# The 2021 Magnonics Roadmap

A Barman, Gianluca Gubbiotti, Sam Ladak, Adekunle Olusola Adeyeye, Maciej Krawczyk, Joachim Gräfe, Christoph Adelmann, Sorin Cotofana, Azad Naeemi, Vitaliy Vasyuchka, et al.

► To cite this version:

A Barman, Gianluca Gubbiotti, Sam Ladak, Adekunle Olusola Adeyeye, Maciej Krawczyk, et al. The 2021 Magnonics Roadmap. Journal of Physics: Condensed Matter, 2021, 33 (41), pp.413001. 10.1088/1361-648X/abec1a. hal-03229528

HAL Id: hal-03229528

https://hal.science/hal-03229528

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<figure><img src="image_1.png" /></figure>
TOPICAL REVIEW • OPEN ACCESS

# The 2021 Magnonics Roadmap

To cite this article: Anjan Barman et al 2021 *J. Phys.: Condens. Matter* **33** 413001

View the [article online](#) for updates and enhancements.

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TOPICAL REVIEW • OPEN ACCESS

# The 2021 Magnonics Roadmap

To cite this article: Anjan Barman et al 2021 *J. Phys.: Condens. Matter* **33** 413001

View the [article online](#) for updates and enhancements.

You may also like

- *Hybrid quantum systems based on magnonics*

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- *Future Physics Programme of BESIII*

M. Ablikim, M. N. Achasov, P. Adlarson et al.

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Topical Review

The 2021 Magnonics Roadmap

Anjan Barman<sup>1,51,*</sup>, Gianluca Gubbiotti<sup>2,51,*</sup>, S Ladak<sup>3</sup>, A O Adeyeye<sup>4</sup>, M Krawczyk<sup>5</sup>, J Gräfe<sup>6</sup>, C Adelmann<sup>7</sup>, S Cotofana<sup>8</sup>, A Naeemi<sup>9</sup>, V I Vasyuchka<sup>10</sup>, B Hillebrands<sup>10</sup>, S A Nikitov<sup>11</sup>, H Yu<sup>12</sup>, D Grundler<sup>13</sup>, A V Sadovnikov<sup>11,14</sup>, A A Grachev<sup>11,14</sup>, S E Sheshukova<sup>11,14</sup>, J-Y Duquesne<sup>15</sup>, M Marangolo<sup>15</sup>, G Csaba<sup>16</sup>, W Porod<sup>17</sup>, V E Demidov<sup>18</sup>, S Urazhdin<sup>19</sup>, S O Demokritov<sup>18</sup>, E Albisetti<sup>20</sup>, D Petti<sup>20</sup>, R Bertacco<sup>20</sup>, H Schultheiss<sup>21,22</sup>, V V Kruglyak<sup>23</sup>, V D Poimanov<sup>24</sup>, S Sahoo<sup>1</sup>, J Sinha<sup>25</sup>, H Yang<sup>26</sup>, M Münzenberg<sup>27</sup>, T Moriyama<sup>28,29</sup>, S Mizukami<sup>29,30</sup>, P Landeros<sup>31,32</sup>, R A Gallardo<sup>31,32</sup>, G Carlotti<sup>33,34</sup>, J-V Kim<sup>35</sup>, R L Stamps<sup>36</sup>, R E Camley<sup>37</sup>, B Rana<sup>38</sup>, Y Otani<sup>38,39</sup>, W Yu<sup>40</sup>, T Yu<sup>41</sup>, G E W Bauer<sup>30,42</sup>, C Back<sup>43</sup>, G S Uhrig<sup>44</sup>, O V Dobrovolskiy<sup>45</sup>, B Budinska<sup>45</sup>, H Qin<sup>46</sup>, S van Dijken<sup>46</sup>, A V Chumak<sup>45</sup>, A Khitun<sup>47</sup>, D E Nikonov<sup>48</sup>, I A Young<sup>48</sup>, B W Zingsem<sup>49</sup> and M Winklhofer<sup>50</sup>

<figure><img src="image_1.png" /></figure>

Original content from this work may be used under the
terms of the Creative Commons Attribution 4.0 licence.

Any further distribution of this work must maintain attribution to the
author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ Department of Condensed Matter Physics and Material Sciences, S N Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700106, India</footnote>

<footnote n="2">² Istituto Officina dei Materiali del Consiglio nazionale delle Ricerche (IOM-CNR), Perugia, Italy</footnote>

<footnote n="3">³ School of Physics and Astronomy, Cardiff University, United Kingdom</footnote>

<footnote n="4">⁴ Department of Physics, University of Durham, United Kingdom</footnote>

<footnote n="5">⁵ Adam Mickiewicz University, Poznan, Poland</footnote>

<footnote n="6">⁶ Max Planck Institute for Intelligent Systems, Stuttgart, Germany</footnote>

<footnote n="7">⁷ Imec, Leuven, Belgium</footnote>

<footnote n="8">⁸ Delft University of Technology, The Netherlands</footnote>

<footnote n="9">⁹ Georgia Institute of Technology, United States of America</footnote>

<footnote n="10">¹⁰ Department of Physics and State Research Center OPTIMAS, Technische Universität Kaiserslautern
(TUK), Kaiserslautern, Germany</footnote>

<footnote n="11">¹¹ Kotelnikov Institute of Radioengineering and Electronics, Moscow, Russia</footnote>

<footnote n="12">¹² Fert Beijing Institute, BDBC, School of Microelectronics, Beijing Advanced Innovation Center for Big Data and Brian Computing, Beihang University, People's Republic of China</footnote>

<footnote n="13">¹³ Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX), Institute of Electrical and Micro Engineering, School of Engineering, École Polytechnique Fédérale de Lausanne
(EPFL), Switzerland</footnote>

<footnote n="14">¹⁴ Laboratory ‘Magnetic Metamaterials’, Saratov State University, Saratov, Russia</footnote>

<footnote n="15">¹⁵ Institut des NanoSciences de Paris, Sorbonne University, CNRS, Paris, France</footnote>

<footnote n="16">¹⁶ Pázmány University, Budapest, Hungary</footnote>

<footnote n="17">¹⁷ University of Notre Dame, IN, United States of America</footnote>

<footnote n="18">¹⁸ Institute for Applied Physics, University of Muenster, Muenster, Germany</footnote>

<footnote n="19">¹⁹ Department of Physics, Emory University, Atlanta, United States of America</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>

<footnote n="51">51 A Barman and G Gubbiotti contributed equally to this work.</footnote>
Topical Review

# The 2021 Magnonics Roadmap

Anjan Barman<sup>1,51,*</sup>, Gianluca Gubbiotti<sup>2,51,*</sup>, S Ladak<sup>3</sup>, A O Adeyeye<sup>4</sup>, M Krawczyk<sup>5</sup>, J Gräfe<sup>6</sup>, C Adelmann<sup>7</sup>, S Cotofana<sup>8</sup>, A Naeemi<sup>9</sup>, V I Vasyuchka<sup>10</sup>, B Hillebrands<sup>10</sup>, S A Nikitov<sup>11</sup>, H Yu<sup>12</sup>, D Grundler<sup>13</sup>, A V Sadovnikov<sup>11,14</sup>, A A Grachev<sup>11,14</sup>, S E Sheshukova<sup>11,14</sup>, J-Y Duquesne<sup>15</sup>, M Marangolo<sup>15</sup>, G Csaba<sup>16</sup>, W Porod<sup>17</sup>, V E Demidov<sup>18</sup>, S Urazhdin<sup>19</sup>, S O Demokritov<sup>18</sup>, E Albisetti<sup>20</sup>, D Petti<sup>20</sup>, R Bertacco<sup>20</sup>, H Schultheiss<sup>21,22</sup>, V V Kruglyak<sup>23</sup>, V D Poimanov<sup>24</sup>, S Sahoo<sup>1</sup>, J Sinha<sup>25</sup>, H Yang<sup>26</sup>, M Münzenberg<sup>27</sup>, T Moriyama<sup>28,29</sup>, S Mizukami<sup>29,30</sup>, P Landeros<sup>31,32</sup>, R A Gallardo<sup>31,32</sup>, G Carlotti<sup>33,34</sup>, J-V Kim<sup>35</sup>, R L Stamps<sup>36</sup>, R E Camley<sup>37</sup>, B Rana<sup>38</sup>, Y Otani<sup>38,39</sup>, W Yu<sup>40</sup>, T Yu<sup>41</sup>, G E W Bauer<sup>30,42</sup>, C Back<sup>43</sup>, G S Uhrig<sup>44</sup>, O V Dobrovolskiy<sup>45</sup>, B Budinska<sup>45</sup>, H Qin<sup>46</sup>, S van Dijken<sup>46</sup>, A V Chumak<sup>45</sup>, A Khitun<sup>47</sup>, D E Nikonov<sup>48</sup>, I A Young<sup>48</sup>, B W Zingsem<sup>49</sup> and M Winklhofer<sup>50</sup>

<figure><img src="image_2.png" /></figure>

<footnote n="1">¹ Department of Condensed Matter Physics and Material Sciences, S N Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700106, India</footnote>

<footnote n="2">² Istituto Officina dei Materiali del Consiglio nazionale delle Ricerche (IOM-CNR), Perugia, Italy</footnote>

<footnote n="3">³ School of Physics and Astronomy, Cardiff University, United Kingdom</footnote>

<footnote n="4">⁴ Department of Physics, University of Durham, United Kingdom</footnote>

<footnote n="5">⁵ Adam Mickiewicz University, Poznan, Poland</footnote>

<footnote n="6">⁶ Max Planck Institute for Intelligent Systems, Stuttgart, Germany</footnote>

<footnote n="7">⁷ Imec, Leuven, Belgium</footnote>

<footnote n="8">⁸ Delft University of Technology, The Netherlands</footnote>

<footnote n="9">⁹ Georgia Institute of Technology, United States of America</footnote>

<footnote n="10">¹⁰ Department of Physics and State Research Center OPTIMAS, Technische Universität Kaiserslautern
(TUK), Kaiserslautern, Germany</footnote>

<footnote n="11">¹¹ Kotelnikov Institute of Radioengineering and Electronics, Moscow, Russia</footnote>

<footnote n="12">¹² Fert Beijing Institute, BDBC, School of Microelectronics, Beijing Advanced Innovation Center for Big Data and Brian Computing, Beihang University, People's Republic of China</footnote>

<footnote n="13">¹³ Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX), Institute of Electrical and Micro Engineering, School of Engineering, École Polytechnique Fédérale de Lausanne
(EPFL), Switzerland</footnote>

<footnote n="14">¹⁴ Laboratory ‘Magnetic Metamaterials’, Saratov State University, Saratov, Russia</footnote>

<footnote n="15">¹⁵ Institut des NanoSciences de Paris, Sorbonne University, CNRS, Paris, France</footnote>

<footnote n="16">¹⁶ Pázmány University, Budapest, Hungary</footnote>

<footnote n="17">¹⁷ University of Notre Dame, IN, United States of America</footnote>

<footnote n="18">¹⁸ Institute for Applied Physics, University of Muenster, Muenster, Germany</footnote>

<footnote n="19">¹⁹ Department of Physics, Emory University, Atlanta, United States of America</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>

<footnote n="51">51 A Barman and G Gubbiotti contributed equally to this work.</footnote>

<footnote>Original content from this work may be used under the
terms of the Creative Commons Attribution 4.0 licence.</footnote>

<footnote>Any further distribution of this work must maintain attribution to the
author(s) and the title of the work, journal citation and DOI.</footnote>
20 Polytechnic University of Milan, Italy

21 Helmholtz-Center Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, Germany

22 Technische Universität Dresden, Germany

23 University of Exeter, Exeter, United Kingdom

24 Donetsk National University, Donetsk, Ukraine

25 Department of Physics and Nanotechnology, SRM Institute of Science and Technology, Kattankulathur, India

26 Department of Electrical and Computer Engineering, National University of Singapore, Singapore

27 Institute of Physics, University of Greifswald, Greifswald, Germany

28 Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, Japan

29 Centre for Spintronics Research Network, Japan

30 Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai, Japan

31 Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile

32 Center for the Development of Nanoscience and Nanotechnology (CEDENNA), Santiago, Chile

33 Dipartimento di Fisica e Geologia, University of Perugia, Perugia, Italy

34 CNR Instituto Nanoscienze, Modena, Italy

35 Centre for Nanosciences and Nanotechnology, CNRS, Université Paris-Saclay, Palaiseau, France

36 Department of Physics and Astronomy, University of Manitoba, Canada

37 Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs, United States of America

38 RIKEN, Japan

39 Institute for Solid State Physics (ISSP), University of Tokyo, Japan

40 Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan

41 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany

42 Zernike Institute for Advanced Materials, Groningen University, The Netherlands

43 Technical University Munich, Germany

44 Technical University Dortmund, Germany

45 Faculty of Physics, University of Vienna, Vienna, Austria

46 Department of Applied Physics, School of Science, Aalto University, Finland

47 University of California Riverside, United States of America

48 Components Research, Intel, Hillsboro, Oregon, United States of America

49 The University of Duisburg-Essen, CENIDE, Germany

50 The Carl von Ossietzky University of Oldenburg, Germany

E-mail: abarman@bose.res.in and gubbiotti@iom.cnr.it

Received 14 August 2020, revised 20 January 2021

Accepted for publication 4 March 2021

Published 18 August 2021

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

Magnonics is a budding research field in nanomagnetism and nanoscience that addresses the use of spin waves (magnons) to transmit, store, and process information. The rapid advancements of this field during last one decade in terms of upsurge in research papers, review articles, citations, proposals of devices as well as introduction of new sub-topics prompted us to present the first roadmap on magnonics. This is a collection of 22 sections written by leading experts in this field who review and discuss the current status besides presenting their vision of future perspectives. Today, the principal challenges in applied magnonics are the excitation of sub-100 nm wavelength magnons, their manipulation on the nanoscale and the creation of sub-micrometre devices using low-Gilbert damping magnetic materials and its interconnections to standard electronics. To this end, magnonics offers lower energy consumption, easier integrability and compatibility with CMOS structure, reprogrammability, shorter wavelength, smaller device features, anisotropic properties, negative group velocity, non-reciprocity and efficient tunability by various external stimuli to name a few. Hence, despite being a young research field, magnonics has come a long way since its early inception. This roadmap asserts a milestone for future emerging research directions in magnonics, and hopefully, it will inspire a series of exciting new articles on the same topic in the coming years.

Keywords: magnonics, spin-waves, magnons, photons, magnetism
20 Polytechnic University of Milan, Italy

21 Helmholtz-Center Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, Germany

22 Technische Universität Dresden, Germany

23 University of Exeter, Exeter, United Kingdom

24 Donetsk National University, Donetsk, Ukraine

25 Department of Physics and Nanotechnology, SRM Institute of Science and Technology, Kattankulathur, India

26 Department of Electrical and Computer Engineering, National University of Singapore, Singapore

27 Institute of Physics, University of Greifswald, Greifswald, Germany

28 Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, Japan

29 Centre for Spintronics Research Network, Japan

30 Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai, Japan

31 Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile

32 Center for the Development of Nanoscience and Nanotechnology (CEDENNA), Santiago, Chile

33 Dipartimento di Fisica e Geologia, University of Perugia, Perugia, Italy

34 CNR Instituto Nanoscienze, Modena, Italy

35 Centre for Nanosciences and Nanotechnology, CNRS, Université Paris-Saclay, Palaiseau, France

36 Department of Physics and Astronomy, University of Manitoba, Canada

37 Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs, United States of America

38 RIKEN, Japan

39 Institute for Solid State Physics (ISSP), University of Tokyo, Japan

40 Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan

41 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany

42 Zernike Institute for Advanced Materials, Groningen University, The Netherlands

43 Technical University Munich, Germany

44 Technical University Dortmund, Germany

45 Faculty of Physics, University of Vienna, Vienna, Austria

46 Department of Applied Physics, School of Science, Aalto University, Finland

47 University of California Riverside, United States of America

48 Components Research, Intel, Hillsboro, Oregon, United States of America

49 The University of Duisburg-Essen, CENIDE, Germany

50 The Carl von Ossietzky University of Oldenburg, Germany

E-mail: abarman@bose.res.in and gubbiotti@iom.cnr.it

Received 14 August 2020, revised 20 January 2021

Accepted for publication 4 March 2021

Published 18 August 2021

<figure><img src="image_3.png" /></figure>

CrossMark

## Abstract

Magnonics is a budding research field in nanomagnetism and nanoscience that addresses the use of spin waves (magnons) to transmit, store, and process information. The rapid advancements of this field during last one decade in terms of upsurge in research papers, review articles, citations, proposals of devices as well as introduction of new sub-topics prompted us to present the first roadmap on magnonics. This is a collection of 22 sections written by leading experts in this field who review and discuss the current status besides presenting their vision of future perspectives. Today, the principal challenges in applied magnonics are the excitation of sub-100 nm wavelength magnons, their manipulation on the nanoscale and the creation of sub-micrometre devices using low-Gilbert damping magnetic materials and its interconnections to standard electronics. To this end, magnonics offers lower energy consumption, easier integrability and compatibility with CMOS structure, reprogrammability, shorter wavelength, smaller device features, anisotropic properties, negative group velocity, non-reciprocity and efficient tunability by various external stimuli to name a few. Hence, despite being a young research field, magnonics has come a long way since its early inception. This roadmap asserts a milestone for future emerging research directions in magnonics, and hopefully, it will inspire a series of exciting new articles on the same topic in the coming years.

Keywords: magnonics, spin-waves, magnons, photons, magnetism
(Some figures may appear in colour only in the online journal)

Introduction

Anjan Barman¹ and Gianluca Gubbiotti²

¹S N Bose National Centre for Basic Sciences, India

²Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Italy

The concept of spin waves was first introduced by F Bloch in 1932 [1] and further developed by Holstein & Primakoff [2] and Dyson [3] who laid the foundation of spin waves theory. Although effects of periodic modulation on wave propagation was known since the late 19th century, and concept of pass and stop bands was introduced for electromagnetic wave prop- agation in 1950s [4], it was introduced in the magnetic system after two decades [5]. Subsequently, different kinds of periodic modulation have been introduced. However, the invention of photonic crystal [6] has fueled intense interest in spin waves in periodic magnetic media in the late 1990s [7] and a new field named magnonics was born. In 2001, the term magnonic crystal (MC) was coined by Gulyaev and Nikitov [8] and later by Puszkarski and Krawczyk as the magnetic counterpart of photonic crystals [9].

Magnonics deals with the excitation, propagation, con-
trol and detection of spin waves (quanta of which are called
magnons) through periodic magnetic media consisting of
either passively (patterned structures, modulated magnetic
properties) or actively (spin texture, electric field, magnetic
field) controlled modulation. It is analogous to photonics or
phononics but it has several advantages over its photonic
or phononic counterparts. These include lower energy con-
sumption, easier integrability and compatibility with com-
plementary metal-oxide semiconductor (CMOS) structure,
programmability, shorter wavelength, smaller device fea-
tures, anisotropic properties, negative group velocity, non-
reciprocity and efficient tunability by various external stimuli
to name a few. Hence, despite being a young research field,
magnonics has come a long way since its early inception. A
flurry of research on propagating and confined spin waves have
unraveled a plethora of fundamental physics, and a handful
of prototype devices and concepts such as magnonic memory,
logic, transistor, transducers, RF components (filters, diodes
and circulators), all-magnon circuits and neuromorphic com-
puting have been unearthed. This brings us to the important
milestone of preparing a roadmap of magnonics research in
the coming years.

This is the first roadmap on magnonics. Three roadmaps
of magnetism have already been published by IOP in 2014,
2017 and 2020, where few aspects of magnonics and spin
dynamics have been discussed. However, a comprehensive
roadmap solely dedicated to magnonics has been long due.
Magnonics community organize the international workshop
titled ‘Magnonics: From Fundamentals to Applications’, every
two years which has become a benchmark conference not

only for the magnonics community but also for researchers
working in related fields like spintronics, microwave elec-
tronics, nanomagnetism, etc. In addition, an ‘*International
Advanced School on Magnonics*’ is organized every two years.
Further, Magnonics sessions regularly feature at the major
international conferences (e.g. MMM, Intermag, JEMS, APS
meetings, etc). The number of research papers and citations
in this field has steadily increased over last three decades and
experienced huge upsurge during the last one decade. Hence,
the time is ripe to prepare a roadmap on magnonics.

