Control of Spin-Wave Propagation using Magnetisation Gradients

We report that in an in-plane magnetised magnetic film the in-plane direction of a propagating spin wave can be changed by up to 90 degrees using an externally induced magnetic gradient field. We have achieved this result using a reconfigurable, laser-induced magnetisation gradient created in a conversion area, in which the backward volume and surface spin-wave modes coexist at the same frequency. Shape and orientation of the gradient control the conversion efficiency. Experimental data and numerical calculations agree very well. Our findings open the way to magnonic circuits with in-plane steering of the spin-wave modes.

| A simple model illustrates the mode conversion process. The saturation magnetisation gradient can be modelled by infinitely thin slices (thickness: ) parallel to the direction. The spin wave is refracted into the direction at each slice.
The change of ⃗ in the direction can be qualitatively understood using the simple model shown in Fig. S1. Next after entering the gradient region, the direction of the group velocity changes depending on the orientation of the interface. As a consequence, the wave propagates into regions with lower saturation magnetisation.
The tangential component of ⃗ -here it is -is conserved at the refraction at an interface of the shown slices. So, only the component can change. How strong this change ∆ will be is determined via the manifold of possible solutions given by the dispersion relations or isofrequency curves, respectively.

Schematic experimental setup and sample design.
Figure S2 | Scheme of the optical experimental setup (a) and the sample consisting of a multilayer system (b).
In contrast to the experimental setup in reference [30] in the main text, no acousto optical modulator was used. A green laser creates temperature or respectively magnetisation gradients (see Fig. S2a): a spatial light modulator changes the local phase fronts of the incoming laser beam to create arbitrary intensity distributions / to reconstruct holograms on the sample which heats up locally. An infrared camera measures the resulting temperature distribution.
The sample used in the experiment is schematically shown in Fig. S1b. It consists of a multilayer system (GGG/YIG/absorber/spacer). The laser light impinges from the GGG (Gadolinium Gallium Garnet) side and is absorbed in YIG (Yttrium Iron Garnet) and the black absorber. The dielectric spacer separates the antennas from the sample in order to minimise the thermal contact.

Additional homogenous light distributions and corresponding local saturation magnetisation.
In our experiments, we also investigated homogenous light distributions. Let us discuss three cases to create the magnetisation landscapes (see Fig. S3): a rectangle, a triangle, and a "triangle at the bottom" (meaning that the horizontal edge of the triangle points away from antenna 2). In all the cases presented in Fig. S3 the sample is heated via a uniform intensity distribution. But, even a uniform intensity distribution will create a non-uniform temperature profile -a temperature gradient ∇ ⃗ ⃗ and, thus, a magnetisation gradient ∇ ⃗ ⃗ S ( ) -due to the intrinsic thermal conductivity of YIG. However, the saturation magnetisation S,2 ( ) at antenna 2 is kept constant via adjusting the hologram laser power holo appropriately.
3 This is done to compare the cases described in Fig. S3. Antenna 1 and antenna 3 are kept far away from the heated area, thus S,1 and S,3 are almost equal and correspond approximately to the value at room temperature. parameter is shown as straight (dotted) line and depicts the transmission of spin waves form antenna 1 to 2 (1 to 3). The light-induced temperature distributions are depicted as colour code (insets, blue: cold, red: hot). max is the maximal temperature of the colour scale and corresponds to the individual laser power holo . The grey area (7.10 -7.15 GHz) is above the ferromagnetic resonance frequency for an unheated sample (see the S 31 parameter) and is not of interest in this work.
For the "triangle at the bottom", one clearly observes a weak 21 transmission in the frequency range from 7.025 to 7.075 GHz. However, magnetostatic spin waves are strongly anisotropic and a change in the propagation direction should occur simultaneously with a change in the type of spin-wave mode. If we rotate the triangle counter clockwise by 90 degrees (triangle case) the detected signal at antenna 2 increases drastically. Even the rectangle shows a weak mode conversion. In the latter case, no spin-wave mode conversion is expected since the symmetry of the system is not broken by the laser intensity distribution in the direction. But regarding the temperature distribution (Fig. S4), the temperature drops down because of von Neumann boundary conditions at the lower edge of the waveguide (at =0.0 mm).
Thus, the symmetry in the saturation magnetisation is broken nonetheless. The spinwave conversion efficiency can be increased further by directly shaping the gradientas shown in the main text. This is easily realised in our experimental setup by modifying the hologram [30].   Fig. S5). Consequently, no isofrequency curves for MSSWs are occurring above the line corresponding to the critical angle c . The spin-wave propagation in the gradient area is the same as in Fig. 4 for waves entering the ∇ ⃗ ⃗ S region below S,FMR . In the case of a rectangular magnetisation gradient area (Fig. S6a), the spin waves propagating at S ≈ 141 kA/m enter the gradient at a certain value S,in defined by the coordinate in the direction perpendicular to the propagation direction. Thus, the spin-wave wavevector, and, respectively, the wavelength change differently in different positions over the waveguide's width resulting in a bending of the phase fronts and in an adjustment of the group velocity's direction.
The corresponding isofrequency curves illustrate how this process is happening: the spin waves at S ≈ 141 kA/m propagate into the gradient region at S,in . The tangential component of ⃗ (with respect to the interface) is conserved and only is changed. After entering ∇ ⃗ ⃗ S ( , ), is conserved and only is modified since the translational symmetry of the system is broken due to the magnetisation gradient in the direction. The change ∆ is due to refraction of the spin waves in the gradient area, which is much larger than the wavelength [29]. As a consequence, ⃗ rotates in the --plane. In the triangular case (Fig. S6b) additional refraction at the interface to the gradient area occurs, which changes the and component of the wavevector.

4) Micromagnetic simulations -qualitative determination of the conversion efficiency.
We used the simulations with no upper waveguide edge (see above) to determine the conversion efficiency qualitatively. Therefore, we calculated the mean value of the precessing dynamic magnetisation's z-compontent in the extended part of the waveguide. We only use the cells of the simulation area which lie above the critical angle (46°, see below). The simulations for a triangular saturation magnetisation distribution (red curve) fit the data very well for frequencies below 6.9 GHz and above 7.05 GHz. In contrast, the orange curve (rectangular magnetisation landscape) corresponds very well for frequencies in between these two values. Since the experimental temperature distribution is not as sharp as in the simulations, the measured curve of the efficiency is a mixture of both simulations. The saturation magnetisation dependency of the critical angle is shown in the following plot for ext = 143 kA m ⁄ (equals µ 0 ext ≈ 180 mT):