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.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** 3D vertical magnonic structures: (a) meander-shaped ferromagnetic film. Reprinted from [78], Copyright (2019), with permission from Elsevier, (b) YIG film grown on GaAs substrate. Reprinted (figure) with permission from [83] Copyright (2019) by the American Physical Society, (c) YIG–metal–semiconductor nonlinear shifter. Reproduced from [84]. © IOP Publishing Ltd. All rights reserved, and (d) MC sandwiched between two ferroelectric layers.</figcaption></figure>

## Acknowledgments

The research of SAN was supported by the Government of the Russian Federation (Agreement 074-02-2018-286) within the laboratory ‘Terahertz spintronics’ of the Moscow Institute of Physics and Technology (National University) and by the State task 0030-2019-0013.

## 7. Short wavelength magnonics

Haiming Yu<fnref n="1" /> and Dirk Grundler<fnref n="2" />

Spin waves are considered to be promising as information carriers in low-power-consuming computing and logics with advantages such as absence of Joule heating, clock frequencies up to the THz frequency regime (see section 14: ‘THz magnonics’) and coherent processing of data across several unit cells (see section 4: ‘magnonic logic, memory and devices’). Spin waves with wavelengths $λ$ at the nanoscale (figure 12) reside in the exchange-dominated regime of the spin-wave dispersion relation $f(k)$ in a ferromagnetic thin film. They exhibit group velocities $v_g = 2π df/dk$ which increase

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Dispersion for short-wavelength spin waves in YIG (20 nm thickness, saturation magnetization 140 kA m$^{-1}$, exchange constant $3 × 10^{-16}$ m$^2$) where the propagation direction is perpendicular to the magnetization. At large $k$ (small wavelength $λ$) the dispersion relation follows a $f ∝ k^2$ behavior dominated by the exchange interaction.</figcaption></figure>

with decreasing $λ$, i.e., increasing wave vector $k = 2π/λ$. Such spin waves (magnons) are particularly attractive for high-speed magnonic devices and data processing. However, excitation and detection of short-wavelength magnons are non-trivial and represent a major challenge. Conventional methods based on microwave antennas such as metallic CPWs are inefficient.

<footnote n="1">¹Beihang University, People’s Republic of China</footnote>

<footnote n="2">²École Polytechnique Fédérale de Lausanne (EPFL), Switzerland</footnote>
<figure><img src="image_14.png" /><figcaption>**Figure 11.** 3D vertical magnonic structures: (a) meander-shaped ferromagnetic film. Reprinted from [78], Copyright (2019), with permission from Elsevier, (b) YIG film grown on GaAs substrate. Reprinted (figure) with permission from [83] Copyright (2019) by the American Physical Society, (c) YIG–metal–semiconductor nonlinear shifter. Reproduced from [84]. © IOP Publishing Ltd. All rights reserved, and (d) MC sandwiched between two ferroelectric layers.</figcaption></figure>

## Acknowledgments

The research of SAN was supported by the Government of the Russian Federation (Agreement 074-02-2018-286) within the laboratory ‘Terahertz spintronics’ of the Moscow Institute of Physics and Technology (National University) and by the State task 0030-2019-0013.

## 7. Short wavelength magnonics

Haiming Yu<fnref n="1" /> and Dirk Grundler<fnref n="2" />

Spin waves are considered to be promising as information carriers in low-power-consuming computing and logics with advantages such as absence of Joule heating, clock frequencies up to the THz frequency regime (see section 14: ‘THz magnonics’) and coherent processing of data across several unit cells (see section 4: ‘magnonic logic, memory and devices’). Spin waves with wavelengths $λ$ at the nanoscale (figure 12) reside in the exchange-dominated regime of the spin-wave dispersion relation $f(k)$ in a ferromagnetic thin film. They exhibit group velocities $v_g = 2π df/dk$ which increase

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Dispersion for short-wavelength spin waves in YIG (20 nm thickness, saturation magnetization 140 kA m$^{-1}$, exchange constant $3 × 10^{-16}$ m$^2$) where the propagation direction is perpendicular to the magnetization. At large $k$ (small wavelength $λ$) the dispersion relation follows a $f ∝ k^2$ behavior dominated by the exchange interaction.</figcaption></figure>

with decreasing $λ$, i.e., increasing wave vector $k = 2π/λ$. Such spin waves (magnons) are particularly attractive for high-speed magnonic devices and data processing. However, excitation and detection of short-wavelength magnons are non-trivial and represent a major challenge. Conventional methods based on microwave antennas such as metallic CPWs are inefficient.

<footnote n="1">¹Beihang University, People’s Republic of China</footnote>

<footnote n="2">²École Polytechnique Fédérale de Lausanne (EPFL), Switzerland</footnote>
This is due to impedance mismatching and ohmic losses when the lateral size of the antennas shrinks down the nanoscale [87]. Several promising methods have been proposed to overcome these challenges e.g. by using resonant and non-resonant magnonic nanogratings [88], ferromagnetic coplanar waveguides (mCPW) [89], parametric pumping [90], spin-transfer torque [91] and spin textures [92].

Grating couplers with multidirectional emission and detection of short-wavelength magnons were proposed and experimentally demonstrated based on both one- and two-dimensional periodically modulated ferromagnets integrated to conventional CPWs. Individual ferromagnetic layers with etched nanotroughs (similar to antidot lattices) as well as bicomponent ferromagnetic systems consisting of arrays of overgrown nanostripes or embedded nanodisks were used to evidence the grating coupler effect on a broad spectrum of metallic ferromagnetic thin films. This versatile approach was used to excite and detect ultrashort magnons with $\lambda = 68$ nm in ferrimagnetic YIG using 2D ferromagnetic nanodisk arrays [88] and $\lambda = 50$ nm using 1D ferromagnetic stripe arrays [93] (figure 13(a)). In case of large magnon velocities thin insulating YIG offers unprecedentedly long decay lengths compared to the metallic ferromagnets [94]. In these periodic lattice-based microwave-to-magnon transducers, the Bloch theorem was exploited and the wave vectors of magnons were defined by reciprocal lattice vectors up to high orders (figure 13(a)). Their amplitudes were on the same order as the ones of long-wavelength spin waves excited directly by the conventional CPWs. The amplitudes were particularly large in case of gratings operated at their own resonance frequencies (resonant grating coupler effect) [88].

Very recently, Che et al [89] reported on magnetic CPWs (mCPWs) to efficiently excite and detect short-wavelength magnons over a broad frequency band (figure 13(b)). Here the localized stray field of an integrated ferromagnetic layer caused an inhomogeneous effective field (see also section 12: ‘graded index magnonics’) and induced a wavelength conversion process in YIG. The authors demonstrated continuous tuning of $\lambda$, supporting multifrequency operation of magnonic circuits. For optimized mCPWs wavelengths below 40 nm are feasible without involving nanofabrication.

Parametric pumping provides a method for spin-wave excitation where wave vectors are without an upper limitation. The wavelengths can be tuned by a magnetic field down to the sub-100 nm scale in that spin waves are excited at half of the frequency of the microwave magnetic field by a non-linear process [90]. Additionally, spin torque nano-oscillators (STNO) enable the generation of exchange-dominated magnons by spin-polarized currents overcoming the intrinsic damping. High current densities are needed and achieved by spatial restrictions. STNOs represented effective spin wave injectors for the top layer of a trilayer stack [such as a magnetic tunnel junction (MTJ)]. The wavelength of spin waves is proportional to the radius of the nanocontact [91]. The high-order modes of propagating spin waves exhibited a wavelength down to about 70 nm, which could be further decreased by a smaller radius of the nanocontacts.

Spin textures, such as domain walls and vortex cores, have also been utilized to generate short-wavelength spin waves [92] (figure 13(c)). Spin waves propagating inside spin textures and nano-sized domain walls have attracted increasing attention (see section 11: ‘interaction of magnons with spin textures’) and promise reconfigurable magnonic devices. For example, short-wavelength spin waves can be generated by applying an alternating magnetic field in domain walls of interlayer exchange-coupled ferromagnetic bilayers. Changing the excitation frequency, the wavelength of spin waves was tuned from 1 µm down to 150 nm in magnon conduits provided by domain walls [92]. In addition, magnetic vortex cores in an antiferromagnetically coupled magnetic heterostructure served as tuneable spin wave emitters driven by an oscillating magnetic field (figure 13(c)) [92]. In a single ferromagnetic layer, high-amplitude dipole-exchange spin waves with $\lambda$ down to 80 nm have been imaged around a vortex core.

Current and future challenges for nanomagnonics.

(a) Magnon excitation: current techniques enable the excitation of magnons in the few 10 GHz frequency regime with wavelengths of a few 10 nm, i.e., about two orders of magnitude larger than the crystal lattice constant. Future transducers should address magnons with $\lambda$ below 10 nm in ferro-, ferri- and antiferromagnetic thin films. Such magnons possess high group velocities which are key to the realization of the high-speed magnonic circuits. Their properties are isotropic, thereby allowing for 3D magnonic circuits. For wave-based logic in particular coherent and phase-controlled magnons are required. A major challenge for low-power consumption lies in thin-film materials supporting the isotropic short-wave magnons in zero external field. The detection of magnons seems less challenging via e.g. spin pumping, spin-Seebeck, inverse spin Hall, or thermoelectric effects.

(b) Direct imaging: near-field BLS already provided a lateral spatial resolution of better than 55 nm [95] and thereby detected standing spin waves beyond the diffraction of light. Recently, synchrotron-based STXM was exploited to image propagating spin waves with high spatial resolution. Laboratory-based techniques offering ultra-high spatial resolution are needed to thoroughly investigate the properties of magnons with $\lambda$ below 10 nm, e.g., the scattering at structural defects and lattice vibrations. Here, magnetometry utilizing nitrogen-vacancy (NV) centers might be particularly powerful.

(c) Antiferromagnetic spin waves: antiferromagnets are immune to external field perturbations. Their spin processional motion is not only right-circularly polarized like in ferromagnets but possesses dual polarizations due to the two-opposite magnetic sublattices. The exchange energy in the antiferromagnetic systems increases the spin wave frequencies. It is still challenging to achieve the efficient excitation and detection of coherent antiferromagnetic spin waves. Recently demonstrated spin-current generation via antiferromagnetic spin waves may
This is due to impedance mismatching and ohmic losses when the lateral size of the antennas shrinks down the nanoscale [87]. Several promising methods have been proposed to overcome these challenges e.g. by using resonant and non-resonant magnonic nanogratings [88], ferromagnetic coplanar waveguides (mCPW) [89], parametric pumping [90], spin-transfer torque [91] and spin textures [92].

Grating couplers with multidirectional emission and detection of short-wavelength magnons were proposed and experimentally demonstrated based on both one- and two-dimensional periodically modulated ferromagnets integrated to conventional CPWs. Individual ferromagnetic layers with etched nanotroughs (similar to antidot lattices) as well as bicomponent ferromagnetic systems consisting of arrays of overgrown nanostripes or embedded nanodisks were used to evidence the grating coupler effect on a broad spectrum of metallic ferromagnetic thin films. This versatile approach was used to excite and detect ultrashort magnons with $\lambda = 68$ nm in ferrimagnetic YIG using 2D ferromagnetic nanodisk arrays [88] and $\lambda = 50$ nm using 1D ferromagnetic stripe arrays [93] (figure 13(a)). In case of large magnon velocities thin insulating YIG offers unprecedentedly long decay lengths compared to the metallic ferromagnets [94]. In these periodic lattice-based microwave-to-magnon transducers, the Bloch theorem was exploited and the wave vectors of magnons were defined by reciprocal lattice vectors up to high orders (figure 13(a)). Their amplitudes were on the same order as the ones of long-wavelength spin waves excited directly by the conventional CPWs. The amplitudes were particularly large in case of gratings operated at their own resonance frequencies (resonant grating coupler effect) [88].

Very recently, Che et al [89] reported on magnetic CPWs (mCPWs) to efficiently excite and detect short-wavelength magnons over a broad frequency band (figure 13(b)). Here the localized stray field of an integrated ferromagnetic layer caused an inhomogeneous effective field (see also section 12: ‘graded index magnonics’) and induced a wavelength conversion process in YIG. The authors demonstrated continuous tuning of $\lambda$, supporting multifrequency operation of magnonic circuits. For optimized mCPWs wavelengths below 40 nm are feasible without involving nanofabrication.

Parametric pumping provides a method for spin-wave excitation where wave vectors are without an upper limitation. The wavelengths can be tuned by a magnetic field down to the sub-100 nm scale in that spin waves are excited at half of the frequency of the microwave magnetic field by a non-linear process [90]. Additionally, spin torque nano-oscillators (STNO) enable the generation of exchange-dominated magnons by spin-polarized currents overcoming the intrinsic damping. High current densities are needed and achieved by spatial restrictions. STNOs represented effective spin wave injectors for the top layer of a trilayer stack [such as a magnetic tunnel junction (MTJ)]. The wavelength of spin waves is proportional to the radius of the nanocontact [91]. The high-order modes of propagating spin waves exhibited a wavelength down to about 70 nm, which could be further decreased by a smaller radius of the nanocontacts.

Spin textures, such as domain walls and vortex cores, have also been utilized to generate short-wavelength spin waves [92] (figure 13(c)). Spin waves propagating inside spin textures and nano-sized domain walls have attracted increasing attention (see section 11: ‘interaction of magnons with spin textures’) and promise reconfigurable magnonic devices. For example, short-wavelength spin waves can be generated by applying an alternating magnetic field in domain walls of interlayer exchange-coupled ferromagnetic bilayers. Changing the excitation frequency, the wavelength of spin waves was tuned from 1 µm down to 150 nm in magnon conduits provided by domain walls [92]. In addition, magnetic vortex cores in an antiferromagnetically coupled magnetic heterostructure served as tuneable spin wave emitters driven by an oscillating magnetic field (figure 13(c)) [92]. In a single ferromagnetic layer, high-amplitude dipole-exchange spin waves with $\lambda$ down to 80 nm have been imaged around a vortex core.

Current and future challenges for nanomagnonics.

(a) Magnon excitation: current techniques enable the excitation of magnons in the few 10 GHz frequency regime with wavelengths of a few 10 nm, i.e., about two orders of magnitude larger than the crystal lattice constant. Future transducers should address magnons with $\lambda$ below 10 nm in ferro-, ferri- and antiferromagnetic thin films. Such magnons possess high group velocities which are key to the realization of the high-speed magnonic circuits. Their properties are isotropic, thereby allowing for 3D magnonic circuits. For wave-based logic in particular coherent and phase-controlled magnons are required. A major challenge for low-power consumption lies in thin-film materials supporting the isotropic short-wave magnons in zero external field. The detection of magnons seems less challenging via e.g. spin pumping, spin-Seebeck, inverse spin Hall, or thermoelectric effects.

(b) Direct imaging: near-field BLS already provided a lateral spatial resolution of better than 55 nm [95] and thereby detected standing spin waves beyond the diffraction of light. Recently, synchrotron-based STXM was exploited to image propagating spin waves with high spatial resolution. Laboratory-based techniques offering ultra-high spatial resolution are needed to thoroughly investigate the properties of magnons with $\lambda$ below 10 nm, e.g., the scattering at structural defects and lattice vibrations. Here, magnetometry utilizing nitrogen-vacancy (NV) centers might be particularly powerful.

(c) Antiferromagnetic spin waves: antiferromagnets are immune to external field perturbations. Their spin processional motion is not only right-circularly polarized like in ferromagnets but possesses dual polarizations due to the two-opposite magnetic sublattices. The exchange energy in the antiferromagnetic systems increases the spin wave frequencies. It is still challenging to achieve the efficient excitation and detection of coherent antiferromagnetic spin waves. Recently demonstrated spin-current generation via antiferromagnetic spin waves may
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 13.** (a) Dispersion of short-wavelength spin waves propagating in a 20 nm-thick YIG film. The frequencies and wave vectors of the data points are extracted from the experiments on three different samples with different spin-wave modes. The red curve is the calculated dispersion $f(k)$ with an applied in-plane field of 1000 Oe. The inset shows the transmission spectra of short-wavelength spin waves with $\lambda \approx 50$ nm. Figure dispersion relations for propagating short-wavelength spin waves. Reproduced from [93]. **CC BY 4.0.** (b) (Left) sketch of micrometer-wide mCPWs consisting of Fe (brown) underneath Au (bright) on YIG. Local detection of magnons was performed through the GGG substrate using BLS. (Right) dispersion relations of magnons are shown for bare YIG (magenta) and close to the mCPW (green). The arrow illustrates the wavelength conversion. Figures schematic diagram and high-frequency modes emitted by mCPWs and wavelength conversion and magnon signal strengths. Reproduced from [89]. **CC BY 4.0.** (c) Dispersion of spin waves propagating in CoFeB/Ru/NiFe trilayer system, where the ferromagnetic layers are antiferromagnetically coupled through a non-magnetic spacer. Full green diamonds: experimental data obtained for spin waves propagating in the domains. Red dots: experimental data obtained for spin waves confined in the domain wall (red dots). The inset shows an STXM image taken for the data point at 0.52 GHz. The calculated dispersion (blue continuous line) and the results of micromagnetic simulations (gray dots) show good agreement [92].</figcaption></figure>

contribute for the potential applications in ultrafast magnonic devices [96].

(d) *Manipulation and control*: an external magnetic field, charge currents or electrical voltages have already been used to modify or guide spin waves. All-magnon-controlled modifications open a novel route for reconfigurable magnonics [97].

(e) *Nonreciprocal transport*: for the steering of signal flow nonreciprocal propagation characteristics are of great interest. A magnetic nanograting induced already unidirectional flow of magnons with $\lambda$ down to 60 nm. Other experimental techniques are required to precisely control the magnon flow at small $\lambda$ e.g. for diodes.

(f) *Magnon logic computing*: so far, magnonic logic gates have been demonstrated mostly with long-wavelength dipolar spin waves. Major challenges exist in the transition from the single logic gate to cascaded architectures giving rise to integrated magnonic circuits. Interferometers form a key component but have not yet been realized based on phase-controlled ultrashort magnons.

(g) *Nanochannels of low damping*: magnon transport in sub-100 nm wide YIG conduits has been reported only very recently [51]. Decay lengths were reduced by roughly two orders of magnitude compared to thin films [94] but magnons were excited by conventional CPWs and exhibited small group velocities. If combined with e.g.
<figure><img src="image_16.png" /><figcaption>**Figure 13.** (a) Dispersion of short-wavelength spin waves propagating in a 20 nm-thick YIG film. The frequencies and wave vectors of the data points are extracted from the experiments on three different samples with different spin-wave modes. The red curve is the calculated dispersion $f(k)$ with an applied in-plane field of 1000 Oe. The inset shows the transmission spectra of short-wavelength spin waves with $\lambda \approx 50$ nm. Figure dispersion relations for propagating short-wavelength spin waves. Reproduced from [93]. CC BY 4.0. (b) (Left) sketch of micrometer-wide mCPWs consisting of Fe (brown) underneath Au (bright) on YIG. Local detection of magnons was performed through the GGG substrate using BLS. (Right) dispersion relations of magnons are shown for bare YIG (magenta) and close to the mCPW (green). The arrow illustrates the wavelength conversion. Figures schematic diagram and high-frequency modes emitted by mCPWs and wavelength conversion and magnon signal strengths. Reproduced from [89]. CC BY 4.0. (c) Dispersion of spin waves propagating in CoFeB/Ru/NiFe trilayer system, where the ferromagnetic layers are antiferromagnetically coupled through a non-magnetic spacer. Full green diamonds: experimental data obtained for spin waves propagating in the domains. Red dots: experimental data obtained for spin waves confined in the domain wall (red dots). The inset shows an STXM image taken for the data point at 0.52 GHz. The calculated dispersion (blue continuous line) and the results of micromagnetic simulations (gray dots) show good agreement [92].</figcaption></figure>

contribute for the potential applications in ultrafast magnonic devices [96].

