Non-Gilbert-damping Mechanism in a Ferromagnetic Heusler Compound Probed by Nonlinear Spin Dynamics

The nonlinear decay of propagating spin waves in the low-Gilbert-damping Heusler film ${\mathrm{Co}}_2{\mathrm{Mn}}_{0.6}{\mathrm{Fe}}_{0.4}\mathrm{Si}$ is reported. Here, two initial magnons with frequency $f_0$ scatter into two secondary magnons with frequencies $f_1$ and $f_2$. The most remarkable observation is that $f_1$ stays fixed if $f_0$ is changed. This indicates, that the $f_1$ magnon mode has the lowest instability threshold, which, however, cannot be understood if only Gilbert damping is present. We show that the observed behavior is caused by interaction of the magnon modes $f_1$ and $f_2$ with the thermal magnon bath. This evidences a significant contribution of the intrinsic magnon-magnon scattering mechanisms to the magnetic damping in high-quality Heusler compounds.

Figure 1

(a) Schematic view of the sample: spin waves are excited in a CMFS waveguide by means of a microstrip antenna. A bias field $H_{\mathrm{ext}}$ is applied perpendicular to the waveguide’s long axis. (b) Brillouin light-scattering intensity $I_{\mathrm{BLS}}$ in the linear regime as a function of the microwave excitation frequency $f_0$.

Figure 2

(a) BLS spectra for $f_0=10.5\mathrm{GHz}$ at different microwave powers. The spectrum at the threshold power ($P=9\mathrm{dBm}$) shows the two unstable mode frequencies $f_1$ and $f_2$ (shaded). The different BLS intensities at $f_1$ and $f_2$ are caused by different detection efficiencies. For supercritical powers ($P>9\mathrm{dBm}$), the intensities at $f_1$ and $f_2$ do not increase further, instead the spectrum becomes continuously broadened. (b) BLS intensity $I_{\mathrm{BLS}}$ for the modes $f_1$, $f_2$, and $f_0$ as a function of the microwave power $P$. Above the threshold power, $I_{\mathrm{BLS}}$ of the unstable modes at $f_1$, $f_2$ rises sharply. (c) Unstable mode frequencies $f_1$ and $f_2$ (circles and squares, respectively) as a function of the microwave frequency $f_0$. $f_1$ is independent of $f_0$ and constant at $f_1\approx 7.1\mathrm{GHz}$, while $f_2$ increases linearly with $f_0$. The blue and green lines indicate the calculated values for $f_1$ and $f_2$ according to Eq. (1) for pure Gilbert-damping (dashed lines) and if an additional non-Gilbert-damping term (Eq. (3) is included (solid lines).

Figure 3

Isofrequency curves for the experimental conditions of Fig. 2. Three possible four-magnon scattering processes are schematically illustrated.

Figure 4

Normalized maximum four-magnon coupling strength $\tilde{W}$ as a function of the frequency $f_1^{\prime }$ of the lower potentially unstable mode [parameters, see Fig. 2]. The dashed line represents the maximum bare four-magnon coupling $W$ without the incorporation of the three-magnon correction $T$.

Figure 5

Individually normalized critical threshold amplitudes $c_{\text{crit}}$ as a function of $f_1^{\prime }$ for pure Gilbert-damping (blue curve) and for different values of the parameter $A$. For $A>A_{\min }\approx 10^{-7}\mathrm{m}{\mathrm{GHz}}^2/\mathrm{rad}$, the threshold is minimal for the experimentally observed unstable mode frequency $f_1\approx 7.1\mathrm{GHz}$.