Modal approach to modeling spin wave scattering

Efficient numerical methods are required for the design of optimized devices. In magnonics, the primary computational tool is micromagnetic simulations, which solve the Landau-Lifshitz equation discretized in time and space. However, their computational cost is high, and the complexity of their output hinders insight into the physics of the simulated system, especially in the case of multimode propagating-wave-based devices. We propose a finite-element modal method allowing an efficient solution of the scattering problem for dipole-exchange spin waves propagating perpendicularly to the magnetization direction. The method gives direct access to the scattering matrix of the whole system and its components. We extend the formula for the power carried by a magnetostatic mode in the Damon-Eshbach configuration to the case with exchange, allowing the scattering coefficients to be normalized to represent the fraction of the input power transferred to each output channel. We apply the method to the analysis of spin wave scattering on a basic functional block of magnonic circuits, consisting of a resonator dynamically coupled to a thin film. The results and the method are validated by comparison with micromagnetic simulations.

Figure 1

Steps required for the solution of a scattering problem with the proposed modal method. Inputs and outputs are denoted with parallelograms, and processes are denoted with rectangles.

Figure 2

An example system whose geometry satisfies the assumptions made in Sec. 2a. The system contains four segments with uniform cross sections in the $xy$ plane. FM, ferromagnetic.

Figure 3

$xz$-plane cross section of the $y$-invariant system analyzed in Sec. 4.

Figure 4

Profiles of the magnetostatic potential $\varphi \left(z\right)$ and the $x$ and $z$ components of the dynamic magnetization $\mathit{m}\left(z\right)$ of the eigenmodes of (a)–(c) the CoFeB film and (d)–(f) the bilayer. Solid lines, right-propagating modes; dashed lines, left-propagating modes. The areas taken by the CoFeB film and the permalloy film are shaded in red and blue, respectively. All plots show only the dominant real or imaginary component; the $L^2$ norm of the other one is over 100 times smaller.

Figure 5

(a)–(d) Dependence of the scattering coefficients of the bilayer, obtained from numerical simulations done with the finite-element modal method and micromagnetic simulations, on the stripe width. All calculations were performed at frequency 17 GHz.

Figure 6

Convergence of (a) the transmittance and (b) the phase shift of the transmitted wave with increasing polynomial degree $p$ of the finite elements used in calculations.

Figure 7

(a)–(d) Comparison of the values of the bilayer's scattering coefficients predicted by the model from Sec. 4e with results of numerical simulations done with the finite-element (FE) modal method. The dashed vertical lines are the positions of resonances predicted with Eq. (38).