<figure><img src="image_33.png" /><figcaption>**Figure 28.** Schematic illustrations of (a) SyAF, (b) RE–TM ferrimagnet, and (c) genuine antiferromagnet, e.g. NiO.</figcaption></figure>

In the early stage of the investigation, the state-of-art spectroscopy with a rather inefficient and weak far-infrared source was employed to investigate various antiferromagnets, such as NiO, CoO, MnO, Fe₂O₃, and Cr₂O₃. Although their high resonant frequencies have been experimentally confirmed, the experimental technique at the time was not sufficiently sensitive to capture more detailed dynamic properties such as relaxation mechanisms, e.g. magnetic damping, and the spatially non-uniform modes. Today, researchers take various experimental approaches for investigating the antiferromagnetic dynamics using more advanced measurement techniques such as the ultrafast measurement techniques with laser optics and THz spectroscopies with much better sensitivity. Due to some difficulties still lying on the measurements for antiferromagnets, materialistic approaches using pseudo-antiferromagnets, such as SyAFs (figure 28(a)) and rare-earth transition-metal (RE–TM) ferrimagnets (figure 28(b)), rather than using genuine antiferromagnets (figure 28(c)) are also in progress.

SyAFs typically consist of two ferromagnetic layers coupled antiferromagnetically via a non-magnetic insertion layer by the Ruderman–Kittel–Kasuya–Yoshida (RKKY) or magnetic dipole interactions. Since the strength of the antiferromagnetic coupling interaction (typically ~1 T) is far smaller than that of genuine antiferromagnets and is tunable by adjusting the insertion layer thickness, the resonant frequency can be in the GHz range where the conventional microwave technology is applicable for measurements. Therefore, SyAFs are a good and easy testbed for investigating antiferromagnetic dynamics. One can observe a clear cross-over between the ferromagnetic dynamics and antiferromagnetic dynamics, the so-called *acoustic* and *optic* modes. Those show strong non-reciprocity in their propagations [176]. The relaxation of these modes and the dynamic coupling of them have been investigated in detail [177]. The mode amplification due to their non-linear coupling was also reported [178]. However, one must be cautious if the knowledge can necessarily extrapolate to genuine antiferromagnets, since most of the cases treat SyAFs as the two macro-spins coupled antiferromagnetically, which

completely differs from genuine antiferromagnets where spins are atomistically coupled.

Another approach is taken by using RE–TM ferrimagnets having both magnetization and angular momentum compensation. RE is typically Gd or Tb and TM is Fe, Co, or their alloy. Due to the slightly different gyromagnetic ratios and their temperature dependences for the RE and TM, these ferrimagnetic alloys exhibit the magnetization and the angular momentum compensation at different temperatures. The intriguing magnetization dynamics emerges at the angular momentum compensation point but not at the magnetization compensation point which seems intuitively relevant to the antiferromagnetism. Because the magnetization dynamics are intrinsically governed by the angular momentum not the magnetic moment, they actually exhibit the antiferromagnetic dynamics at the angular momentum compensation point. The virtue of the RE–TM ferrimagnets is that one can easily access the antiferromagnetic dynamics while the magnetic susceptibility is non-zero, i.e. the static magnetizations are manipulable by external magnetic field. RE–TM ferrimagnets have been a testbed for the laser induced ultra-fast spin switching and ultra-fast DW dynamics [179].

Genuine antiferromagnets are yet the most interesting materials in antiferromagnetic magnonics. Material choices are tremendous ranging from insulators to metals. However, the measurement methodology is always a problem with these materials due to invisible nature of magnetization as well as the quite high resonant frequency. Nickel oxide is one of the most investigated materials since the old age. Recent revisits have revealed the detail magnetization dynamics in both time domain and broad band frequency domain, mostly focused on the localized dynamics with $k = 0$, using advanced technologies [180, 181]. Detail explorations of the antiferromagnetic dynamics, such as relaxation dynamics, magnetic damping, and Q-factors have recently revisited and progressed in accordance with theoretical developments [182]. So far, these ultra-high frequency spectroscopies are limited to bulk materials due mostly to the sensitivity issues. Investigation for antiferromagnetic thin films which are most relevant to the spintronic applications has not yet resolved.
<figure><img src="image_31.png" /><figcaption>**Figure 28.** Schematic illustrations of (a) SyAF, (b) RE–TM ferrimagnet, and (c) genuine antiferromagnet, e.g. NiO.</figcaption></figure>

In the early stage of the investigation, the state-of-art spectroscopy with a rather inefficient and weak far-infrared source was employed to investigate various antiferromagnets, such as NiO, CoO, MnO, Fe₂O₃, and Cr₂O₃. Although their high resonant frequencies have been experimentally confirmed, the experimental technique at the time was not sufficiently sensitive to capture more detailed dynamic properties such as relaxation mechanisms, e.g. magnetic damping, and the spatially non-uniform modes. Today, researchers take various experimental approaches for investigating the antiferromagnetic dynamics using more advanced measurement techniques such as the ultrafast measurement techniques with laser optics and THz spectroscopies with much better sensitivity. Due to some difficulties still lying on the measurements for antiferromagnets, materialistic approaches using pseudo-antiferromagnets, such as SyAFs (figure 28(a)) and rare-earth transition-metal (RE–TM) ferrimagnets (figure 28(b)), rather than using genuine antiferromagnets (figure 28(c)) are also in progress.

SyAFs typically consist of two ferromagnetic layers coupled antiferromagnetically via a non-magnetic insertion layer by the Ruderman–Kittel–Kasuya–Yoshida (RKKY) or magnetic dipole interactions. Since the strength of the antiferromagnetic coupling interaction (typically ~1 T) is far smaller than that of genuine antiferromagnets and is tunable by adjusting the insertion layer thickness, the resonant frequency can be in the GHz range where the conventional microwave technology is applicable for measurements. Therefore, SyAFs are a good and easy testbed for investigating antiferromagnetic dynamics. One can observe a clear cross-over between the ferromagnetic dynamics and antiferromagnetic dynamics, the so-called acoustic and optic modes. Those show strong non-reciprocity in their propagations [176]. The relaxation of these modes and the dynamic coupling of them have been investigated in detail [177]. The mode amplification due to their non-linear coupling was also reported [178]. However, one must be cautious if the knowledge can necessarily extrapolate to genuine antiferromagnets, since most of the cases treat SyAFs as the two macro-spins coupled antiferromagnetically, which

completely differs from genuine antiferromagnets where spins are atomistically coupled.

