# Spin-Wave Channeling in Magnetization-Graded Nanostrips

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## Abstract

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## 1. Introduction

## 2. Results and Discussion

## 3. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Dynamic Matrix Method

## Appendix B. Dipolar Interaction

## Appendix C. Exchange Interaction

## References

- Veerakumar, V.; Camley, R.E. Magnon focusing in thin ferromagnetic films. Phys. Rev. B
**2006**, 74, 214401. [Google Scholar] [CrossRef] - Demidov, V.E.; Demokritov, S.O.; Birt, D.; O’Gorman, B.; Tsoi, M.; Li, X. Radiation of spin waves from the open end of a microscopic magnetic-film waveguide. Phys. Rev. B
**2009**, 80, 014429. [Google Scholar] [CrossRef] - Schneider, T.; Serga, A.A.; Chumak, A.V.; Sandweg, C.W.; Trudel, S.; Wolff, S.; Kostylev, M.P.; Tiberkevich, V.S.; Slavin, A.N.; Hillebrands, B. Nondiffractive Subwavelength Wave Beams in a Medium with Externally Controlled Anisotropy. Phys. Rev. Lett.
**2010**, 104, 197203. [Google Scholar] [CrossRef] [PubMed] - Mansfeld, S.; Topp, J.; Martens, K.; Toedt, J.N.; Hansen, W.; Heitmann, D.; Mendach, S. Spin Wave Diffraction and Perfect Imaging of a Grating. Phys. Rev. Lett.
**2012**, 108, 047204. [Google Scholar] [CrossRef] - Sebastian, T.; Brächer, T.; Pirro, P.; Serga, A.A.; Hillebrands, B.; Kubota, T.; Naganuma, H.; Oogane, M.; Ando, Y. Nonlinear Emission of Spin-Wave Caustics from an Edge Mode of a Microstructured Co
_{2}Mn_{0.6}Fe_{0.4}Si Waveguide. Phys. Rev. Lett.**2013**, 110, 067201. [Google Scholar] [CrossRef] - Gieniusz, R.; Ulrichs, H.; Bessonov, V.D.; Guzowska, U.; Stognii, A.I.; Maziewski, A. Single antidot as a passive way to create caustic spin-wave beams in yttrium iron garnet films. Appl. Phys. Lett.
**2013**, 102, 102409. [Google Scholar] [CrossRef] - Kim, J.V.; Stamps, R.L.; Camley, R.E. Spin Wave Power Flow and Caustics in Ultrathin Ferromagnets with the Dzyaloshinskii-Moriya Interaction. Phys. Rev. Lett.
**2016**, 117, 197204. [Google Scholar] [CrossRef] - Bible, J.J.; Camley, R.E. Focusing of high-wave-vector magnons. Phys. Rev. B
**2017**, 95, 224412. [Google Scholar] [CrossRef] - Krivoruchko, V.N.; Savchenko, A.S.; Kruglyak, V.V. Electric-field control of spin-wave power flow and caustics in thin magnetic films. Phys. Rev. B
**2018**, 98, 024427. [Google Scholar] [CrossRef] - Gallardo, R.A.; Alvarado-Seguel, P.; Kákay, A.; Lindner, J.; Landeros, P. Spin-wave focusing induced by dipole-dipole interaction in synthetic antiferromagnets. Phys. Rev. B
**2021**, 104, 174417. [Google Scholar] [CrossRef] - Winter, J.M. Bloch Wall Excitation. Application to Nuclear Resonance in a Bloch Wall. Phys. Rev.
**1961**, 124, 452–459. [Google Scholar] [CrossRef] - Wagner, K.; Kákay, A.; Schultheiss, K.; Henschke, A.; Sebastian, T.; Schultheiss, H. Magnetic domain walls as reconfigurable spin-wave nanochannels. Nat. Nanotechnol.
**2016**, 11, 432–436. [Google Scholar] [CrossRef] [PubMed] - Sluka, V.; Schneider, T.; Gallardo, R.A.; Kákay, A.; Weigand, M.; Warnatz, T.; Mattheis, R.; Roldán-Molina, A.; Landeros, P.; Tiberkevich, V.; et al. Emission and propagation of 1D and 2D spin waves with nanoscale wavelengths in anisotropic spin textures. Nat. Nanotechnol.
**2019**, 14, 328–333. [Google Scholar] [CrossRef] [PubMed] - Kataoka, M. Spin Waves in Systems with Long Period Helical Spin Density Waves Due to the Antisymmetric and Symmetric Exchange Interactions. J. Phys. Soc. Jpn.
