Effects of disorder on magnetic vortex gyration

A vortex gyrating in a magnetic disk has two regimes of motion in the presence of disorder. At large gyration amplitudes, the vortex core moves quasi-freely through the disorder potential. As the amplitude decreases, the core can become pinned at a particular point in the potential and precess with a significantly increased frequency. In the pinned regime, the amplitude of the gyration decreases more rapidly than it does at larger precession amplitudes in the quasi-free regime. In part, this decreased decay time is due to an increase in the effective damping constant and in part due to geometric distortion of the vortex. A simple model with a single pinning potential illustrates these two contributions.

Figure 1

(a) A typical vortex structure in a Ni${}_{80}$Fe${}_{20}$ disk with 400-nm diameter and 10-nm thickness calculated as described in Sec. 2b. The color indicates the in-plane angle of the magnetization, and the arrows indicate the approximate magnetization direction. (b) Cross section of the $z$-component magnetization along the center of the vortex core in (a).

Figure 2

(a) A typical disorder image with 10-nm spatial correlation length and (b) contour maps of the gyration frequency as a function of position. (a) Part of the saturation magnetization within a Ni${}_{80}$Fe${}_{20}$ disk with a 400-nm diameter and a 10-nm thickness. White indicates maxima in the magnetization and black minima. These results are based on a model of the disorder with $\mathcal{D}=0.05$ and a fixed exchange stiffness constant $A$. (b) The gyration frequency according to the color scale on the right. To move the vortex positions, in-plane static magnetic fields from $-10$ mT to $+10$ mT along the $x$ and $y$ directions were applied. The gyration frequency is obtained by applying a field pulse of 0.1 mT for 200 ps along the $y$ direction. The final location of the vortex core determines the real space position used in the figure.

Figure 3

(a) Time evolution of gyration radius $r$ for the disordered sample in Fig. 2 at points A, B, and C for $\mathcal{D}=0.05$ with fixed $A$. To excite a vortex state, a Gaussian-type field pulse of 20 mT with 1 ns of the full width at half maximum along the $y$ direction is applied. The black line is for the case without disorder. The small amplitude oscillations are due to the excitation procedure, which does not produce purely circular precession. (b) The gyration frequency averaged from $t=2$ ns to 12 ns as a function of field-pulse amplitude at the points A, B, and C.

Figure 4

Time evolution of (a) gyration radius $r$, (b) gyration frequency $f$ and (c) deformation factor $C$ in a single pinning potential of the radius 10 nm for the depth ratio $\delta =0,0.1,0.2,0.3$ and $0.4$ with fixed $A$. The gyration is excited by a 1 ns pulse of 20 mT.

Figure 5

Enhancement ratio of $f\tau$, $C$, and ${\alpha }_{\mathrm{eff}}$ as a function of depth ratio $\delta$ for (a) fixed $A$ and (b) fixed $l_{\mathrm{ex}}$. Subscript 0 indicates values in the free region before trapping.

Figure 6

Frequency dependence of the effective damping for fixed $A$ and various pinning potential radii $r_{\mathrm{pin}}=10$, 20, $\dots$ 70 nm and depth ratios $\delta =0,0.1,0.2,0.3,0.4$. Solid lines indicate values for constant $r_{\mathrm{pin}}$ and dashed lines constant $\delta$.