Another approach is taken by using RE–TM ferrimagnets having both magnetization and angular momentum compensation. RE is typically Gd or Tb and TM is Fe, Co, or their alloy. Due to the slightly different gyromagnetic ratios and their temperature dependences for the RE and TM, these ferrimagnetic alloys exhibit the magnetization and the angular momentum compensation at different temperatures. The intriguing magnetization dynamics emerges at the angular momentum compensation point but not at the magnetization compensation point which seems intuitively relevant to the antiferromagnetism. Because the magnetization dynamics are intrinsically governed by the angular momentum not the magnetic moment, they actually exhibit the antiferromagnetic dynamics at the angular momentum compensation point. The virtue of the RE–TM ferrimagnets is that one can easily access the antiferromagnetic dynamics while the magnetic susceptibility is non-zero, i.e. the static magnetizations are manipulable by external magnetic field. RE–TM ferrimagnets have been a testbed for the laser induced ultra-fast spin switching and ultra-fast DW dynamics [179].

Genuine antiferromagnets are yet the most interesting materials in antiferromagnetic magnonics. Material choices are tremendous ranging from insulators to metals. However, the measurement methodology is always a problem with these materials due to invisible nature of magnetization as well as the quite high resonant frequency. Nickel oxide is one of the most investigated materials since the old age. Recent revisits have revealed the detail magnetization dynamics in both time domain and broad band frequency domain, mostly focused on the localized dynamics with $k = 0$, using advanced technologies [180, 181]. Detail explorations of the antiferromagnetic dynamics, such as relaxation dynamics, magnetic damping, and Q-factors have recently revisited and progressed in accordance with theoretical developments [182]. So far, these ultra-high frequency spectroscopies are limited to bulk materials due mostly to the sensitivity issues. Investigation for antiferromagnetic thin films which are most relevant to the spintronic applications has not yet resolved.
## 15.2. Current and future challenges

Frontiers of the antiferromagnetic magnonics are in various directions. While the most important and impactful part of the antiferromagnetic magnonics is the use of genuine antiferrom agents with the ultra-high frequency, materialistic approaches are still alternative to harness the gaps of the antiferromagnetic measurement techniques. Those materialistic approaches discussed above could of course find their own magnonic applications. However, for understanding the physics of magnons in genuine antiferromagnets, one should not rely too much on these approaches as it is still controversial whether those physics can be simply extrapolated.

In genuine antiferromagnets, comparing to the understanding of the resonant frequency ($\omega_r \approx \gamma\sqrt{2H_EH_A}$), magnetic relaxation mechanisms in antiferromagnets, such as magnetic damping which is typically evaluated from the spectral linewidth of the resonance, have not been investigated or discussed as much as the ferromagnetic cases. In the context of antiferromagnetic spintronics, revisits to the magnetic damping in antiferromagnet and to both experimental and theoretical investigations have recently been active [180, 181]. In both physics and engineering points of views, it is quite important to have a firm foundation of the relaxation mechanisms.

Direct observation of the antiferromagnetic magnon modes ($k \neq 0$) is one of the most challenging business. Short wavelength and high frequency magnon modes inherent to the antiferromagnetic dynamics make those measurements quite difficult. Alternatively, although it is indirect observation, we should pay attention to the recent progress in the magnon spin current transport experiments where the spin current is carried through the antiferromagnets in the form of magnons [183].

So far, most of the investigations have been made with bulk antiferromagnetic materials and not much have been investigated in thin films, their multilayers, and confined structures where more interesting physics and applications should be found. It is interesting to see how the magnons behave in the confined structures which should be quite important for magnonic circuitry applications.

Finally, it is greatly desirable to push forward the development of antiferromagnetic measurement techniques. It is necessary to further increase the sensitivity of the THz spectroscopy to be able to measure the antiferromagnetic thin films. Magnetic imaging techniques using various probes such as magneto-optic effects and x-ray linear dichroism with photo electron emission microscopy are viable solutions. Very recent reports of the antiferromagnetic spin pumping effect could provide a unique perspective for electrical detection of antiferromagnetic dynamics [96, 184]. Advancements of these measurement techniques are certainly the key to progress the antiferromagnetic magnonics.

## 15.3. Concluding remarks

Antiferromagnetic magnonics is one of the most interesting destinations in recently emerging antiferromagnetic spintronics. The THz frequency and the polarization degree

of freedoms certainly fascinate the conventional magnonic technologies and potentially yield novel applications in information processing and tele-communications. However, actual rise of antiferromagnetic magnonics has yet to come as the methodological challenges impede some of the key experimental studies. Further efforts in advancing the measurement technologies for antiferromagnets will open up tremendous opportunities in antiferromagnetic magnonics.

## Acknowledgments

We thank to the Spintronics Research Network of Japan.

# 16. Spin-wave nonreciprocity

Pedro Landeros$^{1,2}$, Rodolfo A Gallardo$^{1,2}$, and Giovanni Carlotti$^{3,4}$

## 16.1. Status

Nonreciprocal propagation of electromagnetic, acoustic or spin waves is attracting more and more interest, due to the possible exploitation in devices for information and communication technology, such as isolators, circulators, phase shifter, etc. Nonreciprocity refers to the case where wave propagation changes, or it is even forbidden, upon inversion of the propagation direction and requires a breaking of the time-reversal symmetry [185]. This can be induced, for instance, by an applied stimulus, such as an external field with a specific bias direction, or by the presence of a substrate providing a suitable interaction. In the case of spin waves, nonreciprocity has been known to appear in different kinds of magnetic systems, leading to differences in either amplitude localization or frequency (the latter may also correspond to different velocity or phase, depending on the situation) for counter-propagating waves. Then, further advances in the field of nonreciprocal magnetic materials is of great scientific importance, since it will provide the technological basis for the realization of nanoscale nonreciprocal devices operating at microwave frequencies, and at the same time it will help us to solve the riddle of the relevance of the different fundamental physical interactions in magnetism.

The most well-known case of spin-wave nonreciprocity is the magnetostatic surface spin wave (DE wave), where the in-plane wavevector **k** is always perpendicular to the in-plane magnetization **M**. In a semi-infinite medium, such surface wave has an amplitude that decays exponentially within the medium and can travel from left to right but not in the opposite direction, thanks to the symmetry breaking introduced by the presence of the surface and the dynamic magnetostatic field. In the case of a ferromagnetic film with finite thickness **d**, the same physical mechanism leads to a nonreciprocity in the amplitude for magnetostatic waves if $kd \gg 1$ (that

<footnote>$^{1}$Universidad Técnica Federico Santa María, Chile</footnote>

<footnote>$^{2}$Center for the Development of Nanoscience and Nanotechnology (CEDENNA), Chile</footnote>

<footnote>$^{3}$University of Perugia, Italy</footnote>

<footnote>$^{4}$CNR Instituto Nanoscienze, Italy</footnote>
## 15.2. Current and future challenges

Frontiers of the antiferromagnetic magnonics are in various directions. While the most important and impactful part of the antiferromagnetic magnonics is the use of genuine antiferrom agents with the ultra-high frequency, materialistic approaches are still alternative to harness the gaps of the antiferromagnetic measurement techniques. Those materialistic approaches discussed above could of course find their own magnonic applications. However, for understanding the physics of magnons in genuine antiferromagnets, one should not rely too much on these approaches as it is still controversial whether those physics can be simply extrapolated.

