# Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii–Moriya Interactions

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## Abstract

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## 1. Introduction

## 2. Analytical Model

#### 2.1. Magnonic Snell’s Law

#### 2.2. Total Internal Reflection

## 3. Micromagnetic Simulations

## 4. Spin-Wave Fiber and Lens

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**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 $\widehat{y}$ 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. ${\theta}_{i,t,r}$ denote their angles with respect to the interface normal. The red double-headed arrow shows the Gaussian distribution AC Magnetic field $\mathbf{h}\left(t\right)$ exciting the spin-wave.

**Figure 2.**(

**a**) Schematic illustrations of reflection and refraction of spin-wave at an interface between two different media in wave vector (${\mathbf{k}}_{x}-{\mathbf{k}}_{y}$) space. The pink and green circles indicate the individual frequency contours of the allowed modes in the same-color-coded media A and B, respectively. The color-coded arrows denote the spin-wave vectors $\mathbf{k}$ propagating in each medium, as indicated by the incident (pink) and refracted (green) rays. The blue arrow denotes the critical angle. (

**b**) Phase diagrams of critical angle ${\theta}_{C}$ in the ${D}_{1}-{D}_{2}$ plane. No TIR exists in the white regions. (

**c**) Critical angle ${\theta}_{c}$ as a function of DMI constants ${D}_{2}$ with a fixed DMI constant ${D}_{1}=4\times {10}^{-3}$ J/m${}^{2}$. The symbols (red squares) are simulation data, and the solid curve represents the analytical results of Equation (5).

**Figure 3.**(

**a**) The refracted angle as a function of the incident angle. Vertical dashed and solid lines correspond to the critical angle ${\theta}_{c}$. (

**b**–

**f**) The micromagnetic simulations results for spin-wave beam reflection and refraction under different incident angles (

**b**) ${\theta}_{i}={17}^{\circ}$, (

**c**) ${\theta}_{i}={41.5}^{\circ}$, (

**d**) ${\theta}_{i}={44}^{\circ}$, (

**e**) ${\theta}_{i}={51.2}^{\circ}$ and (

**f**) ${\theta}_{i}={67}^{\circ}$. The DMI constants in medium A and B are ${D}_{1}=4\times {10}^{-3}$ J/m${}^{2}$ and ${D}_{2}=3.5\times {10}^{-3}$ J/m${}^{2}$, respectively. The color map shows the z component of the magnetization in the snapshot of micromagnetic simulations at some selected time. The black solid lines correspond to the rays of the incident and refractive beams. The red rectangular area is the excitation area of the spin-wave, and the exciting field frequency is $f=100$ GHz.

**Figure 4.**(

**a**) Schematic illustration of a spin-wave fiber. The inset shows the enlarged figure at the interface. (

**b**) Schematic illustration of a spin-wave convex lens. In all of the above figures, the color map shows z component of the magnetization in the snapshot of micromagnetic simulations at some selected time. The spin-wave trajectories are represented by solid red lines with an arrow. The simulated propagation of the spin wave excited by a AC source in blue bars with an exciting frequency $f=100$ GHz.

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**MDPI and ACS Style**

Zhuo, F.; Li, H.; Cheng, Z.; Manchon, A. Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii–Moriya Interactions. *Nanomaterials* **2022**, *12*, 1159.
https://doi.org/10.3390/nano12071159

**AMA Style**

Zhuo F, Li H, Cheng Z, Manchon A. Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii–Moriya Interactions. *Nanomaterials*. 2022; 12(7):1159.
https://doi.org/10.3390/nano12071159

**Chicago/Turabian Style**

Zhuo, Fengjun, Hang Li, Zhenxiang Cheng, and Aurélien Manchon. 2022. "Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii–Moriya Interactions" *Nanomaterials* 12, no. 7: 1159.
https://doi.org/10.3390/nano12071159