This roadmap aims to encompass the state of the art in the
field of magnonics [10, 11] and to move forward in the quest
for new concepts, phenomena and techniques for the ultimate
goal of translational research in the coming years. We search
for ways to conjugate the substantial knowledge-base made
in the conventional and emerging topics of magnonics (e.g.
spin textures, strong coupling, topology, biology, spin-orbit
effects) and lead toward hybrid magnonics, where more than
one effects/stimuli can drive each other or have interconver-
sion for discovering novel phenomena, as well as developing
more robust and efficient devices (figure 1). The roadmap is
an interdisciplinary collection of sections written by leading
experts in the field where classical magnonics merges with
quantum effects, e.g. spin-orbit coupling (SOC), the spin Hall
and spin pumping effect, Dzyaloshinskii-Moriya interaction
(DMI), superconductivity, nonlinearity, topology, etc.

The roadmap contains a discussion of fabrication tech-
niques of planar and three-dimensional magnonic structures
using top-down and bottom-up approaches besides pattern-
ing of spin textures (section 1). The research on planar MC
have matured substantially during the last one decade and
the future challenges lie in extensive research in unconven-
tional structures, quasi-periodic lattices, controllable modula-
tors, vortex and skymion lattices, etc (section 2). Efforts on
the extension of planar MC toward the third dimensions has
just started [12], which promises very complex spin-wave dis-
persion and confinement effects (sections 1, 2 and 6). The
3rd dimension can be exploited to control the in-plane prop-
agation of spin waves, offering a robust route for designing
non-reciprocal magnonic spectra with chiral properties with-
out the need for DMI and sophisticated material growth. 3D
integrations with respect to 2D systems permit to fit more
functionality into a smaller space, allowing a large number
of vertical connections between the layers, and an increase
of the density of elements for the fabrication of scalable and
configurable magnonic networks. Moreover, curved surfaces
and 3D micro- and nano-objects exhibit peculiar and unex-
pected spin textures which are normally not observed in planar
nanostructures and allows for the exploitation of magnon’s
chirality and the resulting non-reciprocity of the magnon dis-
persion (section 6). One of the most fundamental topics of
magnonics is magnon Bose-Einstein condensation (BEC).
Beyond conventional parametric pumping, kinetic instability
(Some figures may appear in colour only in the online journal)

Introduction

Anjan Barman¹ and Gianluca Gubbiotti²

¹S N Bose National Centre for Basic Sciences, India

²Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Italy

The concept of spin waves was first introduced by F Bloch in 1932 [1] and further developed by Holstein & Primakoff [2] and Dyson [3] who laid the foundation of spin waves theory. Although effects of periodic modulation on wave propagation was known since the late 19th century, and concept of pass and stop bands was introduced for electromagnetic wave prop- agation in 1950s [4], it was introduced in the magnetic system after two decades [5]. Subsequently, different kinds of periodic modulation have been introduced. However, the invention of photonic crystal [6] has fueled intense interest in spin waves in periodic magnetic media in the late 1990s [7] and a new field named magnonics was born. In 2001, the term magnonic crystal (MC) was coined by Gulyaev and Nikitov [8] and later by Puszkarski and Krawczyk as the magnetic counterpart of photonic crystals [9].

Magnonics deals with the excitation, propagation, con-
trol and detection of spin waves (quanta of which are called
magnons) through periodic magnetic media consisting of
either passively (patterned structures, modulated magnetic
properties) or actively (spin texture, electric field, magnetic
field) controlled modulation. It is analogous to photonics or
phononics but it has several advantages over its photonic
or phononic counterparts. These include lower energy con-
sumption, easier integrability and compatibility with com-
plementary metal-oxide semiconductor (CMOS) structure,
programmability, shorter wavelength, smaller device fea-
tures, anisotropic properties, negative group velocity, non-
reciprocity and efficient tunability by various external stimuli
to name a few. Hence, despite being a young research field,
magnonics has come a long way since its early inception. A
flurry of research on propagating and confined spin waves have
unraveled a plethora of fundamental physics, and a handful
of prototype devices and concepts such as magnonic memory,
logic, transistor, transducers, RF components (filters, diodes
and circulators), all-magnon circuits and neuromorphic com-
puting have been unearthed. This brings us to the important
milestone of preparing a roadmap of magnonics research in
the coming years.

This is the first roadmap on magnonics. Three roadmaps
of magnetism have already been published by IOP in 2014,
2017 and 2020, where few aspects of magnonics and spin
dynamics have been discussed. However, a comprehensive
roadmap solely dedicated to magnonics has been long due.
Magnonics community organize the international workshop
titled ‘Magnonics: From Fundamentals to Applications’, every
two years which has become a benchmark conference not

only for the magnonics community but also for researchers
working in related fields like spintronics, microwave elec-
tronics, nanomagnetism, etc. In addition, an ‘*International
Advanced School on Magnonics*’ is organized every two years.
Further, Magnonics sessions regularly feature at the major
international conferences (e.g. MMM, Intermag, JEMS, APS
meetings, etc). The number of research papers and citations
in this field has steadily increased over last three decades and
experienced huge upsurge during the last one decade. Hence,
the time is ripe to prepare a roadmap on magnonics.

This roadmap aims to encompass the state of the art in the
field of magnonics [10, 11] and to move forward in the quest
for new concepts, phenomena and techniques for the ultimate
goal of translational research in the coming years. We search
for ways to conjugate the substantial knowledge-base made
in the conventional and emerging topics of magnonics (e.g.
spin textures, strong coupling, topology, biology, spin-orbit
effects) and lead toward hybrid magnonics, where more than
one effects/stimuli can drive each other or have interconver-
sion for discovering novel phenomena, as well as developing
more robust and efficient devices (figure 1). The roadmap is
an interdisciplinary collection of sections written by leading
experts in the field where classical magnonics merges with
quantum effects, e.g. spin-orbit coupling (SOC), the spin Hall
and spin pumping effect, Dzyaloshinskii-Moriya interaction
(DMI), superconductivity, nonlinearity, topology, etc.

The roadmap contains a discussion of fabrication tech-
niques of planar and three-dimensional magnonic structures
using top-down and bottom-up approaches besides pattern-
ing of spin textures (section 1). The research on planar MC
have matured substantially during the last one decade and
the future challenges lie in extensive research in unconven-
tional structures, quasi-periodic lattices, controllable modula-
tors, vortex and skymion lattices, etc (section 2). Efforts on
the extension of planar MC toward the third dimensions has
just started [12], which promises very complex spin-wave dis-
persion and confinement effects (sections 1, 2 and 6). The
3rd dimension can be exploited to control the in-plane prop-
agation of spin waves, offering a robust route for designing
non-reciprocal magnonic spectra with chiral properties with-
out the need for DMI and sophisticated material growth. 3D
integrations with respect to 2D systems permit to fit more
functionality into a smaller space, allowing a large number
of vertical connections between the layers, and an increase
of the density of elements for the fabrication of scalable and
configurable magnonic networks. Moreover, curved surfaces
and 3D micro- and nano-objects exhibit peculiar and unex-
pected spin textures which are normally not observed in planar
nanostructures and allows for the exploitation of magnon’s
chirality and the resulting non-reciprocity of the magnon dis-
persion (section 6). One of the most fundamental topics of
magnonics is magnon Bose-Einstein condensation (BEC).
Beyond conventional parametric pumping, kinetic instability
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The many branches of magnonics.</figcaption></figure>

regime, rapid cooling, spin pumping effect have been explored to attain more efficient magnon-BEC. On the other hand, various indirect confirmation of magnon supercurrent, which is considered to be a key element of information transfer from magnon-BEC, has been obtained, e.g. by observation of quantized vortices in a two-component Bose-Einstein condensate by using Brillouin light scattering (BLS) spectroscopy (section 5).

A very important aspect of modern magnonics is to use exchange-dominated short-wavelength magnons as it promises high-speed magnonic devices and data processing. However, conventional microwave antenna-based excitation is inefficient and new approaches such as resonant and non-resonant magnonic nanogratings, ferromagnetic coplanar waveguides (CPWs), parametric pumping, spin-transfer torque and spin textures will be useful. Some important issues like using topologically protected magnetic states as nanochannels of low damping and magnon-qubit coupling scheme involving short-wavelength magnons for integrated hybrid quantum systems have been discussed in section 7. Optically inspired magnonics, based on spin-wave nano-optics and easy local modification of ‘optical’ properties can be useful for miniaturization of devices using short-wavelength magnons in metallic ferromagnets, but high damping and low dynamic range limit the maximum achievable device size (section 9). Anti-damping torque from the spin-orbit effect can play a crucial role in damping modulation (sections 10 and 13) [13]. Voltage-controlled magnonics can offer energy-efficient alternatives in beyond-CMOS computing and a two-pronged approach can be taken to this end, using magnon straintronics (section 8) and voltage-controlled

magnetic anisotropy (VCMA, section 18). While the former depends on inverse magnetostriction (Villari effect), the latter works on the change in electronic occupation at the ferromagnet/oxide interface and the ensuing orbital hybridization. Both the effects can excite and control spin waves and produce dynamic MC and magnonic nanochannels. SOC effects can play a big role in energy-efficient magnonics, and effects like perpendicular magnetic anisotropy (PMA), spin Hall effect (SHE), Rashba effect, spin pumping, spin caloric effect and DMI have already been extensively used in various branches of magnonics ranging from magnon-BEC to magnonic devices. They have their origin both in the intrinsic and extrinsic mechanism, where the latter can be efficiently engineered to externally control magnonic systems made of SOC materials. The SHE can cause auto-oscillation by completely compensating damping and this auto-oscillation in a nano-notch SHE oscillator have shown to efficiently emit unidirectional propagating spin waves into a magnonic waveguide. The propagation length of emitted spin waves has been enhanced by up to a factor of three by the spin current injected over the entire length of the waveguide. However, nonlinear scattering of propagating spin waves from magnetic fluctuations and spatial self-localization of oscillations, preventing the emission of propagating spin waves both caused by the pure spin current are important issues to be addressed in future (section 10).

Spin wave non-reciprocity is both fundamentally and technologically important and it occurs when spin-wave propagation changes or becomes forbidden upon inversion of the propagation direction and requires a breaking of the time-reversal symmetry (section 16). Various effects such as surface acoustic wave (SAW), dipolar interaction, gradient
<figure><img src="image_4.png" /><figcaption>**Figure 1.** The many branches of magnonics.</figcaption></figure>

regime, rapid cooling, spin pumping effect have been explored to attain more efficient magnon-BEC. On the other hand, various indirect confirmation of magnon supercurrent, which is considered to be a key element of information transfer from magnon-BEC, has been obtained, e.g. by observation of quantized vortices in a two-component Bose-Einstein condensate by using Brillouin light scattering (BLS) spectroscopy (section 5).

A very important aspect of modern magnonics is to use exchange-dominated short-wavelength magnons as it promises high-speed magnonic devices and data processing. However, conventional microwave antenna-based excitation is inefficient and new approaches such as resonant and non-resonant magnonic nanogratings, ferromagnetic coplanar waveguides (CPWs), parametric pumping, spin-transfer torque and spin textures will be useful. Some important issues like using topologically protected magnetic states as nanochannels of low damping and magnon-qubit coupling scheme involving short-wavelength magnons for integrated hybrid quantum systems have been discussed in section 7. Optically inspired magnonics, based on spin-wave nano-optics and easy local modification of ‘optical’ properties can be useful for miniaturization of devices using short-wavelength magnons in metallic ferromagnets, but high damping and low dynamic range limit the maximum achievable device size (section 9). Anti-damping torque from the spin-orbit effect can play a crucial role in damping modulation (sections 10 and 13) [13]. Voltage-controlled magnonics can offer energy-efficient alternatives in beyond-CMOS computing and a two-pronged approach can be taken to this end, using magnon straintronics (section 8) and voltage-controlled

magnetic anisotropy (VCMA, section 18). While the former depends on inverse magnetostriction (Villari effect), the latter works on the change in electronic occupation at the ferromagnet/oxide interface and the ensuing orbital hybridization. Both the effects can excite and control spin waves and produce dynamic MC and magnonic nanochannels. SOC effects can play a big role in energy-efficient magnonics, and effects like perpendicular magnetic anisotropy (PMA), spin Hall effect (SHE), Rashba effect, spin pumping, spin caloric effect and DMI have already been extensively used in various branches of magnonics ranging from magnon-BEC to magnonic devices. They have their origin both in the intrinsic and extrinsic mechanism, where the latter can be efficiently engineered to externally control magnonic systems made of SOC materials. The SHE can cause auto-oscillation by completely compensating damping and this auto-oscillation in a nano-notch SHE oscillator have shown to efficiently emit unidirectional propagating spin waves into a magnonic waveguide. The propagation length of emitted spin waves has been enhanced by up to a factor of three by the spin current injected over the entire length of the waveguide. However, nonlinear scattering of propagating spin waves from magnetic fluctuations and spatial self-localization of oscillations, preventing the emission of propagating spin waves both caused by the pure spin current are important issues to be addressed in future (section 10).

Spin wave non-reciprocity is both fundamentally and technologically important and it occurs when spin-wave propagation changes or becomes forbidden upon inversion of the propagation direction and requires a breaking of the time-reversal symmetry (section 16). Various effects such as surface acoustic wave (SAW), dipolar interaction, gradient
magnetic field and interfacial DMI (iDMI) can give rise to such non-reciprocity. Such non-reciprocal propagation can continuously tune the magnonic band structure and bandgap in MC with iDMI (section 13). The introduction of periodic DMI can cause indirect magnonic gaps, flat bands and complex temporal evolution of the spin waves, and can be regarded as a *chiral* MC, where topological magnons should be observed (section 16). iDMI also plays important role in stabilizing chiral spin textures, which can affect the magnon propagation giving rise to the topological and magnon Hall effect. Spin textures play a key role in magnonics due to their stability and resilience combined with a remarkable degree of tunability and scalability toward nanoscale dimensions. Some prominent examples are spin-wave channelling within domain walls, reconfigurable MC and reprogrammable spin-wave circuits (section 11). A related field is the graded-index magnonics where tailored graded magnonic landscapes from nonuniformity of internal magnetic field can be exploited to create practical devices such as magnonic Luneburg lens. However, development of designer magnonic landscapes with low damping is a major challenge ahead (section 12). Antiferromagnetic and THz magnonics offer ultra-high frequency devices. In antiferromagnets, the THz eigenfrequency stems from the very high sublattice exchange field, while laser-induced spin current in a ferromagnet/nonmagnet thin films can also cause THz spin-wave emission due to the confinement of exchange standing spin waves in nanometer-thick layer. Synthetic antiferromagnets (SyAFs) and angular momentum compensated ferrimagnets also exhibit exotic spin dynamics. Magnetic damping measurements and direct observation of antiferromagnetic magnon modes are crucial future challenges, while magnon current driven by all-electronic on-chip THz sources may enable novel energy-efficient spin memories operating at THz frequencies (sections 14 and 15). Spin-wave power flow and caustics will be important in magnonic computing which may require the creation of multiple spin-wave beams and methods to tune the focusing on a small and local scale. Magnon focusing can occur by playing with the curvature of the isofrequency curve, viz by tuning internal system parameters such as magnetic anisotropy, iDMI, nanostructuring or by external control such as electric field, magnetic field, spin-polarized current etc (section 17). Cavity magnonics is an emerging field that emphasizes strong coupling between cavity photons and magnons because of their coherence over large distances, whereas magnons and phonons also generate indirect interactions between tiny magnets over large distances. A coherent magnon-photon coupling causes photon-magnon mode repulsion. Some challenges in this field are to customize and load cavities, chiral interaction leading toward designing chiral magnonic molecules and increased nonlinearity for creating tripartite entanglement between magnons, photons and phonons (section 19).

Topologically protected magnon modes hold great promises due to their inherent robustness against defects and imperfections and their generic chirality, whereas possible applications in advanced and quantum information processing may occur due to lower dissipation and possible coherent control of magnons. However, their experimental

confirmation is still limited to the magnon Hall effect and observation of topological magnon bands. To this end, one of the major challenges will be to find systems with strong DMI for the generation of a Berry curvature of the magnon bands leading toward a transverse deflection of the exchange dominated magnon current so that it can be detected at higher temperatures and occurred on shorter time scales (section 20). Superconductor/ferromagnet (S/F) hybrid systems at low temperatures offer highly interesting physics, e.g. spin-polarized triplet supercurrents via spin mixing and spin rotation processes in proximity coupled S/F interface and a fluxon-induced MC in proximity decoupled S/F bilayer system in out-of-plane magnetic fields, when the superconductor attains a mixed state having Abrikosov vortices. S/F heterostructures can also allow for highly efficient magnon-photon coupling, which is important for quantum cavity magnonics (section 21). Biologically encoded magnonics is still in its infancy, which takes the approach of using microorganisms having the genetic machinery to form dedicated ensembles of physically separated magnetic nanoparticles (magnetosomes) for orientation by the Earth's magnetic field. Their GHz frequency dynamic response shows coherent oscillation inside each nanoparticle and dipolar spin wave along the particle chain. Magnetotactic bacteria like *magnetotactic spirilla* and *magnetotactic cocci* can be genetically or mechanically modified to form magnetosome chains with sharp kinks, which can be used for developing magnonic gates (section 22). Despite showing initial promises, these systems will require extensive refinements for being used in devices.

The overriding aim of magnonics is to deliver high-frequency and nanoscale on-chip devices and circuits. Magnonic data processing has potential advantages like using the spin-wave bus, miniaturization to the atomic scale, large coherence length, broad bandwidth, reconfigurability, dynamic control and non-linearity, and various classes of magnonic devices have been developed or proposed. These include RF components, e.g. reconfigurable filters, delay lines, phase shifters, Y-circulators, multiplexers, wake-up receivers, signal-to-noise enhancers, spectrum analysers, interference-based Boolean logic and majority gates, all-magnon circuits, unconventional spin-wave computing, neuromorphic and quantum computing (section 4). A realistic magnonic computer, however, will require magnonic logic circuits, interconnects, and magnonic memory but in its current status a competitive replacement of all aspects of the state-of-the-art charge-based computing systems by its magnonic counterpart seems elusive. A hybrid spin-wave-CMOS system with local spin-wave islands embedded in a CMOS periphery seems more realistic if the signal conversion between magnonic and electric domains can be efficient. It also has potential for area reduction. There are major challenges for magnonic logic to be integrated alongside CMOS in practical microelectronic applications. Alongside magnonic logic satisfying all criteria for circuit design, development of energy-efficient scalable transducers and efficient periphery to interface between transducers and magnonic circuits with the larger CMOS part of the chip is required (section 3).
magnetic field and interfacial DMI (iDMI) can give rise to such non-reciprocity. Such non-reciprocal propagation can continuously tune the magnonic band structure and bandgap in MC with iDMI (section 13). The introduction of periodic DMI can cause indirect magnonic gaps, flat bands and complex temporal evolution of the spin waves, and can be regarded as a *chiral* MC, where topological magnons should be observed (section 16). iDMI also plays important role in stabilizing chiral spin textures, which can affect the magnon propagation giving rise to the topological and magnon Hall effect. Spin textures play a key role in magnonics due to their stability and resilience combined with a remarkable degree of tunability and scalability toward nanoscale dimensions. Some prominent examples are spin-wave channelling within domain walls, reconfigurable MC and reprogrammable spin-wave circuits (section 11). A related field is the graded-index magnonics where tailored graded magnonic landscapes from nonuniformity of internal magnetic field can be exploited to create practical devices such as magnonic Luneburg lens. However, development of designer magnonic landscapes with low damping is a major challenge ahead (section 12). Antiferromagnetic and THz magnonics offer ultra-high frequency devices. In antiferromagnets, the THz eigenfrequency stems from the very high sublattice exchange field, while laser-induced spin current in a ferromagnet/nonmagnet thin films can also cause THz spin-wave emission due to the confinement of exchange standing spin waves in nanometer-thick layer. Synthetic antiferromagnets (SyAFs) and angular momentum compensated ferrimagnets also exhibit exotic spin dynamics. Magnetic damping measurements and direct observation of antiferromagnetic magnon modes are crucial future challenges, while magnon current driven by all-electronic on-chip THz sources may enable novel energy-efficient spin memories operating at THz frequencies (sections 14 and 15). Spin-wave power flow and caustics will be important in magnonic computing which may require the creation of multiple spin-wave beams and methods to tune the focusing on a small and local scale. Magnon focusing can occur by playing with the curvature of the isofrequency curve, viz by tuning internal system parameters such as magnetic anisotropy, iDMI, nanostructuring or by external control such as electric field, magnetic field, spin-polarized current etc (section 17). Cavity magnonics is an emerging field that emphasizes strong coupling between cavity photons and magnons because of their coherence over large distances, whereas magnons and phonons also generate indirect interactions between tiny magnets over large distances. A coherent magnon–photon coupling causes photon-magnon mode repulsion. Some challenges in this field are to customize and load cavities, chiral interaction leading toward designing chiral magnonic molecules and increased nonlinearity for creating tripartite entanglement between magnons, photons and phonons (section 19).