In genuine antiferromagnets, comparing to the understanding of the resonant frequency ($\omega_r \approx \gamma\sqrt{2H_EH_A}$), magnetic relaxation mechanisms in antiferromagnets, such as magnetic damping which is typically evaluated from the spectral linewidth of the resonance, have not been investigated or discussed as much as the ferromagnetic cases. In the context of antiferromagnetic spintronics, revisits to the magnetic damping in antiferromagnet and to both experimental and theoretical investigations have recently been active [180, 181]. In both physics and engineering points of views, it is quite important to have a firm foundation of the relaxation mechanisms.

Direct observation of the antiferromagnetic magnon modes ($k \neq 0$) is one of the most challenging business. Short wavelength and high frequency magnon modes inherent to the antiferromagnetic dynamics make those measurements quite difficult. Alternatively, although it is indirect observation, we should pay attention to the recent progress in the magnon spin current transport experiments where the spin current is carried through the antiferromagnets in the form of magnons [183].

So far, most of the investigations have been made with bulk antiferromagnetic materials and not much have been investigated in thin films, their multilayers, and confined structures where more interesting physics and applications should be found. It is interesting to see how the magnons behave in the confined structures which should be quite important for magnonic circuitry applications.

Finally, it is greatly desirable to push forward the development of antiferromagnetic measurement techniques. It is necessary to further increase the sensitivity of the THz spectroscopy to be able to measure the antiferromagnetic thin films. Magnetic imaging techniques using various probes such as magneto-optic effects and x-ray linear dichroism with photo electron emission microscopy are viable solutions. Very recent reports of the antiferromagnetic spin pumping effect could provide a unique perspective for electrical detection of antiferromagnetic dynamics [96, 184]. Advancements of these measurement techniques are certainly the key to progress the antiferromagnetic magnonics.

## 15.3. Concluding remarks

Antiferromagnetic magnonics is one of the most interesting destinations in recently emerging antiferromagnetic spintronics. The THz frequency and the polarization degree

of freedoms certainly fascinate the conventional magnonic technologies and potentially yield novel applications in information processing and tele-communications. However, actual rise of antiferromagnetic magnonics has yet to come as the methodological challenges impede some of the key experimental studies. Further efforts in advancing the measurement technologies for antiferromagnets will open up tremendous opportunities in antiferromagnetic magnonics.

## Acknowledgments

We thank to the Spintronics Research Network of Japan.

## 16. Spin-wave nonreciprocity

Pedro Landeros<fnref n="1" />,<fnref n="2" />, Rodolfo A Gallardo<fnref n="1" />,<fnref n="2" />, and Giovanni Carlotti<fnref n="3" />,<fnref n="4" />

### 16.1. Status

Nonreciprocal propagation of electromagnetic, acoustic or spin waves is attracting more and more interest, due to the possible exploitation in devices for information and communication technology, such as isolators, circulators, phase shifter, etc. Nonreciprocity refers to the case where wave propagation changes, or it is even forbidden, upon inversion of the propagation direction and requires a breaking of the time-reversal symmetry [185]. This can be induced, for instance, by an applied stimulus, such as an external field with a specific bias direction, or by the presence of a substrate providing a suitable interaction. In the case of spin waves, nonreciprocity has been known to appear in different kinds of magnetic systems, leading to differences in either amplitude localization or frequency (the latter may also correspond to different velocity or phase, depending on the situation) for counter-propagating waves. Then, further advances in the field of nonreciprocal magnetic materials is of great scientific importance, since it will provide the technological basis for the realization of nanoscale nonreciprocal devices operating at microwave frequencies, and at the same time it will help us to solve the riddle of the relevance of the different fundamental physical interactions in magnetism.

The most well-known case of spin-wave nonreciprocity is the magnetostatic surface spin wave (DE wave), where the in-plane wavevector **k** is always perpendicular to the in-plane magnetization **M**. In a semi-infinite medium, such surface wave has an amplitude that decays exponentially within the medium and can travel from left to right but not in the opposite direction, thanks to the symmetry breaking introduced by the presence of the surface and the dynamic magnetostatic field. In the case of a ferromagnetic film with finite thickness **d**, the same physical mechanism leads to a nonreciprocity in the amplitude for magnetostatic waves if $kd \gg 1$ (that

<footnote n="1">¹Universidad Técnica Federico Santa María, Chile</footnote>

<footnote n="2">²Center for the Development of Nanoscience and Nanotechnology (CEDENNA), Chile</footnote>

<footnote n="3">³University of Perugia, Italy</footnote>

<footnote n="4">⁴CNR Instituto Nanoscienze, Italy</footnote>
<figure><img src="image_34.png" /><figcaption>**Figure 29.** Top panel: nonreciprocal spin wave propagation in the DE geometry ($k \perp M$) in a film with thickness $d$ comparable to the spin-wave wavelength. It is seen that the wave amplitude is maximum at the bottom (top) interface for propagation along $+x$ ($-x$). Central and bottom panels: nonreciprocal spin wave propagation in an ultrathin ferromagnetic film (FM), whose thickness is much lower than the spin-wave wavelength, in presence of a heavy-metal substrate (HM) that induces DMI. Spin waves propagating along $+x$ and $-x$ have opposite chirality and therefore different frequencies.</figcaption></figure>

corresponds, for spin waves with wavelength in the range of hundreds of nanometers, to film thickness well above one micron). In fact, the dynamic magnetostatic field localizes mostly in the upper or lower surface, depending on the relative orientation between **k** and **M**, as illustrated in figure 29 (top panel). For $kd \approx 1$ (i.e. film thickness in the tens of nanometers range), instead, the DE hybridizes with discrete dipole-exchange spin wave modes, and there is still a preferential

localization at either the upper or the lower interface. In the ultrathin film limit ($kd \ll 1$), this behavior disappears, since the magnetostatic field and the wave amplitude are practically uniform across the film thickness.