(d) *Manipulation and control*: an external magnetic field, charge currents or electrical voltages have already been used to modify or guide spin waves. All-magnon-controlled modifications open a novel route for reconfigurable magnonics [97].

(e) *Nonreciprocal transport*: for the steering of signal flow nonreciprocal propagation characteristics are of great interest. A magnetic nanograting induced already unidirectional flow of magnons with $\lambda$ down to 60 nm. Other experimental techniques are required to precisely control the magnon flow at small $\lambda$ e.g. for diodes.

(f) *Magnon logic computing*: so far, magnonic logic gates have been demonstrated mostly with long-wavelength dipolar spin waves. Major challenges exist in the transition from the single logic gate to cascaded architectures giving rise to integrated magnonic circuits. Interferometers form a key component but have not yet been realized based on phase-controlled ultrashort magnons.

(g) *Nanochannels of low damping*: magnon transport in sub-100 nm wide YIG conduits has been reported only very recently [51]. Decay lengths were reduced by roughly two orders of magnitude compared to thin films [94] but magnons were excited by conventional CPWs and exhibited small group velocities. If combined with e.g.
tailored grating couplers the YIG nanoconduits might support large decay lengths needed for cascaded logic gates. Non-collinear spin structures promising reduced scattering at edges and topologically protected magnon states might give rise to further nanochannels of low damping.

(h) *Quantum effects*: magnons are bosons and room-temperature BEC was realized via magnons (see section 5: ‘magnon-BEC and supercurrent transport’). The coherent coupling between a single magnon in a macroscopic YIG sphere and superconducting qubit in a microwave cavity was already demonstrated and opens the route for investigations on magnonics at the single-quantum level. A magnon-qubit coupling scheme involving a short-wavelength magnon is needed for integrated hybrid quantum systems.

Magnonics is rapidly evolving field and we expect more advanced methods to be proposed and tools to be developed for the coherent excitation and detection of magnons on the nanoscale. These methods are a prerequisite to gain a deep understanding of both the properties and functionalities of exchange-dominated spin waves. The tools will push the frontiers of magnonics and fuel technological prospects.

## Acknowledgments

DG thanks SNSF for financial support via Grants 163016, 177550 and sinergia project 171003 Nanoskyrmionics. HY thanks the financial support from NSF China under Grants No. 11674020 and No. U1801661, and the National Key Research and Development Program of China (254).

# 8. Toward energy efficient magnonics: static and dynamic strain for reconfigurable spin-wave transport

A V Sadovnikov<sup>1,2</sup>, A A Grachev<sup>1</sup>, S E Sheshukova<sup>1</sup>, S A Nikitov<sup>1,2</sup>, J-Y Duquesne<sup>3</sup> and M Marangolo<sup>3</sup>

## 8.1. Status

The electric field controlled transfer of the magnetic moment or spin of an electron instead of charge transport opens up new possibilities to build the element base of devices for encoding and storage of information using the collective precessional motion of ordered magnetic spins-spin waves (SW) with wavelengths ranging from micrometer down to tens of nanometer with the frequencies from gigahertz to terahertz. Emerging field of magnon straintronics [98, 99] opens a promising alternative in beyond-CMOS computing technology with low-level energy consumption and minimization of Joule heating for development of next-generation devices for

sensor applications and energy-saving data processing technologies focused on the generation, propagation, manipulation, channeling and commutation of information carriers.

When strain is applied statically or dynamically to a magnonic device it's possible to generate SW by acoustic waves [100], nonreciprocal propagation of SWs [101], the control of spin-wave generation and routing in lateral array of magnonic stripes and crystals [102]. These phenomena offer new way to envisage spin-wave demultiplexers and can be used simultaneously as an interconnection unit in reconfigurable magnonic networks.

The strain in the magnetic media can be produced locally by static mechanical deformations, local heating or with electric field acting on the one of the phase of the composite magnonic heterostructure which comprises magnetic and piezoelectric phase. The strain-induced physical effects in magnetic structures can be used to engineer energy-efficient complicated 2D and 3D topology of magnonic devices and heterostructures based on multiferroic materials, which exhibit electric field tunability of simultaneous ferroelectric, anti-ferromagnetic (AFM), and ferroelastic properties.

The effect of the electric field on the magnetic configuration results from the modification of the effective internal magnetic field. The latter is changed due to inverse magnetostriction (Villari effect) as a result of the local deformation of the magnetic film. This can induce a FMR frequency shift in YIG/lead zirconate titanate [Pb[Zr<sub>x</sub>Ti<sub>1-x</sub>]O<sub>3</sub> (0 ≤ x ≤ 1) (PZT)] bilayer [103]. The control of the spin-wave transport can be performed via local elastic strains by the fabrication of the system of the electrodes at the YIG/PZT interface.

The magnon straintronic approach can be used to design the magnonic logic devices with voltage control of the spin-wave amplitude and phase. From design point of view this is multistage 3D self-consistent problem. At the first stage, the finite-element calculation of elastic deformations caused by an external electric field in the piezoelectric layer is performed. Next, the obtained profiles of the internal magnetic field in the confined magnonic element are used in the micromagnetic simulation and the eigenmode spin-wave spectra calculation in the magnonic device.

In particular, strain-induced spin-wave routing was demonstrated across three adjacent magnonic stripes, which are strain coupled to a piezoelectric layer (see figure 14(a)). The strain may effectively induce voltage-controlled dipolar spin-wave interactions and results in the electric field induced spin-wave switching in the lateral array of magnonic stripes, which can be probed by means of BLS (figure 15) and microwave spectroscopy.

Strain gives us also a great opportunity to influence dynamically SW transport and even to generate SWs. As anticipated by theoretical research in late 50s the resonant coupling between phonons and magnons in heterostructures composed of piezoelectric and magnetostrictive materials lead to FMR and SW-generation. Indeed, phonons and magnons branches can cross in the (k, ω)-space either at magnetic remanence or when a moderate magnetic field is applied to a ferromagnetic material (FM), leading to modes hybridization engendered by magneto-elastic coupling (MEC) (an example

<footnote n="1"><sup>1</sup>Saratov State University, Russia</footnote>

<footnote n="2"><sup>2</sup>Kotelnikov Institute of Radioengineering and Electronics, Russia</footnote>

<footnote n="3"><sup>3</sup>Sorbonne University, France</footnote>
tailored grating couplers the YIG nanoconductors might support large decay lengths needed for cascaded logic gates. Non-collinear spin structures promising reduced scattering at edges and topologically protected magnon states might give rise to further nanochannels of low damping.

(h) *Quantum effects*: magnons are bosons and room-temperature BEC was realized via magnons (see section 5: ‘magnon-BEC and supercurrent transport’). The coherent coupling between a single magnon in a macroscopic YIG sphere and superconducting qubit in a microwave cavity was already demonstrated and opens the route for investigations on magnonics at the single-quantum level. A magnon-qubit coupling scheme involving a short-wavelength magnon is needed for integrated hybrid quantum systems.

Magnonics is rapidly evolving field and we expect more advanced methods to be proposed and tools to be developed for the coherent excitation and detection of magnons on the nanoscale. These methods are a prerequisite to gain a deep understanding of both the properties and functionalities of exchange-dominated spin waves. The tools will push the frontiers of magnonics and fuel technological prospects.

## Acknowledgments

DG thanks SNSF for financial support via Grants 163016, 177550 and sinergia project 171003 Nanoskyrmionics. HY thanks the financial support from NSF China under Grants No. 11674020 and No. U1801661, and the National Key Research and Development Program of China (254).

# 8. Toward energy efficient magnonics: static and dynamic strain for reconfigurable spin-wave transport

A V Sadovnikov<sup>1,2</sup>, A A Grachev<sup>1</sup>, S E Sheshukova<sup>1</sup>, S A Nikitov<sup>1,2</sup>, J-Y Duquesne<sup>3</sup> and M Marangolo<sup>3</sup>

## 8.1. Status

The electric field controlled transfer of the magnetic moment or spin of an electron instead of charge transport opens up new possibilities to build the element base of devices for encoding and storage of information using the collective precessional motion of ordered magnetic spins-spin waves (SW) with wavelengths ranging from micrometer down to tens of nanometer with the frequencies from gigahertz to terahertz. Emerging field of magnon straintronics [98, 99] opens a promising alternative in beyond-CMOS computing technology with low-level energy consumption and minimization of Joule heating for development of next-generation devices for

sensor applications and energy-saving data processing technologies focused on the generation, propagation, manipulation, channeling and commutation of information carriers.

When strain is applied statically or dynamically to a magnonic device it's possible to generate SW by acoustic waves [100], nonreciprocal propagation of SWs [101], the control of spin-wave generation and routing in lateral array of magnonic stripes and crystals [102]. These phenomena offer new way to envisage spin-wave demultiplexers and can be used simultaneously as an interconnection unit in reconfigurable magnonic networks.

The strain in the magnetic media can be produced locally by static mechanical deformations, local heating or with electric field acting on the one of the phase of the composite magnonic heterostructure which comprises magnetic and piezoelectric phase. The strain-induced physical effects in magnetic structures can be used to engineer energy-efficient complicated 2D and 3D topology of magnonic devices and heterostructures based on multiferroic materials, which exhibit electric field tunability of simultaneous ferroelectric, anti-ferromagnetic (AFM), and ferroelastic properties.

The effect of the electric field on the magnetic configuration results from the modification of the effective internal magnetic field. The latter is changed due to inverse magnetostriction (Villari effect) as a result of the local deformation of the magnetic film. This can induce a FMR frequency shift in YIG/lead zirconate titanate [Pb[Zr<sub>x</sub>Ti<sub>1-x</sub>]O<sub>3</sub> (0 ≤ x ≤ 1) (PZT)] bilayer [103]. The control of the spin-wave transport can be performed via local elastic strains by the fabrication of the system of the electrodes at the YIG/PZT interface.

The magnon straintronic approach can be used to design the magnonic logic devices with voltage control of the spin-wave amplitude and phase. From design point of view this is multistage 3D self-consistent problem. At the first stage, the finite-element calculation of elastic deformations caused by an external electric field in the piezoelectric layer is performed. Next, the obtained profiles of the internal magnetic field in the confined magnonic element are used in the micromagnetic simulation and the eigenmode spin-wave spectra calculation in the magnonic device.

In particular, strain-induced spin-wave routing was demonstrated across three adjacent magnonic stripes, which are strain coupled to a piezoelectric layer (see figure 14(a)). The strain may effectively induce voltage-controlled dipolar spin-wave interactions and results in the electric field induced spin-wave switching in the lateral array of magnonic stripes, which can be probed by means of BLS (figure 15) and microwave spectroscopy.

Strain gives us also a great opportunity to influence dynamically SW transport and even to generate SWs. As anticipated by theoretical research in late 50s the resonant coupling between phonons and magnons in heterostructures composed of piezoelectric and magnetostrictive materials lead to FMR and SW-generation. Indeed, phonons and magnons branches can cross in the (k, ω)-space either at magnetic remanence or when a moderate magnetic field is applied to a ferromagnetic material (FM), leading to modes hybridization engendered by magneto-elastic coupling (MEC) (an example

<footnote n="1"><sup>1</sup>Saratov State University, Russia</footnote>

<footnote n="2"><sup>2</sup>Kotelnikov Institute of Radioengineering and Electronics, Russia</footnote>

<footnote n="3"><sup>3</sup>Sorbonne University, France</footnote>
<figure><img src="image_19.png" /><figcaption>**Figure 14.** The demonstration of magnon straintronics concept: the bilateral YIG stripes ($S_1$, $S_2$, $S_3$) with strain coupled PZT layer (a). The electric voltage is applied between two electrodes ($V_{C1}$ and $V_{C3}$) at the YIG/PZT interface and GND electrode. The distribution of stress tensor component showing a local deformation of the PZT layer and induced stress on the YIG/PZT interface (b). The voltage-induced transformation of the internal magnetic field profile ($H_{int}(x)$) (c). Left panels of (d) show the BLS spin-wave intensity at the frequency of 4.925 GHz at the different values of the applied voltages (denoted in the figure). Right panels of (d) show snapshots of the dynamic out-of-plane component of dynamic magnetization calculated by means of micromagnetic simulations. Edges of stripes are guided with dotted lines. Reprinted (figure) with permission from [99], Copyright (2018) by the American Physical Society.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 15.** Left panel: SAW driven FMR in magnetostrictive element placed between two interdigital transducers (IDTs) on a single crystal piezoelectric substrate lithium niobate (LiNbO₃). Li et al show in reference [100] that the resonant coupling induces SW generation and spin pumping ($J_S$ in the figure is the spin current induced by magnetization precession). Reprinted from [100], with the permission of AIP Publishing. Right panel: dispersion relation of the NiFe/Co magnonic phononic crystal calculated by Graczyk et al in reference [109]. In the color scale, blue and red correspond to acoustic and spin waves, respectively. Green indicates coupled magnetoelastic waves. Reprinted (figure) with permission from [109], Copyright (2017) by the American Physical Society.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 14.** The demonstration of magnon straintronics concept: the bilateral YIG stripes ($S_1$, $S_2$, $S_3$) with strain coupled PZT layer (a). The electric voltage is applied between two electrodes ($V_{C1}$ and $V_{C3}$) at the YIG/PZT interface and GND electrode. The distribution of stress tensor component showing a local deformation of the PZT layer and induced stress on the YIG/PZT interface (b). The voltage-induced transformation of the internal magnetic field profile ($H_{int}(x)$) (c). Left panels of (d) show the BLS spin-wave intensity at the frequency of 4.925 GHz at the different values of the applied voltages (denoted in the figure). Right panels of (d) show snapshots of the dynamic out-of-plane component of dynamic magnetization calculated by means of micromagnetic simulations. Edges of stripes are guided with dotted lines. Reprinted (figure) with permission from [99], Copyright (2018) by the American Physical Society.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 15.** Left panel: SAW driven FMR in magnetostrictive element placed between two interdigital transducers (IDTs) on a single crystal piezoelectric substrate lithium niobate (LiNbO₃). Li et al show in reference [100] that the resonant coupling induces SW generation and spin pumping ($J_S$ in the figure is the spin current induced by magnetization precession). Reprinted from [100], with the permission of AIP Publishing. Right panel: dispersion relation of the NiFe/Co magnonic phononic crystal calculated by Graczyk et al in reference [109]. In the color scale, blue and red correspond to acoustic and spin waves, respectively. Green indicates coupled magnetoelastic waves. Reprinted (figure) with permission from [109], Copyright (2017) by the American Physical Society.</figcaption></figure>
is given in figure 15). The strain control can be imple-
mented locally or globally in a continuous-wave or pulsed
configuration. This interaction may lead to gap opening, fre-
quencies shifts and tunable MC, where periodic modulation of
the magnetic properties occurs at a timescale shorter than the
characteristic time of SWs propagation through the crystal.

Recently, it has been shown that an efficient mean to obtain
resonant MEC is by SAWs, a mature technology that has
proven to tickle magnetization in thin films. By the help of
interdigitated transducers (IDT) exciting SAW in the GHz
and sub-GHz regime in a piezoelectric media (see IDT in
figure 15), Weiler et al [104] and Thevenard et al [105]
observed SAW-FMR in Ni and GaMnAs thin films, respec-
tively. These experiments pave the way to the integration of
SAW-FMR mechanism in spintronic and magnonic devices
where SWs, generated by antennas or by dynamic strain could
be locally handled, triggered, scattered or even suppressed by
SAWs excited, in turn, by remote voltage-driven IDTs.

These SAW-FMR devices are claimed to operate with very
low power, very high tunability and can benefit of the high
directionality of the SW–SAW interaction [106]. Moreover,
since SAWs are widely used in today’s sensors, filters and
microwave circuitry, the possibility to implement magnetiza-
tion dynamics in this mature technology would be a worth-
while asset. An alternative way to excite acoustically SW
by strain has been indicated by Cherepov et al [107]. Here,
an alternating voltage excites strain waves in an multiferroic
magnetoelectric (ME) cell transducers, constituted of a mag-
netostrictive Ni film and a piezoelectric substrate, generating
spin waves.

8.2. Current and future challenges

8.2.1. *Static strain for magnon straintronics.* The electric-field-induced dipolar coupling between magnetic stripes leads to the formation of symmetric and antisymmetric collective modes in the spectra of propagating spin waves in lateral array of MCs, where local strain can tune the frequency of magnonic band gap and control the spin-wave spectrum. This effect can be used as the building block for voltage controlled tunable filters, interferometers, frequency-spatial demultiplexers and directional couplers [102].

The control over spin-wave transport can be also performed
in FM/semiconductor or FM/HM structures [101], where the
strain-induced iDMI gives rise to the nonreciprocal propaga-
tion of SWs. DMI can also induce the magnetic skyrmions,
which are envisioned as ideal candidates as information carri-
ers for future spintronic devices. It was demonstrated that the
uniaxial strain modifies the average DMI constant and intro-
duces anisotropy to the DMI with simultaneous change the
sign of DMI constant for different directions of SW propaga-
tion [101]. Magnon straintronic approach underlie the mech-
anism of the spin-wave scattering and channelling along the
domain walls, since local deformation and electric field local-
ization can be used to control the magnetic topological defects
like nanodomains and skyrmions [108].

8.2.2. *Dynamic strain for magnon straintronics.* The reso-
nant regime of SAW-FMR is usually above 1 GHz and even

below in the case of epitaxied Fe on GaAs [106]. Some points
are worthwhile to be reported: (i) SAW propagate over long
distances in piezoelectric materials, like LiNbO₃ or GaAs,
and magnons for few micrometres in iron and even for few
millimetres in very low-damping YIG. These length scales
are very comfortable for handling these waves in sub-micron
devices. (ii) The penetration depth of an SAW is close to its
wavelength (few microns in the GHz regime) and much larger
than the ferromagnetic film thickness (10–100 nm), but nev-
ertheless efficiency in absorbing the acoustic power is high.
SAW-FMR is efficient enough to enable the excitation of a sin-
gle spin wave mode with an in-plane wave vector *k* matched
to the magnetoelastic wave vector [100].

A smart and elegant way to further enhance
phonon–magnon interaction has been proposed by Graczyk
et al [109] who have shown that in a 1D magnetic periodic
structure (FeNi/Co), the folding back to the first Brillouin
zone, allows for multiple crossings of the spin-wave and
acoustic branches (see right panel in figure 15). This phe-
nomenon accompanied by engineering the SW and acoustic
group velocities provides a powerful tool for SW acoustic
generation.

The interaction of SAW and SW can lead to a non
reciprocal propagation of SAW. This effect is rather small
but developments involving strain-induced iDMI [101, 109]
could greatly enhanced the effect, a step toward practical
non-reciprocal devices (circulators, spin-wave couplers). The
wave nature of SAW and SW also offers the possibility to
modify their frequencies because of inelastic interactions.
Frequencies of SW can be shifted up (down) by the pro-
cess of scattering of on copropagating (counterpropagating)
SAW, i.e. by reverse Doppler effect [109]. This could be
exploited for data processing. SAW can also induce SW
which, in turn, induce spin pumping [100], i.e. spin current
excitation, at a FM/non-magnetic interface (see left panel in
figure 15).

8.3. Concluding remarks

Magnon straintronics excels as an example of powerful
method to manipulate spin-wave transport through either stat-
ically induced reconfigurable magnetization landscape or the
mutual dynamic interaction of acoustic and spin waves in the
GHz and sub-THz. Versatile and energy efficient components
constituted of cascades of frequency-selective all-magnonic
logic units can be envisioned.

Acknowledgments

Work was supported by the Ministry of Education and Science of Russia in the framework of Project No. FSRR-2020-0005. J-Y Duquesne and M Marangolo thank P Rovillain, L Thevenard and C Gourdon four useful discussions.

9. Optical-like processing by magnonic devices

G Csaba<fnref n="1" /> and W Porod<fnref n="2" />

<footnote n="1">¹Pázmány University, Hungary</footnote>

<footnote n="2">²University of Notre Dame, United States of America</footnote>
is given in figure 15). The strain control can be imple-
mented locally or globally in a continuous-wave or pulsed
configuration. This interaction may lead to gap opening, fre-
quencies shifts and tunable MC, where periodic modulation of
the magnetic properties occurs at a timescale shorter than the
characteristic time of SWs propagation through the crystal.

