Anomalous Refraction of Spin Waves as a Way to Guide Signals in Curved Magnonic Multimode Waveguides

Abstract

Phase and amplitude are the fundamental characteristics of waves. The processing of any wave relies on the interference effects, which depend on these characteristics. Thus, the control of spin waves’ (SWs’) phase and amplitude is essential in magnonics to perform both analog and digital SW-based computing [1]. Equally important is the transmission of the information (i.e. guiding and routing of the signal). We present a method for a coherent spin-wave guiding within the narrow magnonic nanostripe.In the magnonic multimode waveguide, scattering between the modes is one of the most important factors for the SW decoherence at the bends. Therefore, the question arises: can we modify the properties of the bending region to block the redistribution of incoming mode into the perpendicularly quantized modes, keeping the transmission as high as possible? Our proposed solution is to fill the bending region with the material of spatially tailored properties, which refract the SW and redirect its propagation strictly along the outgoing section of the waveguide (fig. 1). In other words, we are looking for the particular kind of graded-index (GRIN) element for SWs [2].To design the GRIN element, we proposed an algorithm which achieving desired properties in a few step.In the first step, we developed the analytical theory for the scattering of exchange SWs on the homogeneous ferromagnetic slab of finite width embedded in a ferromagnetic layer. Minimizing the total energy, we derived the boundary conditions on the interfaces between the slab and its surroundings. As a results, we obtained the complete relations between the phases and amplitudes of the incident and scattered SWs.Later, we used our findings to demonstrate both analytically and numerically anomalous refraction for the purely exchange SWs incident from a waveguide to a semi-infinite film through a flat magnonic GRIN slab (fig. 2). We treat the GRIN element as an inhomogeneous slab linking the waveguide's input and output branches at the bend. For anomalous refraction, the wavefronts of refracted waves are tilted at a desirable angle with respect to wavefronts of the incident waves. The effect can be observed even at normal incidence. This phenomenon requires a linear change of the transmitted waves phase alongside the interface, where the refraction takes place. To describe it, we used the generalization of Snell’s law [3]. From previous step, we knew the phase which is acquired at specific positions so we can conduct analytical calculation of generalized Snell’s law and get the refraction angle.In the ultimate step, we could design the magnonic waveguide with the GRIN element at the bend. Knowing the SW bending, we design the properly shaped curved magnonic waveguide with GRIN element and run micromagnetic simulations. Fig. 1 presents a snapshot from simulations, where results are compared to the structure with the neglected GRIN element.We demonstrate that our findings can be used to guide the spin waves smoothly in curved waveguides, even through sharp bends, without reflection and scattering between different waveguide’s modes, preserving the phase, the quantity essential for wave computing.The systemsIn all the simulations we consider SW propagation in 5-nm-thick Co-Fe-B film (MS = 1200 kA/m, Aex = 27 pJ/m, neglected dumping) in the presence of the out-of-plane magnetic field (applied along the z-axis) of value μ0H =0.5 T. The SW frequency was 25 GHz. The width of the slab had 150 nm. In the considerations, we neglected dipolar interaction, however we checked and discussed the impact of this kind of interaction on the proposed system.In the first step, we considered reduced magnetization saturation in the slab in the range of 300–800 kA/m or additional anisotropy with anisotropy constant in the range of 0–490 kJ/m3.Anomalous refraction was demonstrated for two-dimensional system (fig. 2) composed of three sections. 100 nm wide waveguide, GRIN element with induced The gradient of the magnetic parameters was induced along the y-axis. Saturation magnetization changes in the range of 300–800kA/m on the distance 100 nm, with direct contact with waveguide.Finally, using the previous step's knowledge, we design a curved waveguide, which supports coherent SW propagation alongside the waveguide [4]. **