In addition to the above amplitude nonreciprocity, if there are different surface anisotropies $K_s$ at the two faces of a film [186], then it can be also observed a frequency nonreciprocity, i.e. a small frequency difference $\Delta f$ between two
<figure><img src="image_32.png" /><figcaption>**Figure 29.** Top panel: nonreciprocal spin wave propagation in the DE geometry ($k \perp M$) in a film with thickness $d$ comparable to the spin-wave wavelength. It is seen that the wave amplitude is maximum at the bottom (top) interface for propagation along $+x$ ($-x$). Central and bottom panels: nonreciprocal spin wave propagation in an ultrathin ferromagnetic film (FM), whose thickness is much lower than the spin-wave wavelength, in presence of a heavy-metal substrate (HM) that induces DMI. Spin waves propagating along $+x$ and $-x$ have opposite chirality and therefore different frequencies.</figcaption></figure>

corresponds, for spin waves with wavelength in the range of hundreds of nanometers, to film thickness well above one micron). In fact, the dynamic magnetostatic field localizes mostly in the upper or lower surface, depending on the relative orientation between **k** and **M**, as illustrated in figure 29 (top panel). For $kd \approx 1$ (i.e. film thickness in the tens of nanometers range), instead, the DE hybridizes with discrete dipole-exchange spin wave modes, and there is still a preferential

localization at either the upper or the lower interface. In the ultrathin film limit ($kd \ll 1$), this behavior disappears, since the magnetostatic field and the wave amplitude are practically uniform across the film thickness.

In addition to the above amplitude nonreciprocity, if there are different surface anisotropies $K_s$ at the two faces of a film [186], then it can be also observed a frequency nonreciprocity, i.e. a small frequency difference $\Delta f$ between two
counter-propagating waves that is linear with $\Delta K_s$. An explicit formula for $\Delta f$ was derived by Gladii et al (see equation (7) in [186]), showing that the frequency difference is linear with the wavenumber $k$ at small film thickness, and predicting a more complex behavior for larger $d$. Note that such a nonreciprocity in frequency implies a phase-difference for waves covering the same distance in opposite directions.

Another source of both frequency and amplitude non-reciprocity in a magnetic film, which has been widely investigated in the last years, is the bulk or iDMI (see figure 29, central and bottom panels). Experimental evidence, based on BLS measurements, of such a frequency difference due to DMI between counter-propagating spin waves was first reported by Di et al [187]. Shortly after, using the same technique in ferromagnetic/HM interfaces (such as, for instance, Co/Pt), several groups have measured a noticeable frequency asymmetry in the spin-wave dispersion, that is proportional to both the wavenumber $k$ and the strength of the DMI constant $D$, in agreement with the expression [188]:

$$ \Delta f = \frac{2\gamma}{\pi M_s} Dk \sin \phi_k, $$

where $\phi_k$ is the angle between $k$ and $M$ and $\gamma$ the gyromagnetic ratio. These experiments allow to estimate $D$, whose measured values were collected in tables 5.2-3 of reference [188], which also contain, for the sake of comparison, measurements of $D$ obtained with other techniques. It is now well established that the strength of the DMI is larger for interfacial systems than for chiral lattice ferromagnets with bulk DMI and that its strength decreases with the thickness of the ferromagnetic layer, while slightly increases with the thickness of the heavy-metal layer. Therefore, besides the enlargement of damping produced by the heavy-metal, a drawback of iDMI systems is that ferromagnetic layers with very low thickness $d$ are needed (typically in the range between 0.5 and 5 nm) to generate a noticeable nonreciprocity, since $D \sim 1/d$. For bulk DMI systems, such as the chiral lattice ferromagnet with tetragonal symmetry $Cu_2OSeO_3$, $\Delta f = \frac{2\gamma}{\pi M_s} Dk \cos \phi_k$, so the effect is maximum when it is measured for the backward-volume wave configuration ($\phi_k = 0$) [189]. If the magnetization is tipped out an angle $\alpha$ from the plane, a factor $\cos \alpha$ enters in $\Delta f$ for both bulk and interface DMI systems [188].

A third case of spin-wave nonreciprocity, both in amplitude and in frequency, occurs if one considers two oppositely magnetized ferromagnetic layers, separated by a non-magnetic spacer [113, 190], shown in figure 30. In this case, the frequency difference is due to the dynamic volumetric and surface dipolar interactions. For a particular system with two identical layers of thickness $d$, separated by a non-magnetic layer of thickness $s$, the frequency nonreciprocity is given by equation (8) in reference [190]:

$$ \Delta f = \frac{2\gamma}{\pi} \mu_0 M_s \frac{e^{-k(d+s)}}{kd} \sinh^2\left(\frac{kd}{2}\right). $$

This frequency difference presents a non-monotonic behavior with $k$ and $d$, with a well-defined maximum that is obtained for a value of $k$ such that: $\tanh(kd/2) = kd/[1 + k(d+s)]$.

<figure><img src="image_35.png" /><figcaption>**Figure 30.** Top panel: sketch of the dynamic stray fields induced by the surface and volumetric dynamic magnetic charges in a ferromagnetic bilayer with antiparallel magnetizations, similar to that studied in reference [190], where spin waves propagate in the DE configuration. The large arrows depict the orientation of the dynamic magnetization, while the static magnetizations point along the $\pm z$ directions. The distributions of dynamic magnetizations and stray fields (curved arrows) are shown for both $k > 0$ and $k < 0$. Center panel: dispersion curves for spin waves propagating in a magnetic bilayer with antiparallel magnetization similar to that studied in reference [113]. It can be seen that, while the dispersion curve is symmetric around the $k = 0$ vertical axis for the magnetostatic backward configuration ($\phi_k = 0^\circ$), in the DE configuration ($\phi_k = 90^\circ$) the curves account for a marked nonreciprocity. In particular, in the frequency range (shaded area) between $f_0$ and $f_{min}$, only waves with $k > 0$ can exist in the structure. Bottom panel: the above unidirectional propagation is illustrated by three snapshots (top view) of the simulation relative to different times for a spin-wave packet excited in the DE configuration, before and after reaching a 200 nm wide non-magnetic circular defect: no back-reflection is observed, since the counter-propagation is not allowed by the dispersion curve. Scale bars: 500 nm. (Latter panel) [113] John Wiley & Sons. [Copyright © 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim].</figcaption></figure>

Then, given $d$ and $s$, the wavenumber $k$ at which $\Delta f$ is maximum can be easily estimated and one can easily verify that the effect is sizable in a very wide range of $kd$, i.e. for film thickness ranging from a few nanometers to several microns. Recent BLS measurements in a CoFeB (5.7 nm)/Ir (0.6 nm)/NiFe
counter-propagating waves that is linear with $\Delta K_s$. An explicit formula for $\Delta f$ was derived by Gladii et al (see equation (7) in [186]), showing that the frequency difference is linear with the wavenumber $k$ at small film thickness, and predicting a more complex behavior for larger $d$. Note that such a nonreciprocity in frequency implies a phase-difference for waves covering the same distance in opposite directions.

