Design of magnonic waveguides using surface anisotropy-induced Bragg mirrors

Abstract

Waveguides are fundamental components for signal transmission in integrated wave-based processing systems. In this paper, we address the challenges in designing magnonic waveguides, including limitations such as non-uniform demagnetizing fields, reduced group velocity, and restricted operating frequency ranges. We propose a magnonic waveguide design with promising properties that overcome these limitations to a significant extent. Specifically, we investigate a waveguide formed within a uniform ferromagnetic layer (Co$_{20}$Fe$_{60}$B$_{20}$) by applying surface anisotropy in strip regions, thereby creating Bragg mirror structures to confine spin waves and guide them along a single direction. The proposed waveguide enables the propagation of high-frequency spin waves with high velocities in the ferromagnetic layer while minimizing static demagnetizing effects. We developed a model that allows for spin-wave confinement and guidance in two perpendicular directions by spatially modulating the surface anisotropy. The theoretical model was solved using the finite element method to calculate the dispersion relations of the waveguide modes and analyze their spatial profiles. Additionally, we determine the group velocity and localization characteristics, providing a comprehensive understanding of the waveguide's performance.

Figures (5)

• Figure 1: Selected designs of planar magnonic waveguides: (a) Geometric confinement: a strip of ferromagnetic material (violet) embedded in a nonmagnetic matrix. (b) Material parameter modification: a waveguide formed in the central region of a ferromagnetic layer by reducing the saturation magnetization (violet strip) compared to the surrounding ferromagnet (green areas). (c) Voltage-controlled surface anisotropy: a waveguide induced by locally modified anisotropy (red region on the surface).
• Figure 2: (a) Structure under study: spin waves are confined by two Bragg mirrors (red stripes) formed by periodically modulated surface anisotropy on the top and bottom surfaces. Waves are guided along $z$ in the central region (wider strip -- defect). The anisotropy is set by $K_{\rm s}$ and the mirror period by $a$. The static field $H_0$ is applied in-plane, perpendicular to the waveguide. The blue line shows an exemplary out-of-plane profile $|m_y|$. The inset illustrates a possible experimental realization. (b) Dispersion $f(k_z)$ of the waveguide modes (colored lines). Modes are confined when their frequencies fall into the band gaps of the magnonic crystals (white regions). The lower limit is the FMR of a uniform layer with continuous surface anisotropy on both sides (dash-dotted black line); for this design, higher-order modes (e.g., Nos. 5–6) can appear even above the FMR of a uniform layer without surface anisotropy (dotted line). Vertical dashed lines indicate the $k_z$ values used for mode profiles in Fig. \ref{['fgr:profiles']}; for $k_z=10$ rad/$\mu$m we also evaluate a figure of merit for the trade-off between propagation and localization (see Supplemental Material 2).
• Figure 3: Group velocity of the waveguide modes as a function of the wave vector $v_{g}(k_{z})$. Line colors denote successive modes with increasing frequencies (see Fig. \ref{['fgr:dispersion_relation']}(b)).
• Figure 4: Measure of the localization of the waveguide modes: modified inverse participation ratio as a function of the wave vector $\widetilde{\rm IPR}(k_{z})$ -- see Eq.\ref{['eq:IPR']}. Line colors denote successive modes with increasing frequencies (see Fig. \ref{['fgr:dispersion_relation']}(b)).
• Figure 5: Profiles of the out-of-plane component of the dynamic magnetization, $|m_y|$, for successive waveguide modes at two selected wave vectors: $k_{z}=10$ rad/$\mu$m (green) and $k_{z}=60$ rad/$\mu$m (blue). See Fig. \ref{['fgr:dispersion_relation']}(b) for the corresponding dispersion branches. Pink (white) regions indicate sections of the ferromagnetic layer with (without) surface anisotropy. Green dashed lines indicate the isolation distance $L_i$ necessary to prevent crosstalk between adjacent waveguides in integrated circuits.