Nonlinear Emission of Spin-Wave Caustics from an Edge Mode of a Microstructured ${\mathrm{Co}}_2{\mathrm{Mn}}_{0.6}{\mathrm{Fe}}_{0.4}\mathrm{Si}$ Waveguide

Magnetic Heusler materials with very low Gilbert damping are expected to show novel magnonic transport phenomena. We report nonlinear generation of higher harmonics leading to the emission of caustic spin-wave beams in a low-damping microstructured ${\mathrm{Co}}_2{\mathrm{Mn}}_{0.6}{\mathrm{Fe}}_{0.4}\mathrm{Si}$ Heusler waveguide. The source for the higher harmonic generation is a localized edge mode formed by the strongly inhomogeneous field distribution at the edges of the spin-wave waveguide. The radiation characteristics of the propagating caustic waves observed at twice and three times the excitation frequency are described by an analytical calculation based on the anisotropic dispersion of spin waves in a magnetic thin film.

Figure 1

Sketch of the sample design. The shortened end of a coplanar waveguide is used as an antenna structure to excite spin dynamics in a $5\mu \mathrm{m}$ wide CMFS waveguide with a thickness of 30 nm. The waveguide is positioned in the $x\mathrm{\text{−}}y$ plane with its long axis pointing along the $x$ direction. The external magnetic field is applied transversely to the waveguide in the $y$ direction. The figure includes a $\mu \mathrm{BLS}$ spectrum taken at a distance of $4.5\mu \mathrm{m}$ from the antenna in the center of the waveguide (see laser beam in the sketch) for an excitation frequency of $f_e=3.5\mathrm{GHz}$, a microwave power of 20 mW, and an external field ${\mu }_0{H}_{\mathrm{ext}}=48\mathrm{mT}$.

Figure 2

(a) $\mu \mathrm{BLS}$ intensity distribution for the detection frequency $f_d$ being equal to the excitation frequency $f_e=3.5\mathrm{GHz}$. The observed intensity of the edge mode is maximal near the edges of the CMFS waveguide. Note that the exciting antenna is positioned between $x=-1\mu \mathrm{m}$ and $x=0\mu \mathrm{m}$. The blue dot in the graph indicates the position of the measurement presented in Fig. 1. (b) Calculated dispersion relation for the center of the CMFS waveguide according to Ref. 19 as well as excitation frequency and higher harmonics (dashed lines).

Figure 3

Power dependence of the μBLS intensity for the detection frequencies $f_d=3.5$, 7.0, and 10.5 GHz for the fixed excitation frequency $f_e=3.5\mathrm{GHz}$. Please note the log-log presentation of the data. The lines in the graph correspond to fits according to Eq. (1). A least-squares fit of the data yields $s_{1f}=0.9\pm 0.1$, $s_{2f}=2.1\pm 0.1$, and $s_{3f}=2.8\pm 0.3$.

Figure 4

$\mu \mathrm{BLS}$ intensity distribution for (a) $f_d=2f_e=7.0\mathrm{GHz}$ and (b) $f_d=3f_e=10.5\mathrm{GHz}$. Both intensity maps show strongly directed spin-wave beams along the angle $\theta =\angle ({\mathbf{H}}_{\mathbf{e}\mathbf{x}\mathbf{t}},{\mathbf{v}}_{\mathbf{G}})$. The lines in the maps are guides to the eye to identify the propagation angle $\theta$. The blue dots in the graphs indicate the positions of the measurements presented in Fig. 1.

Figure 5

Analytical calculations according to Ref. 19. (a) Isofrequency curves $f_{\mathrm{const}}=f(k_x,k_y)$ for $f=2f_e=7.0\mathrm{GHz}$ and $f=3f_e=10.5\mathrm{GHz}$. On the basis of these calculations, exemplary directions for $\mathbf{k}$, ${\mathbf{v}}_{\mathbf{G}}$, ${\mathbf{H}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$, and the propagation angle $\theta =\angle ({\mathbf{H}}_{\mathbf{e}\mathbf{x}\mathbf{t}},{\mathbf{v}}_{\mathbf{G}})=\arctan (d{k}_y/d{k}_x)$ are shown in the graph. (b) Radiation direction $\theta$ calculated from the isofrequency curves shown in (a). Dash-dotted lines correspond to the angles ${\theta }_{\exp }$ observed in the experiment.