# 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

- Kruglyak, V.V.; Hicken, R.J. Magnonics: Experiment to prove the concept. J. Magn. Magn. Mater.
**2006**, 306, 191. [Google Scholar] [CrossRef] [Green Version] - Kruglyak, V.V.; Demokritov, S.O.; Grundler, D. Magnonics. J. Phys. D Appl. Phys.
**2010**, 43, 264001. [Google Scholar] [CrossRef] - Serga, A.A.; Chumak, A.V.; Hillebrands, B. YIG magnonics. J. Phys. D Appl. Phys.
**2010**, 43, 264002. [Google Scholar] [CrossRef] - Lenk, B.; Ulrichs, H.; Garbs, F.; Münzenberg, M. The Building Blocks of Magnonics. Phys. Rep.
**2011**, 507, 107. [Google Scholar] [CrossRef] [Green Version] - Demokritov, S.O.; Slavin, A.N. Magnonics: From Fundamentals to Applications, 1st ed.; Springer: New York, NY, USA, 2013; p. 125. [Google Scholar]
- Chumak, A.V.; Vasyuchka, V.I.; Serga, A.A.; Hillebrands, B. Magnon spintronics. Nat. Phys.
**2015**, 11, 453. [Google Scholar] [CrossRef] - Baltz, V.; Manchon, A.; Tsoi, M.; Moriyama, T.; Ono, T.; Tserkovnyak, Y. Antiferromagnetic spintronics. Rev. Mod. Phys.
**2018**, 90, 015005. [Google Scholar] [CrossRef] [Green Version] - Barman, A.; Gubbiotti, G.; Ladak, S.; Adeyeye, A.O.; Krawczyk, M.; Gräfe, J.; Adelmann, C.; Cotofana, S.; Naeemi, A.; Vasyuchka, V.I.; et al. The 2021 Magnonics Roadmap. J. Phys. Condens. Matter
**2021**, 33, 413001. [Google Scholar] [CrossRef] [PubMed] - Bonbien, V.; Zhuo, F.; Salimath, A.; Ly, O.; About, A.; Manchon, A. Topological aspects of antiferromagnets. J. Phys. D Appl. Phys.
**2022**, 55, 103002. [Google Scholar] [CrossRef] - Khitun, A.; Bao, M.; Wang, K.L. Magnonic logic circuits. J. Phys. D Appl. Phys.
**2010**, 43, 264005. [Google Scholar] [CrossRef] - Chumak, A.V.; Serga, A.A.; Hillebrands, B. Magnon transistor for all-magnon data processing. Nat. Commun.
**2014**, 5, 4700. [Google Scholar] [CrossRef] [Green Version] - Zhuo, F.; Li, H.; Manchon, A. Topological phase transition and thermal Hall effect in kagome ferromagnets. Phys. Rev. B
**2021**, 104, 144422. [Google Scholar] [CrossRef] - Zhuo, F.; Li, H.; Manchon, A. Topological thermal Hall effect and magnonic edge states in kagome ferromagnets with bond anisotropy. New J. Phys.
**2022**, 24, 023033. [Google Scholar] [CrossRef] - Liu, Z.; Giesen, F.; Zhu, X.; Sydora, R.D.; Freeman, M.R. Spin Wave Dynamics and the Determination of Intrinsic Damping in Locally Excited Permalloy Thin Films. Phys. Rev. Lett.
**2007**, 98, 087201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Serga, A.A.; Demokritov, S.O.; Hillebrands, B.; Slavin, A.N. Self-Generation of Two-Dimensional Spin-Wave Bullets. Phys. Rev. Lett.
**2014**, 92, 117203. [Google Scholar] [CrossRef] [PubMed] - Covington, M.; Crawford, T.M.; Parker, G.J. Time-Resolved Measurement of Propagating Spin Waves in Ferromagnetic Thin Films. Phys. Rev. Lett.
