# Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii-Moriya Interactions
## 3 Micromagnetic Simulations  
To test the validity of these analytical findings in realistic situations, micromagnetic simulations have been proven to be an efficient tool for the investigation of spin-wave dynamics in various magnetic textures and geometries. The simulations here are performed in the GPU-accelerated micromagnetic simulations program MuMax3 [73], which solves the time-dependent LLG Equation (1) based on the finite difference method. In our simulations, we used typical magnetic parameters for YIG at zero temperature [74]: \(M_{\mathrm{s}}=0.194\times 10^{5}\) A/m, \(A=3.8\) pJ/m and \(K=10^{4}\) J/m\({}^{3}\). All simulations presented here were performed for a thin film of size \(L_{x}\times L_{y}\times L_{z}\), which discretized with cuboid meshes of dimensions \(l_{x}\times l_{y}\times l_{z}\). The lateral dimensions of unit mesh (\(l_{x}\) and \(l_{y}\)) and the thickness of the film \(L_{z}\) are all smaller than the exchange length of YIG [3]. The simulations were implemented with the mesh size \(2\times 2\times 2\) nm\({}^{3}\). The simulations were split into two stages: the static and dynamic stage. In the first stage, the static stage, the magnetic configuration is stabilized by minimization of the total energy starting from the random magnetic configuration with a high value of damping (\(\alpha=0.5\)). In the dynamic stage of the simulations, the equilibrium magnetic configuration was used to excite a spin-wave beam that propagates through the film with a small damping parameter (\(\alpha=0.0005\)) to ensure long-distance propagation. During this step, a Gaussian type spin-wave beam was continuously generated by a harmonic dynamic external magnetic field following a Gaussian distribution function in a small rectangular region (red double-headed arrow shown in Figure 1). The detailed description of the Gaussian spin-wave beam generation procedure can be found in Ref. [75; 76; 77]. The Gaussian spin-wave beam is clearly visible and does not change with time after continuously exciting a sufficiently long time, which corresponds to a steady spin-wave propagation. Moreover, to avoid spin-wave reflection at the boundaries of the film, absorbing boundary conditions are applied on all boundaries by assigning a large damping constant (\(\alpha=1\)) near the edges. In order to verify the magnonic Snell's law in Equation (4) for the spin-wave propagation through a DMI step interface, we focus on a 4 um \(\times\) 4 um \(\times\) 2 nm nanowire. The spin-wave beams presented here are all exchange-dominated spin waves with 200 nm beam width and 30 nm wavelength generated by an external AC magnetic field with frequency \(f=100\) GHz. The phase diagram of the critical angle \(\theta_{c}\) in the \(D_{1}-D_{2}\) plane is shown in Figure 2b. As \(D_{1}<D_{2}\) in the white region I, spin waves transmit from a thinner medium with a lower refractive index to a denser medium with a higher refractive index, and thus no TIR happens. A gap falls in between the two isofrequency circles--in other words, TIR occurs in all incident angles when \(\delta\) is chosen to be large enough as shown in the white region II. Figure 2c shows the critical angle as a function of the DMI constant \(D_{2}\) in medium B, where the DMI constant of medium A is fixed at \(D_{1}=4\times 10^{-3}\) J/m\({}^{2}\). All incident angle spin waves are totally reflected at a small \(D_{2}\) corresponding to Region II in Figure 2b. After that, the critical angle increases monotonically with \(D_{2}\) and shows a good agreement with the analytical results. In Figure 3a, we show the refracted angle \(\theta_{i}\) as a function of the incident angle \(\theta_{i}\) from micromagnetic simulations (red triangle) and the prediction from Equation (4) (blue curve) with DMI constants \(D_{1}=4\times 10^{-3}\) J/m\({}^{2}\) and \(D_{2}=3.5\times 10^{-3}\) J/m\({}^{2}\), respectively.