Topologically protected magnon modes hold great promises due to their inherent robustness against defects and imperfections and their generic chirality, whereas possible applications in advanced and quantum information processing may occur due to lower dissipation and possible coherent control of magnons. However, their experimental

confirmation is still limited to the magnon Hall effect and observation of topological magnon bands. To this end, one of the major challenges will be to find systems with strong DMI for the generation of a Berry curvature of the magnon bands leading toward a transverse deflection of the exchange dominated magnon current so that it can be detected at higher temperatures and occurred on shorter time scales (section 20). Superconductor/ferromagnet (S/F) hybrid systems at low temperatures offer highly interesting physics, e.g. spin-polarized triplet supercurrents via spin mixing and spin rotation processes in proximity coupled S/F interface and a fluxon-induced MC in proximity decoupled S/F bilayer system in out-of-plane magnetic fields, when the superconductor attains a mixed state having Abrikosov vortices. S/F heterostructures can also allow for highly efficient magnon–photon coupling, which is important for quantum cavity magnonics (section 21). Biologically encoded magnonics is still in its infancy, which takes the approach of using microorganisms having the genetic machinery to form dedicated ensembles of physically separated magnetic nanoparticles (magnetosomes) for orientation by the Earth’s magnetic field. Their GHz frequency dynamic response shows coherent oscillation inside each nanoparticle and dipolar spin wave along the particle chain. Magnetotactic bacteria like *magnetotactic spirilla* and *magnetotactic cocci* can be genetically or mechanically modified to form magnetosome chains with sharp kinks, which can be used for developing magnonic gates (section 22). Despite showing initial promises, these systems will require extensive refinements for being used in devices.

The overriding aim of magnonics is to deliver high-frequency and nanoscale on-chip devices and circuits. Magnonic data processing has potential advantages like using the spin-wave bus, miniaturization to the atomic scale, large coherence length, broad bandwidth, reconfigurability, dynamic control and non-linearity, and various classes of magnonic devices have been developed or proposed. These include RF components, e.g. reconfigurable filters, delay lines, phase shifters, Y-circulators, multiplexers, wake-up receivers, signal-to-noise enhancers, spectrum analysers, interference-based Boolean logic and majority gates, all-magnon circuits, unconventional spin-wave computing, neuromorphic and quantum computing (section 4). A realistic magnonic computer, however, will require magnonic logic circuits, interconnects, and magnonic memory but in its current status a competitive replacement of all aspects of the state-of-the-art charge-based computing systems by its magnonic counterpart seems elusive. A hybrid spin-wave–CMOS system with local spin-wave islands embedded in a CMOS periphery seems more realistic if the signal conversion between magnonic and electric domains can be efficient. It also has potential for area reduction. There are major challenges for magnonic logic to be integrated alongside CMOS in practical microelectronic applications. Alongside magnonic logic satisfying all criteria for circuit design, development of energy-efficient scalable transducers and efficient periphery to interface between transducers and magnonic circuits with the larger CMOS part of the chip is required (section 3).
Finally, imaging spin waves with high spatial resolution is an important issue. Whereas near-field BLS and time-resolved magneto-optical Kerr effect (TR-MOKE) microscopes can go below 100 nm resolution at the expense of measurement sensitivity, progress in x-ray and photoelectron microscopy and nitrogen vacancy (NV) center magnetome-try may offer better spatial resolution for the measurement of short-wavelength magnons, spin-wave caustics and other nanoscale spin-wave phenomena. Emergent and associated fields like two-dimensional magnonics in van der Waals mag-nets [14], hybrid magnonics including strong magnon-phonon coupling [15], spin-torque and spin-Hall nano-oscillators [16], neuromorphic computing [17], etc are making fast progress and would surely be topics of future magnonics roadmaps.

Thus, despite having great deal of success in both fun-damental and application fronts, magnonics faces many stern challenges in developing new materials and structures, improving energy efficiency, scalability and integration of devices to practical circuits. This roadmap points toward all the major issues that must be addressed in the coming years to make magnonics a competitive technology.

# 1. Novel fabrication techniques for magnonics

S Ladak<fnref n="1" /> and A O Adeyeye<fnref n="2" />

## 1.1. Status

Top-down fabrication processes has yielded a revolution in electronics, enabling a plethora of technological developments associated with computing and communication. Lithographi-cally defined magnetic structures followed suit from the 1970s, providing new avenues for studying solid-state magnetic phe-nomenon. The use of quantized spin-waves to transmit and store information is a paradigm-shifting approach to next gen-eration computing and communication technologies. By the time such developments were first proposed, advanced lithog-raphy and processing was well established as a platform for producing well-defined nanostructures with high quality inter-faces, allowing rapid growth in understanding of how spin-waves and their quasi-particles, can be harnessed within simple devices. Moving beyond simple lithographic processes allows the tuning of magnonic phenomena by nanoscale control of material interfaces, studying their interaction with non-trivial spin textures and by realising complex three-dimensional networks. A simple but powerful means to realise a one-dimensional MC is to produce periodic variations in film thickness via chemical etching. Such processes are not sim-ple to implement with technologically relevant materials such as yttrium iron garnet (YIG). In a pioneering study, Chumak *et al* [18] used orthophosphoric acid in order to realise a struc-tured YIG film, with periodic grooves etched into the sur-face (figure 2(a)). Microstrip antennas, placed 8 mm apart, were used for excitation and detection, in the presence of a

bias field applied along the strip length, allowing study of backward volume magnetostatic waves. Figure 2(b) shows the transmission characteristics of the grating as a function of groove depth. Clear stop bands are observed with the rejec-tion efficiency and the stop band width increasing with groove depth. The results were understood in the context of a sim-ple model, which approximated the grating with a series of transmission line segments, each with different propagation constants. Standard lithographic processes and processing can also be used in original ways in order to realise novel mag-netic systems with bespoke dynamic response. By harness-ing state-of-the-art deep ultraviolet lithography and angled deposition [19] (figure 2(c)), Ding *et al* realised periodic arrays of binary magnetic nanostructures consisting of over-lapping Ni₈₀Fe₂₀–Ni₈₀Fe₂₀ or Ni₈₀Fe₂₀–Ni islands as shown in figure 2(d). Dynamic properties of the arrays were measured using VNA-ferromagnetic resonance (FMR). The Ni₈₀Fe₂₀ and Ni₈₀Fe₂₀–Ni₈₀Fe₂₀ structures exhibited a single resonant frequency while the Ni₈₀Fe₂₀–Ni binary structures showed two distinct frequencies corresponding to a low frequency mode from Ni elements and a high frequency mode from the Ni₈₀Fe₂₀ contribution. The work shows simple variations in electron-beam lithography and processing can be a powerful means to control high frequency response. Another intrig-uing concept that utilises standard processing aims to pro-duce microwave-magnon transducers. Here, Yu *et al* patterned a lattice of Ni₈₁Fe₁₉ nanodisks (350 nm) between a YIG film and CPWs [20]. The higher frequency resonant response of the nanodisks at saturation was then exploited to pro-duce exchange-dominated spin waves with wavelength below 100 nm within the YIG film.

Recent work has shown 3D nanostructured magnetic mate-rials can yield a range of interesting phenomena, not seen in conventional planar geometries [21]. With respect to magnonic structures two aspects are particularly important. Firstly, the realisation of simple 3D cylindrical structures upon the nanoscale is a means to realise spin-Cherenkov effects, allow-ing controlled spin-wave emission. Secondly, the controlled production of 3D lattices upon the nanoscale is a route to bespoke 3D MCs. One means of realising such 3D struc-tures is via two-photon lithography (TPL), a powerful tech-nique which when combined with growth and processing can yield magnetic materials of arbitrary 3D geometry at a resolu-tion of approximately 100 nm. Proof-of-principle has recently been obtained in a pioneering study [22] by Sahoo *et al*, where TPL was used in combination with electrodeposition in order to realise complex 3D structures. Figure 2(e) shows an example 3D Co tetrapod structure, where individual wires had feature sizes of ~500 nm. Optically pumped, TR-MOKE was used to probe the magnetisation dynamics at the tetrapod junction, allowing the identification of three precessional modes at 1 GHz, 10 GHz and 30 GHz. Finite element sim-ulations (FEM) were used to visualise the profiles show-ing a spatially uniform mode (figure 2(f)) at 30 GHz while those at lower frequency were dipolar dominated with nodal planes spreading along two perpendicular directions. The study shows experimentally accessible techniques such as

<footnote n="1">¹Cardiff University, United Kingdom</footnote>

<footnote n="2">²University of Durham, United Kingdom</footnote>
Finally, imaging spin waves with high spatial resolution is an important issue. Whereas near-field BLS and time-resolved magneto-optical Kerr effect (TR-MOKE) microscopes can go below 100 nm resolution at the expense of measurement sensitivity, progress in x-ray and photoelectron microscopy and nitrogen vacancy (NV) center magnetome-try may offer better spatial resolution for the measurement of short-wavelength magnons, spin-wave caustics and other nanoscale spin-wave phenomena. Emergent and associated fields like two-dimensional magnonics in van der Waals mag-nets [14], hybrid magnonics including strong magnon-phonon coupling [15], spin-torque and spin-Hall nano-oscillators [16], neuromorphic computing [17], etc are making fast progress and would surely be topics of future magnonics roadmaps.

Thus, despite having great deal of success in both fun-damental and application fronts, magnonics faces many stern challenges in developing new materials and structures, improving energy efficiency, scalability and integration of devices to practical circuits. This roadmap points toward all the major issues that must be addressed in the coming years to make magnonics a competitive technology.

# 1. Novel fabrication techniques for magnonics

S Ladak<fnref n="1" /> and A O Adeyeye<fnref n="2" />

## 1.1. Status

Top-down fabrication processes has yielded a revolution in electronics, enabling a plethora of technological developments associated with computing and communication. Lithographi-cally defined magnetic structures followed suit from the 1970s, providing new avenues for studying solid-state magnetic phe-nomenon. The use of quantized spin-waves to transmit and store information is a paradigm-shifting approach to next gen-eration computing and communication technologies. By the time such developments were first proposed, advanced lithog-raphy and processing was well established as a platform for producing well-defined nanostructures with high quality inter-faces, allowing rapid growth in understanding of how spin-waves and their quasi-particles, can be harnessed within simple devices. Moving beyond simple lithographic processes allows the tuning of magnonic phenomena by nanoscale control of material interfaces, studying their interaction with non-trivial spin textures and by realising complex three-dimensional networks. A simple but powerful means to realise a one-dimensional MC is to produce periodic variations in film thickness via chemical etching. Such processes are not sim-ple to implement with technologically relevant materials such as yttrium iron garnet (YIG). In a pioneering study, Chumak *et al* [18] used orthophosphoric acid in order to realise a struc-tured YIG film, with periodic grooves etched into the sur-face (figure 2(a)). Microstrip antennas, placed 8 mm apart, were used for excitation and detection, in the presence of a

bias field applied along the strip length, allowing study of backward volume magnetostatic waves. Figure 2(b) shows the transmission characteristics of the grating as a function of groove depth. Clear stop bands are observed with the rejec-tion efficiency and the stop band width increasing with groove depth. The results were understood in the context of a sim-ple model, which approximated the grating with a series of transmission line segments, each with different propagation constants. Standard lithographic processes and processing can also be used in original ways in order to realise novel mag-netic systems with bespoke dynamic response. By harness-ing state-of-the-art deep ultraviolet lithography and angled deposition [19] (figure 2(c)), Ding *et al* realised periodic arrays of binary magnetic nanostructures consisting of over-lapping Ni₈₀Fe₂₀–Ni₈₀Fe₂₀ or Ni₈₀Fe₂₀–Ni islands as shown in figure 2(d). Dynamic properties of the arrays were measured using VNA-ferromagnetic resonance (FMR). The Ni₈₀Fe₂₀ and Ni₈₀Fe₂₀–Ni₈₀Fe₂₀ structures exhibited a single resonant frequency while the Ni₈₀Fe₂₀–Ni binary structures showed two distinct frequencies corresponding to a low frequency mode from Ni elements and a high frequency mode from the Ni₈₀Fe₂₀ contribution. The work shows simple variations in electron-beam lithography and processing can be a powerful means to control high frequency response. Another intrig-uing concept that utilises standard processing aims to pro-duce microwave-magnon transducers. Here, Yu *et al* patterned a lattice of Ni₈₁Fe₁₉ nanodisks (350 nm) between a YIG film and CPWs [20]. The higher frequency resonant response of the nanodisks at saturation was then exploited to pro-duce exchange-dominated spin waves with wavelength below 100 nm within the YIG film.

Recent work has shown 3D nanostructured magnetic mate-rials can yield a range of interesting phenomena, not seen in conventional planar geometries [21]. With respect to magnonic structures two aspects are particularly important. Firstly, the realisation of simple 3D cylindrical structures upon the nanoscale is a means to realise spin-Cherenkov effects, allow-ing controlled spin-wave emission. Secondly, the controlled production of 3D lattices upon the nanoscale is a route to bespoke 3D MCs. One means of realising such 3D struc-tures is via two-photon lithography (TPL), a powerful tech-nique which when combined with growth and processing can yield magnetic materials of arbitrary 3D geometry at a resolu-tion of approximately 100 nm. Proof-of-principle has recently been obtained in a pioneering study [22] by Sahoo *et al*, where TPL was used in combination with electrodeposition in order to realise complex 3D structures. Figure 2(e) shows an example 3D Co tetrapod structure, where individual wires had feature sizes of ~500 nm. Optically pumped, TR-MOKE was used to probe the magnetisation dynamics at the tetrapod junction, allowing the identification of three precessional modes at 1 GHz, 10 GHz and 30 GHz. Finite element sim-ulations (FEM) were used to visualise the profiles show-ing a spatially uniform mode (figure 2(f)) at 30 GHz while those at lower frequency were dipolar dominated with nodal planes spreading along two perpendicular directions. The study shows experimentally accessible techniques such as

<footnote n="1">¹Cardiff University, United Kingdom</footnote>

<footnote n="2">²University of Durham, United Kingdom</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) A structured YIG film with periodic grooves with embedded microstrip antennas. (b) Transmission spectra showing stops bands as function of groove depth. (c) Reprinted from [18], with the permission of AIP Publishing. (d) Novel methodology for producing binary magnetic nanostructure arrays. (e) SEM of binary Ni₈₀Fe₂₀–Ni islands. (f) [19] John Wiley & Sons. [Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]. (g) Co tetrapod structure fabricated with TPL and electrodeposition. (h) Micro-magnetic simulation showing spatially uniform mode at junction. (i) Reproduced from [22]. **CC BY 4.0**. [22]. **CC BY 4.0**. (j) Schematic showing thermally assisted magnetic scanning probe lithography. (k) Spatial map showing spin-wave excitation across a straight Néel domain wall. (l) Reproduced from [23]. **CC BY 4.0**.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (a) A structured YIG film with periodic grooves with embedded microstrip antennas. (b) Transmission spectra showing stops bands as function of groove depth. (c) Reprinted from [18], with the permission of AIP Publishing. (d) Novel methodology for producing binary magnetic nanostructure arrays. (e) SEM of binary Ni₈₀Fe₂₀–Ni islands. (f) [19] John Wiley & Sons. [Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]. (g) Co tetrapod structure fabricated with TPL and electrodeposition. (h) Micro-magnetic simulation showing spatially uniform mode at junction. (i) Reproduced from [22]. **CC BY 4.0**. [22]. **CC BY 4.0**. (j) Schematic showing thermally assisted magnetic scanning probe lithography. (k) Spatial map showing spin-wave excitation across a straight Néel domain wall. (l) Reproduced from [23]. **CC BY 4.0**.</figcaption></figure>
MOKE can be used to measure magnetisation dynamics in complex 3D structures.

An exceptionally novel approach to magnonic waveguide fabrication harnesses the patterning of spin-textures rather than physical structures. A key advantage here is the resolution that can be obtained, which surpasses physical lithography techniques and in addition, the waveguides can be reconfigured. Such a route was demonstrated experimentally by Ablisetti et al who used thermally assisted magnetic scanning probe lithography in order to produce reconfigurable spin-wave channels [24] within an exchange bias magnetic multilayer (figure 2(g)). Here, scanning of a heated tip in the presence of an external field allows the controlled formation of domains and their associated domain walls which were then harnessed for spin-wave transport. By patterning Néel domain walls into a number of configurations and utilising a microstrip antenna for excitation, the authors were able to show via transmission x-ray microscopy, channelled spin-wave excitation (figure 2(h)).

Alternative approaches, including bottom-up fabrication processes have also been pursued to realise magnonic structures. In particular, harnessing self-assembly driven by biological organisms and their associated processes is a promising avenue of research. Zingsem et al utilised such an approach [25], harnessing the natural capability of magnetotactic bacterium to grow magnetite crystal chains (figure 3(a)). A resonant microcavity at 9.1 GHz was used to excite and detect magnetostatic spin waves as a function of external field magnitude and angle (figures 3(b) and (c)). The band deformation and band gap were found to be a complex function of magnetite particle orientation and local arrangement which, in itself, could be controlled by genetic engineering, paving the way to biomagnonics.

The pyrolysis of ferrocene is a promising bottom-up methodology for realising iron filled carbon nanotubes [26] (FeCNT). Micro-manipulation was used to place FeCNTs upon patterned microresonsators in order to measure FMR or onto a substrate with microwave antennas for BLS measurements. A high resolution TEM image of a typical FeCNT is shown in figure 3(d). Both pristine and FIB cut FeCNTs were studied resulting in diameters varying between 20–41 nm and lengths varying between 0.73–14.5 µm. The thermal spin-wave spectrum of longer FeCNTs, as measured by micro-BLS, is shown in figure 3(e). The peaks originating from the FeCNT are found to be discontinuous across its length, indicating there are points with either degraded magnetisation or geometric gaps. Micromagnetic simulations which implemented geometric gaps (20 nm) between segments allowed reproduction of key experimental results. Overall, the single crystal nature of the nanotubes, along with the narrow linewidth, make such systems promising candidates for magnonic applications.

Self-assembly of magnetic nanoparticles is a promising bottom-up approach in realising truly 3D MCs but defining the precise 3D geometry such that it can be implemented into devices is challenging. Okuda et al [27] utilised an approach whereby protein/inorganic Fe nanoparticle composite systems were first prepared by fixing hydrated apoferritin crystals.

Focused ion-beam milling was then used to machine the composites into a micro-cube (figure 3(f)), which was integrated with a co-planar waveguide. Preliminary FMR data showed a peak at approximately 15 GHz, though this was substantially broadened at 5 K due to nanoparticle anisotropy variation.

## 1.2. Current and future challenges

The main challenges in magnonic sample fabrication remains the achievement of high-quality periodic nanostructures, with excellent interfacial characteristics, which can be scaled up and combined with standard lithographic processes to enable mass device production. Techniques that harness top-down processes have the best hope for scaling to device level. Here, there is still a great deal of research to be done upon investigating optimum materials, geometries and then designing optimal device architectures. The realisation of 3D nanostructures, such as nanotubes (figure 3(g)), may be important for harnessing spin-Cherenkov effects [28] in order to realise tunable spin-wave emission. Here a key challenge is both producing high quality 3D structures, with high precision as well as interfacing to on-chip magnonic circuitry. TPL and electrodeposition is a powerful methodology for producing 3D magnetic nanostructures. A key challenge is reaching the relevant length scales. Here, the implementation of shorter wavelength lasers and spatial light modulators allows the writing of sub-100 nm features with multiple foci, allowing rapid manufacture. Bottom-up technologies are also promising for realising 3D structures, though more work is needed to control the precise geometry. Additionally, more efficient means need to be determined for such structures to be implemented onto the relevant electronic chips. For example, micro-manipulation though fine for prototypes is largely inefficient. One approach may start with the relevant device architecture and make use of a suitable ‘seeding’ such that self-assembly only occurs upon the required parts of the planar device.