**2002**, 89, 237202. [Google Scholar] [CrossRef] [PubMed] - Demidov, V.E.; Jersch, J.; Demokritov, S.O.; Rott, K.; Krzysteczko, P.; Reiss, G. Transformation of propagating spin-wave modes in microscopic waveguides with variable width. Phys. Rev. B
**2009**, 79, 054417. [Google Scholar] [CrossRef] - Demidov, V.E.; Kostylev, M.P.; Rott, K.; Münchenberger, J.; Reiss, G.; Demokritov, S.O. Excitation of short-wavelength spin waves in magnonic waveguides. Appl. Phys. Lett.
**2011**, 99, 082507. [Google Scholar] [CrossRef] - Stigloher, J. Snell’s Law for Spin Waves. Phys. Rev. Lett.
**2016**, 117, 037204. [Google Scholar] [CrossRef] - Kim, S.K.; Choi, S.; Lee, K.S.; Han, D.S.; Jung, D.E.; Choi, Y.S. Negative refraction of dipole-exchange spin waves through a magnetic twin interface in restricted geometry. Appl. Phys. Lett.
**2008**, 92, 212501. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Zhang, B.; Cao, Y.; Yan, P. Probing the Dzyaloshinskii–Moriya Interaction via the Propagation of Spin Waves in Ferromagnetic Thin Films. Phys. Rev. Applied
**2018**, 10, 054018. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Cao, Y.; Yan, P. Goos–Hänchen effect of spin waves at heterochiral interfaces. Phys. Rev. B
**2019**, 100, 064421. [Google Scholar] [CrossRef] [Green Version] - Choi, S.K.; Lee, K.S.; Kim, S.K. Spin-wave interference. Appl. Phys. Lett.
**2006**, 89, 062501. [Google Scholar] [CrossRef] [Green Version] - Perzlmaier, K.; Woltersdorf, G.; Back, C.H. Observation of the propagation and interference of spin waves in ferromagnetic thin films. Phys. Rev. B
**2008**, 77, 054425. [Google Scholar] [CrossRef] [Green Version] - Birt, D.R.; Gorman, B.O.; Tsoi, M.; Li, X.; Demidov, V.E.; Demokritov, S.O. Diffraction of spin waves from a submicrometer-size defect in a microwaveguide. Appl. Phys. Lett.
**2009**, 95, 122510. [Google Scholar] [CrossRef] - Demokritov, S.O.; Serga, A.A.; André, A.; Demidov, V.E.; Kostylev, M.P.; Hillebrands, B.; Slavin, A.N. Tunneling of Dipolar Spin Waves through a Region of Inhomogeneous Magnetic Field. Phys. Rev. Lett.
**2004**, 93, 047201. [Google Scholar] [CrossRef] - Stancil, D.D.; Henty, B.E.; Cepni, A.G.; Van’t Hof, J.P. Observation of an inverse Doppler shift from left-handed dipolar spin waves. Phys. Rev. B
**2006**, 74, 060404. [Google Scholar] [CrossRef] [Green Version] - Vlaminck, V.; Bailleul, M. Current-Induced Spin-Wave Doppler Shift. Science
**2008**, 322, 410. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hertel, R.; Wulfhekel, W.; Kirschner, J. Domain-Wall Induced Phase Shifts in Spin Waves. Phys. Rev. Lett.
**2004**, 93, 257202. [Google Scholar] [CrossRef] [Green Version] - Lee, K.S.; Kim, S.K. Conceptual design of spin wave logic gates based on a Mach–Zehnder-type spin wave interferometer for universal logic functions. J. Appl. Phys.
**2008**, 104, 053909. [Google Scholar] [CrossRef] [Green Version] - Sanchez, F.G.; Borys, P.; Soucaille, R.; Adam, J.P.; Stamps, R.L.; Kim, J.V. Narrow Magnonic Waveguides Based on Domain Walls. Phys. Rev. Lett.
**2015**, 114, 247206. [Google Scholar] [CrossRef] - Mulkers, J.; Waeyenberge, B.V.; Milošević, M.V. Effects of spatially engineered Dzyaloshinskii–Moriya interaction in ferromagnetic films. Phys. Rev. B
**2017**, 95, 144401. [Google Scholar] [CrossRef] [Green Version] - Vogt, K.; Fradin, F.Y.; Pearson, J.E.; Sebastian, T.; Bader, S.D.; Hillebrands, B.; Hoffmann, A.; Schultheiss, H. Realization of a spin-wave multiplexer. Nat. Commun.
