Nonlinear ferromagnetic resonance shift in submicron Permalloy ellipses

We report a systematic study of nonlinearity in the ferromagnetic resonance of a series of submicron Permalloy ellipses with varying aspect ratios. At high excitation powers, the resonances are found to shift to higher or lower applied field. We focus here on the sign of the shift and its dependence on the applied field and shape-induced anisotropy of the ellipses. Using ferromagnetic resonance force microscopy, we find that the measured nonlinear coefficient changes sign as a function of anisotropy field and applied field in qualitative agreement with a macrospin analysis. This macrospin analysis also points to origins of the nonlinearity in a combination of hard-axis in-plane anisotropy and precession ellipticity. In comparison of the macrospin predictions with both experimental and micromagnetic modeling results, we measure/model values of the nonlinear coefficient that are more positive than predicted by the macrospin model. The results are useful in understanding nonlinear physics in nanomagnets and applications of spin-torque oscillators.

Figure 1

Power dependent resonance spectra measured with a $560\mathrm{nm}\times 450\mathrm{nm}$ elliptical sample with an aspect ratio of 1.25 and with microwave frequencies of (a) 9 GHz and (b) 12 GHz. The static field is applied in-plane along the long axis and the microwave field is in-plane, perpendicular to the static field.

Figure 2

Calculated precession orbits projected in the $x\text{−}y$ plane with an applied field of (a) $H_{\mathrm{appl}}/H_2=0.10$ and (b) $H_{\mathrm{appl}}/H_2=0.49$. For both (a) and (b), $H_1/H_2=-0.07$. Different trajectories correspond to ${\theta }_x=1^{\circ },2^{\circ },3^{\circ },4^{\circ }$, and $5^{\circ }$, where ${\theta }_x$ is the polar angle as the magnetization sweeps through the $x\text{−}z$ plane, i.e., ${\theta }_x={sin}^{-1}(M_x^{\max }/M_s)$. (c) Zero nonlinear coefficient curves predicted by the macrospin model Eq. (8) (solid line) and thermal nonlinearity Eq. (11) (dashed line).

Figure 3

Frequency as a function of applied field measured in linear regime for elliptical samples with various aspect ratios. The microwave power here is typically one order of magnitude less than in the nonlinear measurements (e.g., Fig. 1). The symbols represent experimental data and the solid lines are the fits using the Kittel equation. The insets are the AFM images using the same 200 nm probe.

Figure 4

Measured (a) and modeled (b) sign of nonlinear coefficient as a function of normalized in-plane anisotropy field ($H_1/H_2$) and applied field ($H_{\mathrm{appl}}/H_2$). The aspect ratios are labeled for each sample, corresponding to a given anisotropy obtained from ferromagnetic resonance measurements. The “near zero” open circles indicate that the fit value of $N^{\prime }$ was smaller than its standard deviation. The solid lines in (a) and (b) are the condition for zero nonlinearity predicted by the macrospin model. The dotted lines are the zero nonlinearity curves from the macrospin model including the effect of nonuniform magnetization. (c) Modeled precession amplitude profiles ($|\Delta M_z|$) of the main mode for the cases where the field is parallel (top, AR = 2) and perpendicular (bottom, AR = 0.5) to the long axis of the ellipses. The contour lines mark where $|\Delta M_z|$ is half of its maximum value.