## 1.3. Concluding remarks

A variety of promising fabrication methodologies are being explored for magnonics each with specific advantages and disadvantages. Further optimisation of individual techniques as well as the possibility of combining different approaches will yield a range of cutting-edge magnonic systems, ultimately paving the way to scalable devices.

## Acknowledgments

Sam Ladak acknowledges funding from the EPSRC (EP/R009147/1).

## 2. Magnonic crystals and quasicrystals

Maciej Krawczyk<fnref n="1" /> and Joachim Gräfe<fnref n="2" />

<footnote n="1">¹Adam Mickiewicz University, Poland</footnote>

<footnote n="2">²Max Planck Institute for Intelligent Systems, Germany</footnote>
MOKE can be used to measure magnetisation dynamics in complex 3D structures.

An exceptionally novel approach to magnonic waveguide fabrication harnesses the patterning of spin-textures rather than physical structures. A key advantage here is the resolution that can be obtained, which surpasses physical lithography techniques and in addition, the waveguides can be reconfigured. Such a route was demonstrated experimentally by Ablisetti et al who used thermally assisted magnetic scanning probe lithography in order to produce reconfigurable spin-wave channels [24] within an exchange bias magnetic multilayer (figure 2(g)). Here, scanning of a heated tip in the presence of an external field allows the controlled formation of domains and their associated domain walls which were then harnessed for spin-wave transport. By patterning Néel domain walls into a number of configurations and utilising a microstrip antenna for excitation, the authors were able to show via transmission x-ray microscopy, channelled spin-wave excitation (figure 2(h)).

Alternative approaches, including bottom-up fabrication processes have also been pursued to realise magnonic structures. In particular, harnessing self-assembly driven by biological organisms and their associated processes is a promising avenue of research. Zingsem et al utilised such an approach [25], harnessing the natural capability of magnetotactic bacterium to grow magnetite crystal chains (figure 3(a)). A resonant microcavity at 9.1 GHz was used to excite and detect magnetostatic spin waves as a function of external field magnitude and angle (figures 3(b) and (c)). The band deformation and band gap were found to be a complex function of magnetite particle orientation and local arrangement which, in itself, could be controlled by genetic engineering, paving the way to biomagnonics.

The pyrolysis of ferrocene is a promising bottom-up methodology for realising iron filled carbon nanotubes [26] (FeCNT). Micro-manipulation was used to place FeCNTs upon patterned microresonsators in order to measure FMR or onto a substrate with microwave antennas for BLS measurements. A high resolution TEM image of a typical FeCNT is shown in figure 3(d). Both pristine and FIB cut FeCNTs were studied resulting in diameters varying between 20–41 nm and lengths varying between 0.73–14.5 µm. The thermal spin-wave spectrum of longer FeCNTs, as measured by micro-BLS, is shown in figure 3(e). The peaks originating from the FeCNT are found to be discontinuous across its length, indicating there are points with either degraded magnetisation or geometric gaps. Micromagnetic simulations which implemented geometric gaps (20 nm) between segments allowed reproduction of key experimental results. Overall, the single crystal nature of the nanotubes, along with the narrow linewidth, make such systems promising candidates for magnonic applications.

Self-assembly of magnetic nanoparticles is a promising bottom-up approach in realising truly 3D MCs but defining the precise 3D geometry such that it can be implemented into devices is challenging. Okuda et al [27] utilised an approach whereby protein/inorganic Fe nanoparticle composite systems were first prepared by fixing hydrated apoferritin crystals.

Focused ion-beam milling was then used to machine the composites into a micro-cube (figure 3(f)), which was integrated with a co-planar waveguide. Preliminary FMR data showed a peak at approximately 15 GHz, though this was substantially broadened at 5 K due to nanoparticle anisotropy variation.

## 1.2. Current and future challenges

The main challenges in magnonic sample fabrication remains the achievement of high-quality periodic nanostructures, with excellent interfacial characteristics, which can be scaled up and combined with standard lithographic processes to enable mass device production. Techniques that harness top-down processes have the best hope for scaling to device level. Here, there is still a great deal of research to be done upon investigating optimum materials, geometries and then designing optimal device architectures. The realisation of 3D nanostructures, such as nanotubes (figure 3(g)), may be important for harnessing spin-Cherenkov effects [28] in order to realise tunable spin-wave emission. Here a key challenge is both producing high quality 3D structures, with high precision as well as interfacing to on-chip magnonic circuitry. TPL and electrodeposition is a powerful methodology for producing 3D magnetic nanostructures. A key challenge is reaching the relevant length scales. Here, the implementation of shorter wavelength lasers and spatial light modulators allows the writing of sub-100 nm features with multiple foci, allowing rapid manufacture. Bottom-up technologies are also promising for realising 3D structures, though more work is needed to control the precise geometry. Additionally, more efficient means need to be determined for such structures to be implemented onto the relevant electronic chips. For example, micro-manipulation though fine for prototypes is largely inefficient. One approach may start with the relevant device architecture and make use of a suitable ‘seeding’ such that self-assembly only occurs upon the required parts of the planar device.

## 1.3. Concluding remarks

A variety of promising fabrication methodologies are being explored for magnonics each with specific advantages and disadvantages. Further optimisation of individual techniques as well as the possibility of combining different approaches will yield a range of cutting-edge magnonic systems, ultimately paving the way to scalable devices.

## Acknowledgments

Sam Ladak acknowledges funding from the EPSRC (EP/R009147/1).

## 2. Magnonic crystals and quasicrystals

Maciej Krawczyk<fnref n="1" /> and Joachim Gräfe<fnref n="2" />

<footnote n="1">¹Adam Mickiewicz University, Poland</footnote>

<footnote n="2">²Max Planck Institute for Intelligent Systems, Germany</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Magnetite crystal chain grown by magnetotactic bacteria. (b) Angular dependence of magnetite crystal chain FMR spectra and (c) two single spectra recorded at 0° and 90°. (a-c) Reproduced from [25]. CC BY 4.0. (d) TEM of a single Fe filled carbon nanotube suspended across micro-resonator. (e) Thermal spin-wave spectrum of FeCNT obtained using BLS. (d), (e) Reproduced from [26]. CC BY 4.0. (f) Schematic of Fe₃O₄ crystal monomer (left), crystals (middle) and an SEM of the machined Fe₃O₄ crystal upon a CPW. Reproduced from [27]. © IOP Publishing Ltd. All rights reserved. (g) Spin-Cherenkov effect in magnetic nanotube. Reprinted from [28], with the permission of AIP Publishing.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) Magnetite crystal chain grown by magnetotactic bacteria. (b) Angular dependence of magnetite crystal chain FMR spectra and (c) two single spectra recorded at 0° and 90°. (a-c) Reproduced from [25]. CC BY 4.0. (d) TEM of a single Fe filled carbon nanotube suspended across micro-resonator. (e) Thermal spin-wave spectrum of FeCNT obtained using BLS. (d), (e) Reproduced from [26]. CC BY 4.0. (f) Schematic of Fe₃O₄ crystal monomer (left), crystals (middle) and an SEM of the machined Fe₃O₄ crystal upon a CPW. Reproduced from [27]. © IOP Publishing Ltd. All rights reserved. (g) Spin-Cherenkov effect in magnetic nanotube. Reprinted from [28], with the permission of AIP Publishing.</figcaption></figure>
## 2.1. Status

The term MC, the spin wave analogon to crystals for electrons or photonic crystals for photons, has first been coined in 2001 independently by Puszkarski *et al* and Nikitov *et al* [29]. Similar to any crystal, the fundamental ingredient for a MC is a periodic modulation of the spin-wave potential. According to the Bragg law and the Bloch theorem, such a periodic modulation of the potential consequently results in the formation of a spin-wave band structure.

While the spin-wave potential can be affected by changes in local field, anisotropy, thickness, damping or strain, the easiest way is a full modulation of the saturation magnetization by removal of the material, i.e. patterning holes and groves [30, 31]. Consequently, such a strong modulation has been utilized to demonstrate gratings and the formation of a full spin-wave band structure in regular crystals [31] in one, and subsequently, in two dimensions. At the same time lithographic capabilities evolved from microstructured YIG [32] to nanopatterned metallic thin films [31]. In the past years, aperiodic structures moved more and more into the spotlight and magnonic quasicrystals were realized. Both in one- and two-dimensional systems [30] (cf figure 4). Thereby, the mode localization of waves in quasicrystals were directly imaged, gaining general insights into this class of materials.

At the same time as experimental techniques like BLS, MOKE microscope, scanning transmission x-ray microscopy (STXM) and electrical spectroscopy evolved and got optimized for magnonics research, analytical and computational techniques were developed to fully capture MCs. Prominently, these include plane-wave theory for two- and three-dimensional systems [33] and extensions were also found for quasiperiodic systems [30]. These proved to be especially powerful, as they allow for modeling of large systems that had been computationally prohibitive for micromagnetic simulations; while the latter focused on ultra-nanoscale systems to deepen fundamental understanding MCs with strong magnonic modulation.

However, research into MCs goes beyond being yet another wave in a periodic potential. Contemporary micro- and nanolithography allows precise spatial patterning on the length scale of wavelength, while observation techniques like BLS [31] or more recently STXM [30] allow direct imaging of the spin waves themselves (cf figure 4). Furthermore, exchange, magnetostatic, anisotropy energies are main factors shaping the dispersion relation and their tunability provide an extensive playground for new physics.

## 2.2. Current and future challenges

Quasiperiodicity offers new dimensionality in design of collective spin-wave excitation spectra. The discrete spectra feature self-similarity provide extended and localized states. Localized waves controlled by the magnetic field orientation were already detected. They promise for short wavelength wave excitation and broadband RF operation, yet, future studies need to demonstrate such quasiperiodic utility. Here, the influence of extended coupling of elements separated by large

distances, negative group velocity, non-reciprocity, and chirality provide a test bed for new ideas, hardly accessible for other types of waves. The combination of these features with quasiperiodicity awaits discovery.

Moving toward the third dimension is a topic in many branches of solid-state physics and technology, from microelectronics, through photonics to magnonics. Thinking beyond planar magnonic structures, the third dimension can be exploited to control the in-plane propagation of spin waves, offering a robust route for designing nonreciprocal magnonic spectra with chiral properties without the need for DMI and sophisticated material growth. In MCs it can be further exploited to control in-plane coupling between the patterned elements, enhancing the group velocity and non-reciprocity; and guiding the waves in circuits for magnon spintronic applications. This bottom-up approach can be continued toward forming full magnonic 3D crystals (see section 6). New deposition approaches, like focused electron beam induced deposition and atomic layer deposition, offer fabrication of various 3D shapes in a periodic arrangement. Due to their shape anisotropy and small size, their magnetic elements are in a monodomain state and can work as discrete magnetic dipole arranged in a 3D lattice [22].

A further 3D design aspect is artificial spin ice systems with frustration at the vertexes of the elements. This is still a big challenge as even in planar artificial spin ice, the collective dynamics and guiding spin waves along controllable paths are in an infancy state of development [34, 35]. A real breakthrough is needed to provide a feasible approach for controlled coupling between spin wave dynamics of nearest, but also next- and next-next-nearest elements to achieve collective excitation in the whole array.

For the realization of sophisticated magnonic lattices with extended tunability, advances in preparation techniques are necessary to move beyond the strong modulation used in the past (see section 1). Thus, mild and gradual techniques for changing saturation magnetization, anisotropy, and damping are needed (see section 12). First steps toward this end have been undertaken in terms of doping or thermal gradients; additional promising approaches include localized strain modulation (see section 8), graded and combinatorial material deposition [36].

Furthermore, magnetic textures could be envisioned to also achieve such mild modulations. To this end, it has been shown that magnetic force microscopy tips and soft x-ray beams can be used to write arbitrary magnetization textures in specific materials [37]. This brings us to another broad area in the future of magnonics, which is the control of spin waves by magnetization textures rather than material structuring. In this case, the spin waves can follow paths designed on demand by a local change of the magnetization texture (see section 11). This is also true for periodic and other types of regular magnetization patterns, like stripe domains, and well forgotten bubble domains. The sign of possible magnonic band structure formation in stripe domain structures in multilayers with PMA has been shown.

Another approach toward reconfigurable MCs are controllable modulators. These prepatterned structures modulate the
## 2.1. Status

The term MC, the spin wave analogon to crystals for electrons or photonic crystals for photons, has first been coined in 2001 independently by Puszkarski *et al* and Nikitov *et al* [29]. Similar to any crystal, the fundamental ingredient for a MC is a periodic modulation of the spin-wave potential. According to the Bragg law and the Bloch theorem, such a periodic modulation of the potential consequently results in the formation of a spin-wave band structure.

While the spin-wave potential can be affected by changes in local field, anisotropy, thickness, damping or strain, the easiest way is a full modulation of the saturation magnetization by removal of the material, i.e. patterning holes and groves [30, 31]. Consequently, such a strong modulation has been utilized to demonstrate gratings and the formation of a full spin-wave band structure in regular crystals [31] in one, and subsequently, in two dimensions. At the same time lithographic capabilities evolved from microstructured YIG [32] to nanopatterned metallic thin films [31]. In the past years, aperiodic structures moved more and more into the spotlight and magnonic quasicrystals were realized. Both in one- and two-dimensional systems [30] (cf figure 4). Thereby, the mode localization of waves in quasicrystals were directly imaged, gaining general insights into this class of materials.

At the same time as experimental techniques like BLS, MOKE microscope, scanning transmission x-ray microscopy (STXM) and electrical spectroscopy evolved and got optimized for magnonics research, analytical and computational techniques were developed to fully capture MCs. Prominently, these include plane-wave theory for two- and three-dimensional systems [33] and extensions were also found for quasiperiodic systems [30]. These proved to be especially powerful, as they allow for modeling of large systems that had been computationally prohibitive for micromagnetic simulations; while the latter focused on ultra-nanoscale systems to deepen fundamental understanding MCs with strong magnonic modulation.

However, research into MCs goes beyond being yet another wave in a periodic potential. Contemporary micro- and nanolithography allows precise spatial patterning on the length scale of wavelength, while observation techniques like BLS [31] or more recently STXM [30] allow direct imaging of the spin waves themselves (cf figure 4). Furthermore, exchange, magnetostatic, anisotropy energies are main factors shaping the dispersion relation and their tunability provide an extensive playground for new physics.

## 2.2. Current and future challenges

Quasiperiodicity offers new dimensionality in design of collective spin-wave excitation spectra. The discrete spectra feature self-similarity provide extended and localized states. Localized waves controlled by the magnetic field orientation were already detected. They promise for short wavelength wave excitation and broadband RF operation, yet, future studies need to demonstrate such quasiperiodic utility. Here, the influence of extended coupling of elements separated by large

distances, negative group velocity, non-reciprocity, and chirality provide a test bed for new ideas, hardly accessible for other types of waves. The combination of these features with quasiperiodicity awaits discovery.

Moving toward the third dimension is a topic in many branches of solid-state physics and technology, from microelectronics, through photonics to magnonics. Thinking beyond planar magnonic structures, the third dimension can be exploited to control the in-plane propagation of spin waves, offering a robust route for designing nonreciprocal magnonic spectra with chiral properties without the need for DMI and sophisticated material growth. In MCs it can be further exploited to control in-plane coupling between the patterned elements, enhancing the group velocity and non-reciprocity; and guiding the waves in circuits for magnon spintronic applications. This bottom-up approach can be continued toward forming full magnonic 3D crystals (see section 6). New deposition approaches, like focused electron beam induced deposition and atomic layer deposition, offer fabrication of various 3D shapes in a periodic arrangement. Due to their shape anisotropy and small size, their magnetic elements are in a monodomain state and can work as discrete magnetic dipole arranged in a 3D lattice [22].

A further 3D design aspect is artificial spin ice systems with frustration at the vertexes of the elements. This is still a big challenge as even in planar artificial spin ice, the collective dynamics and guiding spin waves along controllable paths are in an infancy state of development [34, 35]. A real breakthrough is needed to provide a feasible approach for controlled coupling between spin wave dynamics of nearest, but also next- and next-next-nearest elements to achieve collective excitation in the whole array.

For the realization of sophisticated magnonic lattices with extended tunability, advances in preparation techniques are necessary to move beyond the strong modulation used in the past (see section 1). Thus, mild and gradual techniques for changing saturation magnetization, anisotropy, and damping are needed (see section 12). First steps toward this end have been undertaken in terms of doping or thermal gradients; additional promising approaches include localized strain modulation (see section 8), graded and combinatorial material deposition [36].

Furthermore, magnetic textures could be envisioned to also achieve such mild modulations. To this end, it has been shown that magnetic force microscopy tips and soft x-ray beams can be used to write arbitrary magnetization textures in specific materials [37]. This brings us to another broad area in the future of magnonics, which is the control of spin waves by magnetization textures rather than material structuring. In this case, the spin waves can follow paths designed on demand by a local change of the magnetization texture (see section 11). This is also true for periodic and other types of regular magnetization patterns, like stripe domains, and well forgotten bubble domains. The sign of possible magnonic band structure formation in stripe domain structures in multilayers with PMA has been shown.

Another approach toward reconfigurable MCs are controllable modulators. These prepatterned structures modulate the
<figure><img src="image_6.png" /><figcaption>**Figure 4.** A magnonic quasicrystal, consisting of permalloy stripes arranged in a Fibonacci sequence. (a) Integrated density of states measured by BLS and calculated by FEM. (b) STXM measurement of the spin wave phase and amplitude at an excitation frequency of 4.2 GHz. (c) 3D rendering of a snapshot of the out of plane magnetization component $m_z$ of the STXM movie. Reproduced from [30]. CC BY 4.0.</figcaption></figure>

potential for magnons by a non-magnetic stimulus like a current or voltage (see section 18). First steps in this direction have been taken by utilizing superconductor hybrids and mediation by magnon-fluxon interaction [38] (see section 21). This opened up a vast field of control effects like piezoelectricity or VCMA and control mechanisms that have quasiparticle-magnon interactions for future research.

Chern numbers and Zak phase describe the topology of the crystal's bands. Whenever the sum of the Chern numbers of the bands up to the band gap edge differ from zero, the states localized at the edge of the MC are topologically protected [39]. This property, common for other types of waves, has been predicted theoretically for magnonics, but is awaiting experimental demonstrations. The wealth of magnetization textures in magnetic material makes magnonics most suitable to offer reprogrammable topological properties for guiding waves (see section 20).

Vortex and skyrmion lattices were and are, respectively, hot topics in magnetic community. Their collective excitations can be tuned and shaped in a broad range, however the transmission of spin waves through these complex textures is so far limited due to scattering and damping. This loss is a bottleneck in the study and potential applications of MCs, especially in metallic films. Just recently, the technology for patterning of insulating ferromagnetic samples has been developed offering structure sizes below 100 nm, but still low magnetization saturation limit some potential applications [39]. Traditional antiferromagnetics as well as SyAFs may offer a viable alternatives for RF applications, because they feature higher operation frequencies and CMOS compatible processing, respectively.

However, when adding gain, for instance by external microwave field, these systems can preserve parity-time (PT) symmetry, and the loss starts to play significant role. In photonics the PT-symmetry and breaking PT-symmetry opened the large area of new physics, with fascinating exceptional points were the dispersion branches start to overlap. In magnonics, this research has just started and it can be expected to become some of the leading directions in the coming years. Additionally,

to patterned structures, the gain-loss will be explored in stripe domains and other regular magnetization textures. Moreover, nonlinearities and their periodicity in space and time will be exploited in the context of MCs. Recently, the existence of magnonic space time crystals has been observed [32]. In analogy to conventional MCs, it can be expected that they form a band structure in space and time, that offers additional degrees of freedom.