**1987**, 56, 3635–3647. [Google Scholar] [CrossRef] - Cortés-Ortuño, D.; Landeros, P. Influence of the Dzyaloshinskii–Moriya interaction on the spin-wave spectra of thin films. J. Phys. Condens. Matter
**2013**, 25, 156001. [Google Scholar] [CrossRef] - Iguchi, Y.; Uemura, S.; Ueno, K.; Onose, Y. Nonreciprocal magnon propagation in a noncentrosymmetric ferromagnet LiFe
_{5}O_{8}. Phys. Rev. B**2015**, 92, 184419. [Google Scholar] [CrossRef] - Di, K.; Zhang, V.L.; Lim, H.S.; Ng, S.C.; Kuok, M.H.; Yu, J.; Yoon, J.; Qiu, X.; Yang, H. Direct Observation of the Dzyaloshinskii-Moriya Interaction in a Pt/Co/Ni Film. Phys. Rev. Lett.
**2015**, 114, 047201. [Google Scholar] [CrossRef] - Cho, J.; Kim, N.H.; Lee, S.; Kim, J.S.; Lavrijsen, R.; Solignac, A.; Yin, Y.; Han, D.S.; van Hoof, N.J.J.; Swagten, H.J.M.; et al. Thickness dependence of the interfacial Dzyaloshinskii-Moriya interaction in inversion symmetry broken systems. Nat. Commun.
**2015**, 6, 7635. [Google Scholar] [CrossRef] [PubMed] - Nembach, H.T.; Shaw, J.M.; Weiler, M.; Jue, E.; Silva, T.J. Linear relation between Heisenberg exchange and interfacial Dzyaloshinskii-Moriya interaction in metal films. Nat. Phys.
**2015**, 11, 825–829. [Google Scholar] [CrossRef] - Belmeguenai, M.; Adam, J.P.; Roussigné, Y.; Eimer, S.; Devolder, T.; Kim, J.V.; Cherif, S.M.; Stashkevich, A.; Thiaville, A. Interfacial Dzyaloshinskii-Moriya interaction in perpendicularly magnetized Pt/Co/AlO
_{x}ultrathin films measured by Brillouin light spectroscopy. Phys. Rev. B**2015**, 91, 180405. [Google Scholar] [CrossRef] - Chaurasiya, A.K.; Banerjee, C.; Pan, S.; Sahoo, S.; Choudhury, S.; Sinha, J.; Barman, A. Direct Observation of Interfacial Dzyaloshinskii-Moriya Interaction from Asymmetric Spin-wave Propagation in W/CoFeB/SiO
_{2}Heterostructures Down to Sub-nanometer CoFeB Thickness. Sci. Rep.**2016**, 6, 32592. [Google Scholar] [CrossRef] - Seki, S.; Okamura, Y.; Kondou, K.; Shibata, K.; Kubota, M.; Takagi, R.; Kagawa, F.; Kawasaki, M.; Tatara, G.; Otani, Y.; et al. Magnetochiral nonreciprocity of volume spin wave propagation in chiral-lattice ferromagnets. Phys. Rev. B
**2016**, 93, 235131. [Google Scholar] [CrossRef] - Tacchi, S.; Troncoso, R.E.; Ahlberg, M.; Gubbiotti, G.; Madami, M.; Åkerman, 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] - Weber, T.; Waizner, J.; Tucker, G.S.; Beddrich, L.; Skoulatos, M.; Georgii, R.; Bauer, A.; Pfleiderer, C.; Garst, M.; Böni, P. Non-reciprocal magnons in non-centrosymmetric MnSi. AIP Adv.
**2018**, 8, 101328. [Google Scholar] [CrossRef] - Gallardo, R.A.; Alvarado-Seguel, P.; Schneider, T.; Gonzalez-Fuentes, C.; Roldán-Molina, A.; Lenz, K.; Lindner, J.; Landeros, P. Spin-wave non-reciprocity in magnetization-graded ferromagnetic films. New J. Phys.
**2019**, 21, 033026. [Google Scholar] [CrossRef] - Gallardo, R.; Schneider, T.; Chaurasiya, A.; Oelschlägel, A.; Arekapudi, S.; Roldán-Molina, A.; Hübner, R.; Lenz, K.; Barman, A.; Fassbender, J.; et al. Reconfigurable Spin-Wave Nonreciprocity Induced by Dipolar Interaction in a Coupled Ferromagnetic Bilayer. Phys. Rev. Appl.
**2019**, 12, 034012. [Google Scholar] [CrossRef] - Gallardo, R.A.; Cortés-Ortuño, D.; Troncoso, R.E.; Landeros, P. Three-Dimensional Magnonics, Layered, Micro- and Nanostructures; Jenny Stanford Publishing: Berlin/Heidelberg, Germany, 2019; pp. 121–160. [Google Scholar]
- Sheka, D.D.; Pylypovskyi, O.V.; Landeros, P.; Gaididei, Y.; Kákay, A.; Makarov, D. Nonlocal chiral symmetry breaking in curvilinear magnetic shells. Commun. Phys.
