# Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii-Moriya Interactions
## 2 Analytical Model  
### Magnonic Snell's Law  
We consider a thin magnetic film in the \(x-y\) plane with the thickness much smaller than lateral dimensions of the film (\(L_{z}\ll L_{x},L_{y}\)), whose initial magnetization is homogeneous along the \(\hat{\mathbf{y}}\) direction. The magnetization dynamics are governed by the LLG Equation [68],  
\[\frac{\partial\mathbf{m}}{\partial t}=-\frac{\gamma}{M_{s}}\mathbf{m}\times \mathbf{H}_{eff}+\alpha\mathbf{m}\times\frac{\partial\mathbf{m}}{\partial t}, \tag{1}\]  
where \(\mathbf{m}\) is the unit direction of local magnetization \(\mathbf{M}=M_{s}\mathbf{m}\) with a saturation magnetization \(M_{s}\). \(\alpha\) is the phenomenological [PERSON] damping constant, and \(\gamma\) is the gyromagnetic ratio. Here, \(\mathbf{H}_{eff}=A^{*}\
abla^{2}\mathbf{m}-D^{*}(x)(\hat{\mathbf{z}}\times \
abla)\times\mathbf{m}-K^{*}m_{y}\hat{\mathbf{y}}\) is the effective field [69], and \(A^{*}=2A/\mu_{0}M_{s}\), \(D^{*}(x)=2D(x)/\mu_{0}M_{s}\), \(K^{*}=2K/\mu_{0}M_{s}\). \(A\) is the symmetric exchange constant, \(D(x)\) is the interfacial antisymmetric DMI constant spatially inhomogeneous along the \(x\) direction, \(K\) is the in-plane anisotropy and \(\mu_{0}\) is the permeability of vacuum. Under the perturbative approximation, the small-amplitude spin waves propagating in the \(x-y\) plane take the following form [70]:  
\[\mathbf{m}=\mathbf{\hat{y}}+\delta\mathbf{m}\exp{[i(\mathbf{\hat{k}}\cdot \mathbf{\hat{r}}-wt)]}, \tag{2}\]  
where \(\delta\mathbf{m}=(\delta m_{x},0,\delta m_{z})\) is the spin-wave contribution to magnetization (\(|\delta\mathbf{m}|\ll 1\)). \(\delta\mathbf{k}=(\mathbf{k}_{x},\mathbf{k}_{y},0)\) is the spin-wave wavevector. Considering the system shown in Figure 1, we use a DMI step (i.e., \(D=D_{1}\) in medium A and \(D=D_{2}\) in medium B) to induce a difference in spin-wave dispersion relations between two magnetic domains. Inserting Equation (2) into Equation (1) and neglecting higher order terms, we obtain the spin-wave dispersion relation in each region [71; 72],  
\[\omega(\mathbf{k}_{n})=\gamma\mu_{0}(K^{*}+A^{*}\mathbf{k}_{n}^{2}-D_{n}^{*} \mathbf{k}_{n,y}), \tag{3}\]  
with \(D_{n}^{*}=2D_{n}/\mu_{0}M_{s}\) and \(\mathbf{k}_{n}=\sqrt{\mathbf{k}_{n,x}^{2}+\mathbf{k}_{n,y}^{2}}\). The spin-wave group velocity is \(\mathbf{v}_{\mathbf{g},n}=\partial\omega/\partial\mathbf{k}_{n}=2A^{*} \mathbf{k}_{g,n}\), where \(\mathbf{k}_{g,n}=\mathbf{k}_{n}-\delta_{n}\mathbf{\hat{y}}\) and \(\delta_{n}=D_{n}^{*}/2A^{*}\). To simplify the model, we assume that the group velocity is parallel to the phase velocity at each point of the dispersion relation--that is to say, the dispersion relation is isotropic. Equation (3) represents an isofrequency circle with radius \(\mathbf{k}_{g,n}\) in momentum space, whose center deviates from the origin by \(\delta_{n}\) in \(-\mathbf{\hat{y}}\) direction as illustrated in Figure 2a. Nevertheless, in the magnetic films with in-plane magnetization, the spin-wave dispersion relation is anisotropic at low frequencies, where dipolar contribution dominates. When increasing frequency, the isofrequency contours smoothly transform through elliptical to almost circular. Consequently, the dispersion relation is isotropic as determined by the exchange interactions at high frequencies. In the following simulations, we use quite high frequency spin waves (100 GHz), and thus the spin-wave dynamic is determined by the exchange interactions. Figure 1: Schematic illustration of spin-wave transmission and reflection at an interface between media A and B with different interfacial DMI in a thin YIG film. The interfacial DMI step here is realized by utilizing two different HM layers (HM\({}_{1}\) and HM\({}_{2}\)) below the YIG film. The blue arrows along the \(\hat{y}\) direction denote the magnetization \(\mathbf{m}\). \(\mathbf{k}_{i}\), \(\mathbf{k}_{t}\) and \(\mathbf{k}_{r}\) are the wave vectors of the incident, refracted and reflected spin-wave shown as the yellow and red arrows, respectively.