Direct measurement of nonlinear ferromagnetic damping via the intrinsic foldover effect

An approach to measure precisely nonlinear ferromagnetic damping is demonstrated by using spin dynamos in combination with sensitive electrical probing techniques. The directly measured intrinsic foldover effect unravels a 50-year-old mystery of ferromagnetic metals. Pivotal importance of nonlinear ferromagnetic damping is uncovered via its distinct dependence on the frequency, amplitude, and initial conditions. The experimental results are in excellent agreement with a nonlinear oscillator model, which revises the pioneer work of Anderson and Suhl for nonlinear magnetization dynamics.

Figure 1

(Color online) (a) Galileo’s pendulum with the calculated amplitude $a$, as a function of $\omega -{\omega }_0$ at (b) $f=0.01f_{th}$, (c) $f=f_{th}$, and (d) $f=2f_{th}$. (e) Measurement scheme for photovoltage experiment. Top view micrograph of (f) sample I and (g) sample II selected from the first and second generation spin dynamos, respectively. (h) FMR with the calculated cone angle, $\theta$, as a function of $H-H_0$ at (i) $h=0.01h_{th}$, (j) $h=h_{th}$, and (k) $h=2h_{th}$.

Figure 2

(Color online) (a) Magnetoresistance (main panel) and photoresistance (bottom inset) which detect AMR and FMR, respectively. The magnetic field $H$ is applied perpendicular to the Py strip (top inset). (b) Mappings of the power dependence of FMR detected by the photoresistance measurements. (c) $\Delta R$ as a function of the microwave power measured at 2.2 and 5.2 GHz. The dotted lines in (b) and (c) are calculated based on the relation ${\theta }^2\propto P$, the solid lines are based on ${\theta }^2\propto P^{1/3}$. The arrows in (b) and (c) mark a threshold power of $P_{th}\approx 15\text{mW}$ at 2.2 GHz.

Figure 3

(Color online) Linear and foldover FMR detected by photovoltages measured on (a) sample I and (b) sample II. Power dependence of $H_{up}$ and $H_{down}$ (circles) determined from the foldover FMR of (c) sample I and (d) sample II. Solid lines are theoretical fits. Dotted lines in (c) are calculated via Eq. (3) by using the fitted values of $S$ and $\beta$.

Figure 4

(Color online) (a) Cone-angle (power) dependence of $\Delta H$ measured at different frequencies. Inset shows the measured (circles) $\omega$ dependence of $\beta$ and the empirical relation (solid curve) of $\beta =4.0/\omega$. (b) $\omega$ dependence of $\Delta H$ (solid curves) and $\Delta H_0$ (dotted line) calculated at different cone angles by using Eqs. (2, 1), respectively. Circles show $\Delta H$ measured from foldover FMR. Inset shows two FMR spectra measured below and above 2.2 GHz. (c) $\omega$ dependence of Anderson-Suhl critical field $h_c$ (dotted curve) and the threshold field $h_{th}$ (solid curve). Note the suppressed region for detecting foldover FMR due to nonlinear damping, as confirmed by the photovoltage measurements (symbols).