Engineering the Spectrum of Dipole Field-Localized Spin-Wave Modes to Enable Spin-Torque Antidamping

Auto-oscillation of a ferromagnet due to spin-orbit torques in response to a dc current is of wide interest as a flexible mechanism for generating controllable high-frequency magnetic dynamics. However, degeneracies of the spin-wave modes and nonlinear magnon-magnon scattering impede coherent precession. Discretization of the spin-wave modes can reduce this scattering. Spatial localization of the spin-wave modes by the strongly inhomogeneous dipole magnetic field of a nearby spherical micromagnet provides variable spatial confinement, thus offering the option of systematic tunability of magnon spectrum for studying multimode interactions. Here we demonstrate that field localization generates a discrete spin-wave spectrum observable as a series of well-resolved localized modes in the presence of imposed spin currents arising from the spin Hall effect in a permalloy-platinum ($\mathrm{Py}/\mathrm{Pt}$) microstrip. The observation of linewidth reduction through damping control in this micromagnetically engineered spectrum demonstrates that localized modes can be controlled efficiently, an important step toward continuously tunable spin Hall effect–driven auto-oscillators.

Figure 1

(a) Schematic of STFMR setup, illustrating orientations of the damping torque ${\tau }_{\alpha }$, spin torque ${\tau }_{\mathrm{ST}}$ and torque from the Oersted field ${\tau }_H$. Here, the external field $H_{\mathrm{ext}}$ and $M$ lie in the plane and $\theta$ denotes the in-plane angle between $H_{\mathrm{ext}}$ and the direction of the applied current in the microstrip. (b) Side view of a micromagnetic particle of a diameter $D$, glued onto a spacer of thickness $S$ separating it from the sample, depicted for $H_{\mathrm{ext}}$ at $\theta =90°$. (c) Corresponding to the configuration in (b), the plotted curves are spatial profiles of the $y$ component of dipole field, $H_d$, from a $1\text{−}\mu \mathrm{m}$ Ni particle (with saturated moment, $0.15\text{−}\mu \mathrm{m}$ particle-sample separation) as a function of lateral $x$ (dashed blue) or $y$ (solid blue) position in the $\mathrm{Py}/\mathrm{Pt}$ sample plane under the particle. Dimensions used in (b) and (c) are associated with experiments shown in Fig. 3.

Figure 2

STFMR spectra taken with the $5\text{−}\mu \mathrm{m}$ Ni particle on a $3\text{−}\mu \mathrm{m}$-wide strip, with a $0.45\text{−}\mu \mathrm{m}$ photoresist S1805 spacer, at $I_{\mathrm{dc}}=0\mathrm{mA}$, $H_{\mathrm{ext}}$ applied at 45°, and frequencies from 6 to 10 GHz, at an rf power of 6 dBm. (a) $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ strip. Inset: STFMR spectra of a bare $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ strip at 6 to 10 GHz, at an rf power of 0 dBm. (b) $3\mu \mathrm{m}\times 6\mu \mathrm{m}$ strip.

Figure 3

Evolution of the STFMR spectra with $I_{\mathrm{dc}}$ and frequency, taken with a $1\text{−}\mu \mathrm{m}$ Ni particle separated from the $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ strip by a $0.15\text{−}\mu \mathrm{m}$ PMMA spacer, and $H_{\mathrm{ext}}$ applied at 70°. (a) $f=8\mathrm{GHz}$, for $I_{\mathrm{dc}}$ increasing from 0 to 25 mA, all at an rf power of 0 dBm. The three lowest-energy localized modes (LM) are labeled as LM1, LM2, and LM3. The black curves for spectra above 15 mA are representative multipeak Lorentzian fits. (b) Spectra of well-resolved localized modes at $I_{\mathrm{dc}}=25\mathrm{mA}$, at rf powers of $-3\mathrm{dBm}$ (5, 9, and 10 GHz spectra are rescaled to maximize figure details). (c) Frequency vs resonance field plots for the quasiuniform mode and the first three localized modes. The symbols (circle, square, triangle) show experimental $H_{\mathrm{res}}$ data extracted from Fig. 3, reduced by 50 G to account for the Oersted field and heating from $I_{\mathrm{dc}}$ (determined by the shift of the quasiuniform mode). The curves for the localized modes show the results of micromagnetic simulations, ignoring the effect of spin torque (i.e., for $I_{\mathrm{dc}}=0$): $n=1$ (red), $n=2$ (green), $n=1$, $m=2$ (dotted blue), and $n=3$ (solid blue). Inset: Effective magnetization of the unsaturated Ni particle used at each external field to give the simulation results in Fig. 3, which agrees with the known hysteresis loop of a Ni particle [24]. The solid line is a linear fit, yielding $4\pi M_{\mathrm{Ni}}=2.3H+722.9\mathrm{G}$.

Figure 4

(a),(b) Spatial variation of the internal static field (gray) and mode amplitudes of the $n=1$ (red) and $n=2$ (green) localized mode profiles plotted vs lateral position in the $H_{\mathrm{ext}}$ direction. (c),(d) Spatial profiles of transverse magnetization of the localized modes for a $5\text{−}\mu \mathrm{m}$ particle, $0.45\text{−}\mu \mathrm{m}$ particle-sample separation (c), and a $1\text{−}\mu \mathrm{m}$ particle, $0.15\text{−}\mu \mathrm{m}$ particle-sample separation (d). The micromagnetic simulations assume the Ni particles are saturated; $H_{\mathrm{ext}}=2000\mathrm{G}$ at $\theta =90°$. The mode elongation in the direction perpendicular to the applied field (see text) is evident.

Figure 5

(a),(b) Dependencies of the extracted linewidth $\mathrm{\Delta }H$ and resonance field $H_{\mathrm{res}}$ on $I_{\mathrm{dc}}$ from Fig. 3 for the quasiuniform mode and first three localized modes. The line in (a) is a linear fit. The dc-Oersted field shift is shown by the dashed line in (b). (c) $H_{\mathrm{res}}$ shift relative to resonance value at 15 mA, corrected by the field-shift contribution due to the dc-Oersted field. The gray curve shows the calculated $H_{\mathrm{res}}$ shift for LM1 due to Joule heating.