Exchange-mediated, nonlinear, out-of-plane magnetic field dependence of the ferromagnetic vortex gyrotropic mode frequency driven by core deformation

We have performed micromagnetic simulations of low-amplitude gyrotropic dynamics of magnetic vortices in the presence of spatially uniform out-of-plane magnetic fields. For disks having small lateral dimensions, we observe a frequency drop-off when approaching the disk's out-of-plane saturation field. This nonlinear frequency response is shown to be associated with a vortex core deformation driven by nonuniform demagnetizing fields that act on the shifted core. The deformation results in an increase in the average out-of-plane magnetization of the displaced vortex state (contrasting the effect of gyrofield-driven deformation at low field), which causes the exchange contribution to the vortex stiffness to switch from positive to negative. This generates an enhanced reduction of the core stiffness at high field, leading to a nonlinear field dependence of the gyrotropic mode frequency.

Figure 1

(a) Simulated gyrotropic frequency as a function of uniform out-of-plane field amplitude for disks of thickness $L=30\mathrm{nm}$ and a range of diameters. $f_{\mathrm{G}}$ has been normalized by the simulated frequency in a zero out-of-plane field while the field amplitude has been normalized by the disk saturation field. (b) Thickness-averaged out-of-plane magnetization for a slice through the disk center for a 384-nm-diameter disk in increasing field amplitudes. (c) Comparison of the simulated values of $f_{\mathrm{G}}$ found using mumax3 (for two different cell sizes) with the frequencies found using oommf and finmag (disk diameter 192 nm).

Figure 2

(a) Zoomed-in plot of the out-of-plane magnetization profile close to the vortex core [e.g., see (c)] for 192- and 768-nm disk diameters in three normalized field amplitudes. The solid red lines show the magnetization profile while the core is undergoing gyrotropic motion. The black dashed lines show the magnetization profile when the core has been shifted to a new equilibrium position by a static in-plane magnetic field. The horizontal black lines reference the minimum of the magnetostatic halo when the vortex core is stationary and at the disk center. (b) Schematic representation of how the dynamic one-dimensional profiles shown in (a) have been taken. Note, though, that the displacement of the core in the schematic is strongly exaggerated (actually only $\sim 3\mathrm{nm})$. (c) Full displaced core profiles in the 192-nm disk at zero out-of-plane field. The shaded box shows the approximate region used to plot panel (i). The core radius is $\sim 10$ nm at half maximum (i.e., at $m_z=0.5)$.

Figure 3

The gyrofield close to the vortex core as calculated using Eq. (2) for disk diameters of 192 and 768 nm and three different normalized out-of-plane field values.

Figure 4

(a,b) Slices through the created vortexlike magnetization configuration for the (a) 192- and (b) 768-nm disk diameter. The solid red lines show the profile for a core shifted 10 nm in the positive $x$ direction. The black dashed lines show the profile for a centered vortex core. (c,d) Corresponding demagnetizing field profiles for a displaced and centered vortex core in the (c) 192- and (d) 768-nm disk. (e,f) Demagnetizing field close to the vortex core for the (e) 192- and (f) 768-nm disk. The solid horizontal lines reference the magnetostatic field 25 nm on either side of the core maximum for the case of a displaced vortex core. The dashed horizontal line marks the magnetostatic field 25 nm on either side of the core maximum for the case of a centered vortex core. Note that the profiles in (e,f) have been translated so that they are centered at the same point.

Figure 5

Plots of the evolution of the spatially averaged out-of-plane magnetization as a function of time following the onset of gyrotropic motion for a 192-nm disk (a) and a 768-nm disk (b) for three out-of-plane field values. The gyrotropic motion was driven by an in-plane sinusoidal field with frequency equal to the $f_{\mathrm{G}}$ values given in Fig. 1, which were determined via a sinc pulse excitation as detailed in Sec. 2.

Figure 6

(a) Gyroconstant field dependence (normalized by its value in a zero out-of-plane field). (b) Comparison of the static magnetization profiles of a centered vortex core for the 192- and 768-nm disks in various out-of-plane field amplitudes. The one-dimensional slice has been taken across the disk center for the thickness-averaged magnetization.

Figure 7

(a) Maximum core displacement as a function of sinusoidal driving field frequency for a range of disk geometries in the presence of an external out-of-plane field which is close in magnitude to $H_{\mathrm{S}}$. Vertical dashed lines show the gyrotropic frequency extracted from sinc-pulse-driven resonant dynamics in the same out-of-plane field. (b) Total stiffness coefficient normalized by its value in a zero out-of-plane field as a function of field amplitude for the 192- and 768-nm disks. (c) The contribution to $\kappa$ arising from the magnetostatic energy associated with the in-plane magnetization and (d) the exchange energy. Both of these contributions have been normalized by the total stiffness coefficient in the zero out-of-plane field. (e) The change in exchange energy (relative to that for a centered vortex) as a function of core displacement for the 192-nm disk in $H/H_{\mathrm{S}}=0$ and 0.85.

Figure 8

(a) The gyrotropic frequency associated with the exchange contribution to the stiffness coefficient $\left(f_{\mathrm{G},\mathrm{E}}\right)$ for a range of disk diameters. (b) The simulated gyrotropic frequency $\left(f_{\mathrm{G}}\right)$ minus the frequency associated with the exchange contribution to the stiffness coefficient $\left(f_{\mathrm{G},\mathrm{E}}\right)$ for a range of disk diameters (the frequencies have been normalized by the simulated gyrotropic frequency in zero field).