**2020**, 3, 128. [Google Scholar] [CrossRef] - Albisetti, E.; Tacchi, S.; Silvani, R.; Scaramuzzi, G.; Finizio, S.; Wintz, S.; Rinaldi, C.; Cantoni, M.; Raabe, J.; Carlotti, G.; et al. Optically Inspired Nanomagnonics with Nonreciprocal Spin Waves in Synthetic Antiferromagnets. Adv. Mater.
**2020**, 32, 1906439. [Google Scholar] [CrossRef] - Grassi, M.; Geilen, M.; Louis, D.; Mohseni, M.; Brächer, T.; Hehn, M.; Stoeffler, D.; Bailleul, M.; Pirro, P.; Henry, Y. Slow-Wave-Based Nanomagnonic Diode. Phys. Rev. Appl.
**2020**, 14, 024047. [Google Scholar] [CrossRef] - Gallardo, R.A.; Alvarado-Seguel, P.; Landeros, P. High spin-wave asymmetry and emergence of radial standing modes in thick ferromagnetic nanotubes. Phys. Rev. B
**2022**, 105, 104435. [Google Scholar] [CrossRef] - Neusser, S.; Grundler, D. Magnonics: Spin Waves on the Nanoscale. Adv. Mater.
**2009**, 21, 2927–2932. [Google Scholar] [CrossRef] - 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] - Locatelli, N.; Cros, V.; Grollier, J. Spin-torque building blocks. Nat. Mater.
**2014**, 13, 11–20. [Google Scholar] [CrossRef] - Chumak, A.V.; Vasyuchka, V.I.; Serga, A.A.; Hillebrands, B. Magnon spintronics. Nat. Phys.
**2015**, 11, 453–461. [Google Scholar] [CrossRef] - Bauer, G.E.W.; Saitoh, E.; van Wees, B.J. Spin caloritronics. Nat. Mater.
**2012**, 11, 391–399. [Google Scholar] [CrossRef] - Khitun, A.; Bao, M.; Wang, K.L. Magnonic logic circuits. J. Phys. D Appl. Phys.
**2010**, 43, 264005. [Google Scholar] [CrossRef] - 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] - Lenk, B.; Ulrichs, H.; Garbs, F.; Münzenberg, M. The building blocks of magnonics. Phys. Rep.
**2011**, 507, 107–136. [Google Scholar] [CrossRef] - Krawczyk, M.; Grundler, D. Review and prospects of magnonic crystals and devices with reprogrammable band structure. J. Phys. Condens. Matter
**2014**, 26, 123202. [Google Scholar] [CrossRef] - Chumak, A.V.; Serga, A.A.; Hillebrands, B. Magnonic crystals for data processing. J. Phys. D
**2017**, 50, 244001. [Google Scholar] [CrossRef] - Yu, H.; Xiao, J.; Schultheiss, H. Magnetic texture based magnonics. Phys. Rep.
**2021**, 905, 1–59. [Google Scholar] [CrossRef] - Ríos-Venegas, C.; Brevis, F.; Gallardo, R.A.; Landeros, P. Dynamic origin of conical helix magnetization textures stabilized by Dzyaloshinskii-Moriya interaction. Phys. Rev. B
**2022**, 105, 224403. [Google Scholar] [CrossRef] - Graczyk, P.; Kłos, J.; Krawczyk, M. Broadband magnetoelastic coupling in magnonic-phononic crystals for high-frequency nanoscale spin-wave generation. Phys. Rev. B
**2017**, 95, 104425. [Google Scholar] [CrossRef] - Bozhko, D.A.; Vasyuchka, V.I.; Chumak, A.V.; Serga, A.A. Magnon-phonon interactions in magnon spintronics (Review article). Low Temp. Phys.
**2020**, 46, 383–399. [Google Scholar] [CrossRef] - Gubbiotti, G. Three-Dimensional Magnonics: Layered, Micro-and Nanostructures; CRC Press: Boca Raton, FL, USA, 2019. [Google Scholar]
- Makarov, D.; Sheka, D. Curvilinear Micromagnetism: From Fundamentals to Applications; Topics in Applied Physics; Springer International Publishing: Berlin, Germany, 2022. [Google Scholar]
- Garcia-Sanchez, F.; 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] - Xing, X.; Zhou, Y. Fiber optics for spin waves. NPG Asia Mater.