Another source of both frequency and amplitude non-reciprocity in a magnetic film, which has been widely investigated in the last years, is the bulk or iDMI (see figure 29, central and bottom panels). Experimental evidence, based on BLS measurements, of such a frequency difference due to DMI between counter-propagating spin waves was first reported by Di et al [187]. Shortly after, using the same technique in ferromagnetic/HM interfaces (such as, for instance, Co/Pt), several groups have measured a noticeable frequency asymmetry in the spin-wave dispersion, that is proportional to both the wavenumber $k$ and the strength of the DMI constant $D$, in agreement with the expression [188]:

$$ \Delta f = \frac{2\gamma}{\pi M_s} Dk \sin \phi_k, $$

where $\phi_k$ is the angle between $k$ and $M$ and $\gamma$ the gyromagnetic ratio. These experiments allow to estimate $D$, whose measured values were collected in tables 5.2-3 of reference [188], which also contain, for the sake of comparison, measurements of $D$ obtained with other techniques. It is now well established that the strength of the DMI is larger for interfacial systems than for chiral lattice ferromagnets with bulk DMI and that its strength decreases with the thickness of the ferromagnetic layer, while slightly increases with the thickness of the heavy-metal layer. Therefore, besides the enlargement of damping produced by the heavy-metal, a drawback of iDMI systems is that ferromagnetic layers with very low thickness $d$ are needed (typically in the range between 0.5 and 5 nm) to generate a noticeable nonreciprocity, since $D \sim 1/d$. For bulk DMI systems, such as the chiral lattice ferromagnet with tetragonal symmetry $Cu_2OSeO_3$, $\Delta f = \frac{2\gamma}{\pi M_s} Dk \cos \phi_k$, so the effect is maximum when it is measured for the backward-volume wave configuration ($\phi_k = 0$) [189]. If the magnetization is tipped out an angle $\alpha$ from the plane, a factor $\cos \alpha$ enters in $\Delta f$ for both bulk and interface DMI systems [188].

A third case of spin-wave nonreciprocity, both in amplitude and in frequency, occurs if one considers two oppositely magnetized ferromagnetic layers, separated by a non-magnetic spacer [113, 190], shown in figure 30. In this case, the frequency difference is due to the dynamic volumetric and surface dipolar interactions. For a particular system with two identical layers of thickness $d$, separated by a non-magnetic layer of thickness $s$, the frequency nonreciprocity is given by equation (8) in reference [190]:

$$ \Delta f = \frac{2\gamma}{\pi} \mu_0 M_s \frac{e^{-k(d+s)}}{kd} \sinh^2\left(\frac{kd}{2}\right). $$

This frequency difference presents a non-monotonic behavior with $k$ and $d$, with a well-defined maximum that is obtained for a value of $k$ such that: $\tanh(kd/2) = kd/[1 + k(d+s)]$.

<figure><img src="image_33.png" /><figcaption>**Figure 30.** Top panel: sketch of the dynamic stray fields induced by the surface and volumetric dynamic magnetic charges in a ferromagnetic bilayer with antiparallel magnetizations, similar to that studied in reference [190], where spin waves propagate in the DE configuration. The large arrows depict the orientation of the dynamic magnetization, while the static magnetizations point along the $\pm z$ directions. The distributions of dynamic magnetizations and stray fields (curved arrows) are shown for both $k > 0$ and $k < 0$. Center panel: dispersion curves for spin waves propagating in a magnetic bilayer with antiparallel magnetization similar to that studied in reference [113]. It can be seen that, while the dispersion curve is symmetric around the $k = 0$ vertical axis for the magnetostatic backward configuration ($\phi_k = 0^\circ$), in the DE configuration ($\phi_k = 90^\circ$) the curves account for a marked nonreciprocity. In particular, in the frequency range (shaded area) between $f_0$ and $f_{min}$, only waves with $k > 0$ can exist in the structure. Bottom panel: the above unidirectional propagation is illustrated by three snapshots (top view) of the simulation relative to different times for a spin-wave packet excited in the DE configuration, before and after reaching a 200 nm wide non-magnetic circular defect: no back-reflection is observed, since the counter-propagation is not allowed by the dispersion curve. Scale bars: 500 nm. (Latter panel) [113] John Wiley & Sons. [Copyright © 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim].</figcaption></figure>

Then, given $d$ and $s$, the wavenumber $k$ at which $\Delta f$ is maximum can be easily estimated and one can easily verify that the effect is sizable in a very wide range of $kd$, i.e. for film thickness ranging from a few nanometers to several microns. Recent BLS measurements in a CoFeB (5.7 nm)/Ir (0.6 nm)/NiFe
<table><caption>**Table 1.** Group of nonreciprocal magnonic systems discussed in this paper and their key physical features.</caption><thead><tr><th>Magnonic system</th><th>Source of nonreciprocity</th><th>Limit<sup>a</sup></th><th>Amplitude nonreciprocity</th><th>Frequency nonreciprocity</th><th>References</th></tr></thead><tbody><tr><td>Thick film</td><td>Dipolar coupling</td><td>kd &gt;&gt; 1</td><td>Yes</td><td>No</td><td>[185]</td></tr><tr><td>Thin film with surface anisotropies</td><td>Surface/interface anisotropy</td><td>kd &lt;&lt; 1</td><td>Yes</td><td>Yes</td><td>[186]</td></tr><tr><td>Ultrathin film/HM</td><td>Interfacial Dzyaloshinskii–Moriya coupling</td><td>kd &lt;&lt; 1</td><td>Yes<sup>b</sup></td><td>Yes</td><td>[187, 188]</td></tr><tr><td>Non-centrosymmetric chiral magnet</td><td>Bulk Dzyaloshinskii–Moriya coupling</td><td>c</td><td>Yes<sup>b</sup></td><td>Yes</td><td>[188, 189]</td></tr><tr><td>AFM bilayer</td><td>Dipolar coupling</td><td>kd &lt; 1</td><td></td><td></td><td>[113, 190]</td></tr><tr><td>Graded magnetization film</td><td>Dipolar coupling</td><td>kd &gt; 1</td><td>Yes</td><td>Yes</td><td>[191]</td></tr><tr><td>Curved surfaces</td><td>Dipolar coupling</td><td>kR &lt;&lt; 1</td><td>Yes</td><td>Yes</td><td>[192]</td></tr></tbody></table>

<sup>a</sup>Referring to spin waves detected by BLS, where the *k* value is of the order of 10–20 rad µm⁻¹, the limit *kd* ≪ 1 (*kd* < 1) corresponds to *d* in the range of a few (tens of) nanometers.

<sup>b</sup>In this case the amplitude nonreciprocity has nothing to do with the localization at the top or bottom film surface, typical of thick films, but with the DMI-induced lifting of energy degeneration for ±*k*.

<sup>c</sup>In bulk DMI systems, nonreciprocal propagation has been observed in systems with *kd* ≥ 1, and *kd* ≫ 1, so the nonreciprocal behavior is in principle not limited.

