radius $R=100$ nm and thickness $t=10$ nm, meshed adaptively around its center. (b) Variation of its reduced longitudinal (black) and azimuthal (green) magnetization along an upward perpendicular magnetization cycle (the dashed line is the behavior of the order parameter for a critical exponent, $0.5$). The schematics above show the equilibrium configurations simulated for three increasing values of the perpendicular field $H_z$.

Figure 2

Simulated spin-wave spectra of the nanodisk as a function of the perpendicular field $H_z$. Modes are labeled $\left({\ell }_m\right)$, respectively, the azimuthal and radial indices. The two panels differentiate SW depending on their sense of gyration: (a) $\ell >0$ and (b) $\ell \le 0$. Snapshot images of the eigenvectors in the vortex (${\mu }_0{H}_z=0$ T) and saturated states (2 T) are shown on the left and right sides, respectively (bivariate color code, where amplitude/phase = luminance/hue). (c) Zoomed-in view of the softening of the gyrotropic mode at $H_s$ (the dashed line shows the dependence for a critical exponent, $0.3$).

Figure 3

(a) Dynamical magnetization vector (short arrow) in the local frame of the magnetization texture (long arrow) for the mode with a uniform phase (phase $\phi$ coded with the hue color wheel). (b) Top view of the $(-0_0)$ eigenvector at $H_z=0.0$, 0.6, and 2.0 T using either the amplitude/phase representation or (c) the Cartesian directions (red and blue indicate regions of opposite polarity). (d) The radial profile of the normalized rms amplitude (solid line) compared with the analytical prediction (dashed line); the local value of the static magnetization $M_z$ is shown for comparison using a dotted line.

Figure 4

Zoomed-in view of the level anticrossing between SW modes of different character in the vortex state.