**2016**, 8, e246. [Google Scholar] [CrossRef] - Stamps, R.L.; Kim, J.V.; Garcia-Sanchez, F.; Borys, P.; Gubbiotti, G.; Li, Y.; Camley, R.E. Spin Wave Confinement: Propagating Waves, 2nd ed.; Pan Stanford Publishing: Redwood City, CA, USA, 2017; pp. 219–260. [Google Scholar]
- Hämäläinen, S.J.; Madami, M.; Qin, H.; Gubbiotti, G.; van Dijken, S. Control of spin-wave transmission by a programmable domain wall. Nat. Commun.
**2018**, 9, 4853. [Google Scholar] [CrossRef] - Henry, Y.; Stoeffler, D.; Kim, J.V.; Bailleul, M. Unidirectional spin-wave channeling along magnetic domain walls of Bloch type. Phys. Rev. B
**2019**, 100, 024416. [Google Scholar] [CrossRef] - Lara, A.; Metlushko, V.; Aliev, F.G. Observation of propagating edge spin waves modes. J. Appl. Phys.
**2013**, 114, 213905. [Google Scholar] [CrossRef] - Roldán-Molina, A.; Nunez, A.S.; Fernández-Rossier, J. Topological spin waves in the atomic-scale magnetic skyrmion crystal. New J. Phys.
**2016**, 18, 045015. [Google Scholar] [CrossRef] - Lara, A.; Robledo Moreno, J.; Guslienko, K.Y.; Aliev, F.G. Information processing in patterned magnetic nanostructures with edge spin waves. Sci. Rep.
**2017**, 7, 5597. [Google Scholar] [CrossRef] - Lan, J.; Yu, W.; Wu, R.; Xiao, J. Spin-Wave Diode. Phys. Rev. X
**2015**, 5, 041049. [Google Scholar] [CrossRef] - Reiskarimian, N.; Krishnaswamy, H. Magnetic-free non-reciprocity based on staggered commutation. Nat. Commun.
**2016**, 7, 11217. [Google Scholar] [CrossRef] - Sounas, D.L.; Alù, A. Non-reciprocal photonics based on time modulation. Nat. Photonics
**2017**, 11, 774–783. [Google Scholar] [CrossRef] - Garst, M.; Waizner, J.; Grundler, D. Collective spin excitations of helices and magnetic skyrmions: Review and perspectives of magnonics in non-centrosymmetric magnets. J. Phys. D Appl. Phys.
**2017**, 50, 293002. [Google Scholar] [CrossRef] - Heussner, F.; Serga, A.A.; Brächer, T.; Hillebrands, B.; Pirro, P. A switchable spin-wave signal splitter for magnonic networks. Appl. Phys. Lett.
**2017**, 111, 122401. [Google Scholar] [CrossRef] - Heussner, F.; Talmelli, G.; Geilen, M.; Heinz, B.; Brächer, T.; Meyer, T.; Ciubotaru, F.; Adelmann, C.; Yamamoto, K.; Serga, A.A.; et al. Experimental Realization of a Passive Gigahertz Frequency-Division Demultiplexer for Magnonic Logic Networks. Phys. Status Solidi RRL
**2020**, 14, 1900695. [Google Scholar] [CrossRef] - Xing, X.; Zhou, Y.; Braun, H. Magnetic Skyrmion Tubes as Nonplanar Magnonic Waveguides. Phys. Rev. Appl.
**2020**, 13, 034051. [Google Scholar] [CrossRef] - Levy, U.; Abashin, M.; Ikeda, K.; Krishnamoorthy, A.; Cunningham, J.; Fainman, Y. Inhomogenous Dielectric Metamaterials with Space-Variant Polarizability. Phys. Rev. Lett.
**2007**, 98, 243901. [Google Scholar] [CrossRef] - Hecht, E. Optics, 5e; Pearson Education India: Noida, India, 2002. [Google Scholar]
- Markos, C.; Travers, J.C.; Abdolvand, A.; Eggleton, B.J.; Bang, O. Hybrid photonic-crystal fiber. Rev. Mod. Phys.
**2017**, 89, 045003. [Google Scholar] [CrossRef] - Boonzajer Flaes, D.E.; Stopka, J.; Turtaev, S.; de Boer, J.F.; Tyc, T.; Čižmár, T. Robustness of Light-Transport Processes to Bending Deformations in Graded-Index Multimode Waveguides. Phys. Rev. Lett.
**2018**, 120, 1–5. [Google Scholar] [CrossRef] [PubMed] - Davies, C.S.; Kruglyak, V.V. Graded-index magnonics. Low Temp. Phys.
**2015**, 41, 760–766. [Google Scholar] [CrossRef] - Gruszecki, P.; Krawczyk, M. Spin-wave beam propagation in ferromagnetic thin films with graded refractive index: Mirage effect and prospective applications. Phys. Rev. B
**2018**, 97, 094424. [Google Scholar] [CrossRef] - Jorzick, J.; Demokritov, S.O.; Hillebrands, B.; Bailleul, M.; Fermon, C.; Guslienko, K.Y.; Slavin, A.N.; Berkov, D.V.; Gorn, N.L. Spin Wave Wells in Nonellipsoidal Micrometer Size Magnetic Elements. Phys. Rev. Lett.
