Perpendicular ferromagnetic resonance in soft cylindrical elements: Vortex and saturated states

Linear magnetic excitations in perpendicularly magnetized micrometer-sized disks have been investigated in detail both in the saturated and the vortex states using ferromagnetic resonance spectroscopy and micromagnetic simulations. Broadband ferromagnetic resonance spectra measured in disk arrays reveal a set of discrete resonance lines associated with the dipole-exchange spin-wave modes quantized by the disk edge in the saturated state and several new resonance lines (up to four) with negative slopes for the frequency-field dispersion relation $\omega \left(H_z\right)$ in the vortex state at intermediate magnetic fields. The micromagnetic simulations performed for a Py disk array (regime of negligible coupling between the disks) allow us to identify the four excitations occurring in the deformed vortex state as vortex core, disk edge, and coupled vortex core/disk edge modes, and to reproduce very satisfactorily their experimental $\omega \left(H_z\right)$ curves. In addition, the nonlinear frequency dependence of the resonance linewidth for the predominant coupled vortex core/edge mode experimentally observed is in agreement with the numerical prediction. These findings are finally confirmed by magnetic resonance force microscopy measurements conducted on an isolated NiMnSb disk. The remarkable similarity between the experimental results coming from two magnetic systems and using two different microwave probes demonstrates the robustness of the physical phenomenon.

Figure 1

Experimental in-plane magnetization curves for arrays of Py disks with various diameters $2R=1,2,$ and $5\mu \mathrm{m}$ from MOKE measurements. The interdisk distance is fixed at $x=2\mu \mathrm{m}$ and the disk thickness is $L_z=50\mathrm{nm}$. The inset corresponds to a SEM image of the array with $2R=1\mu \mathrm{m}$. The arrows indicate the sense of the magnetic field variation.

Figure 2

(a) Experimental microwave absorption spectra (derivative form) for the array with $2R=1$ $\mu$m recorded at $f=3.5$ GHz using a broadband FMR spectrometer. The insets are magnifications of the spectrum in the field range associated with the vortex and the saturated states. (b) Frequency dependencies of resonance fields for the main peak in the vortex state. (c) Frequency dependencies of the fundamental peak in the saturated state. For each case, two disk diameters are considered $2R=1$ $\mu$m and $5\mu \mathrm{m}$.

Figure 3

Computed perpendicular magnetization curves (half cycle) for the space-averaged normalized components $\langle m_z\rangle$ and $\langle m_{\phi }\rangle$. The 3D images correspond to the static magnetization configurations for four decreasing values of the magnetic field. The disk array with $2R=1$ $\mu$m is considered.

Figure 4

Computed dynamic susceptibilities in the plane $(H_z,f)$ for a downward field sweep and a positive field branch; (a) ${\chi }_{-}^{\prime \prime }$ and (b) ${\chi }_{+}^{\prime \prime }$. The high values are in black and the low ones in light gray. The resonance lines in the vortex state are labeled using a peak numbering. The array with $2R=1$$\mu$m is considered. A vertical area near the transition between the saturated and vortex states (a) is masked to avoid confusion due to stroboscopic effect.

Figure 5

(a) Experimental (Experiment) and numerical (Simulation) field derivatives of the perpendicular power absorption spectra recorded (resp. computed) at $f=9$ GHz for the disk array with $2R=1$ $\mu$m. The standing in-plane spin-wave resonances are labeled using the radial mode number $n=0,...,6$. (b) Local susceptibilities for the first four spin-wave modes. For each mode, the mode profile ${\chi }_{-}^{\prime \prime }\left(r\right)$ at the positions $\phi =0$ and $z=0$ (midplane) and the ${\chi }_{-}^{\prime \prime }(r,\phi )$ map at $z=0$ are displayed.

Figure 6

Comparison between the experimental (Experiment, full circles), analytical (Analytical, solid green line), and numerical (Simulation, crosses) evolutions of the resonance fields as function of the mode index at the frequency $f=9$ GHz (a) and versus frequency for the first two modes $n=0$ and $n=1$ (b).

Figure 7

(a) Experimental (Experiment) and numerical (Simulation) field derivatives of the perpendicular power absorption spectra recorded (resp. computed) at $f=3.5$ GHz for the disk array with $2R=1$ $\mu$m. (b) Local susceptibilities for the first four detected excitations. As in Fig. 5b, the mode profile ${\chi }_{-}^{\prime \prime }\left(r\right)$ at the positions $\phi =0$ and $z=0$ and the ${\chi }_{-}^{\prime \prime }(r,\phi )$ map at $z=0$ are displayed.

Figure 8

Comparison between the experimental (Experiment, full circles) and numerical (Simulation, crosses) evolutions of the resonance fields as function of frequency for the four resonance lines exhibited in Fig. 7.

Figure 9

Comparison between the experimental (Experiment, full symbols) and numerical (Simulations, crosses and plus) evolutions of the resonance linewidths as function of frequency for the dipole-exchange mode $n=0$ (saturated state) and for mode $1_{-}$ (vortex state). The disk array with $2R=1$ $\mu$m is considered. The dashed lines are fits of experimental data. The solid lines are guides for the eye.

Figure 10

(a) Resonance spectra obtained by magnetic resonance force microscopy (MRFM) on an individual disk of NiMnSb ($L_z=44$ nm, $2R=1\mu$m). The dotted vertical line displays the saturation field $H_s=8.1$ kOe of the NiMnSb disk, separating the vortex state from the saturated state. (b) Dispersion relation of the different spin-wave modes observed experimentally. Solid symbols have been obtained from field-sweep MRFM spectra at fixed frequency [see panel (a)]. Open symbols have been obtained from frequency-sweep MRFM spectra at fixed magnetic field. Solid lines in the saturated ($H_z>H_s$) and vortex states ($H_z<H_s$) are theoretical predictions from Eqs. (1) and (2) of Ref. 29, respectively. Dashed lines are guides to the eye. The inset shows the variation of the full linewidth $\Delta H$ as a function of frequency for the modes $n=0$ and $1_{-}$.