## 2.3. Concluding remarks

While ferromagnetic films with periodic corrugation for steering spin waves were introduced in the seventies of the previous century, but band properties of spin waves were investigated only from the beginning of the 21st century. This strongly contributed to the rise of magnonics as emerging field in physics and technology. The richness of ways for magnetic film modulation—including patterning, modulation with ions or light, complex magnetization textures—make wave dynamics in MCs an intriguing topic of research. Fascinating aspects like negative group velocity, anisotropic dispersion relation, chirality and nonreciprocity, topology in dynamics mixed with topologically protected static magnetization configuration, reconfigurable textures, and different types of interactions can be immersed into one magnonic system. A unique combination of this vast toolbox of parameters for control and direct observability of spin waves exerts a fascination on fundamental researchers and makes MCs and quasicrystals a prospective topic of research.

## Acknowledgments

MK has received financial support from the National Science Center of Poland (Project Number UMO-2018/30/Q/ST3/00416). JG acknowledges financial support by the Federal Ministry of Education and Research of Germany in the framework of DynaMAX (Project Number 05K18EYA).
<figure><img src="image_7.png" /><figcaption>**Figure 4.** A magnonic quasicrystal, consisting of permalloy stripes arranged in a Fibonacci sequence. (a) Integrated density of states measured by BLS and calculated by FEM. (b) STXM measurement of the spin wave phase and amplitude at an excitation frequency of 4.2 GHz. (c) 3D rendering of a snapshot of the out of plane magnetization component $m_z$ of the STXM movie. Reproduced from [30]. CC BY 4.0.</figcaption></figure>

potential for magnons by a non-magnetic stimulus like a current or voltage (see section 18). First steps in this direction have been taken by utilizing superconductor hybrids and mediation by magnon-fluxon interaction [38] (see section 21). This opened up a vast field of control effects like piezoelectricity or VCMA and control mechanisms that have quasiparticle-magnon interactions for future research.

Chern numbers and Zak phase describe the topology of the crystal's bands. Whenever the sum of the Chern numbers of the bands up to the band gap edge differ from zero, the states localized at the edge of the MC are topologically protected [39]. This property, common for other types of waves, has been predicted theoretically for magnonics, but is awaiting experimental demonstrations. The wealth of magnetization textures in magnetic material makes magnonics most suitable to offer reprogrammable topological properties for guiding waves (see section 20).

Vortex and skyrmion lattices were and are, respectively, hot topics in magnetic community. Their collective excitations can be tuned and shaped in a broad range, however the transmission of spin waves through these complex textures is so far limited due to scattering and damping. This loss is a bottleneck in the study and potential applications of MCs, especially in metallic films. Just recently, the technology for patterning of insulating ferromagnetic samples has been developed offering structure sizes below 100 nm, but still low magnetization saturation limit some potential applications [39]. Traditional antiferromagnetics as well as SyAFs may offer a viable alternatives for RF applications, because they feature higher operation frequencies and CMOS compatible processing, respectively.

However, when adding gain, for instance by external microwave field, these systems can preserve parity-time (PT) symmetry, and the loss starts to play significant role. In photonics the PT-symmetry and breaking PT-symmetry opened the large area of new physics, with fascinating exceptional points were the dispersion branches start to overlap. In magnonics, this research has just started and it can be expected to become some of the leading directions in the coming years. Additionally,

to patterned structures, the gain-loss will be explored in stripe domains and other regular magnetization textures. Moreover, nonlinearities and their periodicity in space and time will be exploited in the context of MCs. Recently, the existence of magnonic space time crystals has been observed [32]. In analogy to conventional MCs, it can be expected that they form a band structure in space and time, that offers additional degrees of freedom.

## 2.3. Concluding remarks

While ferromagnetic films with periodic corrugation for steering spin waves were introduced in the seventies of the previous century, but band properties of spin waves were investigated only from the beginning of the 21st century. This strongly contributed to the rise of magnonics as emerging field in physics and technology. The richness of ways for magnetic film modulation—including patterning, modulation with ions or light, complex magnetization textures—make wave dynamics in MCs an intriguing topic of research. Fascinating aspects like negative group velocity, anisotropic dispersion relation, chirality and nonreciprocity, topology in dynamics mixed with topologically protected static magnetization configuration, reconfigurable textures, and different types of interactions can be immersed into one magnonic system. A unique combination of this vast toolbox of parameters for control and direct observability of spin waves exerts a fascination on fundamental researchers and makes MCs and quasicrystals a prospective topic of research.

## Acknowledgments

MK has received financial support from the National Science Center of Poland (Project Number UMO-2018/30/Q/ST3/00416). JG acknowledges financial support by the Federal Ministry of Education and Research of Germany in the framework of DynaMAX (Project Number 05K18EYA).
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Block diagram of a hybrid spin-wave–CMOS system including a spin wave circuit as well as a CMOS periphery to enable the usage of voltage signals for input and output. © [2014] IEEE. Reprinted, with permission, from [45].</figcaption></figure>

# 3. Magnonic circuits

Christoph Adelmann<fnref n="1" />, Sorin Cotofana<fnref n="2" /> and Azad Naeemi<fnref n="3" />

## 3.1. Status

In the last decade, tremendous progress has been made in the field of magnonic logic devices and gates [40]. The current status and the roadmap for devices are summarized in section 4. Various spin-wave-based logic devices have been proposed and experimentally demonstrated, e.g. interferometer-based logic gates [41] or spin-wave majority gates [42–44]. Practical applications of such logic devices and gates will however require their combination in circuits and, ultimately, in computing systems containing both logic and memory. Yet comparatively little work has been devoted to magnonic circuits and systems [22, 42, 45, 46]. Currently, no complete proposal of a realistic ‘magnonic computer’, including magnonic logic circuits, magnonic interconnects, and magnonic memory, exists and it is unclear if magnonics can competitively replace all aspects of state-of-the-art charge-based computing systems.

Beyond logic gates, interconnects are the key elements of any circuit. Interconnects transport signals to cascade different gates or provide clock as well as power. In today’s circuits based on CMOS technology, more energy is typically dissipated in interconnects by moving data around rather than in transistors by processing information. The fraction of the power dissipated in interconnects may be even larger for spin-based logic if communication happens via spin currents, domain wall propagation, or by converting back and forth spin signals to electrical signals [47]. In magnonic circuits, it is natural to employ spin waves as information carriers. However, when compared to electromagnetic waves, spin waves are slow and lossy, and as such communication over large distances cannot be competitive with electric or optical interconnects. On the other hand, in magnonic circuits, the boundary between computation and communication may disappear. If losses can be kept under control, communication may not require extra

energy cost. Unlike optical waveguides, magnonic circuits can scale to nanoscale dimensions. Currently, these considerations point toward a hybrid systems concept with local spin-wave islands embedded in a CMOS periphery (see figure 5).

Previous benchmarking of such hybrid spin-wave–CMOS systems has suggested substantial potential for power reduction with respect to CMOS [45–47] if the signal conversion between magnonic and electric domains can be sufficiently efficient. This indicates that magnonic logic circuits may find application in future ultralow power electric systems. Moreover, such hybrid systems have potential for area reduction even for relaxed lithography specifications, hence potentially reducing cost. The main drawback of magnonic circuits is that they can be expected to be considerably slower than CMOS circuits, although they may still outperform current CMOS counterparts in terms of the area-delay-power product (ADPP) [46].

Currently, no experimental proof-of-concept magnonic circuits have been realized. Missing elements are efficient transducers to convert signals at the inputs and outputs of the magnonic circuit between the spin-wave domain and the charge- or voltage-based periphery, and methods to restore signals in magnonic circuits to compensate for spin-wave amplitude variations. In these fields, substantial advances are required to realize the prospect of hybrid spin-wave–CMOS systems for microelectronic applications.

## 3.2. Current and future challenges

To progress from magnonic logic devices and gates to circuits, many challenges lie ahead that must still be overcome. Challenges exist in the magnonic circuits themselves, at the interface between magnonic and CMOS parts of the system, as well as in the CMOS periphery.

The design of spin wave circuits using a set of logic gates requires that the gates satisfy several criteria. The logic gates need to be cascadable, i.e. the output signal of a gate must be usable as input signal for a subsequent gate. In typical circuits, logic gates need to be connected to several subsequent gates, requiring fan-out as well as gain. The logic levels must be robust and should not degrade within the circuit, i.e. 0 and 1 logic levels should remain clearly separate. Finally, the calculation in a circuit should be unidirectional and spin-wave

<footnote n="1">¹Imec, Leuven, Belgium</footnote>

<footnote n="2">²Delft University of Technology, The Netherlands</footnote>

<footnote n="3">³Georgia Institute of Technology, United States of America</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Block diagram of a hybrid spin-wave–CMOS system including a spin wave circuit as well as a CMOS periphery to enable the usage of voltage signals for input and output. © [2014] IEEE. Reprinted, with permission, from [45].</figcaption></figure>

# 3. Magnonic circuits

Christoph Adelmann<fnref n="1" />, Sorin Cotofana<fnref n="2" /> and Azad Naeemi<fnref n="3" />

## 3.1. Status

In the last decade, tremendous progress has been made in the field of magnonic logic devices and gates [40]. The current status and the roadmap for devices are summarized in section 4. Various spin-wave-based logic devices have been proposed and experimentally demonstrated, e.g. interferometer-based logic gates [41] or spin-wave majority gates [42–44]. Practical applications of such logic devices and gates will however require their combination in circuits and, ultimately, in computing systems containing both logic and memory. Yet comparatively little work has been devoted to magnonic circuits and systems [22, 42, 45, 46]. Currently, no complete proposal of a realistic ‘magnonic computer’, including magnonic logic circuits, magnonic interconnects, and magnonic memory, exists and it is unclear if magnonics can competitively replace all aspects of state-of-the-art charge-based computing systems.

Beyond logic gates, interconnects are the key elements of any circuit. Interconnects transport signals to cascade different gates or provide clock as well as power. In today’s circuits based on CMOS technology, more energy is typically dissipated in interconnects by moving data around rather than in transistors by processing information. The fraction of the power dissipated in interconnects may be even larger for spin-based logic if communication happens via spin currents, domain wall propagation, or by converting back and forth spin signals to electrical signals [47]. In magnonic circuits, it is natural to employ spin waves as information carriers. However, when compared to electromagnetic waves, spin waves are slow and lossy, and as such communication over large distances cannot be competitive with electric or optical interconnects. On the other hand, in magnonic circuits, the boundary between computation and communication may disappear. If losses can be kept under control, communication may not require extra

energy cost. Unlike optical waveguides, magnonic circuits can scale to nanoscale dimensions. Currently, these considerations point toward a hybrid systems concept with local spin-wave islands embedded in a CMOS periphery (see figure 5).

Previous benchmarking of such hybrid spin-wave–CMOS systems has suggested substantial potential for power reduction with respect to CMOS [45–47] if the signal conversion between magnonic and electric domains can be sufficiently efficient. This indicates that magnonic logic circuits may find application in future ultralow power electric systems. Moreover, such hybrid systems have potential for area reduction even for relaxed lithography specifications, hence potentially reducing cost. The main drawback of magnonic circuits is that they can be expected to be considerably slower than CMOS circuits, although they may still outperform current CMOS counterparts in terms of the area-delay-power product (ADPP) [46].

Currently, no experimental proof-of-concept magnonic circuits have been realized. Missing elements are efficient transducers to convert signals at the inputs and outputs of the magnonic circuit between the spin-wave domain and the charge- or voltage-based periphery, and methods to restore signals in magnonic circuits to compensate for spin-wave amplitude variations. In these fields, substantial advances are required to realize the prospect of hybrid spin-wave–CMOS systems for microelectronic applications.

## 3.2. Current and future challenges

To progress from magnonic logic devices and gates to circuits, many challenges lie ahead that must still be overcome. Challenges exist in the magnonic circuits themselves, at the interface between magnonic and CMOS parts of the system, as well as in the CMOS periphery.

The design of spin wave circuits using a set of logic gates requires that the gates satisfy several criteria. The logic gates need to be cascadable, i.e. the output signal of a gate must be usable as input signal for a subsequent gate. In typical circuits, logic gates need to be connected to several subsequent gates, requiring fan-out as well as gain. The logic levels must be robust and should not degrade within the circuit, i.e. 0 and 1 logic levels should remain clearly separate. Finally, the calculation in a circuit should be unidirectional and spin-wave

<footnote n="1">¹Imec, Leuven, Belgium</footnote>

<footnote n="2">²Delft University of Technology, The Netherlands</footnote>

<footnote n="3">³Georgia Institute of Technology, United States of America</footnote>
logic gates should not be influenced by signals propagating
backwards from the output toward the input ports (input/output
isolation).

Currently, concepts of spin-wave logic gates do not sat-
isfy (clearly) all the above criteria and therefore, the real-
ization of practical complex spin-wave circuits has remained
elusive. The main challenges to realize magnonic circuits
concern the propagation of spin waves as well as the cascading
of logic gates. During propagation, spin waves attenuate due
to magnetic damping and therefore the logic signal degrades.
This limits the length of interconnects, increases power dissi-
pation, and ultimately determines how many logic gates can
be cascaded in a circuit. Ideal materials for such logic gates
combine very low losses with fast spin-wave group veloc-
ity and the ability to be co-integrated with CMOS on a Si
wafer. To scale the logic gates footprint, the spin waves have
to propagate in nanoscale waveguides within potentially com-
plex geometries, which may also lead to losses e.g. when spin
waves propagate around bends. If several interconnect layers
are required, spin-wave vias are required that transmit infor-
mation between the different layers. Propagation losses can be
compensated by repeater stages or amplifiers, which however
increase clocking complexity and dissipate additional energy,
therefore increasing the overall energy consumption of the cir-
cuit and system. No ultralow energy repeater or amplifier has
been experimentally demonstrated to date.

In addition, many spin-wave-based logic gate concepts are
not directly cascadable, e.g. due to the usage of hybrid charge-
spin-wave signals [40, 41] or ill-defined output amplitudes for
majority gates using phase encoding [40–42]. While fan-out
can be achieved by suitable gate designs (see figure 6(a)) [48],
spin-wave logic gates are typically passive and do not show
gain. Gain can be achieved by spin-wave amplifiers or in sig-
nal repeater concepts, although at the expense of additional
energy consumption. The development of a complete cascad-
able spin-wave circuit concept is one of the key challenges
in this field.

A second major challenge is the efficient transduction at
the interface between spin-wave- and charge-based domains of
the system. Conventional approaches to generate spin waves
using inductive antennas are neither efficient nor scalable.
Magnetoelectric transducers based on the coupling between
spin waves and voltage signals using multiferroics or piezo-
electric/magnetostrictive compounds have been proposed as
much more efficient transducers but significant experimen-
tal progress is still needed to demonstrate both their scal-
ability as well as their energy efficiency. Such devices are
rendered even more complex due to the targeted nm scales
and GHz frequencies. In addition to generation, also the
detection of spin waves is still a major challenge. Spin-
wave transducers typically generate weak electric signals, and
require sensitive detectors, e.g. sense amplifiers, which are
however power-hungry [49]. In addition, transducers need
to be able to read the result of the magnonic circuit suffi-
ciently rapidly to limit additional delays. This will also require
large transducer signals compatible with CMOS circuits to
improve the signal-to-noise ratio. Alternatively, spin waves
can be detected by switching a metastable nanomagnet (see

figure 6(b)) [50]. Simulations have indicated that this can
also be sensitive to the spin-wave phase even in the presence
of thermal noise. However, major breakthroughs in material
growth and processing are needed before such concepts can
be realized experimentally.

Finally, the design of peripheral CMOS circuits that inter-
face efficiently with magnonic circuits is only emerging. This
work will have to go hand in hand with the development of
transducers and future breakthroughs will be needed to obtain
an efficient environment, in which magnonic circuits can be
embedded. This research topic has been less in the focus
of interest than the magnonic circuits themselves; however,
it will be equally critical for the realization of the vision of
magnonic circuits integrated in commercial ultralow power
applications.

3.3. *Concluding remarks*

Currently, we believe that there are three major challenges that need to be overcome before magnonic logic can be integrated alongside CMOS in practical microelectronic applications. Present concepts for magnonic logic gates need to be completed to satisfy the necessary criteria for circuit design. Energy-efficient scalable transducers need to be demonstrated to enable low-power generation and detection of spin waves in scaled waveguides. Finally, an efficient periphery needs to be developed so that the transducers and magnonic circuits can interface with the larger CMOS part of a chip. Recent benchmarking suggests that such hybrid spin-wave–CMOS circuits and systems show great promise for ultralow power applications, which is a clear motivation for research to overcome these challenges [40, 45, 46].

Acknowledgments

CA's and SC's contributions have been funded by the Euro-
pean Union's Horizon 2020 research and innovation pro-
gram within the FET-OPEN project CHIRON under Grant
Agreement No. 801055. CA acknowledges support by imec's
industrial affiliate program on beyond-CMOS logic. AN's
contributions have been supported by Intel Corp. under
Grant Agreements SRC Intel-MSR 2011-IN-2198 and spin
interconnects-2616.001.

4. Magnonic logic, memory and devices

A V Chumak<fnref n="1" />, A Khitun<fnref n="2" />, D E Nikonov<fnref n="3" /> and I A Young<fnref n="3" />

4.1. *Status*

The increasing challenges with further CMOS scaling down
have stimulated a great deal of interest in novel logic devices
and architectures. Spin-wave data processing offer one of the
promising directions toward charge-less future circuitry [40].

<footnote n="1">¹University of Vienna, Austria</footnote>

<footnote n="2">²University of California Riverside, United States of America</footnote>

<footnote n="3">³Intel, United States of America</footnote>
logic gates should not be influenced by signals propagating backwards from the output toward the input ports (input/output isolation).

Currently, concepts of spin-wave logic gates do not satisfy (clearly) all the above criteria and therefore, the realization of practical complex spin-wave circuits has remained elusive. The main challenges to realize magnonic circuits concern the propagation of spin waves as well as the cascading of logic gates. During propagation, spin waves attenuate due to magnetic damping and therefore the logic signal degrades. This limits the length of interconnects, increases power dissipation, and ultimately determines how many logic gates can be cascaded in a circuit. Ideal materials for such logic gates combine very low losses with fast spin-wave group velocity and the ability to be co-integrated with CMOS on a Si wafer. To scale the logic gates footprint, the spin waves have to propagate in nanoscale waveguides within potentially complex geometries, which may also lead to losses e.g. when spin waves propagate around bends. If several interconnect layers are required, spin-wave vias are required that transmit information between the different layers. Propagation losses can be compensated by repeater stages or amplifiers, which however increase clocking complexity and dissipate additional energy, therefore increasing the overall energy consumption of the circuit and system. No ultralow energy repeater or amplifier has been experimentally demonstrated to date.

In addition, many spin-wave-based logic gate concepts are not directly cascadable, e.g. due to the usage of hybrid charge-spin-wave signals [40, 41] or ill-defined output amplitudes for majority gates using phase encoding [40–42]. While fan-out can be achieved by suitable gate designs (see figure 6(a)) [48], spin-wave logic gates are typically passive and do not show gain. Gain can be achieved by spin-wave amplifiers or in signal repeater concepts, although at the expense of additional energy consumption. The development of a complete cascadable spin-wave circuit concept is one of the key challenges in this field.

A second major challenge is the efficient transduction at the interface between spin-wave- and charge-based domains of the system. Conventional approaches to generate spin waves using inductive antennas are neither efficient nor scalable. Magnetoelectric transducers based on the coupling between spin waves and voltage signals using multiferroics or piezoelectric/magnetostrictive compounds have been proposed as much more efficient transducers but significant experimental progress is still needed to demonstrate both their scalability as well as their energy efficiency. Such devices are rendered even more complex due to the targeted nm scales and GHz frequencies. In addition to generation, also the detection of spin waves is still a major challenge. Spin-wave transducers typically generate weak electric signals, and require sensitive detectors, e.g. sense amplifiers, which are however power-hungry [49]. In addition, transducers need to be able to read the result of the magnonic circuit sufficiently rapidly to limit additional delays. This will also require large transducer signals compatible with CMOS circuits to improve the signal-to-noise ratio. Alternatively, spin waves can be detected by switching a metastable nanomagnet (see

figure 6(b)) [50]. Simulations have indicated that this can also be sensitive to the spin-wave phase even in the presence of thermal noise. However, major breakthroughs in material growth and processing are needed before such concepts can be realized experimentally.