**2002**, 88, 047204. [Google Scholar] [CrossRef] [PubMed] - Park, J.P.; Eames, P.; Engebretson, D.M.; Berezovsky, J.; Crowell, P.A. Spatially Resolved Dynamics of Localized Spin-Wave Modes in Ferromagnetic Wires. Phys. Rev. Lett.
**2002**, 89, 277201. [Google Scholar] [CrossRef] [PubMed] - Kruglyak, V.V.; Barman, A.; Hicken, R.J.; Childress, J.R.; Katine, J.A. Picosecond magnetization dynamics in nanomagnets: Crossover to nonuniform precession. Phys. Rev. B
**2005**, 71, 220409. [Google Scholar] [CrossRef] - Demidov, V.E.; Demokritov, S.O.; Rott, K.; Krzysteczko, P.; Reiss, G. Nano-optics with spin waves at microwave frequencies. Appl. Phys. Lett.
**2008**, 92, 232503. [Google Scholar] [CrossRef] - Fripp, K.G.; Kruglyak, V.V. Spin-wave wells revisited: From wavelength conversion and Möbius modes to magnon valleytronics. Phys. Rev. B
**2021**, 103, 184403. [Google Scholar] [CrossRef] - Mathieu, C.; Jorzick, J.; Frank, A.; Demokritov, S.O.; Slavin, A.N.; Hillebrands, B.; Bartenlian, B.; Chappert, C.; Decanini, D.; Rousseaux, F.; et al. Lateral Quantization of Spin Waves in Micron Size Magnetic Wires. Phys. Rev. Lett.
**1998**, 81, 3968–3971. [Google Scholar] [CrossRef] - Jorzick, J.; Demokritov, S.O.; Mathieu, C.; Hillebrands, B.; Bartenlian, B.; Chappert, C.; Rousseaux, F.; Slavin, A.N. Brillouin light scattering from quantized spin waves in micron-size magnetic wires. Phys. Rev. B
**1999**, 60, 15194–15200. [Google Scholar] [CrossRef] - Mantese, J.V.; Micheli, A.L.; Schubring, N.W.; Hayes, R.W.; Srinivasan, G.; Alpay, S.P. Magnetization-graded ferromagnets: The magnetic analogs of semiconductor junction elements. Appl. Phys. Lett.
**2005**, 87, 082503. [Google Scholar] [CrossRef] - Fallarino, L.; Riego, P.; Kirby, B.J.; Miller, C.W.; Berger, A. Modulation of Magnetic Properties at the Nanometer Scale in Continuously Graded Ferromagnets. Materials
**2018**, 11, 251. [Google Scholar] [CrossRef] [PubMed] - Sudakar, C.; Naik, R.; Lawes, G.; Mantese, J.V.; Micheli, A.L.; Srinivasan, G.; Alpay, S.P. Internal magnetostatic potentials of magnetization-graded ferromagnetic materials. Appl. Phys. Lett.
**2007**, 90, 062502. [Google Scholar] [CrossRef] - Supper, N.; Margulies, D.; Moser, A.; Berger, A.; Do, H.; Fullerton, E. Writability enhancement using exchange spring media. IEEE Trans. Magn.
**2005**, 41, 3238–3240. [Google Scholar] [CrossRef] - Berger, A.; Supper, N.; Ikeda, Y.; Lengsfield, B.; Moser, A.; Fullerton, E.E. Improved media performance in optimally coupled exchange spring layer media. Appl. Phys. Lett.
**2008**, 93, 122502. [Google Scholar] [CrossRef] - Zhou, T.J.; Lim, B.C.; Liu, B. Anisotropy graded FePt–TiO2 nanocomposite thin films with small grain size. Appl. Phys. Lett.
**2009**, 94, 152505. [Google Scholar] [CrossRef] - Kirby, B.J.; Davies, J.E.; Liu, K.; Watson, S.M.; Zimanyi, G.T.; Shull, R.D.; Kienzle, P.A.; Borchers, J.A. Vertically graded anisotropy in Co/Pd multilayers. Phys. Rev. B
**2010**, 81, 100405. [Google Scholar] [CrossRef] - Dumas, R.K.; Fang, Y.; Kirby, B.J.; Zha, C.; Bonanni, V.; Nogués, J.; Åkerman, J. Probing vertically graded anisotropy in FePtCu films. Phys. Rev. B
**2011**, 84, 054434. [Google Scholar] [CrossRef] - Kirby, B.J.; Belliveau, H.F.; Belyea, D.D.; Kienzle, P.A.; Grutter, A.J.; Riego, P.; Berger, A.; Miller, C.W. Spatial Evolution of the Ferromagnetic Phase Transition in an Exchange Graded Film. Phys. Rev. Lett.