**2014**, 5, 3727. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sadovnikov, A.V.; Davies, C.S.; Grishin, S.V.; Kruglyak, V.V.; Romanenko, D.V.; Sharaevskii, Y.P.; Nikitov, S.A. Magnonic beam splitter: The building block of parallel magnonic circuitry. Appl. Phys. Lett.
**2015**, 106, 192406. [Google Scholar] [CrossRef] - Lan, J.; Yu, W.; Wu, R.; Xiao, J. Spin-Wave Diode. Phys. Rev. X
**2015**, 5, 041049. [Google Scholar] [CrossRef] - Seo, S.M.; Lee, K.J.; Yang, H.; Ono, T. Current-Induced Control of Spin-Wave Attenuation. Phys. Rev. Lett.
**2009**, 102, 147202. [Google Scholar] [CrossRef] [PubMed] - Yu, W.; Lan, J.; Wu, R.; Xiao, J. Magnetic Snell’s law and spin-wave fiber with Dzyaloshinskii–Moriya interaction. Phys. Rev. B
**2016**, 94, 140410. [Google Scholar] [CrossRef] [Green Version] - Jamali, M.; Kwon, J.H.; Seo, S.M.; Lee, K.J.; Yang, H. Spin wave nonreciprocity for logic device applications. Sci. Rep.
**2013**, 3, 3160. [Google Scholar] [CrossRef] [PubMed] - Davies, C.S.; Francis, A.; Sadovnikov, A.V.; Chertopalov, S.V.; Bryan, M.T.; Grishin, S.V.; Allwood, D.A.; Sharaevskii, Y.P.; Nikitov, S.A.; Kruglyak, V.V. Towards graded-index magnonics: Steering spin waves in magnonic networks. Phys. Rev. B
**2015**, 92, 020408. [Google Scholar] [CrossRef] [Green Version] - Davies, C.S.; Kruglyak, V.V. Graded-index magnonics. Low Temp. Phys.
**2015**, 41, 760. [Google Scholar] [CrossRef] [Green Version] - Marchand, E.W. Gradient Index Optics, 1st ed.; ScienceDirect: London, UK, 1978; p. 125. [Google Scholar]
- Chen, H.; Chan, C.T.; Sheng, P. Transformation optics and metamaterials. Nat. Mater.
**2010**, 9, 387. [Google Scholar] [CrossRef] [PubMed] - Pendry, J.B.; Domínguez, A.I.F.; Luo, Y.; Zhao, R. Capturing photons with transformation optics. Nat. Phys.
**2013**, 9, 518. [Google Scholar] [CrossRef] - Dadoenkova, Y.S.; Dadoenkova, N.N.; Lyubchanskii, I.L.; Sokolovskyy, M.L.; Kłos, J.W.; Romero-Vivas, J.; Krawczyk, M. Huge Goos–Hänchen effect for spin waves: A promising tool for study magnetic properties at interfaces. Appl. Phys. Lett.
**2012**, 101, 042404. [Google Scholar] [CrossRef] - Xi, H.; Xue, S. Spin-wave propagation and transmission along magnetic nanowires in long wavelength regime. J. Appl. Phys.
**2007**, 101, 123905. [Google Scholar] [CrossRef] - Xi, H.; Wang, X.; Zheng, Y.; Ryan, P.J. Spin wave propagation and coupling in magnonic waveguides. J. Appl. Phys.
**2008**, 104, 063921. [Google Scholar] [CrossRef] - Vogel, M.; Aßmann, R.; Pirro, P.; Chumak, A.V.; Hillebrands, B.; Freymann, G.V. Control of Spin-Wave Propagation using Magnetisation Gradients. Sci. Rep.
