Thermal vortex dynamics in thin circular ferromagnetic nanodisks

The dynamics of gyrotropic vortex motion in a thin circular nanodisk of soft ferromagnetic material is considered. The demagnetization field is calculated using two-dimensional Green's functions for the thin-film problem and fast Fourier transforms. At zero temperature, the dynamics of the Landau-Lifshitz-Gilbert equation is simulated using fourth-order Runge-Kutta integration. Pure vortex initial conditions at a desired position are obtained with a Lagrange multipliers constraint. These methods give accurate estimates of the vortex restoring force constant $k_F$ and gyrotropic frequency, showing that the vortex core motion is described by the Thiele equation to very high precision. At finite temperature, the second-order Heun algorithm is applied to the Langevin dynamical equation with thermal noise and damping. A spontaneous gyrotropic motion takes place without the application of an external magnetic field, driven only by thermal fluctuations. The statistics of the vortex radial position and rotational velocity are described with Boltzmann distributions determined by $k_F$ and by a vortex gyrotropic mass $m_G=G^2/k_F$, respectively, where $G$ is the vortex gyrovector.

Figure 1

Vortex motion with damping, at zero temperature. This is clockwise motion for a vortex with positive ($+\widehat{z}$) gyrovector, starting from the dot on the $x$ axis. The vortex performs gyrotropic motion of decreasing radius and increasing frequency as it moves towards the disk center, $\mathbf{r}=(0,0)$.

Figure 2

For the vortex motion in Fig. 1, the phase relationship between perpendicular components of position and in-plane magnetization.

Figure 3

Typical motions of the vortex core coordinate $x_c\left(\tau \right)$ at zero temperature, for circular disks of thickness $L=10$ nm with different radii (shifted vertically from $x_c=0$ for clarity). The damping $\alpha =0.02$ was turned off at time $\tau =1000$. Periods were calculated from the energy-conserving motion after $\tau >1000$. The motion of $y_c\left(\tau \right)$ is similar but shifted a quarter of a period.

Figure 4

Zero-temperature vortex gyrotropic frequency $f_G$ for various disk radii $R$ versus aspect ratio $L/R$. (For Permalloy, $\frac{{\mu }_0}{4\pi }\gamma M_s\approx 15.1$ GHz.) The computation cell size is $a=2.0$ nm. The vortex state is unstable below a minimum disk thickness, as expected due to the diminished restoring forces from the reduced edge area. The dashed line shows the result [Eq. (59)] from using the linear approximation in Eq. (55) for $k_F$.

Figure 5

Vortex force constant $k_F$ scaled by disk thickness versus disk aspect ratio. These were obtained by assuming a parabolic potential for vortex motion within the disk. The dashed line indicates that the slope of this relationship is close to $1/4$ for some range of parameters, Eq. (55), for disks of adequate thickness. Cell edge is $a=2.0$ nm.

Figure 6

The dimensionless gyrotropic frequencies (found from dynamics) versus force constant scaled by disk thickness. The dashed line of unit slope is Eq. (50). This verifies the dynamics of the Thiele equation, and shows the complete consistency between the static energetics and the dynamics. Cell edge is $a=2.0$ nm.

Figure 7

Vortex motion in Py at room temperature (300 K), starting from an initial displacement of 16 nm from the disk center. The $y$ component of average magnetization in the disk is correlated to the $x$ component of the vortex position.

Figure 8

Spontaneous gyrotropic vortex motion in Py due to thermal fluctuations at 300 K, starting from a vortex at the center of the disk.

Figure 9

Thermal power spectrum of the in-plane magnetization fluctuations due to spontaneous gyrotropic vortex motion in Py at 300 K, for the motion in Fig. 8.

Figure 10

Spontaneous gyrotropic vortex motion, due to thermal fluctuations, in a 20-nm-thick Py disk at 300 K, with the vortex starting at the center of the disk. The natural periodic motion executes 32 revolutions in this time sequence, with period ${\tau }_G\approx 1870$.

Figure 11

For the spontaneous gyrotropic vortex motion in Fig. 10 ($R=120$ nm, $L=20$ nm Py disk at 300 K), details of the motion at earlier times. The vortex started at the center of the disk. There is a high-frequency spin-wave oscillation apparent in the magnetization dynamics, excited together with the gyrotropic motion.

Figure 12

The thermally averaged power spectrum in one component of the magnetization (squared FFT) for the vortex motion in Fig. 10. The low-frequency gyrotropic mode dominates strongly over a much weaker doublet at high frequency. For Permalloy parameters ($f=1336\mathrm{GHz}\times \nu$), the gyrotropic frequency is $f_G=0.71$ GHz while the components of the doublet lie at $f_1=9.3$ GHz and $f_2=11.4$ GHz.

Figure 13

Typical spontaneous fluctuations of the vortex core $x$ coordinate for 30-nm-radius Py disks with various thicknesses, at 300 K. The vortex was initiated at the disk center. Curves are shifted vertically from $x_c=0$ for clarity.

Figure 14

Typical spontaneous fluctuations of the vortex core $x$ coordinate for 120-nm-radius Py disks with various thicknesses, at 300 K. The vortex was initiated at the disk center. Curves are shifted vertically from $x_c=0$ for clarity.

Figure 15

Average squared displacement of the vortex core from the disk center, versus reciprocal force constant. The points come from simulations out to time $\tau =2.5\times {10}^5$; the solid lines are the predictions from the equipartition theorem, Eq. (72), using the parameters for Py.

Figure 16

Probability distributions in Py disks of radius 30 nm at temperatures 300 and 150 K, for the radial position $r$ of the vortex, measured from the disk center, in units of the cell size, $a=2$ nm. Solid curves are the theoretical expression (74) based on a Boltzmann distribution using the static force constants; points are from simulations out to time $\tau =2.5\times {10}^5$.

Figure 17

Probability distributions for vortex radial position in Py disks of radius 120 nm, as explained in Fig. 16.