**2016**, 116, 047203. [Google Scholar] [CrossRef] [PubMed] - Fallarino, L.; Kirby, B.J.; Pancaldi, M.; Riego, P.; Balk, A.L.; Miller, C.W.; Vavassori, P.; Berger, A. Magnetic properties of epitaxial CoCr films with depth-dependent exchange-coupling profiles. Phys. Rev. B
**2017**, 95, 134445. [Google Scholar] [CrossRef] - Kirby, B.J.; Fallarino, L.; Riego, P.; Maranville, B.B.; Miller, C.W.; Berger, A. Nanoscale magnetic localization in exchange strength modulated ferromagnets. Phys. Rev. B
**2018**, 98, 064404. [Google Scholar] [CrossRef] - Guslienko, K.Y.; Demokritov, S.O.; Hillebrands, B.; Slavin, A.N. Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes. Phys. Rev. B
**2002**, 66, 132402. [Google Scholar] [CrossRef] - Davies, C.S.; Poimanov, V.D.; Kruglyak, V.V. Mapping the magnonic landscape in patterned magnetic structures. Phys. Rev. B
**2017**, 96, 094430. [Google Scholar] [CrossRef] - Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Waeyenberge, B.V. The design and verification of MuMax3. AIP Adv.
**2014**, 4, 107133. [Google Scholar] [CrossRef] - Fallarino, L.; Kirby, B.J.; Fullerton, E.E. Graded magnetic materials. J. Phys. D Appl. Phys.
**2021**, 54, 303002. [Google Scholar] [CrossRef] - Green, M.L.; Takeuchi, I.; Hattrick-Simpers, J.R. Applications of high throughput (combinatorial) methodologies to electronic, magnetic, optical, and energy-related materials. J. Appl. Phys.
**2013**, 113, 231101. [Google Scholar] [CrossRef] - Bali, R.; Wintz, S.; Meutzner, F.; Hübner, R.; Boucher, R.; Ünal, A.A.; Valencia, S.; Neudert, A.; Potzger, K.; Bauch, J.; et al. Printing Nearly-Discrete Magnetic Patterns Using Chemical Disorder Induced Ferromagnetism. Nano Lett.
**2014**, 14, 435–441. [Google Scholar] [CrossRef] - Röder, F.; Hlawacek, G.; Wintz, S.; Hübner, R.; Bischoff, L.; Lichte, H.; Potzger, K.; Lindner, J.; Fassbender, J.; Bali, R. Direct Depth- and Lateral- Imaging of Nanoscale Magnets Generated by Ion Impact. Sci. Rep.
**2015**, 5, 16786. [Google Scholar] [CrossRef] - Nord, M.; Semisalova, A.; Kákay, A.; Hlawacek, G.; MacLaren, I.; Liersch, V.; Volkov, O.M.; Makarov, D.; Paterson, G.W.; Potzger, K.; et al. Strain Anisotropy and Magnetic Domains in Embedded Nanomagnets. Small
**2019**, 15, 1904738. [Google Scholar] [CrossRef] - Grimsditch, M.; Giovannini, L.; Montoncello, F.; Nizzoli, F.; Leaf, G.K.; Kaper, H.G. Magnetic normal modes in ferromagnetic nanoparticles: A dynamical matrix approach. Phys. Rev. B
**2004**, 70, 054409. [Google Scholar] [CrossRef] - Giovannini, L.; Montoncello, F.; Nizzoli, F.; Gubbiotti, G.; Carlotti, G.; Okuno, T.; Shinjo, T.; Grimsditch, M. Spin excitations of nanometric cylindrical dots in vortex and saturated magnetic states. Phys. Rev. B
**2004**, 70, 172404. [Google Scholar] [CrossRef] - Henry, Y.; Gladii, O.; Bailleul, M. Propagating spin-wave normal modes: A dynamic matrix approach using plane-wave demagnetizating tensors. arXiv
**2016**, arXiv:1611.06153. [Google Scholar] - Newell, A.J.; Williams, W.; Dunlop, D.J. A generalization of the demagnetizing tensor for nonuniform magnetization. J. Geophys. Res. Solid Earth
**1993**, 98, 9551–9555. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Illustration of the coordinate system and the main geometrical parameters of a magnetic strip. In (