Finally, the design of peripheral CMOS circuits that interface efficiently with magnonic circuits is only emerging. This work will have to go hand in hand with the development of transducers and future breakthroughs will be needed to obtain an efficient environment, in which magnonic circuits can be embedded. This research topic has been less in the focus of interest than the magnonic circuits themselves; however, it will be equally critical for the realization of the vision of magnonic circuits integrated in commercial ultralow power applications.

### 3.3. Concluding remarks

Currently, we believe that there are three major challenges that need to be overcome before magnonic logic can be integrated alongside CMOS in practical microelectronic applications. Present concepts for magnonic logic gates need to be completed to satisfy the necessary criteria for circuit design. Energy-efficient scalable transducers need to be demonstrated to enable low-power generation and detection of spin waves in scaled waveguides. Finally, an efficient periphery needs to be developed so that the transducers and magnonic circuits can interface with the larger CMOS part of a chip. Recent benchmarking suggests that such hybrid spin-wave-CMOS circuits and systems show great promise for ultralow power applications, which is a clear motivation for research to overcome these challenges [40, 45, 46].

### Acknowledgments

CA’s and SC’s contributions have been funded by the European Union’s Horizon 2020 research and innovation program within the FET-OPEN project CHIRON under Grant Agreement No. 801055. CA acknowledges support by imec’s industrial affiliate program on beyond-CMOS logic. AN’s contributions have been supported by Intel Corp. under Grant Agreements SRC Intel-MSR 2011-IN-2198 and spin interconnects-2616.001.

## 4. Magnonic logic, memory and devices

A V Chumak<fnref n="1" />, A Khitun<fnref n="2" />, D E Nikonov<fnref n="3" /> and I A Young<fnref n="3" />

### 4.1. Status

The increasing challenges with further CMOS scaling down have stimulated a great deal of interest in novel logic devices and architectures. Spin-wave data processing offer one of the promising directions toward charge-less future circuitry [40].

<footnote n="1">¹University of Vienna, Austria</footnote>

<footnote n="2">²University of California Riverside, United States of America</footnote>

<footnote n="3">³Intel, United States of America</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** (a) Fan-out enabled spin-wave majority gate with four inputs (I) and two output (O) ports. Reproduced from [48]. (b) Non-volatile clocked spin-wave interconnect using metastable nanomagnets to detect spin waves with phase sensitivity. Reproduced from [50]. CC BY 4.0.</figcaption></figure>

The unique physical and technological aspects inherent to spin waves to be implemented for data processing are listed below.

## 4.2. Key advantages offered by spin waves for data processing

* Spin-wave bus. A spin wave is a collective oscillation of a spin system in a magnetic lattice. It is naturally confined within a magnetic media with zero likelihood to escape or to leak into non-magnetic surroundings. This feature makes it possible to build a spin-wave bus for information transfer using magnetic nano-wires similar to optical waveguides [51–53]. Moreover, since the angle of magnetization precession of a linear spin wave usually does not exceed 1 degree (as opposite to the data-storage concepts relying on the π-switch of the magnetization), a low energy is required to excite spin waves what, consequently, ensures a low energy loss during the data transport.

* Miniaturization down to atomic scale. The smallest magnonics element size is limited by the lattice constant of material and, thus, is comparable to the fundamental limitations of CMOS. Currently, the lateral sizes of the magnonics conduits made of such a complex material as YIG reached 50 nm which is around 40 lattice constants [51].

* Large coherence length and broad frequency range. Spin-waves show relatively large coherence and propagation length $l_{pl}$ even at room temperature (e.g., up to 1 cm in YIG of micrometre thicknesses). These properties enable the intriguing possibility of exploiting spin-wave phase in addition to amplitude for data transfer and processing for RF applications, processing of binary data, and unconventional computing. In nanostructures, the spin-wave propagation length is usually limited to tens of micrometers. Nevertheless, considering the recent progress in the miniaturization of spin-wave wavelength $\lambda$ below 100 nm (see section 7 of this roadmap), $l_{pl}$ might constitute up to 200 µm in YIG [54]. The smallest wavelength of a spin wave is limited by the lattice constant of material and the further miniaturization enables access to the THz magnonics (see section 14).

The increase in the spin-wave frequency results in the decrease of its lifetime (e.g. linearly according to the Gilbert model). However, the group velocity $v_{gr} = d\omega/dk$ of the exchange spin wave $\omega \propto k^2$ increases linearly with the increase in $k$. Thus, the ratio of the spin-wave propagation length $l_{pl}$ to the wavelength $l_{pl}/\lambda$, which is the most important parameter in the wave-based data processing, stays constant at a level of around three thousand for YIG [54] or one hundred for CoFeB [40].

* Manipulation, dynamic control and reconfigurability. Among others, spin-wave dispersion depends on the magnetic field, the magnetization of the material, and on the geometrical sizes of the magnetic element. This fact translates in the possibility of the static and dynamic control of spin waves (see e.g. sections 2 and 12), for the realization of non-volatile reconfigurable magnonic elements using nano-magnets (see figure 7), and for the realization of magnetic bit read-in and read-out [40, 50, 52].

* Energy-efficient spin-wave to voltage converters and magnetoelectric transducers. There is a robust and energy-efficient mechanism for spin-wave to voltage and vice versa conversion, using multiferroic cells [40, 50, 52] (see also sections 3, 8 and 18). This approach is of great importance for integrating spin-wave devices with conventional electronic components and for the integration of memory cells into magnonic circuits (see figure 7(c)).

* Efficient nonlinear magnon phenomena. The processing of data, in general, requires the utilization of elements with nonlinear characteristics that are, e.g., provided by a semiconductor transistor in CMOS. The pronounced benefit of spin waves for the data processing is their pronounced natural nonlinearity that allows for an all-magnon control of one magnonic unit by another to realise integrated magnonic circuits [53].

## 4.3. Classes of magnon devices

* Spin-wave RF applications. Spin-wave frequencies cover the range from sub-GHz up to hundreds of THz. Modern magnonics (see section 14). The increase in the spin-wave frequency results in the decrease of its lifetime (e.g. linearly according to the Gilbert model). However, the group velocity $v_{gr} = d\omega/dk$ of the exchange spin wave $\omega \propto k^2$ increases linearly with the increase in $k$. Thus, the ratio of the spin-wave propagation length $l_{pl}$ to the wavelength $l_{pl}/\lambda$, which is the most important parameter in the wave-based data processing, stays constant at a level of around three thousand for YIG [54] or one hundred for CoFeB [40].
<figure><img src="image_9.png" /><figcaption>**Figure 6.** (a) Fan-out enabled spin-wave majority gate with four inputs (I) and two output (O) ports. Reproduced from [48]. (b) Non-volatile clocked spin-wave interconnect using metastable nanomagnets to detect spin waves with phase sensitivity. Reproduced from [50]. CC BY 4.0.</figcaption></figure>

The unique physical and technological aspects inherent to spin waves to be implemented for data processing are listed below.

## 4.2. Key advantages offered by spin waves for data processing

* Spin-wave bus. A spin wave is a collective oscillation of a spin system in a magnetic lattice. It is naturally confined within a magnetic media with zero likelihood to escape or to leak into non-magnetic surroundings. This feature makes it possible to build a spin-wave bus for information transfer using magnetic nano-wires similar to optical waveguides [51–53]. Moreover, since the angle of magnetization precession of a linear spin wave usually does not exceed 1 degree (as opposite to the data-storage concepts relying on the π-switch of the magnetization), a low energy is required to excite spin waves what, consequently, ensures a low energy loss during the data transport.

* Miniaturization down to atomic scale. The smallest magnonics element size is limited by the lattice constant of material and, thus, is comparable to the fundamental limitations of CMOS. Currently, the lateral sizes of the magnonics conduits made of such a complex material as YIG reached 50 nm which is around 40 lattice constants [51].

* Large coherence length and broad frequency range. Spin-waves show relatively large coherence and propagation length $l_{pl}$ even at room temperature (e.g., up to 1 cm in YIG of micrometre thicknesses). These properties enable the intriguing possibility of exploiting spin-wave phase in addition to amplitude for data transfer and processing for RF applications, processing of binary data, and unconventional computing. In nanostructures, the spin-wave propagation length is usually limited to tens of micrometers. Nevertheless, considering the recent progress in the miniaturization of spin-wave wavelength $\lambda$ below 100 nm (see section 7 of this roadmap), $l_{pl}$ might constitute up to 200 µm in YIG [54]. The smallest wavelength of a spin wave is limited by the lattice constant of material and the further miniaturization enables access to the THz magnonics (see section 14).

The increase in the spin-wave frequency results in the decrease of its lifetime (e.g. linearly according to the Gilbert model). However, the group velocity $v_{gr} = d\omega/dk$ of the exchange spin wave $\omega \propto k^2$ increases linearly with the increase in $k$. Thus, the ratio of the spin-wave propagation length $l_{pl}$ to the wavelength $l_{pl}/\lambda$, which is the most important parameter in the wave-based data processing, stays constant at a level of around three thousand for YIG [54] or one hundred for CoFeB [40].

* Manipulation, dynamic control and reconfigurability. Among others, spin-wave dispersion depends on the magnetic field, the magnetization of the material, and on the geometrical sizes of the magnetic element. This fact translates in the possibility of the static and dynamic control of spin waves (see e.g. sections 2 and 12), for the realization of non-volatile reconfigurable magnonic elements using nano-magnets (see figure 7), and for the realization of magnetic bit read-in and read-out [40, 50, 52].

* Energy-efficient spin-wave to voltage converters and magnetoelectric transducers. There is a robust and energy-efficient mechanism for spin-wave to voltage and vice versa conversion, using multiferroic cells [40, 50, 52] (see also sections 3, 8 and 18). This approach is of great importance for integrating spin-wave devices with conventional electronic components and for the integration of memory cells into magnonic circuits (see figure 7(c)).

* Efficient nonlinear magnon phenomena. The processing of data, in general, requires the utilization of elements with nonlinear characteristics that are, e.g., provided by a semiconductor transistor in CMOS. The pronounced benefit of spin waves for the data processing is their pronounced natural nonlinearity that allows for an all-magnon control of one magnonic unit by another to realise integrated magnonic circuits [53].

## 4.3. Classes of magnon devices

* Spin-wave RF applications. Spin-wave frequencies cover the range from sub-GHz up to hundreds of THz. Modern magnonics (see section 14). The increase in the spin-wave frequency results in the decrease of its lifetime (e.g. linearly according to the Gilbert model). However, the group velocity $v_{gr} = d\omega/dk$ of the exchange spin wave $\omega \propto k^2$ increases linearly with the increase in $k$. Thus, the ratio of the spin-wave propagation length $l_{pl}$ to the wavelength $l_{pl}/\lambda$, which is the most important parameter in the wave-based data processing, stays constant at a level of around three thousand for YIG [54] or one hundred for CoFeB [40].
<figure><img src="image_9.png" /><figcaption>**Figure 7.** A two-bit magnonic holographic memory [57]. Magnonic holographic image of two cobalt magnets on ferrite waveguide. The red markers show experimentally detected inductive voltage in millivolts produced by four interfering spin waves. The cyan surface is a computer reconstructed 3D plot. (b) Schematics of the magnet configuration. The operational frequency is 3.4 GHz. All experiments are done at room temperature. © [2015] IEEE. Reprinted, with permission, from [57]. We utilize spin waves for the parallel read-out of magnetic bits. (c) Illustration of a clocked three-stage cascaded spin-wave device comprising of: a charge to spin (C–S) converter, intermediate spin-wave repeaters and spin-wave interconnects. Reproduced from [50]. CC BY 4.0. The sequential switching of the converter and the repeaters is accomplished via application of clocks 1–4 while the input data is applied in the form of current pulses using the STT clock.</figcaption></figure>

state-of-the-art technology usually covers the frequency range of up to 100 GHz which is of special interest for the upcoming 5G communication systems. Different devices have been proposed, including reconfigurable filters (e.g. based on MCs discussed in section 2), delay lines, phase shifters, Y-circulators, multiplexers, wake-up receivers, signal-to-noise enhancers, and spectrum analysers [55] (see also section 9). The nano-sized spin-wave directional coupler realised recently is also a

universal RF data processing unit with multiple functionalities [53].

* Interference-based Boolean logic and majority gate. Most of the developed spin-wave Boolean logic gates exploit spin-wave phase to perform operations. The Boolean data can be encoded into spin-wave amplitude, and the constructive or destructive interference allows one to obtain logic ‘1’ (a certain amplitude) or ‘0’ (zero amplitude). This approach has recently been used for the realization of XNOR logic gate based on a 54 nm thick YIG interferometer (see figure 8(a)) [56]. Another way is to encode Boolean data into spin-wave phase rather than amplitude. It is especially successful for the realization of logic majority gates (see also section 3). This logic device usually features a three-input combiner with the logic information encoded in a phase of 0 or $\pi$ of and the phase of the output signal represents the majority of the input phase states. Recently, a reconfigurable nanoscale inline design without the spin-wave combiner majority gate was realized at the nano-scale (see figure 8(b)) [44]. It has been shown, that the interference-based computation allows for frequency-division multiplexing as well as the computation of different logic functions in the same device.

* All-magnon circuits. The two-way conversion of magnonic signals to electric is always associated with parasitic loss and the ‘all-magnon’ data processing aims to process as much information purely within magnonic system as possible without its conversion to electric signals. This is the main idea behind the original work on magnon transistors. The converters in this approach are replaced by spin-wave nonlinearity and amplifiers. Recently, a nanoscale magnonic directional coupler has been realized experimentally (see figure 8(c)). This coupler was used to design and investigate numerically a first integrated magnonic circuit in a form of half-adder [53]. The half-adder consists of three magnetic nanowires and spin-wave amplifier, and replaces 12 to 20 transistors in various CMOS implementations.

* Unconventional spin-wave computing. Here we would like to report on the significant progress in spin-wave logic and memory devices prototyping since the first publication [52]. The most notable results include a two-bit magnonic holographic memory where spin waves were exploited for magnetic bit parallel-read-out (see figures 7(a) and (b)) [57]. Multi-terminal spin-wave devices were used for special type data processing in the following studies. The feasibility of data encoding in the phase combination of input spin-waves has been demonstrated. The recognition of the input pattern is accomplished by measuring the inductive voltage produced by the interfering spin waves. The first example of reversible spin-wave logic gates was demonstrated using the unique spin-wave dispersion characteristics [58]. The utilization of phase in addition to amplitude has demonstrated great potential in application to NP-problems. For instance, prime factorization was accomplished for
<figure><img src="image_10.png" /><figcaption>**Figure 7.** A two-bit magnonic holographic memory [57]. Magnonic holographic image of two cobalt magnets on ferrite waveguide. The red markers show experimentally detected inductive voltage in millivolts produced by four interfering spin waves. The cyan surface is a computer reconstructed 3D plot. (b) Schematics of the magnet configuration. The operational frequency is 3.4 GHz. All experiments are done at room temperature. © [2015] IEEE. Reprinted, with permission, from [57]. We utilize spin waves for the parallel read-out of magnetic bits. (c) Illustration of a clocked three-stage cascaded spin-wave device comprising of: a charge to spin (C–S) converter, intermediate spin-wave repeaters and spin-wave interconnects. Reproduced from [50]. CC BY 4.0. The sequential switching of the converter and the repeaters is accomplished via application of clocks 1–4 while the input data is applied in the form of current pulses using the STT clock.</figcaption></figure>

state-of-the-art technology usually covers the frequency range of up to 100 GHz which is of special interest for the upcoming 5G communication systems. Different devices have been proposed, including reconfigurable filters (e.g. based on MCs discussed in section 2), delay lines, phase shifters, Y-circulators, multiplexers, wake-up receivers, signal-to-noise enhancers, and spectrum analysers [55] (see also section 9). The nano-sized spin-wave directional coupler realised recently is also a

universal RF data processing unit with multiple functionalities [53].

* Interference-based Boolean logic and majority gate. Most of the developed spin-wave Boolean logic gates exploit spin-wave phase to perform operations. The Boolean data can be encoded into spin-wave amplitude, and the constructive or destructive interference allows one to obtain logic ‘1’ (a certain amplitude) or ‘0’ (zero amplitude). This approach has recently been used for the realization of XNOR logic gate based on a 54 nm thick YIG interferometer (see figure 8(a)) [56]. Another way is to encode Boolean data into spin-wave phase rather than amplitude. It is especially successful for the realization of logic majority gates (see also section 3). This logic device usually features a three-input combiner with the logic information encoded in a phase of 0 or $\pi$ of and the phase of the output signal represents the majority of the input phase states. Recently, a reconfigurable nanoscale inline design without the spin-wave combiner majority gate was realized at the nano-scale (see figure 8(b)) [44]. It has been shown, that the interference-based computation allows for frequency-division multiplexing as well as the computation of different logic functions in the same device.

* All-magnon circuits. The two-way conversion of magnonic signals to electric is always associated with parasitic loss and the ‘all-magnon’ data processing aims to process as much information purely within magnonic system as possible without its conversion to electric signals. This is the main idea behind the original work on magnon transistors. The converters in this approach are replaced by spin-wave nonlinearity and amplifiers. Recently, a nanoscale magnonic directional coupler has been realized experimentally (see figure 8(c)). This coupler was used to design and investigate numerically a first integrated magnonic circuit in a form of half-adder [53]. The half-adder consists of three magnetic nanowires and spin-wave amplifier, and replaces 12 to 20 transistors in various CMOS implementations.

* Unconventional spin-wave computing. Here we would like to report on the significant progress in spin-wave logic and memory devices prototyping since the first publication [52]. The most notable results include a two-bit magnonic holographic memory where spin waves were exploited for magnetic bit parallel-read-out (see figures 7(a) and (b)) [57]. Multi-terminal spin-wave devices were used for special type data processing in the following studies. The feasibility of data encoding in the phase combination of input spin-waves has been demonstrated. The recognition of the input pattern is accomplished by measuring the inductive voltage produced by the interfering spin waves. The first example of reversible spin-wave logic gates was demonstrated using the unique spin-wave dispersion characteristics [58]. The utilization of phase in addition to amplitude has demonstrated great potential in application to NP-problems. For instance, prime factorization was accomplished for
$N = 15$ using spin-wave interferometer [59]. In general, spin wave approach is promising for finding the period (~100) of a given function. Another approach of the unconventional spin-wave data processing is Fourier magnonics described in section 9.

* Inverse-design magnonics. Many magnonic devices were demonstrated, but the development of each of them requires specialized investigations and, usually, one device design is suitable for one function only. Inspired by the recent progress in photonics, an inverse-design magnonics was proposed by two groups independently [60, 61]. In this concept, any functionality can be specified first, and a feedback-based computational algorithm is used to obtain the device design. The proof-of-concept prototype [60] is based on a rectangular ferromagnetic area which can be patterned using square shaped voids. To demonstrate the universality of inverse-design magnonics, linear, nonlinear and nonreciprocal magnonic functionalities were explored and used to realize a magnonic (de-)multiplexer, a nonlinear switch and a circulator. The prototype [61] represents a neural network, where all neuromorphic computing functions, including signal routing and nonlinear activation are performed by spin-wave propagation and interference.

* Neuromorphic magnon computing. It is expected that spin-wave circuits will manifest advantages compared to nanoelectronic devices outside of traditional Boolean computing architectures especially in neuromorphic computing [61, 62]. Spin waves possess all key ingredients: (i) a possibility to construct complex 2D and 3D networks [40, 53, 56, 57] (see also sections 3 and 6), (ii) offer a variety of nonlinear phenomena [53], and (iii) computing units can be complemented by memory cells [50]. A magnon adder, which integrates over incoming spin-wave pulses in an analog fashion, was investigated numerically [62] and represents one of the key building blocks for future magnonic neuromorphic networks. Recently, a nanoscale neural network based on the utilization of nonlinear spin-wave interference was developed by means of inverse-design approach [61].

* Magnon quantum computing. One of the main advantages of magnonics is that novel efficient data processing concepts can be realized at room temperature. Nevertheless, the decrease in temperature down to <100 mK freezes out thermal magnons and allows for the data operations with single magnons. This opens an access to the entangled magnon states and to quantum computing. Currently, the field of magnon quantum physics is at the very beginning of its way (see section 19), but the group of Nakamura from the University of Tokyo has reported on the operations with single magnons in a set of recent publications. Further aspects of cryo-magnonics are overviewed in section 21.