Recently, it has been shown that an efficient mean to obtain
resonant MEC is by SAWs, a mature technology that has
proven to tickle magnetization in thin films. By the help of
interdigitated transducers (IDT) exciting SAW in the GHz
and sub-GHz regime in a piezoelectric media (see IDT in
figure 15), Weiler et al [104] and Thevenard et al [105]
observed SAW-FMR in Ni and GaMnAs thin films, respec-
tively. These experiments pave the way to the integration of
SAW-FMR mechanism in spintronic and magnonic devices
where SWs, generated by antennas or by dynamic strain could
be locally handled, triggered, scattered or even suppressed by
SAWs excited, in turn, by remote voltage-driven IDTs.

These SAW-FMR devices are claimed to operate with very
low power, very high tunability and can benefit of the high
directionality of the SW–SAW interaction [106]. Moreover,
since SAWs are widely used in today’s sensors, filters and
microwave circuitry, the possibility to implement magnetiza-
tion dynamics in this mature technology would be a worth-
while asset. An alternative way to excite acoustically SW
by strain has been indicated by Cherepov et al [107]. Here,
an alternating voltage excites strain waves in an multiferroic
magnetoelectric (ME) cell transducers, constituted of a mag-
netostrictive Ni film and a piezoelectric substrate, generating
spin waves.

8.2. Current and future challenges

8.2.1. *Static strain for magnon straintronics.* The electric-field-induced dipolar coupling between magnetic stripes leads to the formation of symmetric and antisymmetric collective modes in the spectra of propagating spin waves in lateral array of MCs, where local strain can tune the frequency of magnonic band gap and control the spin-wave spectrum. This effect can be used as the building block for voltage controlled tunable filters, interferometers, frequency-spatial demultiplexers and directional couplers [102].

The control over spin-wave transport can be also performed
in FM/semiconductor or FM/HM structures [101], where the
strain-induced iDMI gives rise to the nonreciprocal propaga-
tion of SWs. DMI can also induce the magnetic skyrmions,
which are envisioned as ideal candidates as information carri-
ers for future spintronic devices. It was demonstrated that the
uniaxial strain modifies the average DMI constant and intro-
duces anisotropy to the DMI with simultaneous change the
sign of DMI constant for different directions of SW propaga-
tion [101]. Magnon straintronic approach underlie the mech-
anism of the spin-wave scattering and channelling along the
domain walls, since local deformation and electric field local-
ization can be used to control the magnetic topological defects
like nanodomains and skyrmions [108].

8.2.2. *Dynamic strain for magnon straintronics.* The reso-
nant regime of SAW-FMR is usually above 1 GHz and even

below in the case of epitaxied Fe on GaAs [106]. Some points
are worthwhile to be reported: (i) SAW propagate over long
distances in piezoelectric materials, like LiNbO₃ or GaAs,
and magnons for few micrometres in iron and even for few
millimetres in very low-damping YIG. These length scales
are very comfortable for handling these waves in sub-micron
devices. (ii) The penetration depth of an SAW is close to its
wavelength (few microns in the GHz regime) and much larger
than the ferromagnetic film thickness (10–100 nm), but nev-
ertheless efficiency in absorbing the acoustic power is high.
SAW-FMR is efficient enough to enable the excitation of a sin-
gle spin wave mode with an in-plane wave vector *k* matched
to the magnetoelastic wave vector [100].

A smart and elegant way to further enhance
phonon–magnon interaction has been proposed by Graczyk
et al [109] who have shown that in a 1D magnetic periodic
structure (FeNi/Co), the folding back to the first Brillouin
zone, allows for multiple crossings of the spin-wave and
acoustic branches (see right panel in figure 15). This phe-
nomenon accompanied by engineering the SW and acoustic
group velocities provides a powerful tool for SW acoustic
generation.

The interaction of SAW and SW can lead to a non
reciprocal propagation of SAW. This effect is rather small
but developments involving strain-induced iDMI [101, 109]
could greatly enhanced the effect, a step toward practical
non-reciprocal devices (circulators, spin-wave couplers). The
wave nature of SAW and SW also offers the possibility to
modify their frequencies because of inelastic interactions.
Frequencies of SW can be shifted up (down) by the pro-
cess of scattering of on copropagating (counterpropagating)
SAW, i.e. by reverse Doppler effect [109]. This could be
exploited for data processing. SAW can also induce SW
which, in turn, induce spin pumping [100], i.e. spin current
excitation, at a FM/non-magnetic interface (see left panel in
figure 15).

8.3. Concluding remarks

Magnon straintronics excels as an example of powerful
method to manipulate spin-wave transport through either stat-
ically induced reconfigurable magnetization landscape or the
mutual dynamic interaction of acoustic and spin waves in the
GHz and sub-THz. Versatile and energy efficient components
constituted of cascades of frequency-selective all-magnonic
logic units can be envisioned.

Acknowledgments

Work was supported by the Ministry of Education and Science of Russia in the framework of Project No. FSRR-2020-0005. J-Y Duquesne and M Marangolo thank P Rovillain, L Thevenard and C Gourdon four useful discussions.

**9. Optical-like processing by magnonic devices**

G Csaba<fnref n="1" /> and W Porod<fnref n="2" />

<footnote n="1">¹Pázmány University, Hungary</footnote>

<footnote n="2">²University of Notre Dame, United States of America</footnote>
<figure><img src="image_21.png" /><figcaption>**Figure 16.** Examples of magnon-based optical processing (a) x-ray-based magnetic image of a spin wave lens, realized in a SyAF ([113] John Wiley & Sons. [Copyright © 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]), (b) near-field coupled waveguides in YIG, with frequency-dependent coupling [115] and (c) conceptual view of a Rowland-circle-based microwave spectrum analyser [117].</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 16.** Examples of magnon-based optical processing (a) x-ray-based magnetic image of a spin wave lens, realized in a SyAF ([113] John Wiley & Sons. [Copyright © 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]), (b) near-field coupled waveguides in YIG, with frequency-dependent coupling [115] and (c) conceptual view of a Rowland-circle-based microwave spectrum analyser [117].</figcaption></figure>
## 9.1. Status

Wave phenomena are omnipresent in physics and waves of very different physical origins often show strikingly similar behavior. These similarities perhaps are nowhere more obvious than between magnons and photons: while electromagnetic waves and spin excitations are of fundamentally different physical origin, occur at very different frequency, wavelength, and energy scales, they both display very similar reflection, refraction, and interference effects.

While the possibility of optically-inspired magnonic devices had been proposed and explored in simulations [110], their experimental realization was long hindered by the relatively high damping of metallic ferromagnets. Spin waves in lithographically patterned metallic thin films typically propagate less than ten times of the spin-wave wavelength, limiting the scale and complexity of spin-wave-optics demonstrations. In recent years, convincing experiments were done in relatively simple devices: one such study is the experimental demonstration of Snell's law for magnons [111].

The availability of low-damping YIG films [112] and the development of novel magnetic patterning techniques turned out to be a game-changer and paved the way for a number of very recent spectacular experiments in this area. To give one example, figure 16(a) shows a nanoscale spin-wave lens with textbook-perfect wavefronts mimicking an optical lens [113]. The lens and the magnon-generating structure (i.e. the entire 'optical' setup) is defined by a SyAF that provides local magnetic fields that change the effective index of refraction experienced by the magnons. Similar complex lens-like devices can be realized by locally changing the index of refraction in a YIG layer by local heating, as shown in [114].

Guided-wave magnonic devices are another recent development. For example, near-field optical coupling between YIG waveguides was demonstrated in [115] and a snapshot if this device is shown figure 16(b). In fact, unlike its optical counterpart, this device may function as a logic switch. The weak nonlinearity of spin-wave propagation manifests itself as amplitude-dependent spin-wave wavelength. Since the coupling between the nearby waveguides depends on the wavelength relative to the waveguide length, this results in an intensity-dependent coupling between the couplers. Nonlinearity is not the only plus spin waves posses compared to optical devices: the anisotropic nature of spin waves in in-plane magnetized films results in caustic beams [116] with potential device applications.

Figure 16(c) shows a proposal for an optically inspired device to provide a sought-after signal processing functionality, namely microwave spectrum analysis [117]. It is based on the Rowland-circle spectrometer, an optical device that is well known in x-ray physics. The device has a single waveguide input, generating spin waves on a curved diffraction grating. Different temporal frequency components in the waveguide generate spin waves of correspondingly different wavelength in the magnetic film, which then are focused to the corresponding points on the Rowland circle. This device has the advantage of providing relatively high functionality while utilizing only a single waveguide input. There are other proposals for

achieving high functionality by mimicking optical computing, including wave-based algorithms for NP hard problems [59].

It is probably safe to say that only imagination limits what kind of optically—inspired devices can be realized by magnons. The question is whether these devices will have any practical use besides the fact that they are interesting demonstrations of the photon–magnon analogy. In contrast to photons, magnons are relatively strongly damped, which perhaps is the most important handicap facing optically-inspired magnonic devices. The challenge is to identify possible applications in computing and signal processing where magnonic devices will be competitive so that these ideas do not remain only an academic curiosity.

## 9.2. Current and future challenges

Integrated photonic devices have their own challenges (and they found much fewer applications than integrated electronics). Magnons have obvious advantages and disadvantages compared to photons. The short wavelength of a magnon (potentially going all the way down to few nanometers) allows device densities comparable to deeply scaled electronics, while photonic device scaling stops at 500 nm. Optically-inspired magnonic devices might be a potential solution to one of the most important drawbacks of photonics. Magnon frequency ranges (GHz to THz) also are a very good fit to frequencies in high-speed microelectronic circuits. Magnetic thin films are a planar technology, compatible with microelectronic fabrication. The 'optical' properties of magnetic thin films can be locally modified by several ways (doping, lithography, local magnetic fields, focused ion beams), and some of these techniques offer reconfigurability (i.e., fast re-programming of the function), which is difficult to achieve in optics.

All of the above benefits come with challenges. Short wavelengths are certainly possible in the exchange-dominated spin-wave regime, but such short wavelengths are difficult to probe experimentally, so the vast majority of experiments has been done with dipole-dominated waves, often at much longer-than-optical wavelengths. While metallic ferromagnets can indeed be microelectronics fabrication-friendly, the best known medium for magnon propagation (high-quality YIG) and the highest-quality films are grown on special garnet substrates, while it is a challenge to grow YIG on silicon, but progress is being made in this area [118].

The damping in magnetic materials limits spin-wave-propagation distances to about thousand times the wavelength at best [88]—this likely means that devices (with scattering/refracting structures) are limited to a size of at most a few hundred wavelengths. Damping in photonic systems is not significant so photonic systems are straightforwardly scalable to large sizes. Photonic systems also have high dynamic range, i.e., very low and very high energy photons can be present (and detected) in the same device. This is not the case for spin-waves as high-intensity waves amplitude-dependent (nonlinear) behavior, while low-energy magnons may be lost in the thermal noise. The damping and the low dynamic range of magnons together limit the maximum achievable device size, which is a major challenge.
## 9.1. Status

Wave phenomena are omnipresent in physics and waves of very different physical origins often show strikingly similar behavior. These similarities perhaps are nowhere more obvious than between magnons and photons: while electromagnetic waves and spin excitations are of fundamentally different physical origin, occur at very different frequency, wavelength, and energy scales, they both display very similar reflection, refraction, and interference effects.

While the possibility of optically-inspired magnonic devices had been proposed and explored in simulations [110], their experimental realization was long hindered by the relatively high damping of metallic ferromagnets. Spin waves in lithographically patterned metallic thin films typically propagate less than ten times of the spin-wave wavelength, limiting the scale and complexity of spin-wave-optics demonstrations. In recent years, convincing experiments were done in relatively simple devices: one such study is the experimental demonstration of Snell's law for magnons [111].

The availability of low-damping YIG films [112] and the development of novel magnetic patterning techniques turned out to be a game-changer and paved the way for a number of very recent spectacular experiments in this area. To give one example, figure 16(a) shows a nanoscale spin-wave lens with textbook-perfect wavefronts mimicking an optical lens [113]. The lens and the magnon-generating structure (i.e. the entire 'optical' setup) is defined by a SyAF that provides local magnetic fields that change the effective index of refraction experienced by the magnons. Similar complex lens-like devices can be realized by locally changing the index of refraction in a YIG layer by local heating, as shown in [114].

Guided-wave magnonic devices are another recent development. For example, near-field optical coupling between YIG waveguides was demonstrated in [115] and a snapshot if this device is shown figure 16(b). In fact, unlike its optical counterpart, this device may function as a logic switch. The weak nonlinearity of spin-wave propagation manifests itself as amplitude-dependent spin-wave wavelength. Since the coupling between the nearby waveguides depends on the wavelength relative to the waveguide length, this results in an intensity-dependent coupling between the couplers. Nonlinearity is not the only plus spin waves posses compared to optical devices: the anisotropic nature of spin waves in in-plane magnetized films results in caustic beams [116] with potential device applications.

Figure 16(c) shows a proposal for an optically inspired device to provide a sought-after signal processing functionality, namely microwave spectrum analysis [117]. It is based on the Rowland-circle spectrometer, an optical device that is well known in x-ray physics. The device has a single waveguide input, generating spin waves on a curved diffraction grating. Different temporal frequency components in the waveguide generate spin waves of correspondingly different wavelength in the magnetic film, which then are focused to the corresponding points on the Rowland circle. This device has the advantage of providing relatively high functionality while utilizing only a single waveguide input. There are other proposals for

achieving high functionality by mimicking optical computing, including wave-based algorithms for NP hard problems [59].

It is probably safe to say that only imagination limits what kind of optically—inspired devices can be realized by magnons. The question is whether these devices will have any practical use besides the fact that they are interesting demonstrations of the photon–magnon analogy. In contrast to photons, magnons are relatively strongly damped, which perhaps is the most important handicap facing optically-inspired magnonic devices. The challenge is to identify possible applications in computing and signal processing where magnonic devices will be competitive so that these ideas do not remain only an academic curiosity.

## 9.2. Current and future challenges

Integrated photonic devices have their own challenges (and they found much fewer applications than integrated electronics). Magnons have obvious advantages and disadvantages compared to photons. The short wavelength of a magnon (potentially going all the way down to few nanometers) allows device densities comparable to deeply scaled electronics, while photonic device scaling stops at 500 nm. Optically-inspired magnonic devices might be a potential solution to one of the most important drawbacks of photonics. Magnon frequency ranges (GHz to THz) also are a very good fit to frequencies in high-speed microelectronic circuits. Magnetic thin films are a planar technology, compatible with microelectronic fabrication. The 'optical' properties of magnetic thin films can be locally modified by several ways (doping, lithography, local magnetic fields, focused ion beams), and some of these techniques offer reconfigurability (i.e., fast re-programming of the function), which is difficult to achieve in optics.

All of the above benefits come with challenges. Short wavelengths are certainly possible in the exchange-dominated spin-wave regime, but such short wavelengths are difficult to probe experimentally, so the vast majority of experiments has been done with dipole-dominated waves, often at much longer-than-optical wavelengths. While metallic ferromagnets can indeed be microelectronics fabrication-friendly, the best known medium for magnon propagation (high-quality YIG) and the highest-quality films are grown on special garnet substrates, while it is a challenge to grow YIG on silicon, but progress is being made in this area [118].

The damping in magnetic materials limits spin-wave-propagation distances to about thousand times the wavelength at best [88]—this likely means that devices (with scattering/refracting structures) are limited to a size of at most a few hundred wavelengths. Damping in photonic systems is not significant so photonic systems are straightforwardly scalable to large sizes. Photonic systems also have high dynamic range, i.e., very low and very high energy photons can be present (and detected) in the same device. This is not the case for spin-waves as high-intensity waves amplitude-dependent (nonlinear) behavior, while low-energy magnons may be lost in the thermal noise. The damping and the low dynamic range of magnons together limit the maximum achievable device size, which is a major challenge.
Transmitting and processing signals in the magnonic domain is extremely energy-efficient: magnons propagating in a micron-wide waveguide carry information at a gigahertz bandwidth while using only a few nanowatts of power. However, detecting nanowatt signals at such frequencies requires significant microwave circuitry for amplification and filtering [49], while Johnson–Nyquist noise at room temperature significantly limits data pickup rate. Overall, it requires several milliwatts of electrical power to convert a spin-wave signal to the electrical domain. Inputs are likely to consume the same order of magnitude power.

Magnonic devices, to be competitive, must be scalable to large sizes in order to achieve energy-efficient operation. As discussed above, magnons can propagate at very low power, and the main power requirements are due to generating and detecting magnons. Magnonic devices must be designed to maximize the functionality in the magnon domain and to minimize input and output.

### 9.3. Concluding remarks

The nine orders-of-magnitude energy overhead of magneto-electric interfaces makes it an imperative to maximize the complexity of processing done entirely in the spin wave domain and to minimize the number of inputs and outputs. The challenge is, how to achieve this with the given scalability constraints discussed above. Some of the device constructions (such as the ones described in [116, 117]) may exhibit sufficient internal complexity to overcome the I/O bottleneck. A promising route to increase internal complexity of magnonic devices would be the utilization of efficient magnonic amplifiers—more work is needed in this area.

## Acknowledgments

We are grateful to George Bourianoff (formerly at Intel Corp.) for numerous discussions over the years and also for seed funding for our work. We gratefully acknowledge discussions with Riccardo Bertacco (Politecnico di Milano), Andrii Chumak (Univ. Vienna), and Philipp Pirro (TU Kaiserslautern). We benefited greatly from original research done by Ádam Papp (Pazmany University) and Markus Becherer (TU Munich). This work was partially supported by the DARPA NAC (Nature as Computer) program.

# 10. Pure spin current driven magnonics

VE Demidov<fnref n="1" />, S Urazhdin<fnref n="2" /> and S O Demokritov<fnref n="1" />

## 10.1. Status

Several unique properties of spin waves make them a promising signal carrier for transmission and processing of information at nanoscale. Despite their potential advantages, spin wave-based (magnonic) devices currently suffer from two major drawbacks. First, the inductive mechanism commonly

utilized to convert electrical signals into spin waves, and vice versa, is characterized by relatively low conversion efficiency, especially in microscopic devices. Second, spin waves in microscopic devices based on magnetic films with nanometer-scale thickness are characterized by large propagation losses, resulting in short decay lengths [119]. To make microscopic magnonic devices technologically competitive, it is necessary to overcome these two main drawbacks. This may become possible thanks to the advent of magnetic damping control by pure spin currents in spatially extended regions, which can facilitate the implementation of decay-free propagation of spin waves and even their true amplification. Furthermore, complete compensation of magnetic damping, and the resulting onset of magnetic auto-oscillations, makes it possible to generate coherent spin waves by dc electrical currents, which can emerge as a high-efficiency nanoscale alternative to the traditional inductive excitation mechanism.

Over the last few years, the possibility to compensate spin-wave propagation losses by pure spin currents generated by the SHE was intensively explored in a variety of magnonic systems. The largest effect reported so far was achieved in microscopic magnonic waveguides based on nanometer-thick YIG films [120], where a nearly tenfold increase of spin wave propagation length was demonstrated. Excitation of propagating spin waves by spin currents seems to be a straightforward extension of the damping compensation. However, it turned out to be challenging to implement due to several conflicting requirements. Specifically, efficient generation of current-induced magnetic auto-oscillations requires that the oscillation mode is confined to a nano-scale region. Moreover, since the spin torque effect underlying the spin current-induced dynamics is exerted only at the magnetic interfaces, the thickness of the active magnetic layer should not exceed a few nanometers to maximize the effects of spin current. However, spin waves rapidly decay in such thin magnetic films. Thus, relatively thick active magnetic layers must be utilized to achieve the propagation length scales of several micrometers acceptable for integrated magnonic circuits. For a while, these difficulties did not allow one to achieve spin-wave generation by pure spin-currents, even though the possibility to excite localized coherent dynamics was demonstrated for a variety of device geometries [121–123].