**b**), a schematic representation of the dynamic matrix approach is shown, where the system is divided into many sub-strips of width b, which allows including magnetic graduation along the width. The color graduation represents the variation of the saturation magnetization along the width of the system.

**Figure 2.**Spin-wave dispersion in a homogeneous ferromagnetic nanostrip. In (

**a**–

**d**), different values of the strip width, w, have been considered. The open circles show the SW dispersion of a ferromagnetic thin film, whereas the lines are the spin-wave modes of a magnetic nanostrip. Upper 3D graphics correspond to the spin-wave profiles along the strip’s width evaluated at ${k}_{z}=0$, where the dynamic magnetization components are calculated in arbitrary units. In all cases, SM${}_{0}^{\left(\mathrm{hom}\right)}$ and SM${}_{1}^{\left(\mathrm{hom}\right)}$ are edge modes, while other modes correspond to magnetic excitations with high SW amplitude around the strip center. In (

**d**), the first five low-frequency modes have been calculated.

**Figure 3.**(

**a**) Spin-wave dispersion of a magnetization-graded strip. The magnetic profile is shown in (

**b**), where a notable reduction of ${M}_{\mathrm{s}}$ along the width is assumed ($\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}=0.5$) and $\xi =100$ nm. (

**c**, (

**d**) and (

**e**) depict the SW orbits along the strip width for modes SM${}_{0}^{\left(\mathrm{grad}\right)}$, SM${}_{1}^{\left(\mathrm{grad}\right)}$ and SM${}_{2}^{\left(\mathrm{grad}\right)}$, respectively. The dynamic magnetization components ${m}_{z}$ and ${m}_{y}$ are calculated with arbitrary units.