(6.7 nm) bilayer give Δ*f* about 1–2 GHz [190], in agreement with theory. These values are comparable with typical ones for iDMI systems. Nonetheless, from the above formula for Δ*f*, even larger values are found if the thickness *d* is increased, as in the recent study of CoFeB (45 nm)/Ru (0.6 nm)/CoFeB (45 nm) [113]. Remarkably, the frequency nonreciprocity can be turned off by changing the relative magnetic orientation from antiparallel to parallel alignment, i.e. just by simple switching, without any rotation of the field, as it is required in the case of DMI systems. Moreover, when the frequency of excitation is lower than *f*(*k* = 0), but larger than the low-lying frequency of the dispersion curve (figure 30, central panel), spin waves can travel only in one direction, as a spin current diode, leading to the phenomenon of resilience to back reflection [113], making unidirectional magnon propagation very robust under imperfections and defects (figure 30, bottom panel).

Nonreciprocity from dipolar coupling also appears in films with graded magnetization along the thickness [191] and curved systems [192]. Spin-polarized electric currents, as well as the flexoelectric interaction, also induce a frequency shift of two counterpropagating spin waves (references can be found in [188]).

Another interesting behavior arises when spin waves propagate in a system with a *periodic* DMI [193], which may be obtained by patterning an array of Pt wires on top of an ultra-thin ferromagnetic film. In such film, the DMI is activated only underneath the heavy-metal wires, where nonreciprocal spin waves are expected. It has been shown from calculations and simulations that periodic DMI causes indirect magnonic gaps, flat bands and a complex temporal evolution of the spin waves [193]. Such a system with periodic DMI can be regarded as a *chiral* MC, where topological magnons should be observed.

## 16.2. Current and future challenges

Although the frequency nonreciprocity of the spin waves could in principle lead to the design and the fabrication of new kinds of devices (isolators, circulators, phase shifters, etc), this task requires proper material properties and the ability to

manipulate spin waves. In our opinion, among the different systems leading to spin-wave nonreciprocity, the oppositely magnetized bilayer, or SyAF, deserves special attention and is presently the best candidate to realize real devices. First, frequency nonreciprocity is easily reconfigurable just by switching the bilayer magnetization from parallel to antiparallel without any bias field rotation, as occurs in the case of DMI systems where reconfigurability requires an in-plane or out-of-plane magnetization rotation. Second, there is no need of ultrathin layers and expensive HMs that are necessary to achieve a large DMI at the interface. Note also that the HM in iDMI systems increases the magnetic damping, which imposes a problem for magnonic applications. Moreover, the predicted frequency nonreciprocity in the SyAF [113, 190] can be even larger than the usual values measured for iDMI systems, paving the way to possible applications (Table 1).

## 16.3. Concluding remarks

The phenomenon of frequency nonreciprocity provides additional features for the manipulation and control of the spin-wave spectrum. Frequency nonreciprocity can be easily reconfigurable with an external field, depending on the nonreciprocal magnonic system. Among them, the magnetic bilayer seems to be one of the best candidates for the realization of nonreciprocal magnonic devices, since is easily reconfigurable, it provides a large frequency nonreciprocity without ultrathin layers, and it only requires ordinary magnetic materials that are fully compatible with CMOS technologies.

## Acknowledgments

PL and RAG acknowledge funding from FONDECYT 11170736, and 1201153, and Financiamiento Basal para Cientos Científicos y Tecnológicos de Excelencia (CEDENNA), under project AFB180001. GC acknowledges financial support from the EMPIR program 17FUN08-TOPS, co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program.
<table><caption>**Table 1.** Group of nonreciprocal magnonic systems discussed in this paper and their key physical features.</caption><thead><tr><th>Magnonic system</th><th>Source of nonreciprocity</th><th>Limit<sup>a</sup></th><th>Amplitude nonreciprocity</th><th>Frequency nonreciprocity</th><th>References</th></tr></thead><tbody><tr><td>Thick film</td><td>Dipolar coupling</td><td>kd &gt;&gt; 1</td><td>Yes</td><td>No</td><td>[185]</td></tr><tr><td>Thin film with surface anisotropies</td><td>Surface/interface anisotropy</td><td>kd &lt;&lt; 1</td><td>Yes</td><td>Yes</td><td>[186]</td></tr><tr><td>Ultrathin film/HM</td><td>Interfacial Dzyaloshinskii–Moriya coupling</td><td>kd &lt;&lt; 1</td><td>Yes<sup>b</sup></td><td>Yes</td><td>[187, 188]</td></tr><tr><td>Non-centrosymmetric chiral magnet</td><td>Bulk Dzyaloshinskii–Moriya coupling</td><td>c</td><td>Yes<sup>b</sup></td><td>Yes</td><td>[188, 189]</td></tr><tr><td>AFM bilayer</td><td>Dipolar coupling</td><td>kd &lt; 1</td><td></td><td></td><td>[113, 190]</td></tr><tr><td>Graded magnetization film</td><td>Dipolar coupling</td><td>kd &gt; 1</td><td>Yes</td><td>Yes</td><td>[191]</td></tr><tr><td>Curved surfaces</td><td>Dipolar coupling</td><td>kR &lt;&lt; 1</td><td>Yes</td><td>Yes</td><td>[192]</td></tr></tbody></table>

<sup>a</sup>Referring to spin waves detected by BLS, where the *k* value is of the order of 10–20 rad µm⁻¹, the limit *kd* ≪ 1 (*kd* < 1) corresponds to *d* in the range of a few (tens of) nanometers.

<sup>b</sup>In this case the amplitude nonreciprocity has nothing to do with the localization at the top or bottom film surface, typical of thick films, but with the DMI-induced lifting of energy degeneration for ±*k*.

<sup>c</sup>In bulk DMI systems, nonreciprocal propagation has been observed in systems with *kd* ≥ 1, and *kd* ≫ 1, so the nonreciprocal behavior is in principle not limited.

(6.7 nm) bilayer give Δ*f* about 1–2 GHz [190], in agreement with theory. These values are comparable with typical ones for iDMI systems. Nonetheless, from the above formula for Δ*f*, even larger values are found if the thickness *d* is increased, as in the recent study of CoFeB (45 nm)/Ru (0.6 nm)/CoFeB (45 nm) [113]. Remarkably, the frequency nonreciprocity can be turned off by changing the relative magnetic orientation from antiparallel to parallel alignment, i.e. just by simple switching, without any rotation of the field, as it is required in the case of DMI systems. Moreover, when the frequency of excitation is lower than *f*(*k* = 0), but larger than the low-lying frequency of the dispersion curve (figure 30, central panel), spin waves can travel only in one direction, as a spin current diode, leading to the phenomenon of resilience to back reflection [113], making unidirectional magnon propagation very robust under imperfections and defects (figure 30, bottom panel).