**2018**, 8, 11099. [Google Scholar] [CrossRef] - Xing, X.; Zhou, Y. Fiber optics for spin waves. NPG Asia Mater.
**2016**, 8, e246. [Google Scholar] [CrossRef] [Green Version] - Perez, N.; Diaz, L.L. Magnetic field induced spin-wave energy focusing. Phys. Rev. B
**2015**, 92, 014408. [Google Scholar] [CrossRef] [Green Version] - Houshang, A.; Iacocca, E.; Dürrenfeld, P.; Sani, S.; Akerman, J.; Dumas, R. Spin-wave-beam driven synchronization of nanocontact spin-torque oscillators. Nat. Nanotechnol.
**2016**, 11, 280. [Google Scholar] [CrossRef] [PubMed] - Gruszecki, P.; Krawczyk, M. Spin wave beam propagation in ferromagnetic thin film with graded refractive index: Mirage effect and prospective applications. Phys. Rev. B
**2018**, 97, 094424. [Google Scholar] - Wang, S.; Guan, X.; Cheng, X.; Lian, C.; Huang, T.; Miao, X. Spin-wave propagation steered by electric field modulated exchange interaction. Sci. Rep.
**2016**, 6, 31783. [Google Scholar] [CrossRef] [Green Version] - Kakizakai, H.; Yamada, K.; Ando, F.; Kawaguchi, M.; Koyama, T.; Kim, S.; Moriyama, T.; Chiba, D.; Ono, T. Influence of sloped electric field on magnetic-field-induced domain wall creep in a perpendicularly magnetized Co wire. Jpn. J. Appl. Phys.
**2017**, 56, 050305. [Google Scholar] [CrossRef] - Vogel, M.; Chumak, A.V.; Waller, E.H.; Langner, T.; Vasyuchka, V.I.; Hillebrands, B.; Freymann, G.V. Optically reconfigurable magnetic materials. Nat. Phys.
**2015**, 11, 487. [Google Scholar] [CrossRef] [Green Version] - Busse, F.; Mansurova, M.; Lenk, B.; Ehe, M.; Münzenberg, M. A scenario for magnonic spin-wave traps. Sci. Rep.
**2015**, 5, 12824. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dzyaloshinskii, I.E. A thermodynamic theory of “weak”ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids
**1958**, 4, 241. [Google Scholar] [CrossRef] - Moriya, T. Anisotropic Superexchange Interaction and Weak Ferromagnetism. Phys. Rev.
**1960**, 120, 91. [Google Scholar] [CrossRef] - Mühlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; Böni, P. Skyrmion Lattice in a Chiral Magnet. Science
**2009**, 323, 915. [Google Scholar] [CrossRef] [Green Version] - Huang, S.X.; Chien, C.L. Extended Skyrmion Phase in Epitaxial FeGe(111) Thin Films. Phys. Rev. Lett.
**2012**, 108, 267201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhuo, F.; Sun, Z.Z. Field-driven Domain Wall Motion in Ferromagnetic Nanowires with Bulk Dzyaloshinskii–Moriya Interaction. Sci. Rep.
**2012**, 6, 25122. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fert, A.; Cros, V.; Sampaio, J. Skyrmions on the track. Nat. Nanotechnol.
**2013**, 8, 152. [Google Scholar] [CrossRef] [PubMed] - Chen, G.; Zhu, J.; Quesada, A.; Li, J.; N’Diaye, A.T.; Huo, Y.; Ma, T.P.; Chen, Y.; Kwon, H.Y.; Won, C.; et al. Novel Chiral Magnetic Domain Wall Structure in Fe/Ni/Cu(001) Films. Phys. Rev. Lett.
**2013**, 110, 177204. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, G.; Ma, T.; N’Diaye, A.T.; Kwon, H.; Won, C.; Wu, Y.; Schmid, A.K. Tailoring the chirality of magnetic domain walls by interface engineering. Nat. Commun.
**2013**, 4, 2671. [Google Scholar] [CrossRef] [Green Version] - Torrejon, J.; Kim, J.; Sinha, J.; Mitani, S.; Hayashi, M.; Yamanouchi, M.; Ohno, H. Interface control of the magnetic chirality in CoFeB/MgO heterostructures with heavy-metal underlayers. Nat. Commun.