**Figure 4.**(

**a**) The absolute value of the out-of-plane dynamic magnetization component, for mode SM${}_{0}^{\left(\mathrm{grad}\right)}$, as a function of x is shown for $w=200$ nm and $\xi =100$ nm. Different values of the fractional reduction of the saturation magnetization, $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}$, have been accounted for. (

**b**) The frequency of the modes SM${}_{\nu}^{\left(\mathrm{grad}\right)}$ (with $\nu =0$, 1 and 2) is illustrated as a function of $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}$ for ${k}_{z}=0$.

**Figure 5.**(

**a**) Modes SM${}_{\nu}^{\left(\mathrm{grad}\right)}$ (with $\nu =0$, 1 and 2) as a function of $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}$. The modes are evaluated at ${k}_{z}=0$ and $\xi =100$ nm, for a wider strip with $w=1000$ nm. The crossing between modes reveals the weak coupling caused by the larger strip width w. The absolute value of the normal magnetization component for $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}=0.05$ and 0.45 is illustrated in (

**b**) for modes SM${}_{2}^{\left(\mathrm{grad}\right)}$ and SM${}_{0}^{\left(\mathrm{grad}\right)}$, respectively. (

**c**) Modes SM${}_{\nu}^{\left(\mathrm{grad}\right)}$ (with $\nu =0$, 1, 2 and 3) as a function of $\xi $. The case ${k}_{z}=0$, $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}=0.4$, and $w=1000$ nm is considered. The magnetization profiles of modes SM${}_{0}^{\left(\mathrm{grad}\right)}$ and SM${}_{1}^{\left(\mathrm{grad}\right)}$ are shown in (

**d**) for $\xi =175$ nm.

**Figure 6.**Calculated spin-wave dispersion and its respective profiles obtained from the micromagnetic simulations. In (

**a**) and (

**b**), the cases $w=200$ nm and $w=1000$ nm are respectively illustrated, where $\xi =100$ nm and $\Delta {M}_{\mathrm{s}}/{M}_{\mathrm{s}}=0.5$ are assumed in both cases. The different curves depict the low-frequency spin-wave modes. The SW propagation is simulated for $f=17$ GHz and 18 GHz (dotted horizontal lines). At $f=17$ GHz, the SWs are conducted along the nanostrip center, being this the unique excited mode, as the calculations predict. If the frequency of the field ${\mathbf{h}}_{\mathrm{rf}}$ is 18 GHz, both the edge and the channelized modes are excited.

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**MDPI and ACS Style**

Gallardo, R.A.; Alvarado-Seguel, P.; Brevis, F.; Roldán-Molina, A.; Lenz, K.; Lindner, J.; Landeros, P.
Spin-Wave Channeling in Magnetization-Graded Nanostrips. *Nanomaterials* **2022**, *12*, 2785.
https://doi.org/10.3390/nano12162785

**AMA Style**

Gallardo RA, Alvarado-Seguel P, Brevis F, Roldán-Molina A, Lenz K, Lindner J, Landeros P.
Spin-Wave Channeling in Magnetization-Graded Nanostrips. *Nanomaterials*. 2022; 12(16):2785.
https://doi.org/10.3390/nano12162785

**Chicago/Turabian Style**

Gallardo, Rodolfo A., Pablo Alvarado-Seguel, Felipe Brevis, Alejandro Roldán-Molina, Kilian Lenz, Jürgen Lindner, and Pedro Landeros.
2022. "Spin-Wave Channeling in Magnetization-Graded Nanostrips" *Nanomaterials* 12, no. 16: 2785.
https://doi.org/10.3390/nano12162785