Nonreciprocity from dipolar coupling also appears in films with graded magnetization along the thickness [191] and curved systems [192]. Spin-polarized electric currents, as well as the flexoelectric interaction, also induce a frequency shift of two counterpropagating spin waves (references can be found in [188]).

Another interesting behavior arises when spin waves propagate in a system with a *periodic* DMI [193], which may be obtained by patterning an array of Pt wires on top of an ultra-thin ferromagnetic film. In such film, the DMI is activated only underneath the heavy-metal wires, where nonreciprocal spin waves are expected. It has been shown from calculations and simulations that periodic DMI causes indirect magnonic gaps, flat bands and a complex temporal evolution of the spin waves [193]. Such a system with periodic DMI can be regarded as a *chiral* MC, where topological magnons should be observed.

## 16.2. Current and future challenges

Although the frequency nonreciprocity of the spin waves could in principle lead to the design and the fabrication of new kinds of devices (isolators, circulators, phase shifters, etc), this task requires proper material properties and the ability to

manipulate spin waves. In our opinion, among the different systems leading to spin-wave nonreciprocity, the oppositely magnetized bilayer, or SyAF, deserves special attention and is presently the best candidate to realize real devices. First, frequency nonreciprocity is easily reconfigurable just by switching the bilayer magnetization from parallel to antiparallel without any bias field rotation, as occurs in the case of DMI systems where reconfigurability requires an in-plane or out-of-plane magnetization rotation. Second, there is no need of ultrathin layers and expensive HMs that are necessary to achieve a large DMI at the interface. Note also that the HM in iDMI systems increases the magnetic damping, which imposes a problem for magnonic applications. Moreover, the predicted frequency nonreciprocity in the SyAF [113, 190] can be even larger than the usual values measured for iDMI systems, paving the way to possible applications (Table 1).

## 16.3. Concluding remarks

The phenomenon of frequency nonreciprocity provides additional features for the manipulation and control of the spin-wave spectrum. Frequency nonreciprocity can be easily reconfigurable with an external field, depending on the nonreciprocal magnonic system. Among them, the magnetic bilayer seems to be one of the best candidates for the realization of nonreciprocal magnonic devices, since is easily reconfigurable, it provides a large frequency nonreciprocity without ultrathin layers, and it only requires ordinary magnetic materials that are fully compatible with CMOS technologies.

## Acknowledgments

PL and RAG acknowledge funding from FONDECYT 11170736, and 1201153, and Financiamiento Basal para Cientos Científicos y Tecnológicos de Excelencia (CEDENNA), under project AFB180001. GC acknowledges financial support from the EMPIR program 17FUN08-TOPS, co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program.
# 17. Power flow and caustics in thin magnetic materials

Joo-Von Kim¹, Robert L Stamps², and Robert E Camley³

¹CNRS and Université Paris-Saclay, France

²University of Manitoba, Canada

³University of Colorado, United States of America

## 17.1. Status

Energy flows radially outward from an oscillating point source placed in water, i.e. we see circular ripples whose phase and group velocities point in the same direction. The situation is significantly different for anisotropic elastic crystals. In an anisotropic solid, the phase and group velocities do not point in the same direction. As a result, even though the phase velocities from the oscillating point source may go out equally in all directions, the energy propagation may be narrowly focused into beams in only a small number of directions. Highly focused beams are known as caustics (or burning rays). Studies of phonon focusing, both experimental and theoretical [194], began in the 1980s and continue to this day as they give information on the elastic properties of materials and on heat transport in solids.

It was not until around 2005 that magnon focusing began to be studied and understood [195, 196]. This delay was due, in part, to there being no simple experimental methods to measure the effect. This changed with the emergence of microfocused BLS. Interestingly, magnon focusing displays important features that phonon systems do not. Focusing depends on the existence of some kind of anisotropy, and the inherent anisotropy in elastic systems is basically fixed by the crystal structure. In contrast, magnon focusing has multiple sources for inducing anisotropic behavior. This includes, for example, magnetocrystalline anisotropy, dipolar coupling and anisotropic exchange interactions. Most significantly, the anisotropy can be induced by tunable factors such as the magnitude and direction of an applied magnetic field or spin current.

We illustrate the origin of focusing in figure 31. Here we plot isofrequency curves in k-space for a thin permalloy film. For example, the phase velocity in the *x* direction, can be calculated from the dispersion relation by $v_{p,x} = \omega/k_x$ where $\omega$ is the angular frequency and $k_x$ is the wavevector in the *x* direction. If one is interested in power flow, however, the situation is quite different. The transport of energy is instead given by the group velocity, i.e. $v_g = \nabla_k \omega$. This is reminiscent of the equation relating force, *F*, and potential energy, *U*, in mechanics, $F = -\nabla U$, and one recalls that the force is in the direction perpendicular to curves of constant potential energy. Thus in the case of energy propagation in anisotropic materials, the direction of energy flow is the direction perpendicular to curves of constant frequency. As is illustrated in figure 31(a), this means that regions with high curvature result in the energy spread out over many directions. In contrast, flat regions in the isofrequency curves lead to the energy being sent in one direction only. In principle, caustics occur when the curvature

<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /><figcaption>**Figure 31.** (a) Example of isofrequency curves and focusing for a 50 nm permalloy film with an applied field of 1 kOe. The numbers indicate the frequency in GHz. The blue arrows indicate directions of strong focusing. Small black arrows indicate regions with no focusing. (b) Experimental observation of focused beams in YIG from a small opening on left side. Reprinted (figure) with permission from [196], Copyright (2010) by the American Physical Society.</figcaption></figure>

is exactly zero. As indicated by the large blue arrows, showing the direction of large energy flow, different frequencies are focused into different directions.

An experimental demonstration of focusing in a thin YIG film which is 7.7 microns thick is shown in figure 31(b). The source for the waves is a small channel on the left side. The energy flow is observed using microfocus BLS. It is immediately obvious that the energy leaving the channel does not display a diffractive type behavior as expected for an opening that is not small compared to the wavelength. Instead, one sees two narrow beams leaving the source, in agreement with what one expects from an analysis of the group velocities. It also interesting to note the reflection properties of these beams when they hit the edge of the sample. For waves in isotropic media, one expects the incident angle to be the reflected angle. Whereas the position where the wave hits the boundary acts like a new source as one expects, the beams produced propagate in one of the preferred directions, rather than producing a reflected wave front.
# 17. Power flow and caustics in thin magnetic materials

Joo-Von Kim¹, Robert L Stamps², and Robert E Camley³

¹CNRS and Université Paris-Saclay, France

²University of Manitoba, Canada

³University of Colorado, United States of America

## 17.1. Status

Energy flows radially outward from an oscillating point source placed in water, i.e. we see circular ripples whose phase and group velocities point in the same direction. The situation is significantly different for anisotropic elastic crystals. In an anisotropic solid, the phase and group velocities do not point in the same direction. As a result, even though the phase velocities from the oscillating point source may go out equally in all directions, the energy propagation may be narrowly focused into beams in only a small number of directions. Highly focused beams are known as caustics (or burning rays). Studies of phonon focusing, both experimental and theoretical [194], began in the 1980s and continue to this day as they give information on the elastic properties of materials and on heat transport in solids.