The first demonstration of the excitation of coherent propagating spin waves by pure spin currents was reported in [124]. In this work, we experimentally demonstrated a spin current-driven nanomagnonic system that simultaneously satisfied the conflicting requirements described above. This was accomplished by hybridizing two magnetic subsystems with different dynamic characteristics: the active subsystem in which a spatially confined dynamical mode is excited by the spin current, and the spin-wave guiding subsystem that facilitates efficient propagation of spin waves, as schematically shown in figure 17(a). The active part consists of a nonlocal spin-injection (NLSI) nano-oscillator [122] based on the permalloy (Py) (5 nm)/Cu (20 nm)/CoFe (8 nm) multilayer with a 60 nm circular nanocontact fabricated on the CoFe side. The oscillator exhibits spatially localized spin current-induced auto-oscillations in the Py layer above the nanocontact. To

<footnote n="1">¹University of Muenster, Germany</footnote>

<footnote n="2">²Emory University, United States of America</footnote>
Transmitting and processing signals in the magnonic domain is extremely energy-efficient: magnons propagating in a micron-wide waveguide carry information at a gigahertz bandwidth while using only a few nanowatts of power. However, detecting nanowatt signals at such frequencies requires significant microwave circuitry for amplification and filtering [49], while Johnson–Nyquist noise at room temperature significantly limits data pickup rate. Overall, it requires several milliwatts of electrical power to convert a spin-wave signal to the electrical domain. Inputs are likely to consume the same order of magnitude power.

Magnonic devices, to be competitive, must be scalable to large sizes in order to achieve energy-efficient operation. As discussed above, magnons can propagate at very low power, and the main power requirements are due to generating and detecting magnons. Magnonic devices must be designed to maximize the functionality in the magnon domain and to minimize input and output.

### 9.3. Concluding remarks

The nine orders-of-magnitude energy overhead of magneto-electric interfaces makes it an imperative to maximize the complexity of processing done entirely in the spin wave domain and to minimize the number of inputs and outputs. The challenge is, how to achieve this with the given scalability constraints discussed above. Some of the device constructions (such as the ones described in [116, 117]) may exhibit sufficient internal complexity to overcome the I/O bottleneck. A promising route to increase internal complexity of magnonic devices would be the utilization of efficient magnonic amplifiers—more work is needed in this area.

## Acknowledgments

We are grateful to George Bourianoff (formerly at Intel Corp.) for numerous discussions over the years and also for seed funding for our work. We gratefully acknowledge discussions with Riccardo Bertacco (Politecnico di Milano), Andrii Chumak (Univ. Vienna), and Philipp Pirro (TU Kaiserslautern). We benefited greatly from original research done by Ádam Papp (Pazmany University) and Markus Becherer (TU Munich). This work was partially supported by the DARPA NAC (Nature as Computer) program.

# 10. Pure spin current driven magnonics

VE Demidov<fnref n="1" />, S Urazhdin<fnref n="2" /> and S O Demokritov<fnref n="1" />

## 10.1. Status

Several unique properties of spin waves make them a promising signal carrier for transmission and processing of information at nanoscale. Despite their potential advantages, spin wave-based (magnonic) devices currently suffer from two major drawbacks. First, the inductive mechanism commonly

utilized to convert electrical signals into spin waves, and vice versa, is characterized by relatively low conversion efficiency, especially in microscopic devices. Second, spin waves in microscopic devices based on magnetic films with nanometer-scale thickness are characterized by large propagation losses, resulting in short decay lengths [119]. To make microscopic magnonic devices technologically competitive, it is necessary to overcome these two main drawbacks. This may become possible thanks to the advent of magnetic damping control by pure spin currents in spatially extended regions, which can facilitate the implementation of decay-free propagation of spin waves and even their true amplification. Furthermore, complete compensation of magnetic damping, and the resulting onset of magnetic auto-oscillations, makes it possible to generate coherent spin waves by dc electrical currents, which can emerge as a high-efficiency nanoscale alternative to the traditional inductive excitation mechanism.

Over the last few years, the possibility to compensate spin-wave propagation losses by pure spin currents generated by the SHE was intensively explored in a variety of magnonic systems. The largest effect reported so far was achieved in microscopic magnonic waveguides based on nanometer-thick YIG films [120], where a nearly tenfold increase of spin wave propagation length was demonstrated. Excitation of propagating spin waves by spin currents seems to be a straightforward extension of the damping compensation. However, it turned out to be challenging to implement due to several conflicting requirements. Specifically, efficient generation of current-induced magnetic auto-oscillations requires that the oscillation mode is confined to a nano-scale region. Moreover, since the spin torque effect underlying the spin current-induced dynamics is exerted only at the magnetic interfaces, the thickness of the active magnetic layer should not exceed a few nanometers to maximize the effects of spin current. However, spin waves rapidly decay in such thin magnetic films. Thus, relatively thick active magnetic layers must be utilized to achieve the propagation length scales of several micrometers acceptable for integrated magnonic circuits. For a while, these difficulties did not allow one to achieve spin-wave generation by pure spin-currents, even though the possibility to excite localized coherent dynamics was demonstrated for a variety of device geometries [121–123].

The first demonstration of the excitation of coherent propagating spin waves by pure spin currents was reported in [124]. In this work, we experimentally demonstrated a spin current-driven nanomagnonic system that simultaneously satisfied the conflicting requirements described above. This was accomplished by hybridizing two magnetic subsystems with different dynamic characteristics: the active subsystem in which a spatially confined dynamical mode is excited by the spin current, and the spin-wave guiding subsystem that facilitates efficient propagation of spin waves, as schematically shown in figure 17(a). The active part consists of a nonlocal spin-injection (NLSI) nano-oscillator [122] based on the permalloy (Py) (5 nm)/Cu (20 nm)/CoFe (8 nm) multilayer with a 60 nm circular nanocontact fabricated on the CoFe side. The oscillator exhibits spatially localized spin current-induced auto-oscillations in the Py layer above the nanocontact. To

<footnote n="1">¹University of Muenster, Germany</footnote>

<footnote n="2">²Emory University, United States of America</footnote>
<figure><img src="image_22.png" /><figcaption>**Figure 17.** (a) Layout of the magnonic system enabling excitation of propagating spin waves by pure spin currents generated by the NLSI mechanism. Reproduced from [124]. CC BY 4.0. (b) Color-coded maps demonstrating excitation and propagation of a 3 ns long spin-wave pulse. Reprinted from [125], with the permission of AIP Publishing.</figcaption></figure>

convert these localized magnetization oscillations into a propagating spin wave with a sufficiently large propagation length, a 20 nm-thick and 500 nm-wide Py strip is fabricated on the surface of the extended Py film. The waveguide is terminated at the distance of 150 nm from the center of the nanocontact. This distance is sufficiently small to ensure efficient dynamic coupling between the current-induced magnetic auto-oscillations in the thin film and the magnetization in the strip.

We used imaging of spin waves with sub-micrometer spatial resolution, enabled by micro-focus BLS spectroscopy [119], to show that this system is capable of generation of spin waves in a broad frequency range by the dc driving electric current (figure 17(b)). The generated spin waves exhibit a large decay length of several micrometers, facilitated by the relatively large thickness of the spin-wave guiding subsystem. Moreover, analysis of the power consumption showed

that the power efficiency of these devices is superior to the magnonic systems that utilize traditional inductive spin-wave excitation using microwave currents generated by the external microwave sources. Additionally, the excitation mechanism was found to be intrinsically fast, which allows generation of ultra-short spin-wave packets with the duration of a few nanoseconds [125]. This possibility is particularly important for the implementation of high-speed magnonic circuits.

While NLSI-driven magnonic devices offer numerous benefits, they do not use some of the important advantages that are provided by pure spin currents. In particular, since these devices rely on the local current injection through a nano-contact, they do not allow one to use the same driving current for long-range enhancement of propagation of the generated spin waves, as was demonstrated for spin currents generated by SHE [120]. After an intensive search for a suitable
<figure><img src="image_20.png" /><figcaption>**Figure 17.** (a) Layout of the magnonic system enabling excitation of propagating spin waves by pure spin currents generated by the NLSI mechanism. Reproduced from [124]. CC BY 4.0. (b) Color-coded maps demonstrating excitation and propagation of a 3 ns long spin-wave pulse. Reprinted from [125], with the permission of AIP Publishing.</figcaption></figure>

convert these localized magnetization oscillations into a propagating spin wave with a sufficiently large propagation length, a 20 nm-thick and 500 nm-wide Py strip is fabricated on the surface of the extended Py film. The waveguide is terminated at the distance of 150 nm from the center of the nanocontact. This distance is sufficiently small to ensure efficient dynamic coupling between the current-induced magnetic auto-oscillations in the thin film and the magnetization in the strip.

We used imaging of spin waves with sub-micrometer spatial resolution, enabled by micro-focus BLS spectroscopy [119], to show that this system is capable of generation of spin waves in a broad frequency range by the dc driving electric current (figure 17(b)). The generated spin waves exhibit a large decay length of several micrometers, facilitated by the relatively large thickness of the spin-wave guiding subsystem. Moreover, analysis of the power consumption showed

that the power efficiency of these devices is superior to the magnonic systems that utilize traditional inductive spin-wave excitation using microwave currents generated by the external microwave sources. Additionally, the excitation mechanism was found to be intrinsically fast, which allows generation of ultra-short spin-wave packets with the duration of a few nanoseconds [125]. This possibility is particularly important for the implementation of high-speed magnonic circuits.

While NLSI-driven magnonic devices offer numerous benefits, they do not use some of the important advantages that are provided by pure spin currents. In particular, since these devices rely on the local current injection through a nano-contact, they do not allow one to use the same driving current for long-range enhancement of propagation of the generated spin waves, as was demonstrated for spin currents generated by SHE [120]. After an intensive search for a suitable
<figure><img src="image_23.png" /><figcaption>**Figure 18.** (a) Schematic of the SHE-driven device based on the notched Pt/Py bilayer nano-waveguide. (b) Color-coded spatial maps of spin-wave intensity. Dashed lines on the maps show the outlines of the waveguide and of the nano-notch. [126] John Wiley & Sons. [Copyright © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim].</figcaption></figure>

geometry, in [126], we demonstrated an SHE-driven system enabling excitation of spin waves, and simultaneous enhancement of their propagation by pure spin currents. The system utilized a new concept of nano-notch SHE oscillators directly incorporated into a magnonic nano-waveguide. These devices (figure 18(a)) were based on 180 nm-wide nano-waveguides patterned from a Py (15 nm)/Pt (4 nm) bilayer. Ion milling was used to pattern a rectangular 200 nm-wide and 10 nm-deep notch in the top Py layer of the waveguide. When electric current *I* flows through the device, SHE in Pt injects pure spin current *I*<sub>S</sub> into the Py layer, producing anti-damping spin torque. The thickness-averaged magnitude of this torque is inversely proportional to the thickness of the magnetic layer. Thus, the effects of spin torque on the 5 nm-thick Py layer in the nano-notch area are significantly larger than on the

15 nm-thick Py waveguide. As the current *I* is increased, damping becomes completely compensated in the nano-notch region, resulting in the local excitation of magnetization auto-oscillations serving as a source of propagating spin waves. Simultaneously, the damping is only partially compensated in the rest of the waveguide, resulting in the enhancement of spin-wave propagation.

Micro-focus BLS measurements showed that the auto-
oscillations in the nano-notch can efficiently emit propa-
gating spin waves into the waveguide (figure 18(b)). This
emission was found to be strongly unidirectional, with the
preferential direction controlled by the direction of the static
magnetic field. Additionally, it was shown that the propaga-
tion length of emitted spin waves is enhanced by up to a factor
of three by the spin current injected over the entire length
<figure><img src="image_21.png" /><figcaption>**Figure 18.** (a) Schematic of the SHE-driven device based on the notched Pt/Py bilayer nano-waveguide. (b) Color-coded spatial maps of spin-wave intensity. Dashed lines on the maps show the outlines of the waveguide and of the nano-notch. [126] John Wiley & Sons. [Copyright © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim].</figcaption></figure>

geometry, in [126], we demonstrated an SHE-driven system enabling excitation of spin waves, and simultaneous enhancement of their propagation by pure spin currents. The system utilized a new concept of nano-notch SHE oscillators directly incorporated into a magnonic nano-waveguide. These devices (figure 18(a)) were based on 180 nm-wide nano-waveguides patterned from a Py (15 nm)/Pt (4 nm) bilayer. Ion milling was used to pattern a rectangular 200 nm-wide and 10 nm-deep notch in the top Py layer of the waveguide. When electric current *I* flows through the device, SHE in Pt injects pure spin current *I*<sub>S</sub> into the Py layer, producing anti-damping spin torque. The thickness-averaged magnitude of this torque is inversely proportional to the thickness of the magnetic layer. Thus, the effects of spin torque on the 5 nm-thick Py layer in the nano-notch area are significantly larger than on the

15 nm-thick Py waveguide. As the current *I* is increased, damping becomes completely compensated in the nano-notch region, resulting in the local excitation of magnetization auto-oscillations serving as a source of propagating spin waves. Simultaneously, the damping is only partially compensated in the rest of the waveguide, resulting in the enhancement of spin-wave propagation.

Micro-focus BLS measurements showed that the auto-
oscillations in the nano-notch can efficiently emit propa-
gating spin waves into the waveguide (figure 18(b)). This
emission was found to be strongly unidirectional, with the
preferential direction controlled by the direction of the static
magnetic field. Additionally, it was shown that the propaga-
tion length of emitted spin waves is enhanced by up to a factor
of three by the spin current injected over the entire length
of the waveguide. Therefore, this system combines all the
advantages provided by spin currents to locally excite prop-
agating spin waves, and to simultaneously enhance their prop-
agation characteristics. Additionally, the proposed approach
can be easily scaled to chains of SHE nano-oscillators cou-
pled via propagating spin waves, facilitating the development
of novel nanoscale signal processing circuits such as logic and
neuromorphic computing networks.

10.2. Current and future challenges

One of the main challenges in the ongoing development
of spin current-driven magnonics is associated with the
nonlinear phenomena stimulated by pure spin currents, which
drastically limit the operational regimes and device geome-
tries, as well as the possibilities for device integration. In par-
ticular, nonlinear scattering of propagating spin waves from
magnetic fluctuations enhanced by pure spin current has
been identified as the main mechanism responsible for the
saturation of the anti-damping effect, which has so far pre-
vented the development of devices where the propagation
losses of spin waves are completely compensated [120],
and has imposed strict limitations on the spin current-
driven nano-oscillator geometries [122]. Additionally, the
nonlinear frequency shift of large-amplitude current-induced
auto-oscillations is known to be responsible for the spatial
self-localization of oscillations, preventing the emission of
propagating spin waves [122]. Recently, it was shown that
these limitations can be overcome by using materials with
PMA. PMA controls the nonlinear frequency shift, allow-
ing one to design systems where self-localization is avoided
[127, 128]. Additionally, it was recently shown that by tailor-
ing the PMA strength, one can suppress the nonlinear scatter-
ing [129], which can open the possibility to achieve decay-free
propagation of spin waves, and perhaps even their true
amplification. This approach is particularly promising thanks
to the recent progress in the fabrication of nanometer-
thick low-damping YIG films with PMA, which has already
enabled highly efficient current-driven generation of coherent
propagating spin waves in this archetypal low-loss magnetic
insulator [127].

10.3. Concluding remarks

Pure spin currents offer novel opportunities for the develop-
ment of nano-magnonics, and provide the means to address
the main challenges associated with the downscaling of
magnonic devices, their performance enhancement and inte-
gration. We expect that spin currents will be essential to
making magnonic nano-systems a competitive alternative to
conventional CMOS-based microelectronics.

Acknowledgments

This work was supported in part by the Deutsche Forschungs-
gemeinschaft (Project No. 423113162) and the NSF Grant No.
ECCS-1804198.

11. Interaction of magnons with spin textures

Edoardo Albisetti<fnref n="1" />, Daniela Petti<fnref n="1" />, Riccardo Bertacco<fnref n="1" />,
Helmut Schultheiss<fnref n="2" />,<fnref n="3" />

11.1. Status

Spin textures, such as domain walls, vortices and skyrmions,
are nonuniform configurations in the arrangement of spins
in magnetic materials, which are raising interest as active
components in data storage and signal processing. One of
the most relevant key features, which makes spin textures so
appealing, is the fact that they combine stability and resilience
with a remarkable degree of tunability and scalability toward
nanoscale dimensions. Such features can be the basis for devel-
oping new device concepts where the properties and dimen-
sionality of the spin textures ultimately determine the device
functionality and scalability. Notable examples of applications
are represented by magnetic vortex oscillators, which exploit
the GHz dynamics of nanoscale magnetic vortices for emitting
RF signals when excited by a dc current, or racetrack mem-
ories, where domain walls or skyrmions representing mag-
netic bits are electrically created, displaced and detected within
magnetic nanowires. Recently, the possibility of harnessing the
potential of spin textures in the field of magnonics has sparkled
a new wave of theoretical and experimental efforts, aiming to
study the rich phenomenology of the interaction of spin waves
with spin textures, to demonstrate new device concepts, or to
ultimately overcome some of the long-standing challenges of
the field.

11.2. Current and future challenges

One of such challenges is the efficient spatial confine-
ment and waveguiding in 2D circuits. The conventional
method, based on geometrically patterning magnetic micro
or nanoconduits, has important limitations as it is hardly
scalable and not suitable to realize complex spin config-
urations. For this purpose, the use of spin textures such
as magnetic domain walls and tailored domain structures
represents a promising avenue (see figure 19). In [80] it was
proposed that domain walls in ultrathin films act as magnonic
nano-waveguides, supporting confined modes, called Winter
magnons, which are not present in the bulk of the film, and
which can be guided in curved geometries. The spin-wave
channelling within domain walls was demonstrated experi-
mentally in in-plane magnetized films featuring 180° Néel
domain walls, where spin waves were excited by using an
oscillating RF magnetic field generated by a metallic stripline
[130]. The spin-wave localization at the wall was demon-
strated by acquiring the spatially-resolved map of the spin
wave intensity via BLS, which showed the presence of low-
frequency propagating modes, which are not supported by
uniform domains.

<footnote n="1">¹Polytechnic University of Milan, Italy</footnote>

<footnote n="2">²Helmholtz-Center Dresden–Rossendorf, Institute of Ion
Beam Physics and Materials Research, Germany</footnote>

<footnote n="3">³ Technische Universität Dresden, Germany</footnote>
of the waveguide. Therefore, this system combines all the
advantages provided by spin currents to locally excite prop-
agating spin waves, and to simultaneously enhance their prop-
agation characteristics. Additionally, the proposed approach
can be easily scaled to chains of SHE nano-oscillators cou-
pled via propagating spin waves, facilitating the development
of novel nanoscale signal processing circuits such as logic and
neuromorphic computing networks.

10.2. Current and future challenges

One of the main challenges in the ongoing development
of spin current-driven magnonics is associated with the
nonlinear phenomena stimulated by pure spin currents, which
drastically limit the operational regimes and device geome-
tries, as well as the possibilities for device integration. In par-
ticular, nonlinear scattering of propagating spin waves from
magnetic fluctuations enhanced by pure spin current has
been identified as the main mechanism responsible for the
saturation of the anti-damping effect, which has so far pre-
vented the development of devices where the propagation
losses of spin waves are completely compensated [120],
and has imposed strict limitations on the spin current-
driven nano-oscillator geometries [122]. Additionally, the
nonlinear frequency shift of large-amplitude current-induced
auto-oscillations is known to be responsible for the spatial
self-localization of oscillations, preventing the emission of
propagating spin waves [122]. Recently, it was shown that
these limitations can be overcome by using materials with
PMA. PMA controls the nonlinear frequency shift, allow-
ing one to design systems where self-localization is avoided
[127, 128]. Additionally, it was recently shown that by tailor-
ing the PMA strength, one can suppress the nonlinear scatter-
ing [129], which can open the possibility to achieve decay-free
propagation of spin waves, and perhaps even their true
amplification. This approach is particularly promising thanks
to the recent progress in the fabrication of nanometer-
thick low-damping YIG films with PMA, which has already
enabled highly efficient current-driven generation of coherent
propagating spin waves in this archetypal low-loss magnetic
insulator [127].