**2014**, 5, 4655. [Google Scholar] [CrossRef] - Tacchi, S.; Troncoso, R.E.; Ahlberg, M.; Gubbiotti, G.; Madami, M.; Akerman, J.; Landeros, P. Interfacial Dzyaloshinskii–Moriya Interaction in Pt/CoFeB Films: Effect of the Heavy-Metal Thickness. Phys. Rev. Lett.
**2017**, 118, 147201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Belabbes, A.; Bihlmayer, G.; Bechstedt, F.; Blügel, S.; Manchon, A. Hund’s Rule-Driven Dzyaloshinskii–Moriya Interaction at 3d-5d Interfaces. Phys. Rev. Lett.
**2016**, 117, 247202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nawaoka, K.; Miwa, S.; Shiota, Y.; Mizuochi, N.; Suzuki, Y. Voltage induction of interfacial Dzyaloshinskii–Moriya interaction in Au/Fe/MgO artificial multilayer. Appl. Phys. Express
**2015**, 8, 063004. [Google Scholar] [CrossRef] - Gilbert, T.L. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Magn.
**2004**, 40, 3443–3449. [Google Scholar] [CrossRef] - Bogdanov, A.N.; Rößler, U.K. Chiral Symmetry Breaking in Magnetic Thin Films and Multilayers. Phys. Rev. Lett.
**2001**, 87, 037203. [Google Scholar] [CrossRef] - Moon, J.H.; Seo, S.M.; Lee, K.J.; Kim, K.W.; Ryu, J.; Lee, H.W.; McMichael, R.D.; Stiles, M.D. Spin-wave propagation in the presence of interfacial Dzyaloshinskii–Moriya interaction. Phys. Rev. B
**2013**, 88, 184404. [Google Scholar] [CrossRef] [Green Version] - Di, K.; Zhang, V.L.; Lim, H.S.; Ng, S.C.; Kuok, M.H.; Qiu, X.; Yang, H. Asymmetric spin-wave dispersion due to Dzyaloshinskii–Moriya interaction in an ultrathin Pt/CoFeB film. Appl. Phys. Lett.
**2015**, 106, 052403. [Google Scholar] [CrossRef] - Manchon, A.; Ndiaye, P.; Moon, J.H.; Lee, H.W.; Lee, K.J. Magnon-mediated Dzyaloshinskii–Moriya torque in homogeneous ferromagnets. Phys. Rev. B
**2014**, 90, 224403. [Google Scholar] [CrossRef] [Green Version] - Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Van Waeyenberge, B. The design and verification of MuMax3. AIP Adv.
**2014**, 4, 107133. [Google Scholar] [CrossRef] [Green Version] - Klingler, S.; Chumak, A.V.; Mewes, T.; Khodadadi, B.; Mewes, C.; Dubs, C.; Surzhenko, O.; Hillebrands, B.; Conca, A. Measurements of the exchange stiffness of YIG films using broadband ferromagnetic resonance techniques. J. Phys. D Appl. Phys.
**2015**, 48, 015001. [Google Scholar] [CrossRef] - Gruszecki, P.; Romero-Vivas, J.; Dadoenkova, Y.S.; Dadoenkova, N.N.; LyubchanskiiI, L.; Krawczyk, M. Goos–Hänchen effect and bending of spin wave beams in thin magnetic films. Appl. Phys. Lett.
**2014**, 105, 242406. [Google Scholar] [CrossRef] - Gruszecki, P.; Dadoenkova, Y.S.; Dadoenkova, N.N.; Lyubchanskii, I.L.; Vivas, J.R.; Guslienko, K.Y.; Krawczyk, M. Influence of magnetic surface anisotropy on spin wave reflection from the edge of ferromagnetic film. Phys. Rev. B
**2015**, 92, 054427. [Google Scholar] [CrossRef] [Green Version] - Gruszecki, P.; Mailyan, M.; Gorobets, O.; Krawczyk, M. Goos–Hänchen shift of a spin-wave beam transmitted through anisotropic interface between two ferromagnets. Phys. Rev. B
**2017**, 95, 014421. [Google Scholar] [CrossRef] [Green Version] - Klos, J.W.; Gruszecki, P.; Serebryannikov, A.E.; Krawczyk, M. All-Angle Collimation for Spin Waves. IEEE Magn. Lett.