### 4.4. Benchmarking of spin-wave circuits

The theoretical framework for benchmarking beyond-CMOS devices and circuits [63] included spin-wave devices. One of

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Micrograph of three-port logic XNOR gate using forward volume spin wave interference in a 54 nm thick YIG structure. Reproduced from [56]. CC BY 4.0. Two spin waves are injected by ports 1 and 2. The output signal is defined by the interference of these two waves and is measured at the port 3. (b) Scanning-electron micrograph of a in-line spin wave majority gate with an 850 nm-wide $Co_{40}Fe_{40}B_{20}$ waveguide, three input antennas, and one output antenna. From [44]. Reprinted with permission from AAAS. Operation frequency is 13.9 GHz, magnetic bias field 80 mT. The input signals are injected by the antennas $P_1, P_2$ and $P_3$. The interference of these three waves is measured at the antenna ‘O’. The functionality and reconfigurability of the device are shown using all-electrical spin wave spectroscopy. (c) Nonlinear switch functionality of nano-scale directional coupler. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Electronics] [53] (2020). The directional coupler is depicted with a dashed line and is based on 350 nm wide and 80 nm thick YIG waveguides. The color maps represent two-dimensional spin-wave intensity measured by BLS spectroscopy for different input powers of 2 dBm (top) and 10 dBm (bottom). The nonlinear switch of the spin wave path to output 1 or output 2 by the change in the spin wave amplitude is clearly visible. This functionality allows for the realization of complex all-magnon circuits.</figcaption></figure>

its conclusions applicable to spin-based devices in general is that they can be competitive with nanoelectronic devices when relying on magnetoelectric transduction (rather than spin torque or magnetic field). The transduction-based approach has advantage that it can be combined with existing CMOS circuits and does not require the development of a novel type of
$N = 15$ using spin-wave interferometer [59]. In general, spin wave approach is promising for finding the period (~100) of a given function. Another approach of the unconventional spin-wave data processing is Fourier magnonics described in section 9.

* Inverse-design magnonics. Many magnonic devices were demonstrated, but the development of each of them requires specialized investigations and, usually, one device design is suitable for one function only. Inspired by the recent progress in photonics, an inverse-design magnonics was proposed by two groups independently [60, 61]. In this concept, any functionality can be specified first, and a feedback-based computational algorithm is used to obtain the device design. The proof-of-concept prototype [60] is based on a rectangular ferromagnetic area which can be patterned using square shaped voids. To demonstrate the universality of inverse-design magnonics, linear, nonlinear and nonreciprocal magnonic functionalities were explored and used to realize a magnonic (de-)multiplexer, a nonlinear switch and a circulator. The prototype [61] represents a neural network, where all neuromorphic computing functions, including signal routing and nonlinear activation are performed by spin-wave propagation and interference.

* Neuromorphic magnon computing. It is expected that spin-wave circuits will manifest advantages compared to nanoelectronic devices outside of traditional Boolean computing architectures especially in neuromorphic computing [61, 62]. Spin waves possess all key ingredients: (i) a possibility to construct complex 2D and 3D networks [40, 53, 56, 57] (see also sections 3 and 6), (ii) offer a variety of nonlinear phenomena [53], and (iii) computing units can be complemented by memory cells [50]. A magnon adder, which integrates over incoming spin-wave pulses in an analog fashion, was investigated numerically [62] and represents one of the key building blocks for future magnonic neuromorphic networks. Recently, a nanoscale neural network based on the utilization of nonlinear spin-wave interference was developed by means of inverse-design approach [61].

* Magnon quantum computing. One of the main advantages of magnonics is that novel efficient data processing concepts can be realized at room temperature. Nevertheless, the decrease in temperature down to <100 mK freezes out thermal magnons and allows for the data operations with single magnons. This opens an access to the entangled magnon states and to quantum computing. Currently, the field of magnon quantum physics is at the very beginning of its way (see section 19), but the group of Nakamura from the University of Tokyo has reported on the operations with single magnons in a set of recent publications. Further aspects of cryo-magnonics are overviewed in section 21.

### 4.4. Benchmarking of spin-wave circuits

The theoretical framework for benchmarking beyond-CMOS devices and circuits [63] included spin-wave devices. One of

<figure><img src="image_11.png" /></figure>

**Figure 8.** Micrograph of three-port logic XNOR gate using forward volume spin wave interference in a 54 nm thick YIG structure. Reproduced from [56]. CC BY 4.0. Two spin waves are injected by ports 1 and 2. The output signal is defined by the interference of these two waves and is measured at the port 3. (b) Scanning-electron micrograph of a in-line spin wave majority gate with an 850 nm-wide $Co_{40}Fe_{40}B_{20}$ waveguide, three input antennas, and one output antenna. From [44]. Reprinted with permission from AAAS. Operation frequency is 13.9 GHz, magnetic bias field 80 mT. The input signals are injected by the antennas $P_1, P_2$ and $P_3$. The interference of these three waves is measured at the antenna ‘O’. The functionality and reconfigurability of the device are shown using all-electrical spin wave spectroscopy. (c) Nonlinear switch functionality of nano-scale directional coupler. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Electronics] [53] (2020). The directional coupler is depicted with a dashed line and is based on 350 nm wide and 80 nm thick YIG waveguides. The color maps represent two-dimensional spin-wave intensity measured by BLS spectroscopy for different input powers of 2 dBm (top) and 10 dBm (bottom). The nonlinear switch of the spin wave path to output 1 or output 2 by the change in the spin wave amplitude is clearly visible. This functionality allows for the realization of complex all-magnon circuits.

its conclusions applicable to spin-based devices in general is that they can be competitive with nanoelectronic devices when relying on magnetoelectric transduction (rather than spin torque or magnetic field). The transduction-based approach has advantage that it can be combined with existing CMOS circuits and does not require the development of a novel type of
circuitry. One option for it is the piezoelectric-magnetostrictive
transduction. The all-magnon data processing approach, was
benchmarked in reference [53]. It was found that 30 nm-
based magnonic circuit is comparable to 7 nm-based CMOS in
terms of footprint, has 10 times smaller energy consumption,
but is slower. The further magnonic circuits benchmarking is
discussed in section 3 (figure 8).

4.5. Current and future challenges

Although we observe drastic progress in the evolution of the
spin-wave circuitry and, in particular, its scaling down from
mm to sub-100 nm sizes, many challenges still have to be
addressed. Spin-wave damping remains one of the major prob-
lems for the spin-wave circuitry. Most of the above-mentioned
prototypes were realized on the base of YIG material grown on
GGG substrate, as the material with the smallest damping. The
attempts to switch to YIG on silicon have not been successful
as the damping increases significantly. The utilization of new
magnetic materials, e.g. CoFeB or Heusler compounds, grown
on Si attracts interest.

A cross talk between input and output ports is a big chal-
lenge for spin-wave devices using conducting contours for
excitation/detection. Another reason staying behind the low
efficiency of generation and detection of spin waves, is the
impedance mismatch between the electromagnetic waves and
spin waves related to momentum mismatch. The mismatch
suggests using a periodic magnetic or piezoelectric transduc-
tion structure, or possibly acoustic-wave mediated transduc-
tion. Novel transduction geometries are required here. In this
context, the demonstration of switching of non-volatile mag-
netic elements by spin waves as proposed e.g. in [50] is highly
an important result (see figure 7(c)). Moreover, there are many
approaches which allow for the unidirectional propagation of
spin waves (see e.g. section 16) and can ensure input/output
isolation in circuits.

The all-magnon approach does not require converters but
relies on the voltage controlled magnetic anisotropy (VCMA)
parametric amplifiers (see section 18) that were studied the-
oretically but not yet realized experimentally. Moreover, the
simulated half-adder appeared to be slower compared to
CMOS [53] suggesting the need in the increase of operating
spin-wave frequencies up to the sub-THz or THz ranges.

4.6. Concluding remarks

It was shown numerically that ME-cell based spin-wave logic
circuits can overcome a 10 nm CMOS implementation by
up to 100 times in terms of the ADPP (see section 3).
This is the reason why the main attention of the magnonics
research community is focused nowadays on the realization
of high-efficiency magnon-to-current converters. The all-
magnon circuits approach also has shown that can be compet-
itive to CMOS in terms of footprint and energy consumptions.
These achievements are very encouraging and suggest the uti-
lization of the developed concepts and approaches for their
further utilization in much more powerful unconventional,
neuromorphic and quantum computing concepts instead of
Boolean ones. As was shown, unconventional spin-wave logic

circuits promise an intriguing possibility of constructing multi-
functional logic architectures with built-in memory, pattern
recognition, prime factorization and others. Particularly these
unconventional computing concepts at the nano-scale utiliz-
ing the powerful nonlinear spin-wave physics have a large
potential to compete with CMOS and to occupy its niche in
specialized types of computing applications.

Acknowledgments

AVC acknowledges financial support by the European
Research Council Starting Grant 678309 MagnonCircuits.

5. Magnon Bose–Einstein condensation and
supercurrent transport

Vitaliy I Vasyuchka and Burkard Hillebrands

Department of Physics and State Research Center OPTIMAS,
Technische Universität Kaiserslautern (TUK), Kaiserslautern,
Germany

5.1. Status

Novel physical phenomena constitute a driving force for the successful development of the fascinating and dynamic field of magnonics. One very promising direction is the investigation and utilization of magnon macroscopic quantum states. Magnons are bosons, and thus they are able to spontaneously form a spatially extended, coherent macroscopic quantum state—magnon Bose–Einstein condensate (BEC) [64]. The magnon BEC in a magnetic medium is established as one single quantum state with the lowest energy as a result of the thermalization process in an overpopulated magnon system. The condensation of magnons can occur even at room temperature due to specific properties of magnons as such as the small effective mass, nonlinearity, a large number of thermal excitations at non-zero temperatures, the simplicity of the magnon injection into a magnetic system and control of their population.

During the last decades, many efforts have been made to
observe the magnon condensation in solids. One of the best
examples of a magnetic medium, where the magnon BEC
was achieved, is the insulating ferrimagnet YIG possessing
an extremely low magnetic damping. The main approach to
reach the magnon BEC in YIG was based on the increase of the
magnon density above the thermal equilibrium level. Just a few
percent excess of magnons over their thermal level is sufficient
to achieve the condensation in YIG. For this, various meth-
ods of magnon injection into YIG have been employed (see
figure 9). Here, the most interesting studies related to magnon
BEC formation are listed.

5.1.1. Microwave parametric pumping. The most effective and popular way to increase the density of magnons to reach the magnon BEC is parametric pumping. When the applied strong microwave electromagnetic field exceeds a threshold value, the conversion of microwave pumping photons into
circuitry. One option for it is the piezoelectric-magnetostrictive
transduction. The all-magnon data processing approach, was
benchmarked in reference [53]. It was found that 30 nm-
based magnonic circuit is comparable to 7 nm-based CMOS in
terms of footprint, has 10 times smaller energy consumption,
but is slower. The further magnonic circuits benchmarking is
discussed in section 3 (figure 8).

4.5. Current and future challenges

Although we observe drastic progress in the evolution of the
spin-wave circuitry and, in particular, its scaling down from
mm to sub-100 nm sizes, many challenges still have to be
addressed. Spin-wave damping remains one of the major prob-
lems for the spin-wave circuitry. Most of the above-mentioned
prototypes were realized on the base of YIG material grown on
GGG substrate, as the material with the smallest damping. The
attempts to switch to YIG on silicon have not been successful
as the damping increases significantly. The utilization of new
magnetic materials, e.g. CoFeB or Heusler compounds, grown
on Si attracts interest.

A cross talk between input and output ports is a big chal-
lenge for spin-wave devices using conducting contours for
excitation/detection. Another reason staying behind the low
efficiency of generation and detection of spin waves, is the
impedance mismatch between the electromagnetic waves and
spin waves related to momentum mismatch. The mismatch
suggests using a periodic magnetic or piezoelectric transduc-
tion structure, or possibly acoustic-wave mediated transduc-
tion. Novel transduction geometries are required here. In this
context, the demonstration of switching of non-volatile mag-
netic elements by spin waves as proposed e.g. in [50] is highly
an important result (see figure 7(c)). Moreover, there are many
approaches which allow for the unidirectional propagation of
spin waves (see e.g. section 16) and can ensure input/output
isolation in circuits.

The all-magnon approach does not require converters but
relies on the voltage controlled magnetic anisotropy (VCMA)
parametric amplifiers (see section 18) that were studied the-
oretically but not yet realized experimentally. Moreover, the
simulated half-adder appeared to be slower compared to
CMOS [53] suggesting the need in the increase of operating
spin-wave frequencies up to the sub-THz or THz ranges.

4.6. Concluding remarks

It was shown numerically that ME-cell based spin-wave logic
circuits can overcome a 10 nm CMOS implementation by
up to 100 times in terms of the ADPP (see section 3).
This is the reason why the main attention of the magnonics
research community is focused nowadays on the realization
of high-efficiency magnon-to-current converters. The all-
magnon circuits approach also has shown that can be compet-
itive to CMOS in terms of footprint and energy consumptions.
These achievements are very encouraging and suggest the uti-
lization of the developed concepts and approaches for their
further utilization in much more powerful unconventional,
neuromorphic and quantum computing concepts instead of
Boolean ones. As was shown, unconventional spin-wave logic

circuits promise an intriguing possibility of constructing multi-
functional logic architectures with built-in memory, pattern
recognition, prime factorization and others. Particularly these
unconventional computing concepts at the nano-scale utiliz-
ing the powerful nonlinear spin-wave physics have a large
potential to compete with CMOS and to occupy its niche in
specialized types of computing applications.

Acknowledgments

AVC acknowledges financial support by the European
Research Council Starting Grant 678309 MagnonCircuits.

5. Magnon Bose–Einstein condensation and
supercurrent transport

Vitaliy I Vasyuchka and Burkard Hillebrands

Department of Physics and State Research Center OPTIMAS,
Technische Universität Kaiserslautern (TUK), Kaiserslautern,
Germany

5.1. Status

Novel physical phenomena constitute a driving force for the successful development of the fascinating and dynamic field of magnonics. One very promising direction is the investigation and utilization of magnon macroscopic quantum states. Magnons are bosons, and thus they are able to spontaneously form a spatially extended, coherent macroscopic quantum state—magnon Bose–Einstein condensate (BEC) [64]. The magnon BEC in a magnetic medium is established as one single quantum state with the lowest energy as a result of the thermalization process in an overpopulated magnon system. The condensation of magnons can occur even at room temperature due to specific properties of magnons as such as the small effective mass, nonlinearity, a large number of thermal excitations at non-zero temperatures, the simplicity of the magnon injection into a magnetic system and control of their population.

During the last decades, many efforts have been made to
observe the magnon condensation in solids. One of the best
examples of a magnetic medium, where the magnon BEC
was achieved, is the insulating ferrimagnet YIG possessing
an extremely low magnetic damping. The main approach to
reach the magnon BEC in YIG was based on the increase of the
magnon density above the thermal equilibrium level. Just a few
percent excess of magnons over their thermal level is sufficient
to achieve the condensation in YIG. For this, various meth-
ods of magnon injection into YIG have been employed (see
figure 9). Here, the most interesting studies related to magnon
BEC formation are listed.

5.1.1. Microwave parametric pumping. The most effective and popular way to increase the density of magnons to reach the magnon BEC is parametric pumping. When the applied strong microwave electromagnetic field exceeds a threshold value, the conversion of microwave pumping photons into
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Schematic illustration of different mechanisms leading to the formation of magnon Bose-Einstein condensates in magnetic solids.</figcaption></figure>

magnons at half of the pumping frequency occurs. The majority of the magnon BEC experimental investigations were conducted using the parametric pumping technique (see e.g. [64–66]). It is important to note that the magnon condensation in an overpopulated magnon gas is also a threshold process. Typically, it happens in YIG films, when the applied pumping power exceeds the threshold of parametric pumping by two or three orders of magnitude.

**5.1.2. Kinetic instability regime.** Usually, the formation of the magnon condensate is accompanied by multistep cascade processes of magnon–magnon scattering events, which transfer parametrically pumped magnons down to the bottom of a magnon spectrum. Recently, it was demonstrated that the kinetic instability process provides favourable conditions for a more efficient magnon condensation process compared to the cascade-only scenario [67]. In this regime, the direct one-step transfer of the parametrically injected magnons to the lowest energy states surpasses the multistep scattering process and is followed by a thermalization of low-energy magnons into the BEC state.

**5.1.3. Spin pumping via spin-Seebeck effect.** A novel method of magnon injection, based on the process of spin pumping, when a spin angular momentum is transferred over the interface between a nonmagnetic metal and a magnetic material, is very promising for the formation of magnon BEC [68]. The case of the spin Seebeck effect is especially interesting [69], when a magnon flow is generated by a temperature gradient across the interface between a nonmagnetic metallic layer and a magnetic insulator. It has been demonstrated for the case of YIG/Pt nanowires, cooled to cryogenic temperatures, that the magnon current induced by the spin Seebeck effect leads to excitation of YIG magnetisation auto-oscillations and generation of coherent microwave radiation [69]. This achievement paves the way for the spin-caloritronic-based magnon BEC.

**5.1.4. Rapid cooling mechanism.** Very recently, a new and universal approach to enable BEC of magnons by rapid cooling has been demonstrated [70]. For this, a disequilibrium of magnons with the phonon bath is introduced via heating of a magnetic sample to an elevated temperature with a following rapid decrease in the phonon temperature. This decrease is very fast compared to the relaxation time of the magnon system. It results in a large excess of incoherent magnons in the system and in subsequent Bose-Einstein magnon condensation.

## 5.2. Current and future challenges

An extraordinary potential for the field of magnonics is the use of macroscopic quantum phenomena such as the magnon BEC for information transfer and processing. The following non-comprehensive list of items highlights the most promising directions dealing with the current challenges and simultaneously defining new problems for the future:

(a) Magnon supercurrents: due to its zero group velocity, the magnon BEC cannot be directly utilized for the transport of a spin information in space. However, the information transfer can be realised by means of magnon supercurrents, which constitute the transport of angular momentum, driven by a phase gradient in the magnon-condensate wave function. The creation of such a supercurrent was experimentally successful by introducing a time-dependent spatial phase gradient into the wave function of the magnon BEC [66]. The temporal evolution of the BEC formed in a parametrically populated magnon gas was studied via BLS spectroscopy in room-temperature YIG films. It has been found that local heating in the focal point of a probing laser beam leads to the excessive decay of the freely evolving BEC, which is a fingerprint of the supercurrent efflux of condensed
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Schematic illustration of different mechanisms leading to the formation of magnon Bose-Einstein condensates in magnetic solids.</figcaption></figure>

magnons at half of the pumping frequency occurs. The majority of the magnon BEC experimental investigations were conducted using the parametric pumping technique (see e.g. [64–66]). It is important to note that the magnon condensation in an overpopulated magnon gas is also a threshold process. Typically, it happens in YIG films, when the applied pumping power exceeds the threshold of parametric pumping by two or three orders of magnitude.

**5.1.2. Kinetic instability regime.** Usually, the formation of the magnon condensate is accompanied by multistep cascade processes of magnon–magnon scattering events, which transfer parametrically pumped magnons down to the bottom of a magnon spectrum. Recently, it was demonstrated that the kinetic instability process provides favourable conditions for a more efficient magnon condensation process compared to the cascade-only scenario [67]. In this regime, the direct one-step transfer of the parametrically injected magnons to the lowest energy states surpasses the multistep scattering process and is followed by a thermalization of low-energy magnons into the BEC state.

**5.1.3. Spin pumping via spin-Seebeck effect.** A novel method of magnon injection, based on the process of spin pumping, when a spin angular momentum is transferred over the interface between a nonmagnetic metal and a magnetic material, is very promising for the formation of magnon BEC [68]. The case of the spin Seebeck effect is especially interesting [69], when a magnon flow is generated by a temperature gradient across the interface between a nonmagnetic metallic layer and a magnetic insulator. It has been demonstrated for the case of YIG/Pt nanowires, cooled to cryogenic temperatures, that the magnon current induced by the spin Seebeck effect leads to excitation of YIG magnetisation auto-oscillations and generation of coherent microwave radiation [69]. This achievement paves the way for the spin-caloritronic-based magnon BEC.