10.3. Concluding remarks

Pure spin currents offer novel opportunities for the develop-
ment of nano-magnonics, and provide the means to address
the main challenges associated with the downscaling of
magnonic devices, their performance enhancement and inte-
gration. We expect that spin currents will be essential to
making magnonic nano-systems a competitive alternative to
conventional CMOS-based microelectronics.

Acknowledgments

This work was supported in part by the Deutsche Forschungs-
gemeinschaft (Project No. 423113162) and the NSF Grant No.
ECCS-1804198.

11. Interaction of magnons with spin textures

Edoardo Albisetti¹, Daniela Petti¹, Riccardo Bertacco¹,
Helmut Schultheiss²,³

¹Polytechnic University of Milan, Italy

²Helmholtz-Center Dresden–Rossendorf, Institute of Ion
Beam Physics and Materials Research, Germany

³Technische Universität Dresden, Germany

11.1. Status

Spin textures, such as domain walls, vortices and skyrmions,
are nonuniform configurations in the arrangement of spins
in magnetic materials, which are raising interest as active
components in data storage and signal processing. One of
the most relevant key features, which makes spin textures so
appealing, is the fact that they combine stability and resilience
with a remarkable degree of tunability and scalability toward
nanoscale dimensions. Such features can be the basis for devel-
oping new device concepts where the properties and dimen-
sionality of the spin textures ultimately determine the device
functionality and scalability. Notable examples of applications
are represented by magnetic vortex oscillators, which exploit
the GHz dynamics of nanoscale magnetic vortices for emitting
RF signals when excited by a dc current, or racetrack mem-
ories, where domain walls or skyrmions representing mag-
netic bits are electrically created, displaced and detected within
magnetic nanowires. Recently, the possibility of harnessing the
potential of spin textures in the field of magnonics has sparkled
a new wave of theoretical and experimental efforts, aiming to
study the rich phenomenology of the interaction of spin waves
with spin textures, to demonstrate new device concepts, or to
ultimately overcome some of the long-standing challenges of
the field.

11.2. Current and future challenges

One of such challenges is the efficient spatial confine-
ment and waveguiding in 2D circuits. The conventional
method, based on geometrically patterning magnetic micro
or nanoconduits, has important limitations as it is hardly
scalable and not suitable to realize complex spin config-
urations. For this purpose, the use of spin textures such
as magnetic domain walls and tailored domain structures
represents a promising avenue (see figure 19). In [80] it was
proposed that domain walls in ultrathin films act as magnonic
nano-waveguides, supporting confined modes, called Winter
magnons, which are not present in the bulk of the film, and
which can be guided in curved geometries. The spin-wave
channelling within domain walls was demonstrated experi-
mentally in in-plane magnetized films featuring 180° Néel
domain walls, where spin waves were excited by using an
oscillating RF magnetic field generated by a metallic stripline
[130]. The spin-wave localization at the wall was demon-
strated by acquiring the spatially-resolved map of the spin
wave intensity via BLS, which showed the presence of low-
frequency propagating modes, which are not supported by
uniform domains.
<figure><img src="image_24.png" /><figcaption>**Figure 19.** (a) Simulated spin-wave channelling by means of curved domain walls. In the inset, a cross section of the spin configuration is shown. Reproduced from [80]. CC BY 4.0. (b) Spin-wave intensity measured across a domain wall for different externally applied magnetic fields highlighting the localization of spin waves within the domain wall and the tunability of the channel position. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [130] (2016). (c) Experimental spin-wave spectra measured across nanopatterned stripe domains, showing strong spatial modulation of the spin-wave intensity. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [131] (2016).</figcaption></figure>

Tailored magnetic domains can be also used for spatially controlling the spin-wave excitation and propagation within continuous films. In [131], localized heating with a nanoscopic atomic force microscope tip was used for nanopatterning magnetic domains with tailored shape and magnetization direction in an exchange biased ferromagnetic thin film. This allowed both to spatially modulate the intensity of the spin waves and to select different propagating modes, by exploiting the anisotropic excitation efficiency and dispersion of spin waves with respect to the magnetization direction. Furthermore, the use of periodic arrays of narrow magnetic domains was proposed as the basis for building MCs with tunable bandgap.

One of the most appealing features of employing domains and domain walls as building blocks for magnonic devices is their reconfigurability. In fact, the volatile or non-volatile modulation of the spin textures via external stimuli allows the implementation of a dynamically reconfigurable functionality, such as the precise displacement of domain wall-based waveguides, the real-time modulation of the bandstructure in MCs, or the reversible control of the spin-wave transmission across domain walls [132].

A promising application which exploits these unique capabilities is the realization of reprogrammable spin-wave circuits where propagating spin waves are confined at the nanoscale, steered and let interfere in a controlled way. In [133], it is shown theoretically that it is possible to design a reprogrammable spin-wave circuitry providing unidirectional spin-wave transport using domain walls in presence of DMI. Experimentally, a prototypic nanoscale spin-wave circuit based on spin textures has been demonstrated using nanopatterned Néel domain walls, where the spatial superposition and interference of confined modes was controlled by reconfiguring the waveguides in real-time [24].

The dynamics of spin textures is also interesting for the generation of propagating, short-wavelength spin waves. In fact, as the wavelength approaches sub-micrometric

dimension, conventional generation methods based on microstructured antennas become inefficient. Recently, it was demonstrated that magnetic vortex cores and domain walls emit spin waves when driven in oscillation around their equilibrium position by an RF magnetic field [134] (see figure 20(a)). Since the frequency and wavelength of the emitted spin waves can be easily tuned by controlling the excitation field, this method represents a promising avenue for developing highly flexible nanoscale spin-wave sources. In [113] this concept was used as basis for developing an optically-inspired platform, where magnonic nanoantennas based on domain walls and vortices allow for wavefront engineering, focusing, and interference of short-wavelength spin waves in SyAFs.

Two recent developments in the field of spin-orbit torques (SOT) opened the door to unify the generation-manipulation of spin textures and the excitations of magnons therein in a single device based on a simple bilayer of a HM and a ferromagnet. A dc current injected into such a bilayer proved not only to be an efficient method for initializing domain walls or skyrmions with well-defined chirality [135], but also to drive high frequency magnon auto-oscillations within nano-sized domain walls even in the absence of an externally applied magnetic field [136]. This renders spin textures a powerful nano-interface between electronics and magnonics.

## 11.3. Concluding remarks

A futuristic outlook on applications of spin textures for magnonics is featured by recent reports on simulations of magnon transport in antiferromagnets or more exotic systems such 2D ferromagnets. Antiferromagnets are appealing because they offer a new degree of freedom given by the magnon polarization, where domain walls can act as polarizing elements [137] (see figure 20(b)). Truly 2D ferromagnets offer great potential for magnonics due to protected, non-reciprocal edge states emerging from the topology of the magnonic band
<figure><img src="image_22.png" /><figcaption>**Figure 19.** (a) Simulated spin-wave channelling by means of curved domain walls. In the inset, a cross section of the spin configuration is shown. Reproduced from [80]. CC BY 4.0. (b) Spin-wave intensity measured across a domain wall for different externally applied magnetic fields highlighting the localization of spin waves within the domain wall and the tunability of the channel position. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [130] (2016). (c) Experimental spin-wave spectra measured across nanopatterned stripe domains, showing strong spatial modulation of the spin-wave intensity. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [131] (2016).</figcaption></figure>

Tailored magnetic domains can be also used for spatially controlling the spin-wave excitation and propagation within continuous films. In [131], localized heating with a nanoscopic atomic force microscope tip was used for nanopatterning magnetic domains with tailored shape and magnetization direction in an exchange biased ferromagnetic thin film. This allowed both to spatially modulate the intensity of the spin waves and to select different propagating modes, by exploiting the anisotropic excitation efficiency and dispersion of spin waves with respect to the magnetization direction. Furthermore, the use of periodic arrays of narrow magnetic domains was proposed as the basis for building MCs with tunable bandgap.

One of the most appealing features of employing domains and domain walls as building blocks for magnonic devices is their reconfigurability. In fact, the volatile or non-volatile modulation of the spin textures via external stimuli allows the implementation of a dynamically reconfigurable functionality, such as the precise displacement of domain wall-based waveguides, the real-time modulation of the bandstructure in MCs, or the reversible control of the spin-wave transmission across domain walls [132].

A promising application which exploits these unique capabilities is the realization of reprogrammable spin-wave circuits where propagating spin waves are confined at the nanoscale, steered and let interfere in a controlled way. In [133], it is shown theoretically that it is possible to design a reprogrammable spin-wave circuitry providing unidirectional spin-wave transport using domain walls in presence of DMI. Experimentally, a prototypic nanoscale spin-wave circuit based on spin textures has been demonstrated using nanopatterned Néel domain walls, where the spatial superposition and interference of confined modes was controlled by reconfiguring the waveguides in real-time [24].

The dynamics of spin textures is also interesting for the generation of propagating, short-wavelength spin waves. In fact, as the wavelength approaches sub-micrometric

dimension, conventional generation methods based on microstructured antennas become inefficient. Recently, it was demonstrated that magnetic vortex cores and domain walls emit spin waves when driven in oscillation around their equilibrium position by an RF magnetic field [134] (see figure 20(a)). Since the frequency and wavelength of the emitted spin waves can be easily tuned by controlling the excitation field, this method represents a promising avenue for developing highly flexible nanoscale spin-wave sources. In [113] this concept was used as basis for developing an optically-inspired platform, where magnonic nanoantennas based on domain walls and vortices allow for wavefront engineering, focusing, and interference of short-wavelength spin waves in SyAFs.

Two recent developments in the field of spin-orbit torques (SOT) opened the door to unify the generation-manipulation of spin textures and the excitations of magnons therein in a single device based on a simple bilayer of a HM and a ferromagnet. A dc current injected into such a bilayer proved not only to be an efficient method for initializing domain walls or skyrmions with well-defined chirality [135], but also to drive high frequency magnon auto-oscillations within nano-sized domain walls even in the absence of an externally applied magnetic field [136]. This renders spin textures a powerful nano-interface between electronics and magnonics.

## 11.3. Concluding remarks

A futuristic outlook on applications of spin textures for magnonics is featured by recent reports on simulations of magnon transport in antiferromagnets or more exotic systems such 2D ferromagnets. Antiferromagnets are appealing because they offer a new degree of freedom given by the magnon polarization, where domain walls can act as polarizing elements [137] (see figure 20(b)). Truly 2D ferromagnets offer great potential for magnonics due to protected, non-reciprocal edge states emerging from the topology of the magnonic band
<figure><img src="image_25.png" /><figcaption>**Figure 20.** (a) Experimental x-ray microscopy imaging of the tunable emission of short-wavelength spin waves by vortex cores. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [134] (2016).</figcaption></figure>

(b) Antiferromagnetic domain walls as spin wave polarizer and retarder. Reproduced from [137]. CC BY 4.0. (c) Proposal of topological magnonic circuitry based on the unidirectional propagation of topologically protected spin-wave modes along sample edges and domain walls. Reprinted (figure) with permission from [138], Copyright (2018) by the American Physical Society.

structure [138] (see figure 20(c)). These special magnonic states are the magnonic analogy to surface states in topological insulators and exist even in domain walls. Even though these examples are far from experimental realization they indicate the vast possibilities and phenomena still to be discovered and used in the field of magnonics.

## Acknowledgments

Funding from the European Union's Horizon 2020 research and innovation program is gratefully acknowledged under Grant Agreements Number 705326, project SWING, and number 948225, project B3YOND. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged within program SCHU2922/1-1.

## 12. Graded index magnonics

V V Kruglyak<fnref n="1" /> and V D Poimanov<fnref n="2" />

### 12.1. Status

The topic of wave excitation, propagation, and confinement in graded (i.e. continuously inhomogeneous) media arises naturally for any kind of waves, and spin waves are not an exception. The relevant part of spin wave research is known as ‘graded index magnonics’ [139]. Yet, the concept of a refractive index is not as useful in magnonics as it is e.g. in optics and acoustics. Indeed, thanks to the gapless, linear, and

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

<footnote n="2">²Donetsk National University, Ukraine</footnote>
<figure><img src="image_23.png" /><figcaption>**Figure 20.** (a) Experimental x-ray microscopy imaging of the tunable emission of short-wavelength spin waves by vortex cores. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Nanotechnology] [134] (2016).</figcaption></figure>

(b) Antiferromagnetic domain walls as spin wave polarizer and retarder. Reproduced from [137]. CC BY 4.0. (c) Proposal of topological magnonic circuitry based on the unidirectional propagation of topologically protected spin-wave modes along sample edges and domain walls. Reprinted (figure) with permission from [138], Copyright (2018) by the American Physical Society.

structure [138] (see figure 20(c)). These special magnonic states are the magnonic analogy to surface states in topological insulators and exist even in domain walls. Even though these examples are far from experimental realization they indicate the vast possibilities and phenomena still to be discovered and used in the field of magnonics.

## Acknowledgments

Funding from the European Union's Horizon 2020 research and innovation program is gratefully acknowledged under Grant Agreements Number 705326, project SWING, and number 948225, project B3YOND. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged within program SCHU2922/1-1.

## 12. Graded index magnonics

V V Kruglyak<fnref n="1" /> and V D Poimanov<fnref n="2" />

### 12.1. Status

The topic of wave excitation, propagation, and confinement in graded (i.e. continuously inhomogeneous) media arises naturally for any kind of waves, and spin waves are not an exception. The relevant part of spin wave research is known as ‘graded index magnonics’ [139]. Yet, the concept of a refractive index is not as useful in magnonics as it is e.g. in optics and acoustics. Indeed, thanks to the gapless, linear, and

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

<footnote n="2">²Donetsk National University, Ukraine</footnote>
<figure><img src="image_26.png" /><figcaption>**Figure 21.** Numerically simulated snapshots of the dynamic magnetisation component $m_x$ in units of the saturation magnetisation are shown for a packet of forward volume magnetostatic spin waves propagating through a magnonic Luneburg lens (black circle) for time moments of (a) 16 ns, (b) 45 ns, (c) 80 ns, and (d) 106 ns after excitation. The graded magnonic index profile corresponding to the Luneburg lens is formed by a gradual local increase of the saturation magnetisation in an out-of-plane magnetised thin film of YIG. Reprinted from [146], with the permission of AIP Publishing.</figcaption></figure>

isotropic dispersion relations conventionally found for light and sound, the optical and acoustic refractive indices (defined as ratios of the wave speeds in the reference and graded media) hold for wide ranges of frequencies. In contrast, these properties of the dispersion relation are not usually found for spin waves, and so, the conventional definition of the refractive index has limited value in magnonics. Instead, we use term ‘magnonic index’ for a quantity that scales with the wave number that the spin wave of a given frequency acquires in a given point of space under assumption that it propagates in the same direction in a uniform medium with magnetic properties in that point and under equivalent conditions. The latter must include appropriate static and dynamic magnetic fields, such as the bias and demagnetising fields due to the sample’s structure. Inevitably, the non-locality of the magneto-dipole interaction makes this (and in fact, any) definition of the refractive index for spin waves either limited or ambiguous. Yet, it is often useful in interpretation of experimental and numerical results and when adapting design recipes from e.g. graded index optics.

Schlömann was one of the first to exploit graded profiles of the magnonic index, pointing out that the non-uniform static internal magnetic field can mediate coupling between spin waves and essentially uniform (on the scale of the spin wave wavelength) dynamic magnetic fields [140]. This, now known as ‘Schlömann’ mechanism of spin wave excitation extends to effective magnetic fields of any origin, including not only static but also dynamic non-uniform demagnetising fields [141], inherent to patterned magnetic structures. Morgenthaler was one of the first to apply graded magnonic index to confine [142], to focus [143] and to steer [144] spin waves propagating in bulk YIG samples, which were also considered by Schlömann [140]. Most recently, similar ideas

have been systematically developed to demonstrate spin wave confinement [145], focusing (figure 21 [146]) and steering (figure 22 [147]) by graded profiles of the magnonic index in thin film magnonic structures.

Experimentally, graded index magnonics has been the primary beneficiary from development of time-resolved imaging techniques [147]. Importantly, these were recently extended to spin waves with sub-micrometre wavelengths in low-damping YIG [148]. Theoretically, it has mostly benefitted from existing concepts and methods developed in and then adopted from other areas of physics. For instance, the analogy between the static internal field for exchange spin waves and the potential energy for a quantum-mechanical electron was exploited in references [140, 145] along with the Wentzel–Kramers–Brillouin (WKB) method (quasi-classical approximation). In reference [142], the latter was applied to purely magneto-dipole spin waves. The magnonic Luneburg lens shown in figure 21 [146] is designed using the principles from geometrical optics and ultimately classical mechanics. However, as in the case of the refractive index, such analogies have limited applicability. Hence, numerical micromagnetic simulations have been increasingly often used in graded index magnonics both as a stand-alone tool and as a part of hybrid theoretical analyses [141, 145–147].

Several factors will continue driving research in graded index magnonics in the foreseeable future. Firstly, various magnetic non-uniformities and the associated graded magnonic index are ubiquitous in realistic magnonic and more generally magnetic structures [139, 141, 145, 147], making studies and thorough understanding of the associated dynamics a necessity of high-frequency magnetism and spintronics. The same applies to magnonics, including nearly every topic in this review. Secondly, deliberately created graded magnonic
<figure><img src="image_24.png" /><figcaption>**Figure 21.** Numerically simulated snapshots of the dynamic magnetisation component $m_x$ in units of the saturation magnetisation are shown for a packet of forward volume magnetostatic spin waves propagating through a magnonic Luneburg lens (black circle) for time moments of (a) 16 ns, (b) 45 ns, (c) 80 ns, and (d) 106 ns after excitation. The graded magnonic index profile corresponding to the Luneburg lens is formed by a gradual local increase of the saturation magnetisation in an out-of-plane magnetised thin film of YIG. Reprinted from [146], with the permission of AIP Publishing.</figcaption></figure>

isotropic dispersion relations conventionally found for light and sound, the optical and acoustic refractive indices (defined as ratios of the wave speeds in the reference and graded media) hold for wide ranges of frequencies. In contrast, these properties of the dispersion relation are not usually found for spin waves, and so, the conventional definition of the refractive index has limited value in magnonics. Instead, we use term ‘magnonic index’ for a quantity that scales with the wave number that the spin wave of a given frequency acquires in a given point of space under assumption that it propagates in the same direction in a uniform medium with magnetic properties in that point and under equivalent conditions. The latter must include appropriate static and dynamic magnetic fields, such as the bias and demagnetising fields due to the sample’s structure. Inevitably, the non-locality of the magneto-dipole interaction makes this (and in fact, any) definition of the refractive index for spin waves either limited or ambiguous. Yet, it is often useful in interpretation of experimental and numerical results and when adapting design recipes from e.g. graded index optics.

Schlömann was one of the first to exploit graded profiles of the magnonic index, pointing out that the non-uniform static internal magnetic field can mediate coupling between spin waves and essentially uniform (on the scale of the spin wave wavelength) dynamic magnetic fields [140]. This, now known as ‘Schlömann’ mechanism of spin wave excitation extends to effective magnetic fields of any origin, including not only static but also dynamic non-uniform demagnetising fields [141], inherent to patterned magnetic structures. Morgenthaler was one of the first to apply graded magnonic index to confine [142], to focus [143] and to steer [144] spin waves propagating in bulk YIG samples, which were also considered by Schlömann [140]. Most recently, similar ideas

have been systematically developed to demonstrate spin wave confinement [145], focusing (figure 21 [146]) and steering (figure 22 [147]) by graded profiles of the magnonic index in thin film magnonic structures.