**2015**, 6, 3500804. [Google Scholar] [CrossRef] [Green Version] - Sanchez, F.G.; Borys, P.; Vansteenkiste, A.; Kim, J.V.; Stamps, R.L. Nonreciprocal spin-wave channeling along textures driven by the Dzyaloshinskii–Moriya interaction. Phys. Rev. B
**2014**, 89, 224408. [Google Scholar] [CrossRef] [Green Version] - Lee, S.J.; Moon, J.H.; Lee, H.W.; Lee, K.J. Spin-wave propagation in the presence of inhomogeneous Dzyaloshinskii–Moriya interactions. Phys. Rev. B
**2017**, 96, 184433. [Google Scholar] [CrossRef] - Korner, H.S.; Stigloher, J.; Back, C.H. Excitation and tailoring of diffractive spin-wave beams in NiFe using nonuniform microwave antennas. Phys. Rev. B
**2017**, 96, 100401. [Google Scholar] [CrossRef] [Green Version] - Wang, H.; Flacke, L.; Wei, W.; Liu, S.; Jia, H.; Chen, J.; Sheng, L.; Zhang, J.; Zhao, M.; Guo, C.; et al. Sub-50 nm wavelength spin waves excited by low-damping Co
_{25}Fe_{75}nanowires. Appl. Phys. Lett.**2021**, 119, 152402. [Google Scholar] [CrossRef] - Janson, O.; Rousochatzakis, I.; Tsirlin, A.A.; Belesi, M.; Leonov, A.A.; Rößler, U.K.; van den Brink, J.; Rosner, H. The quantum nature of skyrmions and half-skyrmions in Cu
_{2}OSeO_{3}. Nat. Commun.**2014**, 5, 5376. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qian, F.; Bannenberg, L.; Wilhelm, H.; Chaboussant, G.; Debeer-Schmitt, L.M.; Schmidt, M.P.; Aqeel, A.; Palstra, T.T.M.; Brück, E.; Lefering, A.J.E.; et al. New magnetic phase of the chiral skyrmion material Cu
_{2}OSeO_{3}. Sci. Adv.**2018**, 4, eaat7323. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Deng, L.; Wu, H.C.; Litvinchuka, A.P.; Yuan, N.F.Q.; Lee, J.-J.; Dahal, R.; Berger, H.; Yang, H.-D.; Chu, C.-W. Room-temperature skyrmion phase in bulk Cu
_{2}OSeO_{3}under high pressures. Proc. Natl. Acad. Sci. USA**2020**, 117, 8783–8787. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Avci, C.O.; Rosenberg, E.; Caretta, L.; Büttner, F.; Mann, M.; Marcus, C.; Bono, D.; Ross, C.A.; Beach, G.S.D. Interface-driven chiral magnetism and current-driven domain walls in insulating magnetic garnets. Nat. Nanotechnol.
**2019**, 14, 561–566. [Google Scholar] [CrossRef] [PubMed] - Caretta, L.; Rosenberg, E.; Büttner, F.; Fakhrul, T.; Gargiani, P.; Valvidares, M.; Chen, Z.; Reddy, P.; Muller, D.A.; Ross, C.A.; et al. Interfacial Dzyaloshinskii–Moriya interaction arising from rare-earth orbital magnetism in insulating magnetic oxides. Nat. Commun.
**2020**, 11, 1090. [Google Scholar] [CrossRef] [PubMed] - Ding, S.; Baldrati, L.; Ross, A.; Ren, Z.; Wu, R.; Becker, S.; Yang, J.; Jakob, G.; Brataas, A.; Kläui, M. Identifying the origin of the nonmonotonic thickness dependence of spin–orbit torque and interfacial Dzyaloshinskii–Moriya interaction in a ferrimagnetic insulator heterostructure. Phys. Rev. B
**2020**, 102, 054425. [Google Scholar] [CrossRef]

**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