**5.1.4. Rapid cooling mechanism.** Very recently, a new and universal approach to enable BEC of magnons by rapid cooling has been demonstrated [70]. For this, a disequilibrium of magnons with the phonon bath is introduced via heating of a magnetic sample to an elevated temperature with a following rapid decrease in the phonon temperature. This decrease is very fast compared to the relaxation time of the magnon system. It results in a large excess of incoherent magnons in the system and in subsequent Bose-Einstein magnon condensation.

## 5.2. Current and future challenges

An extraordinary potential for the field of magnonics is the use of macroscopic quantum phenomena such as the magnon BEC for information transfer and processing. The following non-comprehensive list of items highlights the most promising directions dealing with the current challenges and simultaneously defining new problems for the future:

(a) Magnon supercurrents: due to its zero group velocity, the magnon BEC cannot be directly utilized for the transport of a spin information in space. However, the information transfer can be realised by means of magnon supercurrents, which constitute the transport of angular momentum, driven by a phase gradient in the magnon-condensate wave function. The creation of such a supercurrent was experimentally successful by introducing a time-dependent spatial phase gradient into the wave function of the magnon BEC [66]. The temporal evolution of the BEC formed in a parametrically populated magnon gas was studied via BLS spectroscopy in room-temperature YIG films. It has been found that local heating in the focal point of a probing laser beam leads to the excessive decay of the freely evolving BEC, which is a fingerprint of the supercurrent efflux of condensed
magnons. An additional indirect confirmation of the existence of magnon supercurrents was obtained through BLS observations of quantized vortices in a two-component Bose–Einstein condensate [65].

(b) *Excitations in magnon condensate and second sound*: investigations of wave-like excitations that propagate in condensed and in normal phases of a magnon gas pose another intriguing challenge. They may pave the way to a complementary approach for spin information transfer. Recently, it was demonstrated that the condensed magnons being pushed out of the locally heated area form compact density humps, which propagate over long distances through a thermally homogeneous magnetic medium [71]. They were understood as a superposition of Bogoliubov waves with oscillations of both the amplitude and the phase of the magnon BEC's wave function. These waves are described by a linear dispersion law in the long-wavelength limit and can be considered as a magnon second sound of the condensed magnon phase.

(c) *Magnonic Josephson effects*: by analogy to the Josephson effects in superconductors, both the alternating current (ac) and the direct current (dc) magnonic Josephson effects were proposed and studied theoretically [72]. A junction connecting two weakly coupled quasi-equilibrium magnon BECs is an essential component for the observation of these fascinating phenomena. The first step in the experimental realisation of the ac magnonic Josephson effect was recently reported in the system of two room-temperature magnon condensates separated by a magnetic trench [73]. Josephson oscillations of the magnon BEC density in the trench were observed, which are induced by the coherent phase shift between the two magnon condensates from the left and right zones of this magnetic inhomogeneity. Further developments within this challenge would facilitate the broad application of magnonic Josephson effects for data processing in magnonic devices.

(d) *Electric control of magnon supercurrents*: the possibility of electric control of magnon supercurrents using the magnonic Aharonov–Cashier effect, which has so far been analyzed only theoretically [72], is extremely appealing. It is predicted that this effect will lead to the generation of a persistent magnon supercurrent in a magnetic ring subjected to an electric field with the radial symmetry due to the accumulation of an induced Aharonov–Cashier phase in the magnon condensate. The long-distance coherency of the magnon condensate is of paramount importance to the observation of the magnonic Aharonov–Cashier effect.

(e) *Computing with magnonic macroscopic quantum states:* nowadays, an exceptional and promising challenge is the utilization of magnonic macroscopic quantum states for information processing and computing. Activities in this direction, in particular those related to experimental implementation, are in their initial stages and research in this area is in great demand. The question of quantum

computing with magnon condensates remains especially interesting.

The following scientific and technological advances can help to overcome many of the specified challenges: magnetic materials with exceptionally low damping, involvement of the interface effects particularly for the nanometer-thick magnetic films, better understanding of the quantum nature of magnon BEC, and the realisation of the magnon-condensate-based qubit.

### 5.3. Concluding remarks

Although investigations of the magnon macroscopic quantum states are in their initial stages, this exciting research field demonstrates a continuing progress in the formation, control and usage of magnon Bose–Einstein condensates and supercurrents. Among the latter achievements, we can recognise the different ways of magnon BEC formation, spin information transfer using magnon supercurrents and second sound, novel magnonic effects, etc. Considering the richness of magnon phenomena, the wealth of nonlinear effects, the scalability and reconfigurability of magnon systems, further fruitful developments in this new branch of magnonics are expected.

## Acknowledgments

Financial support of the European Research Council within the Advanced Grant 694709 SuperMagnonics ‘Supercurrents of Magnon Condensates for Advanced Magnonics’ is gratefully acknowledged.

# 6. Three-Dimensional Magnonics

Gianluca Gubbiotti¹ and Sergej A Nikitov²

¹Istituto Officina dei Materiali del CNR (CNR-IOM), Perugia, Italy

²Kotelnikov Institute of Radioengineering and Electronics, Russia

## 6.1. Status

Magnonics refers to the research field that uses spin waves, the collective excitations of ordered magnetic materials, or their quanta (magnons) as a tool for signal processing, communication and computation [74]. This field has been rapidly growing during the last decade, with magnonic systems investigated up to now mainly represented by periodic one- and two-dimensional planar systems in the form of arrays of nanowires, dots, antidots and width modulated waveguides based on insulating YIG (Y₃Fe₅O₁₂), metallic permalloy (Py), CoFeB, as well as coupled structures containing also pure Ni, Co, Fe and Heusler alloys.

Very recently, due to developments in instrumentation, nanofabrication and modeling tools, researchers have devoted their attention to the fabrication 3D magnetic systems where novel physical effects comprising geometry, topology, chirality, frustration and unconventional spin textures emerge [75]. Examples of three-dimensional (3D) magnonic nanostructures
magnons. An additional indirect confirmation of the existence of magnon supercurrents was obtained through BLS observations of quantized vortices in a two-component Bose–Einstein condensate [65].

(b) *Excitations in magnon condensate and second sound*: investigations of wave-like excitations that propagate in condensed and in normal phases of a magnon gas pose another intriguing challenge. They may pave the way to a complementary approach for spin information transfer. Recently, it was demonstrated that the condensed magnons being pushed out of the locally heated area form compact density humps, which propagate over long distances through a thermally homogeneous magnetic medium [71]. They were understood as a superposition of Bogoliubov waves with oscillations of both the amplitude and the phase of the magnon BEC's wave function. These waves are described by a linear dispersion law in the long-wavelength limit and can be considered as a magnon second sound of the condensed magnon phase.

(c) *Magnonic Josephson effects*: by analogy to the Josephson effects in superconductors, both the alternating current (ac) and the direct current (dc) magnonic Josephson effects were proposed and studied theoretically [72]. A junction connecting two weakly coupled quasi-equilibrium magnon BECs is an essential component for the observation of these fascinating phenomena. The first step in the experimental realisation of the ac magnonic Josephson effect was recently reported in the system of two room-temperature magnon condensates separated by a magnetic trench [73]. Josephson oscillations of the magnon BEC density in the trench were observed, which are induced by the coherent phase shift between the two magnon condensates from the left and right zones of this magnetic inhomogeneity. Further developments within this challenge would facilitate the broad application of magnonic Josephson effects for data processing in magnonic devices.

(d) *Electric control of magnon supercurrents*: the possibility of electric control of magnon supercurrents using the magnonic Aharonov–Cashier effect, which has so far been analyzed only theoretically [72], is extremely appealing. It is predicted that this effect will lead to the generation of a persistent magnon supercurrent in a magnetic ring subjected to an electric field with the radial symmetry due to the accumulation of an induced Aharonov–Cashier phase in the magnon condensate. The long-distance coherency of the magnon condensate is of paramount importance to the observation of the magnonic Aharonov–Cashier effect.

(e) *Computing with magnonic macroscopic quantum states:* nowadays, an exceptional and promising challenge is the utilization of magnonic macroscopic quantum states for information processing and computing. Activities in this direction, in particular those related to experimental implementation, are in their initial stages and research in this area is in great demand. The question of quantum

computing with magnon condensates remains especially interesting.

The following scientific and technological advances can help to overcome many of the specified challenges: magnetic materials with exceptionally low damping, involvement of the interface effects particularly for the nanometer-thick magnetic films, better understanding of the quantum nature of magnon BEC, and the realisation of the magnon-condensate-based qubit.

### 5.3. Concluding remarks

Although investigations of the magnon macroscopic quantum states are in their initial stages, this exciting research field demonstrates a continuing progress in the formation, control and usage of magnon Bose–Einstein condensates and supercurrents. Among the latter achievements, we can recognise the different ways of magnon BEC formation, spin information transfer using magnon supercurrents and second sound, novel magnonic effects, etc. Considering the richness of magnon phenomena, the wealth of nonlinear effects, the scalability and reconfigurability of magnon systems, further fruitful developments in this new branch of magnonics are expected.

## Acknowledgments

Financial support of the European Research Council within the Advanced Grant 694709 SuperMagnonics ‘Supercurrents of Magnon Condensates for Advanced Magnonics’ is gratefully acknowledged.

# 6. Three-Dimensional Magnonics

Gianluca Gubbiotti¹ and Sergej A Nikitov²

¹Istituto Officina dei Materiali del CNR (CNR-IOM), Perugia, Italy

²Kotelnikov Institute of Radioengineering and Electronics, Russia

## 6.1. Status

Magnonics refers to the research field that uses spin waves, the collective excitations of ordered magnetic materials, or their quanta (magnons) as a tool for signal processing, communication and computation [74]. This field has been rapidly growing during the last decade, with magnonic systems investigated up to now mainly represented by periodic one- and two-dimensional planar systems in the form of arrays of nanowires, dots, antidots and width modulated waveguides based on insulating YIG (Y₃Fe₅O₁₂), metallic permalloy (Py), CoFeB, as well as coupled structures containing also pure Ni, Co, Fe and Heusler alloys.

Very recently, due to developments in instrumentation, nanofabrication and modeling tools, researchers have devoted their attention to the fabrication 3D magnetic systems where novel physical effects comprising geometry, topology, chirality, frustration and unconventional spin textures emerge [75]. Examples of three-dimensional (3D) magnonic nanostructures
<figure><img src="image_12.png" /><figcaption>**Figure 10.** (a) Schematic drawing of the permalloy (10 nm)/Cu (10 nm)/permalloy (30 nm) trilayer nanowire array with width of *w* = 280 nm and edge-to-edge spacing of *s* = 80 nm. (b) Measured longitudinal hysteresis loop for the nanowire array. The vertical red dashed line indicates the field value of +150 Oe, which following either the descending or the ascending branch of the hysteresis loop, corresponds to a parallel (P) or antiparallel (AP) alignment between the magnetization in the two permalloy layers. Comparison between the measured BLS (points) and calculated (lines) magnonic band structure for the nanowire array in the P and AP states (*H* = +150 Oe). Reprinted (figure) with permission from [77], Copyright (2018) by the American Physical Society. The external magnetic field is applied along the nanowire length both in the MOKE and in the BLS measurements.</figcaption></figure>

can also be found in sections 1, 11 and 12. The advantage of magnonic systems magnetic materials include the possibility to overcome the fundamental physical limit in heat generation associated with electrical current in silicon-based technology, potentially small magnon wavelengths in the range of tens of nanometers, and high-frequency operation in the terahertz regime. In addition, the interest in 3D and vertically coupled magnonic structures is based on a similar trend in CMOS electronics, as it enables a transition from two-dimensional to three-dimensional functional design for integrated structures [12].

Following this trend, researchers are proposing different strategies to build the next generation of 3D magnonic systems. The first of them is an extension of planar patterned nanostructures, where arrays of patterned magnetic dots or antidots have a layered structure. In this case, the vertical stacking of ferromagnetic materials, placed in direct contact

or separated by a non-magnetic spacer, adds new functionality and an additional degree of freedom to control the spin-wave band structure based on the interplay between exchange [76] and dipolar interactions [77] as well as on the phase relation (in-phase or out-of-phase) between the dynamic magnetizations in the different layers. These also offer a valuable tool for tuning the static magnetization configuration (parallel and anti-parallel) in a reconfigurable manner and therefore reprogram the magnonic band structure on demand, as shown in figure 10 for a dense array of Py (30 nm)/Cu (10 nm)/Py (30 nm) nanowire array.

On the other hand, 3D meander-like single- and multilayer films (see figure 11(a)) grown at the top of the initially structured substrate and magnetized along the thickness steps have been proposed as a candidate for such 3D magnonics crystals for vertical steering of spin-wave beams thus allowing to overcome the difficulties of bending spin-wave beams for in-plane
<figure><img src="image_13.png" /><figcaption>**Figure 10.** (a) Schematic drawing of the permalloy (10 nm)/Cu (10 nm)/permalloy (30 nm) trilayer nanowire array with width of *w* = 280 nm and edge-to-edge spacing of *s* = 80 nm. (b) Measured longitudinal hysteresis loop for the nanowire array. The vertical red dashed line indicates the field value of +150 Oe, which following either the descending or the ascending branch of the hysteresis loop, corresponds to a parallel (P) or antiparallel (AP) alignment between the magnetization in the two permalloy layers. Comparison between the measured BLS (points) and calculated (lines) magnonic band structure for the nanowire array in the P and AP states (*H* = +150 Oe). Reprinted (figure) with permission from [77], Copyright (2018) by the American Physical Society. The external magnetic field is applied along the nanowire length both in the MOKE and in the BLS measurements.</figcaption></figure>

can also be found in sections 1, 11 and 12. The advantage of magnonic systems magnetic materials include the possibility to overcome the fundamental physical limit in heat generation associated with electrical current in silicon-based technology, potentially small magnon wavelengths in the range of tens of nanometers, and high-frequency operation in the terahertz regime. In addition, the interest in 3D and vertically coupled magnonic structures is based on a similar trend in CMOS electronics, as it enables a transition from two-dimensional to three-dimensional functional design for integrated structures [12].

Following this trend, researchers are proposing different strategies to build the next generation of 3D magnonic systems. The first of them is an extension of planar patterned nanostructures, where arrays of patterned magnetic dots or antidots have a layered structure. In this case, the vertical stacking of ferromagnetic materials, placed in direct contact

or separated by a non-magnetic spacer, adds new functionality and an additional degree of freedom to control the spin-wave band structure based on the interplay between exchange [76] and dipolar interactions [77] as well as on the phase relation (in-phase or out-of-phase) between the dynamic magnetizations in the different layers. These also offer a valuable tool for tuning the static magnetization configuration (parallel and anti-parallel) in a reconfigurable manner and therefore reprogram the magnonic band structure on demand, as shown in figure 10 for a dense array of Py (30 nm)/Cu (10 nm)/Py (30 nm) nanowire array.

On the other hand, 3D meander-like single- and multilayer films (see figure 11(a)) grown at the top of the initially structured substrate and magnetized along the thickness steps have been proposed as a candidate for such 3D magnonics crystals for vertical steering of spin-wave beams thus allowing to overcome the difficulties of bending spin-wave beams for in-plane
magnetized systems due to the anisotropic dispersion of dipolar spin waves [78]. Spin waves propagate in film's segments located at right angles with respect to each other thus making possible the wave propagation in three dimensions without significant losses in the junction region [79].

Moreover, domain walls together with DMI can be considered as elements of 3D magnonics for propagation of spin waves in curved geometries and in close proximity to other channels [80]. Because the spin structure of domain walls is primarily governed by intrinsic magnetic properties, they are less sensitive to issues related to lithography or nanofabrication such as edge roughness or sample-to-sample reproducibility.

A very promising and alternative method to physical patterning is based on the use of hybrid heterostructures in the form of bilayer systems in which the non-magnetic component induces periodic modulation of either the static or the dynamic internal magnetic field of the magnetic film itself [12]. These include ferromagnetic (FM) metal-insulator, FM metal-heavy metal (HM), FM metal-antiferromagnet, FM metal-ferroelectric and FM metal-superconductor bilayers (section 21), for which new properties of spin waves such as confining and filtering, guiding and steering, non-reciprocity and reconfigurability, a direct and an indirect gap in the magnonic band structure, have been proved (section 16).

Finally, curved surfaces and 3D micro and nano-objects exhibit peculiar and unexpected spin textures which are normally not observed in planar nanostructures and allows for the exploitation of magnons chirality and the resulting non-reciprocity of the magnon dispersion. 3D nano-objects with micrometer size can be fabricated to study their dynamics and to couple with macroscopic microwave field. The curvature induced spin-wave non-reciprocity offers several advantages over the iDMI since it is not limited to ultrathin ferromagnetic layers and does not increase the spin-wave damping. Moreover, freestanding 3D magnonic structures are obtained by using a complex lithography process including room-temperature deposition and lift-off of amorphous YIG as well as subsequent crystallization by annealing [81].

## 6.2. Current and future challenges

In terms of potential applications, 3D magnonic structures might be exploited to realize complex vertical interconnection elements for efficient spin-wave transport between multiple layers inside magnonic networks. This could lead to the new development of 3D magnonics circuits mainly used for the multiple-layer architecture of signal processing devices comprising the neuromorphic principles of data processing [82]. In 3D circuits, shorter interconnects reduce the average wire length whereas the vertical dimension adds a higher order of connectivity and offers new design possibilities. Increasing the density of elements will be a crucial aspect for the realization of scalable and configurable magnonic networks and will lead to a decrease in spin-wave propagation and energy losses.

The perspectives of the practical use of 3D magnonic structures can also be based on a combination of electronics components using both semiconductor and magnonic technologies. Thus, the opportunity appears to integrate magnonic structures with spin transfer and semiconductor structures with electric charge transfer. The main problem here is to superpose at high technical level material parameters (e.g., YIG) and semiconductors (e.g. Si, GaAs, GaN, etc). Recent experiments have shown that YIG film grown onto GaAs substrates can be a good candidate for the integration of magnonics and semiconductor electronics (figure 11(b)). GaAs has a large band gap in 1.4 eV and high mobility of electrons and holes up to 8600 cm² V⁻¹ s⁻¹ and 400 cm² V⁻¹ s⁻¹, respectively. Making 3D structures with these materials and superimposing light from the laser one can govern the properties of spin waves and information processing [83].

Nonlinear processes originating during spin-wave propagation in ferromagnetic materials are of particular interest, since they occur at sufficiently low power levels and new types of spin-wave excitations can be implemented. The main nonlinear effect in a MC is the nonlinear frequency shift (and, respective shift of the band structure) with increasing input signal power. In 3D magnonic structures, the particular effect of 'nonlinear switching of the MC' is possible, in which periodic structure allows to pass the signal of high power at frequencies within the bandgap (figure 11(c)). This feature allows considering such 3D based on MCs as a nonlinear phase shift modulus and the noise limiters [84].

Another important aspect for 3D magnonics is the integration with piezoelectrical materials where propagating SAWs and electromagnetic waves can both interact with spin waves (figure 11(d)). Such voltage-controlled anisotropy and current-induced magnetization dynamics in magnonic/piezoelectric (and/or ferroelectric) heterostructures can be a subject for future 3D magnonics applications [85].

In the case of FM-HM multilayers, it is possible to take advantage of the unidirectional coupling induced by iDMI to transfer the information carried by the spin waves between thin ferromagnetic layers in only one direction of propagation. In this particular system, the functionality of spin-wave diode and circulator to steer and manipulate spin waves over a wide range of frequency has been demonstrated [86].

## 6.3. Concluding remarks

Here we have reviewed the most recent advances in 3D and vertical stacked magnonic structures and discussed some of the most interesting perspectives. We highlighted several open questions that still need addressing, such as smart methods for excitation, amplification and detection of spin waves and efficient interconnections between magnonic and electric circuits. Being aware of the difficulties and challenges related to the fabrication of 3D magnonic structure, we believe that the exploitation of the third dimension will bring to additional and specific features (e.g. vertical magnon transport, nonreciprocal coupling) while in CMOS technology the vertical stacking only gives higher storage density with the disadvantage of the standard CMOS heat problem.