Experimentally, graded index magnonics has been the primary beneficiary from development of time-resolved imaging techniques [147]. Importantly, these were recently extended to spin waves with sub-micrometre wavelengths in low-damping YIG [148]. Theoretically, it has mostly benefitted from existing concepts and methods developed in and then adopted from other areas of physics. For instance, the analogy between the static internal field for exchange spin waves and the potential energy for a quantum-mechanical electron was exploited in references [140, 145] along with the Wentzel–Kramers–Brillouin (WKB) method (quasi-classical approximation). In reference [142], the latter was applied to purely magneto-dipole spin waves. The magnonic Luneburg lens shown in figure 21 [146] is designed using the principles from geometrical optics and ultimately classical mechanics. However, as in the case of the refractive index, such analogies have limited applicability. Hence, numerical micromagnetic simulations have been increasingly often used in graded index magnonics both as a stand-alone tool and as a part of hybrid theoretical analyses [141, 145–147].

Several factors will continue driving research in graded index magnonics in the foreseeable future. Firstly, various magnetic non-uniformities and the associated graded magnonic index are ubiquitous in realistic magnonic and more generally magnetic structures [139, 141, 145, 147], making studies and thorough understanding of the associated dynamics a necessity of high-frequency magnetism and spintronics. The same applies to magnonics, including nearly every topic in this review. Secondly, deliberately created graded magnonic
<figure><img src="image_27.png" /><figcaption>**Figure 22.** Effect of the anisotropy of the magnonic dispersion is illustrated for the case of spin wave propagation in an asymmetrically magnetised permalloy T-junction. (a) The calculated distributions of the static magnetisation (arrows) and the projection of the internal magnetic field onto the magnetisation (color scale) are shown for the magnetic field of $H_B = 500$ Oe applied at 15° to the vertical symmetry axis. Each arrow represents the average over 5 × 5 mesh cells. (b) $k_x$ spectra of the dynamic magnetisation distributions across the leg (gray line in (a)) and along the arms (blue line in (a), amplified ×5) of the T-junction excited at 7.52 GHz are shown by the gray and blue curves, respectively. The $k_x$ value of 0.94 $\mu$m$^{-1}$ indicated by a dashed line corresponds to maximum spin-wave Fourier amplitude. (c) A construction illustrating the extraction of the group velocities of the incident (index 'i') and reflected (index 'r') beams is shown for the white boxed pixel in (a). For the specific $k_x$ value of 0.94 $\mu$m$^{-1}$ (which is indicated by the vertical dashed lines here and in panel (b)), the group velocities are perpendicular to the characteristic isofrequency curves (purple). The $k_y$ value is given by the crossing of the dashed line and the isofrequency curve. Here and in panel (b), the region shaded yellow has the same size and represents the range of forbidden $k_x$ values, as calculated for the white boxed pixel and for pixels on the gray line in (a), respectively. (d) The extracted directional unit vectors of the group velocities and wave vectors are shown for $k_x$ values shown by the vertical dashed lines in (b) and (c). Reprinted (figure) with permission from [147], Copyright (2015) by the American Physical Society.</figcaption></figure>

landscapes could enable some important yet elusive concepts from other areas of physics [146,149,150] to be tested in the laboratory. Finally, innovative graded magnonic profiles will

prove indispensable in eventual practical magnonic devices, either broadening their functionalities or even underpinning new ones.
<figure><img src="image_25.png" /><figcaption>**Figure 22.** Effect of the anisotropy of the magnonic dispersion is illustrated for the case of spin wave propagation in an asymmetrically magnetised permalloy T-junction. (a) The calculated distributions of the static magnetisation (arrows) and the projection of the internal magnetic field onto the magnetisation (color scale) are shown for the magnetic field of $H_B = 500$ Oe applied at 15° to the vertical symmetry axis. Each arrow represents the average over 5 × 5 mesh cells. (b) $k_x$ spectra of the dynamic magnetisation distributions across the leg (gray line in (a)) and along the arms (blue line in (a), amplified ×5) of the T-junction excited at 7.52 GHz are shown by the gray and blue curves, respectively. The $k_x$ value of 0.94 µm$^{-1}$ indicated by a dashed line corresponds to maximum spin-wave Fourier amplitude. (c) A construction illustrating the extraction of the group velocities of the incident (index 'i') and reflected (index 'r') beams is shown for the white boxed pixel in (a). For the specific $k_x$ value of 0.94 µm$^{-1}$ (which is indicated by the vertical dashed lines here and in panel (b)), the group velocities are perpendicular to the characteristic isofrequency curves (purple). The $k_y$ value is given by the crossing of the dashed line and the isofrequency curve. Here and in panel (b), the region shaded yellow has the same size and represents the range of forbidden $k_x$ values, as calculated for the white boxed pixel and for pixels on the gray line in (a), respectively. (d) The extracted directional unit vectors of the group velocities and wave vectors are shown for $k_x$ values shown by the vertical dashed lines in (b) and (c). Reprinted (figure) with permission from [147], Copyright (2015) by the American Physical Society.</figcaption></figure>

landscapes could enable some important yet elusive concepts from other areas of physics [146,149,150] to be tested in the laboratory. Finally, innovative graded magnonic profiles will

prove indispensable in eventual practical magnonic devices, either broadening their functionalities or even underpinning new ones.
## 12.2. Current and future challenges

As no other sub-field of magnonics, experimental studies into graded index magnonics suffer from excessively high magnetic damping. The major challenge is then to learn to create designer magnonic landscapes in low damping magnonic materials. To illustrate this challenge, we consider the realisation of the magnonic Luneburg lens shown in figure 21 [146]. A 2 µm thick film of YIG is used as a magnonic medium in the micromagnetic model, and the required lens is created by modestly increasing the saturation magnetisation to the required profile. This increase would however be difficult to realise experimentally. Indeed, local ion implantation or heating would typically decrease (rather than increase) the magnetisation, while inevitably also increasing the damping. The use of miniaturised magnets to create a required bias field profile or Peltier elements to locally cool the film is probably possible but would obscure the optical access necessary for spin wave imaging. A graded decrease of the film thickness by a factor of $\sqrt{2}$ is an analytically exact solution of the problem [146] but would most likely be challenging technologically for YIG if we were to preserve its low damping.

The results shown in figure 21 were produced in the forward volume magnetostatic spin wave geometry. This geometry in itself is a significant practical limitation due to the need to magnetise the film to saturation normal to its plane. However, the use of magnonic media with in-plane magnetisation raises challenges associated with the anisotropy of the spin wave dispersion, as illustrated in figure 22 [147]. Due to the Schlömann mechanism of spin wave emission, a uniform microwave magnetic field leads to emission of spin waves from the T-junction of two permalloy stripes into the horizontal stripe. For a modest tilt of the in-plane bias magnetic field, the highly non-uniform distributions of the internal magnetic field and magnetisation (figure 22(a)) and the associated non-uniform distribution of the directions of the group velocity vector (figures 22(b) and (c)) lead to steering of the spin wave beam into just one of the two branches of the T-junction (figure 22(d)). Found as a surprise in reference [147], this perfect steering of the spin wave energy illustrates the opportunities that could emerge once we resolve this challenge of an anisotropic magnonic dispersion.

The goal of mastering the spin wave control in graded magnonic media urges us to search for a universal description of their magnonic properties. Indeed, we have already commented on the limitations associated with the use of the refractive index concept [146]. The analogy with the quantum-mechanical electron [140] is also feeble since the concept of the electronic potential breaks beyond the exchange approximation. Even in the exchange approximation, the concept is generally invalidated (as is the concept of the wave impedance) by the ellipticity of the magnetic precession and emergence of evanescent spin wave modes in inhomogeneous anisotropic magnetic media [151]. The concept of the local magnetic susceptibility has not been demonstrated beyond the problem of spin wave excitation [141], while suffering from the nonlocality of the magneto-dipole interaction, similarly to

the refractive index concept and the WKB method for spin waves [142, 145]. The natural abundance of non-reciprocal magnonic phenomena presents even greater and more fundamental challenges for development of a universal formalism for graded index magnonics.

Even when used for approximate or qualitative analyses, theoretical concepts and methods of graded index magnonics need to be developed and applied systematically, so as to account for (rather than to avoid) peculiarities associated with physics of spin waves. This applies to more traditional studies of spin waves in permalloy [141, 145, 147] and YIG [140, 142–144, 146, 148, 150] media as well as to studies expanding the range of addressed phenomena to spin waves and hybrid magnetic excitations (e.g. magneto-acoustic waves and magnetic polaritons) in more exotic material systems. The graded index magnonics of antiferromagnetic media seems particularly challenging, both experimentally and theoretically. Even as far as permalloy and YIG samples are concerned, the physics of nonlinear spin waves in graded media (nonlinear graded index magnonics) is still at its infancy. The fusion of topics of graded magnonic index and MCs [150] and the use of metamaterial approaches to design the graded magnonic index both look very promising yet hinging on our ability to resolve the challenge of the magnetic damping. Finally, we note that the topic of spin waves in samples with non-uniform magnetisation (spin textures) lends itself readily to the field of graded index magnonics [151], see the relevant section of this review.

## 12.3. Concluding remarks

Among its sister-areas of wave physics, magnonics is arguably the most complex and challenging. These challenges culminate in both experimental and theoretical studies of spin waves propagating in inhomogeneous magnetic media—those of graded index magnonics. However, as shown above as well as in other sections of this review, there are also great rewards and rich opportunities for those who will rise to the challenges and master the physics of spin waves propagating in graded magnonic media.

## Acknowledgments

The research leading to these results has received funding from the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom under Grant No. EP/L015331/1 and from the European Union's Horizon 2020 research and innovation program under Marie Skłodowska-Curie Grant No. 644348 (MagIC).

## 13. Spin-orbit effects in magnonics

Anjan Barman<fnref n="1" />, Sourav Sahoo<fnref n="1" /> and Jaivardhan Sinha<fnref n="2" />

<footnote n="1">¹S N Bose National Centre for Basic Sciences, India</footnote>

<footnote n="2">²SRM Institute of Science and Technology, India</footnote>
## 12.2. Current and future challenges

As no other sub-field of magnonics, experimental studies into graded index magnonics suffer from excessively high magnetic damping. The major challenge is then to learn to create designer magnonic landscapes in low damping magnonic materials. To illustrate this challenge, we consider the realisation of the magnonic Luneburg lens shown in figure 21 [146]. A 2 µm thick film of YIG is used as a magnonic medium in the micromagnetic model, and the required lens is created by modestly increasing the saturation magnetisation to the required profile. This increase would however be difficult to realise experimentally. Indeed, local ion implantation or heating would typically decrease (rather than increase) the magnetisation, while inevitably also increasing the damping. The use of miniaturised magnets to create a required bias field profile or Peltier elements to locally cool the film is probably possible but would obscure the optical access necessary for spin wave imaging. A graded decrease of the film thickness by a factor of $\sqrt{2}$ is an analytically exact solution of the problem [146] but would most likely be challenging technologically for YIG if we were to preserve its low damping.

The results shown in figure 21 were produced in the forward volume magnetostatic spin wave geometry. This geometry in itself is a significant practical limitation due to the need to magnetise the film to saturation normal to its plane. However, the use of magnonic media with in-plane magnetisation raises challenges associated with the anisotropy of the spin wave dispersion, as illustrated in figure 22 [147]. Due to the Schlömann mechanism of spin wave emission, a uniform microwave magnetic field leads to emission of spin waves from the T-junction of two permalloy stripes into the horizontal stripe. For a modest tilt of the in-plane bias magnetic field, the highly non-uniform distributions of the internal magnetic field and magnetisation (figure 22(a)) and the associated non-uniform distribution of the directions of the group velocity vector (figures 22(b) and (c)) lead to steering of the spin wave beam into just one of the two branches of the T-junction (figure 22(d)). Found as a surprise in reference [147], this perfect steering of the spin wave energy illustrates the opportunities that could emerge once we resolve this challenge of an anisotropic magnonic dispersion.

The goal of mastering the spin wave control in graded magnonic media urges us to search for a universal description of their magnonic properties. Indeed, we have already commented on the limitations associated with the use of the refractive index concept [146]. The analogy with the quantum-mechanical electron [140] is also feeble since the concept of the electronic potential breaks beyond the exchange approximation. Even in the exchange approximation, the concept is generally invalidated (as is the concept of the wave impedance) by the ellipticity of the magnetic precession and emergence of evanescent spin wave modes in inhomogeneous anisotropic magnetic media [151]. The concept of the local magnetic susceptibility has not been demonstrated beyond the problem of spin wave excitation [141], while suffering from the nonlocality of the magneto-dipole interaction, similarly to

the refractive index concept and the WKB method for spin waves [142, 145]. The natural abundance of non-reciprocal magnonic phenomena presents even greater and more fundamental challenges for development of a universal formalism for graded index magnonics.

Even when used for approximate or qualitative analyses, theoretical concepts and methods of graded index magnonics need to be developed and applied systematically, so as to account for (rather than to avoid) peculiarities associated with physics of spin waves. This applies to more traditional studies of spin waves in permalloy [141, 145, 147] and YIG [140, 142–144, 146, 148, 150] media as well as to studies expanding the range of addressed phenomena to spin waves and hybrid magnetic excitations (e.g. magneto-acoustic waves and magnetic polaritons) in more exotic material systems. The graded index magnonics of antiferromagnetic media seems particularly challenging, both experimentally and theoretically. Even as far as permalloy and YIG samples are concerned, the physics of nonlinear spin waves in graded media (nonlinear graded index magnonics) is still at its infancy. The fusion of topics of graded magnonic index and MCs [150] and the use of metamaterial approaches to design the graded magnonic index both look very promising yet hinging on our ability to resolve the challenge of the magnetic damping. Finally, we note that the topic of spin waves in samples with non-uniform magnetisation (spin textures) lends itself readily to the field of graded index magnonics [151], see the relevant section of this review.

## 12.3. Concluding remarks

Among its sister-areas of wave physics, magnonics is arguably the most complex and challenging. These challenges culminate in both experimental and theoretical studies of spin waves propagating in inhomogeneous magnetic media—those of graded index magnonics. However, as shown above as well as in other sections of this review, there are also great rewards and rich opportunities for those who will rise to the challenges and master the physics of spin waves propagating in graded magnonic media.

## Acknowledgments

The research leading to these results has received funding from the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom under Grant No. EP/L015331/1 and from the European Union's Horizon 2020 research and innovation program under Marie Skłodowska-Curie Grant No. 644348 (MagIC).

## 13. Spin-orbit effects in magnonics

Anjan Barman¹, Sourav Sahoo¹ and Jaivardhan Sinha²

¹S N Bose National Centre for Basic Sciences, India

²SRM Institute of Science and Technology, India
## 13.1. Status

Though the theoretical concepts of SOC and magnons were both formulated long time back, controlling and redesigning the exciting fields of magnonics and spintronics using SOC have drawn attention recently due to its various advantages [152, 153]. Emerging concepts related to the SOC effects are considered crucial for improving the performance of spin-based devices by markedly reducing the power consumption. The SOC effects can be primarily categorized into intrinsic and extrinsic phenomena. While the intrinsic SOC in simplified assumption can be imagined as a local magnetic field related to band structure, the extrinsic SOC is caused by defects and impurities which act as scattering centers [154, 155]. The intriguing phenomena, namely PMA, SHE, Rashba effect, spin pumping and DMI are all known to have origin in SOC [156–158]. In recent times, the experimentally realized SOT originating from pure spin current is expected to play crucial roles in controlling the magnetization as well as spin wave (SW). In comparison to conventional spin-transfer torque, utilizing SOT in magnonic devices is simple and dramatically more efficient. In SOT spin-polarization arises from the carrier velocity difference and a spin filter is not required to generate spin current. The ability of SOT to compensate Gilbert damping over extended regions has opened up a new avenue for on-chip SW communication devices [152, 153]. In order to develop ultrahigh-speed beyond–CMOS SW-based technology for signal processing, it is crucial to achieve long-range enhancement of SW propagation in a variety of magnonic devices. Development of integrated magnonic circuits requires enhanced SW coherence and propagation length. Both these can be addressed by utilizing SOT, which provides the ability to electrically control damping, resulting in significantly enhanced SW propagation. The complete compensation of damping allows coherent local SW generation. Furthermore, enhanced SW transmission can be achieved if the concept of SOT is integrated in the magnonic devices. SOT-driven SWs can be directly applicable to additional nonconventional computing schemes such as neuromorphic-based computing as well as next generation magnetic memory devices with almost unlimited endurance.

SW dispersion in magnonic antidot waveguides has been theoretically investigated in symmetric and asymmetric structures to understand the modification in magnonic band gaps [159]. The effects of intrinsic symmetry breaking factors were found to be compensable by careful adjustment of extrinsic factor. The underlying principle is based upon the translational and mirror symmetries associated with the crystal structure. Thus, the idea of correcting an intrinsic defect by extrinsic means needs experimental realization in MC developed using thin film heterostructures where ferromagnetic thin film layer is placed adjacent to large spin–orbit scattering centers. Further, two-dimensional antidot MCs carved in magnetic multilayers with PMA have shown localized and collective SW excitation, where sharp decrease of SW frequencies is driven by a dynamical coupling between the localized modes within the shells, caused by tunneling and exchange

interactions [160]. The localized shell modes are efficiently tunable by antidot shape through the control of domain structure inside the shells [161].

Observation of chiral spin textures such as spin spirals, helices and skyrmions are mainly related to the stabilization of iDMI in thin film heterostructures where a ferromagnetic layer is placed adjacent to a HM layer or 2D material, e.g. graphene. Chiral MCs, where interplay of SW with DMI can be tailored, have been theoretically predicted to exhibit exceptional features, e.g. ability to transfer energy unidirectionally. Further, dynamical control of magnetic damping by SOT allows tunable control of SW propagation. To this end spin-charge conversion efficiency, namely, spin Hall angle (SHA) plays an important role. An unambiguous technique based on all-optical TR-MOKE have been implemented to estimate SHA in HM/ferromagnet bilayers along with deciphering the effect of spin current generated by SHE and spin pumping [156, 158].

On the other hand, influence of iDMI on the SW dispersion has become a topic of intense interest. The DMI-induced SW asymmetry has been theoretically predicted, which proposed an estimation of the strength of DMI by measuring the frequency difference of counter propagating SWs. In particular, by means of BLS, nonreciprocal SWs were observed in HM/ferromagnet thin film heterostructures [157]. The frequency asymmetry in the SW dispersion of counter propagating Damon–Eshbach (DE) SWs was confirmed in various combinations of heterostructures due to the presence of iDMI. Furthermore, it was also predicted that in systems with bulk DMI a frequency difference could be observed in the backward-volume geometry. Some of the chiral spin textures have manifested this effect [152].

## 13.2. Current and future challenges

The current and future challenges can be divided into two classes: (a) scientific challenges and (b) technical challenges. The overriding goal of the scientific challenges is to combine a number of spin–orbit effects with the MC and subsequently to the magnonic devices. For example, simultaneous existence of PMA and DMI as well as patterning of two-dimensional MC can help in stabilizing topological magnetic objects like skyrmions or merons and more efficient control of their dynamics. Development of one- and two-dimensional MCs on FM/NM heterostructures with iDMI will lead to novel magnonic band structures with the evolution of new bands and bandgaps due to the asymmetric DE SW dispersion in presence of iDMI (an example of SW dispersion in a one-dimensional magnetic antidot waveguide without and with DMI is shown in figures 23(a)–(c)).

