**1. Introduction**

Magnonics (or magnon spintronics) is an emerging field concentrating on the generation, detection and manipulation of magnons, the quanta of spin-wave, in ferromagnetic or antiferromagnetic metals and insulators [1–9]. As spin waves in magnetic insulators exhibit both low energy dissipation and long coherence length, these constitute a competitive alternative to electronic devices and are deemed to be a promising candidate as a high-quality information carrier [10–13]. Over the past decades, many properties of spin waves have been demonstrated experimentally, in analogy with electromagnetic waves: excitation and propagation [14–18], reflection and refraction [19–22], interference and diffraction [23–25] and tunneling and the Doppler effect [26–28].

Thus, far, based on recent progress in the fabrication of magnetic nanostructures, various device concepts have been proposed, such as spin-wave logic gates and circuits [10,29,30], waveguides [31,32], multiplexors [33], splitter [34] and diodes [35]. The implementations of those devices is usually achieved by the application of external local magnetic fields [26], spin current [28,36] and magnetic textures (for example, the chiral domain wall) [29,31,37] to control the dispersion relation of spin waves, thereby, steering the spin-wave propagation properties. Despite the soundness of the concepts, however, there are some inherent drawbacks and obstacles to applications. First, generating a local high-frequency magnetic field on micro-sized devices complicates the structure design, and the local field is often spatially inhomogeneous, which can inhibit the benefits of the device [38]. In addition, unstable magnetic textures under external excitation and at room temperature may give

**Citation:** 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

Academic Editor: José Antonio Sánchez-Gil

Received: 17 February 2022 Accepted: 28 March 2022 Published: 31 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

rise to poor reliability and high bit-error rates. Therefore, it is desirable to find a new method to manipulate the propagation of spin waves.

Recent discoveries in graded-index magnonics and magnonic metamaterials provide a new way to manipulate spin-wave propagation [39,40], which is inspired by the fields of graded-index photonics (or photonic metamaterials) [41–43]. The core idea of gradedindex magnonics is to manipulate spin-wave propagation by designing a spatially varied magnonic refractive index. In magnetic thin films with in-plane magnetization, the spinwave dispersion relation described by the Landau–Lifshitz–Gilbert (LLG) equation exhibits a much more complex structure compared to the isotropic dispersion relation of light. This offers extremely rich opportunities to modulate the magnonic refractive index.

Up to now, it has been shown that the graded magnonic refractive index can be created by modification of the material properties, such as non-uniform saturation magnetization or exchange constant [44–47], the magnetic anisotropy [19,20] or the internal magnetic field [37,48]. This index can be also achieved by utilizing a non-uniform external magnetic field [39,49–51], electric field (voltage) [52,53] or temperature [54,55]. Therefore, gradedindex magnonics are expected to overcome the current limitation of magnonics and pave feasible routes for the implementation of spin-wave devices.

In this paper, we theoretically propose a magnonic metamaterial, in which we modulate the refractive index of spin waves with the inhomogeneous Dzyaloshinskii–Moriya interaction (DMI) to avoid a barely controllable local magnetic field and unstable magnetic textures. The DMI is an antisymmetric exchange interaction arising from the lack of structural inversion symmetry in magnetic films [56,57]. It has been found both for bulk materials [58–60] and magnetic interfaces [61].

Here, we focus on a spatial inhomogeneous interfacial DMI present in ferromagnet/heavy metal (FM/HM) bilayers realized by tuning the thickness of ferromagnetic layer or HM layer [62–65], the degree of hybridization between 3d-5d states [66] or utilizing a local gating [67]. We begin our work by rapidly deriving the spin-wave dispersion relation with spatially modulated DMI. Then, we further study spin-wave refraction and reflection at the interface between two magnetic media with different DMI and build a generalized Snell's law of spin waves, similar to Snell's law in optics.

According to the magnonic Snell's law, spin-wave can also experience total internal reflection (TIR) at the DMI step interface when their incident angle is larger than a critical value (i.e., the critical angle). Moreover, magnonic Snell's law and TIR are observed and confirmed by micromagnetic simulations. Utilizing the artificial magnonic metamaterials based on spatially modulated DMI, a spin-wave fiber owing to TIR (which can transmit spin waves over a long distance) and a spin-wave lens holding tremendous possibility to build spin-wave circuits are proposed as proofs of concept.

The paper is organized as follows. In Section 2, we introduce our theoretical model and method. Detailed results of micromagnetic simulations are presented in Section 3. Then, we discuss the realization of spin-wave fibers and lenses in Section 4. Finally, we end the paper with a summary in Section 5.

### **2. Analytical Model**
