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

We present a method for efficient spin-wave guiding within the magnonic nanostructures. Our technique is based on the anomalous refraction in the metamaterial flat slab. The gradual change of the material parameters (saturation magnetization or magnetic anisotropy) across the slab allows tilting the wavefronts of the transmitted spin waves and controlling the refraction. Numerical studies of the spin-wave refraction are preceded by the analytical calculations of the phase shift acquired by the spin wave due to the change of material parameters in a confined area. 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.

Figure 1

Scattering of the SWs on a ferromagnetic layer (dark area) embedded in a ferromagnetic matrix (light area). The uniaxial anisotropy field $H_a$ and the static magnetization $M$ for ferromagnetic matrix ($A$) and layer ($B$) are parallel to each other and tangential to the plane of the layer. The layer $B$ of thickness $d$ is exchange coupled to the matrix $A$ by the interfaces of thickness $\delta$.

Figure 2

Geometry of the simulation system with the GRIN element. The system consists of the ferromagnetic slab ($B$) characterized by the gradient of magnetic parameters (e.g., saturation magnetization $M_S$ or anisotropy field $H_a$), which links the straight section of the waveguide and the semi-infinite plane made of the homogeneous material $A$. The magnetic parameters in the slab $B$ are changing in the $y$ direction. At each $x$-$z$ cross section (see Fig. 1), the phase shift of transmitted wave is different, which allows refracting the plane wave propagating initially in the $x$ direction. The absorbing material is placed at the sides of the simulated system, in order to avoid the impact of boundaries. The external magnetic field $\mathbf{H}$ is applied along the $z$ direction; the static magnetization and the anisotropy field are aligned with the external field.

Figure 3

SWs’ wavelength (blue color) as a function of frequency. Red dots indicate fulfilled resonant conditions by Eq. (10). Enhancement of transmittance at these frequencies can be observed. Values on the horizontal axis represent the resonant frequencies. The transmittance (black color) and the phase shift (green color) for SWs traveling through the 150-nm-wide slab with respect to the frequency. Dots and squares represent the values obtained in the numerical simulations, while the solid lines represent the analytical results. Dashed lines represent the case when the reflection in the system is neglected. The value of the external field is equal to 0.5 T, reduced $M_S$ in the slab is equal to 800 kA/m and reduced $A_{\mathrm{ex}}$ to 20 pJ/m.

Figure 4

SWs’ wavelength (blue color) as a function of frequency. Red dots indicate fulfilled resonant conditions, Eq. (10), where we observe an enhancement of the transmittance. Values on the horizontal axis represent the resonant conditions. Transmittance (black color) and phase shift (green color) for SWs propagating through the 150-nm-wide slab with respect to the frequency. Dots and squares represent the values obtained in numerical simulation, while the solid lines represent the analytical results. Dashed lines represent the case when the reflection in the system is neglected. The value of the external field is equal to 0.5 T and the uniaxial anisotropy constant $K=5\mathrm{kJ}/{\mathrm{m}}^3$ within the slab.

Figure 5

Transmittance (black color) and phase shift (green color) for SWs traveling through the 150-nm-wide slab with respect to (a) $M_S$ and (b) the anisotropy field in the slab. Dots and squares represent the values obtained in numerical simulation, while the solid lines represent the analytical results. Dashed lines represent the nonresonant case. The constant frequency equal to 25 GHz and external field equal to 0.5 T are considered. (a) The reduced exchange $A_{\mathrm{ex}}$ is equal to 20 pJ/m. Resonant peaks are visible for the specific values of $M_S$. (b) $M_S$ and $A_{\mathrm{ex}}$ remain the same as in the matrix, i.e., 1200 kA/m and 27 pJ/m, respectively.

Figure 6

The squared amplitude of SWs propagating in the system shown in Fig. 2, at the frequency 25 GHz. (a) When SWs reach the semi-infinite medium after the slab, the new wavefront starts to form due to the gradient of magnetic parameters. The bending of SWs reaches the angle of ca. $36{}^{\circ }$. (b) The color palette of SW. The color indicates the phase, while the intensity indicates the amplitude. (c) The bending of SWs ca. ${36}^{\circ }$ obtained according to the Huygens-Fresnel formula [Eq. (7)]. Source points of cylindrical SWs are located on the right border of the slab—at the position $d$. The amplitude of the transmittance and the phase shift are obtained from the boundary-condition problem, like in Fig. 5.

Figure 7

(a) Propagation of SWs in the $\mathrm{Co}\text{−}\mathrm{Fe}\text{−}\mathrm{B}$ waveguide, where the GRIN slab is placed at the bend, releases the anomalous refraction. The GRIN slab has the same gradient of $M_S$ as used in Fig. 6. The color map represents a dynamical component of the magnetization in the $y$ direction. A snapshot is taken at the moment when a steady state is reached. At the beginning of the horizontal branch of the waveguide, the microwave antenna excites the SWs and at the end of the tilted branch, the SWs are damped to avoid any reflection. After taking a turn, the SWs propagate smoothly. (b) SWs’ propagation in the waveguide without a gradient of magnetic parameters. After taking a turn, the SWs show a complex behavior due to interference between different modes of the waveguide. The width of the waveguide is 100 nm. The material parameters are the same as for the system presented in Fig. 6.

Figure 8

SW propagation in the bent waveguide of the same geometry, as in Fig. 7. The GRIN slab placed at the bend is characterized by the gradient of anisotropy field $H_a$. The gradient of $H_a$ is chosen to obtain refraction at the same angle, as in Fig. 7. In(a) the dipolar interaction is included and in(b) it is neglected in the micromagnetic simulations.

Figure 9

Normalized spatial profile of dynamical component of the magnetization for the five lowest resonances (a) $14.6$ GHz, (b) $16.4$ GHz, (c) $19.6$ GHz, (d) $23.8$ GHz, and (e) $29.4$ GHz. Width of the slab is $150$ nm, $M_S$ in the matrix is $1200$ kA/m and $A_{\mathrm{ex}}=28\mathrm{pJ}/\mathrm{m}$. External field is $0.5$ T, reduced $M_S$ within the slab is $800$ kA/m and reduced exchange stiffness constant $A_{\mathrm{ex}}$ is $20$ pJ/m. The presented resonances correspond to Fig. 3.