The wave-number dependent skew scattering of magnons from skyrmions gives rise to the topological magnon Hall effect, while interaction of skyrmions with current flow leads to skymion Hall effect. However, a combined effect of current flow and magnon on skyrmions may lead to novel effects due to the competition between thermally-driven radial magnon current, transverse magnon current and electron flow. Scattering of magnons from a skyrmion lattice can open
## 13.1. Status

Though the theoretical concepts of SOC and magnons were both formulated long time back, controlling and redesigning the exciting fields of magnonics and spintronics using SOC have drawn attention recently due to its various advantages [152, 153]. Emerging concepts related to the SOC effects are considered crucial for improving the performance of spin-based devices by markedly reducing the power consumption. The SOC effects can be primarily categorized into intrinsic and extrinsic phenomena. While the intrinsic SOC in simplified assumption can be imagined as a local magnetic field related to band structure, the extrinsic SOC is caused by defects and impurities which act as scattering centers [154, 155]. The intriguing phenomena, namely PMA, SHE, Rashba effect, spin pumping and DMI are all known to have origin in SOC [156–158]. In recent times, the experimentally realized SOT originating from pure spin current is expected to play crucial roles in controlling the magnetization as well as spin wave (SW). In comparison to conventional spin-transfer torque, utilizing SOT in magnonic devices is simple and dramatically more efficient. In SOT spin-polarization arises from the carrier velocity difference and a spin filter is not required to generate spin current. The ability of SOT to compensate Gilbert damping over extended regions has opened up a new avenue for on-chip SW communication devices [152, 153]. In order to develop ultrahigh-speed beyond-CMOS SW-based technology for signal processing, it is crucial to achieve long-range enhancement of SW propagation in a variety of magnonic devices. Development of integrated magnonic circuits requires enhanced SW coherence and propagation length. Both these can be addressed by utilizing SOT, which provides the ability to electrically control damping, resulting in significantly enhanced SW propagation. The complete compensation of damping allows coherent local SW generation. Furthermore, enhanced SW transmission can be achieved if the concept of SOT is integrated in the magnonic devices. SOT-driven SWs can be directly applicable to additional nonconventional computing schemes such as neuromorphic-based computing as well as next generation magnetic memory devices with almost unlimited endurance.

SW dispersion in magnonic antidot waveguides has been theoretically investigated in symmetric and asymmetric structures to understand the modification in magnonic band gaps [159]. The effects of intrinsic symmetry breaking factors were found to be compensable by careful adjustment of extrinsic factor. The underlying principle is based upon the translational and mirror symmetries associated with the crystal structure. Thus, the idea of correcting an intrinsic defect by extrinsic means needs experimental realization in MC developed using thin film heterostructures where ferromagnetic thin film layer is placed adjacent to large spin-orbit scattering centers. Further, two-dimensional antidot MCs carved in magnetic multilayers with PMA have shown localized and collective SW excitation, where sharp decrease of SW frequencies is driven by a dynamical coupling between the localized modes within the shells, caused by tunneling and exchange

interactions [160]. The localized shell modes are efficiently tunable by antidot shape through the control of domain structure inside the shells [161].

Observation of chiral spin textures such as spin spirals, helices and skyrmions are mainly related to the stabilization of iDMI in thin film heterostructures where a ferromagnetic layer is placed adjacent to a HM layer or 2D material, e.g. graphene. Chiral MCs, where interplay of SW with DMI can be tailored, have been theoretically predicted to exhibit exceptional features, e.g. ability to transfer energy unidirectionally. Further, dynamical control of magnetic damping by SOT allows tunable control of SW propagation. To this end spin-charge conversion efficiency, namely, spin Hall angle (SHA) plays an important role. An unambiguous technique based on all-optical TR-MOKE have been implemented to estimate SHA in HM/ferromagnet bilayers along with deciphering the effect of spin current generated by SHE and spin pumping [156, 158].

On the other hand, influence of iDMI on the SW dispersion has become a topic of intense interest. The DMI-induced SW asymmetry has been theoretically predicted, which proposed an estimation of the strength of DMI by measuring the frequency difference of counter propagating SWs. In particular, by means of BLS, nonreciprocal SWs were observed in HM/ferromagnet thin film heterostructures [157]. The frequency asymmetry in the SW dispersion of counter propagating Damon-Eshbach (DE) SWs was confirmed in various combinations of heterostructures due to the presence of iDMI. Furthermore, it was also predicted that in systems with bulk DMI a frequency difference could be observed in the backward-volume geometry. Some of the chiral spin textures have manifested this effect [152].

## 13.2. Current and future challenges

The current and future challenges can be divided into two classes: (a) scientific challenges and (b) technical challenges. The overriding goal of the scientific challenges is to combine a number of spin-orbit effects with the MC and subsequently to the magnonic devices. For example, simultaneous existence of PMA and DMI as well as patterning of two-dimensional MC can help in stabilizing topological magnetic objects like skyrmions or merons and more efficient control of their dynamics. Development of one- and two-dimensional MCs on FM/NM heterostructures with iDMI will lead to novel magnonic band structures with the evolution of new bands and bandgaps due to the asymmetric DE SW dispersion in presence of iDMI (an example of SW dispersion in a one-dimensional magnetic antidot waveguide without and with DMI is shown in figures 23(a)–(c)).

The wave-number dependent skew scattering of magnons from skyrmions gives rise to the topological magnon Hall effect, while interaction of skyrmions with current flow leads to skymion Hall effect. However, a combined effect of current flow and magnon on skyrmions may lead to novel effects due to the competition between thermally-driven radial magnon current, transverse magnon current and electron flow. Scattering of magnons from a skyrmion lattice can open
<figure><img src="image_28.png" /><figcaption>**Figure 23.** (a) Simulated spin-wave dispersion curves for a one-dimensional Ni₈₀Fe₂₀ antidot waveguide with 3 µm length and other dimensions as shown in the schematic at the inset. The spin-wave dispersion is calculated in the DE geometry in presence of a bias magnetic field of 1.01 T. Further application of iDMI strength of (b) *D* = −2 mJ m⁻² and (c) *D* = 2 mJ m⁻² have dramatically modified the spin-wave dispersion curves. (d) Application of a current density of 1 × 10¹¹ A m⁻² on (c) has reduced the spin-wave amplitude significantly. The color map of the spin-wave dispersion is jet, i.e. red corresponds to maximum and blue corresponds to minimum spin-wave amplitude.</figcaption></figure>

magnonic bandgap, leading toward dynamic MC, which can be continuously tuned by a dc magnetic field. One of the primary bottlenecks of magnonics is the large damping of SWs in metallic ferromagnets which limits the SW propagation distance and device efficiency. Antidamping properties of SOT may dramatically reduce damping, but it has not been investigated extensively in MCs. Spin current may be used globally or locally to selectively amplify or attenuate propagating SWs (an example is shown by numerical simulation in figure 23(d), where SW amplitude has been dramatically reduced by application of spin current) leading toward development of magnon logic or transistor. Finally, controlled magnon auto-oscillation at zero magnetic field can lead to extremely energy-efficient magnonic devices.

The technological challenges primarily concern about the fabrication of devices, precision measurements and their application. The advancement of devices hinges on the capability to downscale the individual circuit elements. A conceptual magnonic device in figure 24 shows the excitation can be done by various means such as microwave antenna, voltage controlled magnetic anisotropy, SOT by SHE or spin pumping, all of those throw challenges in downscaling the excitation element by keeping the relevant property intact. The fabrication

<figure><img src="image_29.png" /><figcaption>**Figure 24.** Schematic of a conceptual magnonic device composed of antidots on magnetic thin film heterostructures having different SOC related properties with electrical excitation and optical as well as electrical detection technique.</figcaption></figure>

of nanoscale MCs by electron-beam lithography or focused ion-beam milling on thin film heterostructures by keeping the spin-orbit effects intact is also a great challenge [160, 161]
<figure><img src="image_26.png" /><figcaption>**Figure 23.** (a) Simulated spin-wave dispersion curves for a one-dimensional Ni₈₀Fe₂₀ antidot waveguide with 3 µm length and other dimensions as shown in the schematic at the inset. The spin-wave dispersion is calculated in the DE geometry in presence of a bias magnetic field of 1.01 T. Further application of iDMI strength of (b) *D* = −2 mJ m⁻² and (c) *D* = 2 mJ m⁻² have dramatically modified the spin-wave dispersion curves. (d) Application of a current density of 1 × 10¹¹ A m⁻² on (c) has reduced the spin-wave amplitude significantly. The color map of the spin-wave dispersion is jet, i.e. red corresponds to maximum and blue corresponds to minimum spin-wave amplitude.</figcaption></figure>

magnonic bandgap, leading toward dynamic MC, which can be continuously tuned by a dc magnetic field. One of the primary bottlenecks of magnonics is the large damping of SWs in metallic ferromagnets which limits the SW propagation distance and device efficiency. Antidamping properties of SOT may dramatically reduce damping, but it has not been investigated extensively in MCs. Spin current may be used globally or locally to selectively amplify or attenuate propagating SWs (an example is shown by numerical simulation in figure 23(d), where SW amplitude has been dramatically reduced by application of spin current) leading toward development of magnon logic or transistor. Finally, controlled magnon auto-oscillation at zero magnetic field can lead to extremely energy-efficient magnonic devices.

The technological challenges primarily concern about the fabrication of devices, precision measurements and their application. The advancement of devices hinges on the capability to downscale the individual circuit elements. A conceptual magnonic device in figure 24 shows the excitation can be done by various means such as microwave antenna, voltage controlled magnetic anisotropy, SOT by SHE or spin pumping, all of those throw challenges in downscaling the excitation element by keeping the relevant property intact. The fabrication

<figure><img src="image_27.png" /><figcaption>**Figure 24.** Schematic of a conceptual magnonic device composed of antidots on magnetic thin film heterostructures having different SOC related properties with electrical excitation and optical as well as electrical detection technique.</figcaption></figure>

of nanoscale MCs by electron-beam lithography or focused ion-beam milling on thin film heterostructures by keeping the spin-orbit effects intact is also a great challenge [160, 161]
and extensive research is required to overcome this. To this end topological magnetic texture-based dynamic MC originated from spin-orbit effects could be a solution but controlling ordered arrays of such textures over large area would be challenging. The electrical detection based on inductive techniques or inverse SHE faces similar challenges. Alternative detections like optical, x-ray or electron microscope are useful but not viable for practical application. Furthermore, given the intriguing physics involved in the three-dimensional MCs, it will be important to understand the spin-orbit effects on magnons in such systems for their applications in magnonic devices.

### 13.3. Concluding remarks

The ability to implement spin-orbit effects to modify the magnon properties in tailored magnetic materials holds the key for future energy-efficient high-frequency nanoelectronic devices. These will be the subject of future investigation from fundamental interests as the complexities involved need further in-depth exploration using experimental and theoretical studies. We envision the possibility of new information processing devices in the future based on new classes of MC where spin-orbit effects will be crucial ingredients.

## 14. THz magnonics

Hyunsoo Yang<fnref n="1" /> and Markus Münzenberg<fnref n="2" />

### 14.1. Status

Achieving a faster and most energy efficient data processing has been a major task in modern spintronics. To bring magnonics to THz frequency scale, we need three things: THz excitation, as an ultrafast signal input, THz frequency spin-excitation and THz detection of spin-wave signals. From early on, ultrafast spin-dynamics aimed to develop picosecond stimuli by using Auston switches or femtosecond laser pulses for excitation directly. In early ultrafast demagnetization in ferromagnets, the role of THz high-energy-spin waves on the 100 fs to ps time scales driven by infrared femtosecond lasers was clear. However, they could not be exploited so far to generate high energy spin waves in the THz frequency spectrum directly by the laser pulse heating. A breakthrough on THz stimuli was the generation of THz current and field pulses via laser driven picosecond charge and spin-currents. One such central scheme is to utilize spin to charge conversion. The possibility of laser-induced spin to charge conversion at THz frequency was first demonstrated in 2013 [162]. In ferromagnet (FM)/HM heterostructures, laser excitation lifts the electrons in the ferromagnet by the photon energy into different states in the spin-split band structure, thereby generating a non-equilibrium electron distribution. The laser-excited majority spin electrons have a mainly sp-like character and much higher velocity than the excited minority spin electrons with a d-type character. Subsequently, the diffusion of spin current toward

the adjacent HM layer results in a charge current generation due to the spin to charge conversion effect such as the inverse spin Hall and inverse Rashba-Edelstein effects. The ultrafast transient charge current thus induces a THz electromagnetic radiation as schematically shown in figure 25(a). Three independent groups recently reported using optimized emitter structures that the intensity of the THz radiation competes with standard THz emitters [163–165]. The emitter spectrum of the spintronic THz emitter is compared to a commercial interdigitated GaAs switch and a ZnTe emitter. The bandwidth is much larger covering the low frequency (0.1–3 THz) as well as high frequency up to 30 THz without any gap. This indicates that at the same time shorter current pulses, with ultrafast rise times of ~0.1 ps can be generated. Such spin-based THz emitters are insensitive to the polarization and helicity of an incident laser beam which indicates the noise resistive feature. In contrast, the amplitude and polarization of THz waves are fully controllable by an external magnetic field. Spin-based THz emitters can be fabricated on flexible substrates, and driven by a fiber based low power laser [164]. Together with the low cost and mass productive sputtering growth method for the magnetic heterostructure stacks, the spin-based THz emitters can be readily applied to a wide range of THz applications. This opens up a route not only to novel THz emitters, but also potential spintronic devices manipulating spin current bursts on a THz timescale.

It was demonstrated that THz spin waves can be directly excited by the laser driven spin-polarized current pulses in Fe/Au/Fe multilayers [166]. The authors have shown that standing spin wave modes can be excited in a 15 nm thick Fe film due to ultrashort laser-induced spin current bunches as shown in figure 25(b). Laser pump pulse impinging on the first Fe layer (emitter) excites hot electrons at elevated energies. Due to the unequal transmittance of the Fe/Au interface for the majority (green) and minority (red) hot electrons, spin polarized hot electrons are emitted into Au. The spin polarized electrons reach the second ferromagnet (FM2) ballistically and the injected spin current absorbed on a length scale of well below 2 nm allows an asymmetric spin dynamics excitation due to the spin transfer torque effect. Here spin waves for 0.1–0.6 THz are excited and probed by the Kerr rotation. Those THz spin-wave generators could be used to inject THz frequency spin waves into magnonic structures and devices in the future. This is an important step toward increasing the speed and minimization of the device size for possible magnonic applications. In the following we will discuss and compare the three ways THz frequency spin-excitation in magnetically ordered system: (i) homogeneous Kittel modes $k = 0$ excited off resonant, in large applied or internal anisotropy fields, (ii) spin waves with $k \neq 0$ with strong exchange fields, found for standing spin waves and (iii) strong exchange fields in antiferromagnets and ferrimagnets.

For ferromagnetic systems, typical spin dynamics is in the GHz range which increases with the applied magnetic field. Resonances of the Kittel modes may reach THz frequencies with 20 T, posing a serious challenge without special high field facilities. However, if strong internal fields are generated by a high anisotropy or the exchange-interaction, one can reach the

<footnote n="1">¹National University of Singapore, Singapore</footnote>

<footnote n="2">²University of Greifswald, Germany</footnote>
and extensive research is required to overcome this. To this end topological magnetic texture-based dynamic MC originated from spin-orbit effects could be a solution but controlling ordered arrays of such textures over large area would be challenging. The electrical detection based on inductive techniques or inverse SHE faces similar challenges. Alternative detections like optical, x-ray or electron microscope are useful but not viable for practical application. Furthermore, given the intriguing physics involved in the three-dimensional MCs, it will be important to understand the spin-orbit effects on magnons in such systems for their applications in magnonic devices.

### 13.3. Concluding remarks

The ability to implement spin-orbit effects to modify the magnon properties in tailored magnetic materials holds the key for future energy-efficient high-frequency nanoelectronic devices. These will be the subject of future investigation from fundamental interests as the complexities involved need further in-depth exploration using experimental and theoretical studies. We envision the possibility of new information processing devices in the future based on new classes of MC where spin-orbit effects will be crucial ingredients.

## 14. THz magnonics

Hyunsoo Yang<fnref n="1" /> and Markus Münzenberg<fnref n="2" />

### 14.1. Status

Achieving a faster and most energy efficient data processing has been a major task in modern spintronics. To bring magnonics to THz frequency scale, we need three things: THz excitation, as an ultrafast signal input, THz frequency spin-excitation and THz detection of spin-wave signals. From early on, ultrafast spin-dynamics aimed to develop picosecond stimuli by using Auston switches or femtosecond laser pulses for excitation directly. In early ultrafast demagnetization in ferromagnets, the role of THz high-energy-spin waves on the 100 fs to ps time scales driven by infrared femtosecond lasers was clear. However, they could not be exploited so far to generate high energy spin waves in the THz frequency spectrum directly by the laser pulse heating. A breakthrough on THz stimuli was the generation of THz current and field pulses via laser driven picosecond charge and spin-currents. One such central scheme is to utilize spin to charge conversion. The possibility of laser-induced spin to charge conversion at THz frequency was first demonstrated in 2013 [162]. In ferromagnet (FM)/HM heterostructures, laser excitation lifts the electrons in the ferromagnet by the photon energy into different states in the spin-split band structure, thereby generating a non-equilibrium electron distribution. The laser-excited majority spin electrons have a mainly sp-like character and much higher velocity than the excited minority spin electrons with a d-type character. Subsequently, the diffusion of spin current toward

the adjacent HM layer results in a charge current generation due to the spin to charge conversion effect such as the inverse spin Hall and inverse Rashba-Edelstein effects. The ultrafast transient charge current thus induces a THz electromagnetic radiation as schematically shown in figure 25(a). Three independent groups recently reported using optimized emitter structures that the intensity of the THz radiation competes with standard THz emitters [163–165]. The emitter spectrum of the spintronic THz emitter is compared to a commercial interdigitated GaAs switch and a ZnTe emitter. The bandwidth is much larger covering the low frequency (0.1–3 THz) as well as high frequency up to 30 THz without any gap. This indicates that at the same time shorter current pulses, with ultrafast rise times of ~0.1 ps can be generated. Such spin-based THz emitters are insensitive to the polarization and helicity of an incident laser beam which indicates the noise resistive feature. In contrast, the amplitude and polarization of THz waves are fully controllable by an external magnetic field. Spin-based THz emitters can be fabricated on flexible substrates, and driven by a fiber based low power laser [164]. Together with the low cost and mass productive sputtering growth method for the magnetic heterostructure stacks, the spin-based THz emitters can be readily applied to a wide range of THz applications. This opens up a route not only to novel THz emitters, but also potential spintronic devices manipulating spin current bursts on a THz timescale.

It was demonstrated that THz spin waves can be directly excited by the laser driven spin-polarized current pulses in Fe/Au/Fe multilayers [166]. The authors have shown that standing spin wave modes can be excited in a 15 nm thick Fe film due to ultrashort laser-induced spin current bunches as shown in figure 25(b). Laser pump pulse impinging on the first Fe layer (emitter) excites hot electrons at elevated energies. Due to the unequal transmittance of the Fe/Au interface for the majority (green) and minority (red) hot electrons, spin polarized hot electrons are emitted into Au. The spin polarized electrons reach the second ferromagnet (FM2) ballistically and the injected spin current absorbed on a length scale of well below 2 nm allows an asymmetric spin dynamics excitation due to the spin transfer torque effect. Here spin waves for 0.1–0.6 THz are excited and probed by the Kerr rotation. Those THz spin-wave generators could be used to inject THz frequency spin waves into magnonic structures and devices in the future. This is an important step toward increasing the speed and minimization of the device size for possible magnonic applications. In the following we will discuss and compare the three ways THz frequency spin-excitation in magnetically ordered system: (i) homogeneous Kittel modes $k = 0$ excited off resonant, in large applied or internal anisotropy fields, (ii) spin waves with $k \neq 0$ with strong exchange fields, found for standing spin waves and (iii) strong exchange fields in antiferromagnets and ferrimagnets.

