Experimental and numerical understanding of localized spin wave mode behavior in broadly tunable spatially complex magnetic configurations

Spin wave modes confined in a ferromagnetic film by the spatially inhomogeneous magnetic field generated by a scanned micromagnetic tip of a ferromagnetic resonance force microscope (FMRFM) enable microscopic imaging of the internal fields and spin dynamics in nanoscale magnetic devices. Here we report a detailed study of spin wave modes in a thin ferromagnetic film localized by magnetic field configurations frequently encountered in FMRFM experiments, including geometries in which the probe magnetic moment is both parallel and antiparallel to the applied uniform magnetic field. We demonstrate that characteristics of the localized modes, such as resonance field and confinement radius, can be broadly tuned by controlling the orientation of the applied field relative to the film plane. Micromagnetic simulations accurately reproduce our FMRFM spectra allowing quantitative understanding of the localized modes. Our results reveal a general method of generating tightly confined spin wave modes in various geometries with excellent spatial resolution that significantly facilitates the broad application of FMRFM. This paves the way to imaging of magnetic properties and spin wave dynamics in a variety of contexts for uncovering new physics of nanoscale spin excitations.

Figure 1

FMRFM spectra taken at $f_{\mathrm{rf}}=2.157\mathrm{GHz}$ and at various probe-sample separations $a$ when the probe magnetic moment ${\mathit{m}}_p$ is (a) antiparallel and (b) parallel to the applied uniform field ${\mathit{H}}_0$ at ${\theta }_{\mathrm{H}}=0^{\circ }$ $({\mathit{H}}_0$ normal to the film plane). Insets: Schematics of experimental configurations of FMRFM measurements, where the black and green curves represent the spatial profiles of the magnitude of probe field ${\mathit{H}}_{\mathrm{p}}$ and the amplitude of the first localized mode. Spectra are offset for clarity.

Figure 2

(a) Selected FMRFM spectra as a function of $H_0-H_0^{\mathrm{unif}}$ at ${\theta }_{\mathrm{H}}=0{}^{\circ }$, 4°, and 6° for both parallel $\left(a=1000\mathrm{nm}\right)$ and antiparallel $\left(a=2500\mathrm{nm}\right)$ configurations. Inset: Resonance field $H_0^{\mathrm{loc}}$ of the first localized mode as a function of $a$ at ${\theta }_{\mathrm{H}}=0{}^{\circ }$ and 4° in the antiparallel geometry. (b) Shift in resonance field between the first localized mode and the uniform mode $H_0^{\mathrm{loc}}$ − $H_0^{\mathrm{unif}}$ as a function of $a$ at various ${\theta }_{\mathrm{H}}$ in the antiparallel (solid squares) and parallel (solid circles) configurations. The solid curves are micromagnetic modeling results which agree well with the experimental data. (c) Characteristic size of the first localized mode at various ${\theta }_{\mathrm{H}}$ in the antiparallel and parallel configurations obtained by micromagnetic modeling, which represents the radius of the short axis of the mode in different measurement geometries (for ${\theta }_{\mathrm{H}}=0{}^{\circ }$ in the parallel configuration with a doughnut shape mode, we use the difference between the outer and inner radius, see Fig. 5). 3D dependences of characteristic size of the first localized mode as a function of probe-sample separation and angle for (d) the parallel and (e) antiparallel geometries, respectively, emphasizing the strong contrast of the angular dependencies of characteristic mode sizes.

Figure 3

Spatial variation of the out-of-plane components of ${\mathit{H}}_{\mathrm{demag}}\left(\mathit{r}\right)$ and ${\mathit{H}}_{\mathrm{p}}\left(\mathit{r}\right)$ at (a) ${\theta }_{\mathrm{H}}=0{}^{\circ }$ and (b) ${\theta }_{\mathrm{H}}=6{}^{\circ }$ as well as (c) $H_{\mathrm{stat}}\left(\mathit{r}\right)$ for both angles in the antiparallel configuration. Corresponding plots for the parallel configuration are shown in (d), (e), and (f). Insets: Schematics of the spatial profiles of the equilibrium orientation of magnetization $\mathit{M}$ in that particular configuration. Note the two $y$ axes in each plot are offset relative to each other for comparison. It demonstrates that the spatial profile of $H_{\mathrm{stat}}\left(\mathit{r}\right)$ in the antiparallel configuration in (c) only changes slightly with ${\theta }_{\mathrm{H}}$ while the depth of the field well in $H_{\mathrm{stat}}\left(\mathit{r}\right)$ in the parallel case in (f) changes significantly.

Figure 4

Comparison of $H_{\mathrm{eff}}^{\mathrm{unif}}$ and $H_{\mathrm{eff}}^{\mathrm{loc}}$ with the corresponding $H_{\mathrm{stat}}\left(\mathit{r}\right)$ numerically calculated for ${\theta }_{\mathrm{H}}=0{}^{\circ }$ and 6° at probe-sample separation $a=1000\mathrm{nm}$ in the parallel configuration. The external field is set to $H_0^{\mathrm{unif}}$, the field at which the uniform mode is resonant with the effective rf field ${\omega }_{\mathrm{rf}}/\gamma$. The static field profile $H_{\mathrm{stat}}\left(\mathit{r}\right)$, peak effective dynamic fields $H_{\mathrm{dyn}}^{\mathrm{loc}}$, and $H_{\mathrm{dyn}}^{\mathrm{unif}}$ change significantly with ${\theta }_{\mathrm{H}}.$ For example, $H_{\mathrm{dyn}}^{\mathrm{unif}}$ (0°) ≈ 0 at ${\theta }_{\mathrm{H}}=0{}^{\circ }$ changes to $H_{\mathrm{dyn}}^{\mathrm{unif}}$ (6°) ≈ 83 Oe at ${\theta }_{\mathrm{H}}=6{}^{\circ }$. These changes are the origin of the strong angular dependence of the localized mode resonance in the parallel configuration observed in the experiment.

Figure 5

Spatial profile, modulus of the transverse component of the dynamic magnetization, of the first localized mode calculated for (a) the antiparallel $\left(a=2500\mathrm{nm}\right)$ and (b) parallel $\left(a=1000\mathrm{nm}\right)$ configurations for ${\theta }_{\mathrm{H}}=0{}^{\circ }$ and 6°. The lateral size of the mode in the antiparallel case essentially remains unchanged with ${\theta }_{\mathrm{H}}$, while mode size in the parallel configuration is greatly reduced as ${\theta }_{\mathrm{H}}$ increases from 0° to 6°. For all panels the probe is located at (0, 0).

Figure 6

Micromagnetic modeling of the characteristic dimensions of the first localized spin wave mode as a function of the probe-sample separation in a 25-nm YIG thin film at ${\theta }_{\mathrm{H}}=9^{\circ }$ generated by a commercial MFM cantilever in the parallel configuration. Variable density mesh approximation with the cell size as small as $10\times 10$ nm was used for simulations. The rf frequency is 2.157 GHz and the MFM probe is assumed to be a magnetic sphere with a radius of 50 nm and saturation magnetization $4\pi M_{\mathrm{s}}=15000\mathrm{Oe}$. The elliptical mode shape is characterized by the long $\left(R_{\mathrm{long}}\right)$ and short $\left(R_{\mathrm{short}}\right)$ radii as indicated in the inset calculated for 10 nm probe-sample separation.