For ferromagnetic systems, typical spin dynamics is in the GHz range which increases with the applied magnetic field. Resonances of the Kittel modes may reach THz frequencies with 20 T, posing a serious challenge without special high field facilities. However, if strong internal fields are generated by a high anisotropy or the exchange-interaction, one can reach the

<footnote n="1">¹National University of Singapore, Singapore</footnote>

<footnote n="2">²University of Greifswald, Germany</footnote>
<figure><img src="image_30.png" /><figcaption>**Figure 25.** (a) A typical ferromagnet (FM)/non-magnet (NM) sample structure and the THz emission geometry. The femtosecond laser beam is used for pumping the sample. The external magnetic field ($H_{ext}$) is along the x-axis. In (b) and (c) two strategies are shown to excite spin waves using THz pulses in the THz frequency range. (b) Laser-induced excitation of spin dynamics via spin currents pulses. Due to the spatial confinement of the STT perturbation (blue shaded area), spin waves with a broad spectrum of nonzero wavevectors are excited and can be probed by the MOKE in the collector. The exchange standing spin waves are confined in nanometer thick layer, where the frequency increase as ~$k^2$. Reproduced from [165]. **CC BY 4.0.** (c) Crystal lattice of NiO with antiferromagnetically ordered spins (blue arrows) in the (111) planes (light blue) and the direction of the THz magnetic field *B* (double-ended red arrow). An intense free-space THz interacts with the electron spins of a sample to launch a coherent magnon wave. Here the intrinsic modes are in the THz frequency range. A femtosecond near-infrared (NIR) probe pulse (green curves) measures the induced net magnetization by the Faraday effect. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Photonics] [165] (2011).</figcaption></figure>

THz frequency branches of the spin-wave dispersion. In case of ferromagnetic systems, the strategy can be to decrease the spacing of a MC or waveguide to the ~nm length scale and thereby to increase the *k* wave-vector to ~2π/nm, since the dispersion increase with ~$k^2$, which is clearly challenging. However, one may reach >100 GHz by using self-organization, domain walls or networks, skyrmion lattices of nanometer periodicity and full or hollow nanowires. In case of antiferromagnetic systems, the exchange interaction to the neighboring spin of opposite orientation results in a very strong exchange field which can be up to 100 T. Thus in antiferromagnetic and ferrimagnetic systems, intrinsic natural frequencies of the spin-waves modes are in the THz range. Because of their insulating character and thus the absence of conduction electrons, oxides generally have very low damping. THz excitation of antiferromagnets by THz field pulses has been demonstrated in NiO and different orthoferrites $XFeO_3$ ($X$ = Nd, Pr, Y, Dy, Er, and Ho). The phase control of the exciting magnetic field pulse

with THz frequency allows a coherent excitation and thus a phase control of the excited spin-wave as shown in figure 25(c) [167]. Current status and perspectives focused on the THz antiferromagnetic dynamics and magnonics are discussed in detail in the next section on ‘antiferromagnetic magnonics’ (section 15). Not much is known about the propagation of THz spin-wave so far. Generally, the challenge is that high frequency spin-waves are considerably damped since the intrinsic dissipation (viscous damping) results in first order, a signal decay proportional to ~1/ω. The above examples show how to generate THz spin currents or THz spin dynamics which can be utilized as input channels for THz magnonics.

## 14.2. Current and future challenges

The writing speed of ferromagnets has a physical limit in the GHz range due to their inherent slower magnetization dynamics, of the coherent switching of the Kittel mode or
<figure><img src="image_28.png" /><figcaption>**Figure 25.** (a) A typical ferromagnet (FM)/non-magnet (NM) sample structure and the THz emission geometry. The femtosecond laser beam is used for pumping the sample. The external magnetic field ($H_{ext}$) is along the x-axis. In (b) and (c) two strategies are shown to excite spin waves using THz pulses in the THz frequency range. (b) Laser-induced excitation of spin dynamics via spin currents pulses. Due to the spatial confinement of the STT perturbation (blue shaded area), spin waves with a broad spectrum of nonzero wavevectors are excited and can be probed by the MOKE in the collector. The exchange standing spin waves are confined in nanometer thick layer, where the frequency increase as ~$k^2$. Reproduced from [165]. CC BY 4.0. (c) Crystal lattice of NiO with antiferromagnetically ordered spins (blue arrows) in the (111) planes (light blue) and the direction of the THz magnetic field *B* (double-ended red arrow). An intense free-space THz interacts with the electron spins of a sample to launch a coherent magnon wave. Here the intrinsic modes are in the THz frequency range. A femtosecond near-infrared (NIR) probe pulse (green curves) measures the induced net magnetization by the Faraday effect. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Photonics] [165] (2011).</figcaption></figure>

THz frequency branches of the spin-wave dispersion. In case of ferromagnetic systems, the strategy can be to decrease the spacing of a MC or waveguide to the ~nm length scale and thereby to increase the *k* wave-vector to ~2π/nm, since the dispersion increase with ~$k^2$, which is clearly challenging. However, one may reach >100 GHz by using self-organization, domain walls or networks, skyrmion lattices of nanometer periodicity and full or hollow nanowires. In case of antiferromagnetic systems, the exchange interaction to the neighboring spin of opposite orientation results in a very strong exchange field which can be up to 100 T. Thus in antiferromagnetic and ferrimagnetic systems, intrinsic natural frequencies of the spin-waves modes are in the THz range. Because of their insulating character and thus the absence of conduction electrons, oxides generally have very low damping. THz excitation of antiferromagnets by THz field pulses has been demonstrated in NiO and different orthoferrites $XFeO_3$ ($X$ = Nd, Pr, Y, Dy, Er, and Ho). The phase control of the exciting magnetic field pulse

with THz frequency allows a coherent excitation and thus a phase control of the excited spin-wave as shown in figure 25(c) [167]. Current status and perspectives focused on the THz antiferromagnetic dynamics and magnonics are discussed in detail in the next section on ‘antiferromagnetic magnonics’ (section 15). Not much is known about the propagation of THz spin-wave so far. Generally, the challenge is that high frequency spin-waves are considerably damped since the intrinsic dissipation (viscous damping) results in first order, a signal decay proportional to ~1/ω. The above examples show how to generate THz spin currents or THz spin dynamics which can be utilized as input channels for THz magnonics.

## 14.2. Current and future challenges

The writing speed of ferromagnets has a physical limit in the GHz range due to their inherent slower magnetization dynamics, of the coherent switching of the Kittel mode or
<figure><img src="image_31.png" /><figcaption>**Figure 26.** (a) Spin and lattice structure of TmFeO₃. The ferromagnetic moment (*F*), antiferromagnetic vector (*G*), wave vector of the probe pulse (*k*<sub>NIR</sub>), and external magnetic field (*B*<sub>ext</sub>) are shown. On THz excitation, the magnetic potential *W*(*φ*) is abruptly modified. For a THz near field of *E* = 10 MV cm<sup>-1</sup>, the spins can be switched coherently by a single free space THz pulse (red trajectory) at *φ*₁ ~ *φ*₀ + 90°. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature] [167] (2019). (b) CuMnAs device image. The multilevel memory signal as a function of the number of applied picosecond free space THz pulses. Reproduced from [168]. CC BY 4.0.</figcaption></figure>

linear switching, connected to a thermal quenching. Anti-
ferromagnets and ferrimagnets with antiparallel exchange
coupling exhibit faster spin dynamics and could potentially
overcome these limitations. Recently, it has been shown that
not only femtosecond laser pulses, but also intense THz pulses
can manipulate the magnetization, which could be a more
energy efficient and fast way of controlling the magnetiza-
tion. Within one picosecond, intense free space THz pulses
abruptly change the magnetic anisotropy in antiferromag-
netic TmFeO3 at 83 K, causing a full and coherent large
angle (180°) switching, beyond small angle excitations, as
shown in figure 26(a) [168]. The switchable states can be
selected by an external magnetic bias. The authors also showed
that the low dissipation and the antenna's subwavelength
spatial definition could facilitate scalable spin devices

operating at THz rates. In a different report of figure 26(b),
spin-transfer torques mediated by THz-driven electric fields
(~3 × 10⁹ A cm⁻²) have induced switching of antiferromag-
netic domains in CuMnAs, a prototypical antiferromagnet at
room temperature [169]. In addition, the multilevel neuron-
like characteristics allow the integration of memory and logic
within the antiferromagnetic bit cell. However, detecting the
magnetization of antiferromagnets is challenging. In partic-
ular, the readout signal obtained from 90° rotation of above
antiferromagnets via anisotropic magnetoresistance is not suf-
ficient for fast reading and is not fully compatible with a MTJ
based scheme. Although the device can be switched using a
picosecond pulse input, the pulse duration does not necessarily
correspond to the real switching time. Therefore, the tempo-
ral switching profiles need to be confirmed by time resolved
<figure><img src="image_29.png" /><figcaption>**Figure 26.** (a) Spin and lattice structure of TmFeO₃. The ferromagnetic moment (*F*), antiferromagnetic vector (*G*), wave vector of the probe pulse (*k*<sub>NIR</sub>), and external magnetic field (*B*<sub>ext</sub>) are shown. On THz excitation, the magnetic potential *W*(*φ*) is abruptly modified. For a THz near field of *E* = 10 MV cm<sup>-1</sup>, the spins can be switched coherently by a single free space THz pulse (red trajectory) at *φ*₁ ~ *φ*₀ + 90°. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature] [167] (2019). (b) CuMnAs device image. The multilevel memory signal as a function of the number of applied picosecond free space THz pulses. Reproduced from [168]. CC BY 4.0.</figcaption></figure>

linear switching, connected to a thermal quenching. Anti-
ferromagnets and ferrimagnets with antiparallel exchange
coupling exhibit faster spin dynamics and could potentially
overcome these limitations. Recently, it has been shown that
not only femtosecond laser pulses, but also intense THz pulses
can manipulate the magnetization, which could be a more
energy efficient and fast way of controlling the magnetiza-
tion. Within one picosecond, intense free space THz pulses
abruptly change the magnetic anisotropy in antiferromag-
netic TmFeO3 at 83 K, causing a full and coherent large
angle (180°) switching, beyond small angle excitations, as
shown in figure 26(a) [168]. The switchable states can be
selected by an external magnetic bias. The authors also showed
that the low dissipation and the antenna's subwavelength
spatial definition could facilitate scalable spin devices

operating at THz rates. In a different report of figure 26(b),
spin-transfer torques mediated by THz-driven electric fields
(~3 × 10⁹ A cm⁻²) have induced switching of antiferromag-
netic domains in CuMnAs, a prototypical antiferromagnet at
room temperature [169]. In addition, the multilevel neuron-
like characteristics allow the integration of memory and logic
within the antiferromagnetic bit cell. However, detecting the
magnetization of antiferromagnets is challenging. In partic-
ular, the readout signal obtained from 90° rotation of above
antiferromagnets via anisotropic magnetoresistance is not suf-
ficient for fast reading and is not fully compatible with a MTJ
based scheme. Although the device can be switched using a
picosecond pulse input, the pulse duration does not necessarily
correspond to the real switching time. Therefore, the tempo-
ral switching profiles need to be confirmed by time resolved
measurements in order to evaluate the real switching speed of the device.

Both the effect of picosecond charge current pulses and the effect of ultrafast heating on the magnetization can be combined to build a THz magnetic memory. It was demonstrated in a ferrimagnetic material GdFeCo that a laser pulse application to the Austin switch generates a single sub-10 ps electrical pulse, which rapidly excites conduction electrons in magnetic metals [170]. This material is known to show magnetization reversal by a linearly polarized laser heat pulse that acts as an ultrafast heating stimulus. Here the current pulse generated by the Austin switch is directly injected into the GdFeCo with current densities of ~$10^9$ A cm$^{-2}$ and a pulse length below 9 ps. This pulse current injection into the smaller cross section and higher resistive GdFeCo element results in a deterministic switching event, measured by the magneto-optical contrast changing. The above examples show how THz inputs can excite spin dynamics by a small angle and even manipulate the magnetization by a full coherent 180° switching, which builds the basis of a THz magnonics concept.

The emitted THz waves from magnetic heterostructures not only show the potential as THz emitters, but also help to characterize various materials beyond conventional ones, such as ferrimagnets, 2D materials, topological insulators, and Weyl semimetal. The electrical spin torque, involving moving charges, suffers from unavoidable Joule heat and corresponding power dissipation, as well as a short spin propagation length. These fundamental obstacles can be overcome by the magnon torque [171]. In the future, the magnon current driven by all-electronic on-chip THz sources without a fs laser may enable novel spin memories operating at THz clock with least energy dissipation.

## Acknowledgments

Please include any acknowledgments and funding information as appropriate. The work is supported partially by A\*STAR AME-IRG (Grant No. A1983c0037), NUS Hybrid-Integrated Flexible (Stretchable) Electronic Systems Program, SpOT-LITE program (A\*STAR Grant, A18A6b0057) through RIE2020 funds, the Deutsche Forschungsgemeinschaft (DFG), Priority Program ‘Topological Insulators: Materials–Fundamental Properties–Devices SPP 1666’ and Priority Program ‘Skyrmionics Topological Spin Phenomena in Real-Space for Applications SPP 3137’.

# 15. Antiferromagnetic magnonics

Takahiro Moriyama<fnref n="1" />,<fnref n="2" /> and Shigemi Mizukami<fnref n="2" />,<fnref n="3" />

## 15.1. Status

Antiferromagnets have multiple magnetic sublattices and the magnetic moments on the sublattices are compensated by each

<figure><img src="image_32.png" /><figcaption>**Figure 27.** (a) The right-handed mode and (b) the left-handed mode of the antiferromagnetic dynamics.</figcaption></figure>

other. Negligible net magnetization, small magnetic susceptibility, and ultrafast magnetization dynamics are key characteristics of antiferromagnets [172] and are considered fruitful in the field of emerging antiferromagnetic spintronics [173]. Recent active studies have led to various interesting discoveries on interactions between the spin transport and the spin dynamics in antiferromagnets, which could make them alternative to the ferromagnets conventionally used in the present spintronics. Yet, there are still many open issues especially on the ultrafast magnetization dynamics in both physics and technological viewpoints which potentially fascinates the magnonics at the THz frequency leading to novel applications in information processing and tele-communications [174].

In the context of antiferromagnetic spintronics, the antiferromagnetic magnetization dynamics has recently been revisited long after early studies caught on in the late 50s by Kittel and others. Since there are more than one magnetic sublattices, multiple dynamic modes are generally observed. The eigenfrequency of the uniform resonance mode with the wavenumber $k = 0$ can be written by $\omega_t \approx \gamma\sqrt{2H_EH_A}$ where $\gamma$ is the gyromagnetic ratio, $H_E$ is an exchange field, and $H_A$ is a uniaxial anisotropy field. Since $H_E$, typically ~1000 T, comes into play in the eigenfrequency, the antiferromagnetic resonance occurs at much higher frequency, i.e. ~THz, than the FMR, typically ~GHz. For the simplest case of a collinear easy-axis antiferromagnet with two magnetic sublattices, the dynamics is degenerated by two modes with opposite circular polarizations as illustrated in figure 27, which is in contrast to the ferromagnetic cases where only the right-handed mode is allowed. This is also true for non-uniform modes, or magnons with $k \neq 0$, which are relevant to the magnonics. Therefore, comparing to the ferromagnetic cases, the antiferromagnetic magnons possess the additional freedom, i.e. the polarization, which is advantageous for information processing [175].

<footnote n="1">¹Kyoto University, Japan</footnote>

<footnote n="2">²Centre for Spintronics Research Network, Japan</footnote>

<footnote n="3">³Tohoku University, Japan</footnote>
measurements in order to evaluate the real switching speed of the device.

Both the effect of picosecond charge current pulses and the effect of ultrafast heating on the magnetization can be combined to build a THz magnetic memory. It was demonstrated in a ferrimagnetic material GdFeCo that a laser pulse application to the Austin switch generates a single sub-10 ps electrical pulse, which rapidly excites conduction electrons in magnetic metals [170]. This material is known to show magnetization reversal by a linearly polarized laser heat pulse that acts as an ultrafast heating stimulus. Here the current pulse generated by the Austin switch is directly injected into the GdFeCo with current densities of ~$10^9$ A cm$^{-2}$ and a pulse length below 9 ps. This pulse current injection into the smaller cross section and higher resistive GdFeCo element results in a deterministic switching event, measured by the magneto-optical contrast changing. The above examples show how THz inputs can excite spin dynamics by a small angle and even manipulate the magnetization by a full coherent 180° switching, which builds the basis of a THz magnonics concept.

The emitted THz waves from magnetic heterostructures not only show the potential as THz emitters, but also help to characterize various materials beyond conventional ones, such as ferrimagnets, 2D materials, topological insulators, and Weyl semimetal. The electrical spin torque, involving moving charges, suffers from unavoidable Joule heat and corresponding power dissipation, as well as a short spin propagation length. These fundamental obstacles can be overcome by the magnon torque [171]. In the future, the magnon current driven by all-electronic on-chip THz sources without a fs laser may enable novel spin memories operating at THz clock with least energy dissipation.

## Acknowledgments

Please include any acknowledgments and funding information as appropriate. The work is supported partially by A\*STAR AME-IRG (Grant No. A1983c0037), NUS Hybrid-Integrated Flexible (Stretchable) Electronic Systems Program, SpOT-LITE program (A\*STAR Grant, A18A6b0057) through RIE2020 funds, the Deutsche Forschungsgemeinschaft (DFG), Priority Program ‘Topological Insulators: Materials–Fundamental Properties–Devices SPP 1666’ and Priority Program ‘Skyrmionics Topological Spin Phenomena in Real-Space for Applications SPP 3137’.

# 15. Antiferromagnetic magnonics

Takahiro Moriyama<sup>1,2</sup> and Shigemi Mizukami<sup>2,3</sup>

## 15.1. Status

Antiferromagnets have multiple magnetic sublattices and the magnetic moments on the sublattices are compensated by each

<figure><img src="image_30.png" /><figcaption>**Figure 27.** (a) The right-handed mode and (b) the left-handed mode of the antiferromagnetic dynamics.</figcaption></figure>

other. Negligible net magnetization, small magnetic susceptibility, and ultrafast magnetization dynamics are key characteristics of antiferromagnets [172] and are considered fruitful in the field of emerging antiferromagnetic spintronics [173]. Recent active studies have led to various interesting discoveries on interactions between the spin transport and the spin dynamics in antiferromagnets, which could make them alternative to the ferromagnets conventionally used in the present spintronics. Yet, there are still many open issues especially on the ultrafast magnetization dynamics in both physics and technological viewpoints which potentially fascinates the magnonics at the THz frequency leading to novel applications in information processing and tele-communications [174].

In the context of antiferromagnetic spintronics, the antiferromagnetic magnetization dynamics has recently been revisited long after early studies caught on in the late 50s by Kittel and others. Since there are more than one magnetic sublattices, multiple dynamic modes are generally observed. The eigenfrequency of the uniform resonance mode with the wavenumber $k = 0$ can be written by $\omega_t \approx \gamma\sqrt{2H_EH_A}$ where $\gamma$ is the gyromagnetic ratio, $H_E$ is an exchange field, and $H_A$ is a uniaxial anisotropy field. Since $H_E$, typically ~1000 T, comes into play in the eigenfrequency, the antiferromagnetic resonance occurs at much higher frequency, i.e. ~THz, than the FMR, typically ~GHz. For the simplest case of a collinear easy-axis antiferromagnet with two magnetic sublattices, the dynamics is degenerated by two modes with opposite circular polarizations as illustrated in figure 27, which is in contrast to the ferromagnetic cases where only the right-handed mode is allowed. This is also true for non-uniform modes, or magnons with $k \neq 0$, which are relevant to the magnonics. Therefore, comparing to the ferromagnetic cases, the antiferromagnetic magnons possess the additional freedom, i.e. the polarization, which is advantageous for information processing [175].

<footnote n="1"><sup>1</sup>Kyoto University, Japan</footnote>

<footnote n="2"><sup>2</sup>Centre for Spintronics Research Network, Japan</footnote>

<footnote n="3"><sup>3</sup>Tohoku University, Japan</footnote>