It was not until around 2005 that magnon focusing began to be studied and understood [195, 196]. This delay was due, in part, to there being no simple experimental methods to measure the effect. This changed with the emergence of microfocused BLS. Interestingly, magnon focusing displays important features that phonon systems do not. Focusing depends on the existence of some kind of anisotropy, and the inherent anisotropy in elastic systems is basically fixed by the crystal structure. In contrast, magnon focusing has multiple sources for inducing anisotropic behavior. This includes, for example, magnetocrystalline anisotropy, dipolar coupling and anisotropic exchange interactions. Most significantly, the anisotropy can be induced by tunable factors such as the magnitude and direction of an applied magnetic field or spin current.

We illustrate the origin of focusing in figure 31. Here we plot isofrequency curves in k-space for a thin permalloy film. For example, the phase velocity in the *x* direction, can be calculated from the dispersion relation by $v_{p,x} = \omega/k_x$ where $\omega$ is the angular frequency and $k_x$ is the wavevector in the *x* direction. If one is interested in power flow, however, the situation is quite different. The transport of energy is instead given by the group velocity, i.e. $v_g = \nabla_k \omega$. This is reminiscent of the equation relating force, *F*, and potential energy, *U*, in mechanics, $F = -\nabla U$, and one recalls that the force is in the direction perpendicular to curves of constant potential energy. Thus in the case of energy propagation in anisotropic materials, the direction of energy flow is the direction perpendicular to curves of constant frequency. As is illustrated in figure 31(a), this means that regions with high curvature result in the energy spread out over many directions. In contrast, flat regions in the isofrequency curves lead to the energy being sent in one direction only. In principle, caustics occur when the curvature

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /><figcaption>**Figure 31.** (a) Example of isofrequency curves and focusing for a 50 nm permalloy film with an applied field of 1 kOe. The numbers indicate the frequency in GHz. The blue arrows indicate directions of strong focusing. Small black arrows indicate regions with no focusing. (b) Experimental observation of focused beams in YIG from a small opening on left side. Reprinted (figure) with permission from [196], Copyright (2010) by the American Physical Society.</figcaption></figure>

is exactly zero. As indicated by the large blue arrows, showing the direction of large energy flow, different frequencies are focused into different directions.

An experimental demonstration of focusing in a thin YIG film which is 7.7 microns thick is shown in figure 31(b). The source for the waves is a small channel on the left side. The energy flow is observed using microfocus BLS. It is immediately obvious that the energy leaving the channel does not display a diffractive type behavior as expected for an opening that is not small compared to the wavelength. Instead, one sees two narrow beams leaving the source, in agreement with what one expects from an analysis of the group velocities. It also interesting to note the reflection properties of these beams when they hit the edge of the sample. For waves in isotropic media, one expects the incident angle to be the reflected angle. Whereas the position where the wave hits the boundary acts like a new source as one expects, the beams produced propagate in one of the preferred directions, rather than producing a reflected wave front.
<figure><img src="image_38.png" /><figcaption>**Figure 32.** Isofrequency curves and the resulting focusing pattern for a thin ferromagnetic film influenced by iDMI. There is a magnetic field (vertically) parallel to the *xy* plane. The figures in the top panel show the isofrequency curves and the bottom sets of panels present the focusing patterns calculated analytically and numerically. Reprinted (figure) with permission from [200], Copyright (2016) by the American Physical Society.</figcaption></figure>

Spin systems have recently been investigated with the idea that spin waves could encode and process information with significant reductions in energy consumption when compared to electronic circuits. The fact that spin waves of different frequencies will be sent in different directions from a point source offers multiple practical advantages. For example, one can: (1) envision a magnonic de-multiplexer [197, 198] where multiple frequencies enter an antenna and are automatically sent in different directions according to their individual frequencies; (2) create narrow spin wave beams without the need for physical waveguides, and (3) tune the direction of a spin wave beam by changing the magnitude or direction of a static magnetic field or electric field [199], thereby allowing logic devices to be developed.

Up to this point, we have discussed the focusing of spin waves in ferromagnetic films with thicknesses ranging from 10 nm to a few microns. For these materials, internal dipolar fields provide the anisotropy required to cause focusing and caustics. However, for ultrathin ferromagnetic films only 1–2 nm in thickness, the anisotropy caused by dipolar effects is minimal and the constant frequency curves are nearly circular. Interestingly, there are several methods to produce and control caustic beams in this limit. One example occurs in thin ferromagnetic films which lie on top of a substrate that induces an iDMI. In this case, the combination of the dipolar

effects with the DMI can not only produce caustic beams, but create nonreciprocal caustic beams [200] where beam in the *x* direction does not have an equivalent beam pointing in the −*x* direction.

An example of nonreciprocal caustics is shown in figure 32 for a 2 nm thick film in an external field of $\mu_0H_0 = 0.8$ T. The top panels show isofrequency curves for two different frequencies. The middle panels show the focusing pattern calculated analytically, and the bottom panels present micro-magnetic simulations illustrating the spin wave energy as a function of position, calculated from the time average of an oscillating component of the magnetization, $m_z^2(r, t)$, over two periods following 150 periods of oscillation. The results are dramatic. For example, at 4.2 GHz the isofrequency curve has a very flat region on the left side and a rounded region on the right side. This leads to a single beam propagating in the −*x* direction.

A second method to create and modify caustics in ultra-thin films has recently been investigated [201]. All of the previous examples dealt with wavelengths that are large compared to a lattice constant. When one looks as high wavevector spin waves, the lattice structure itself can create anisotropic isofrequency curves, with the resulting focusing of energy flows.

## 17.2. Current and future challenges

Large scale applications for ferromagnetic spin waves in magnonic computing may require creation of multiple beams and methods to tune the focusing on a small, local scale. It will be necessary to investigate a number of options, including tuning the focusing by local magnetic fields and/or by polarized spin currents.

Recent advances in microscopy techniques may allow focusing effects to be probed on extremely short length scales. This might require artificial point sources such as anti-dots [202]. Progress in spin wave imaging with NV-center magnetometry allows caustics to be imaged with high spatial resolution [203], which opens up new avenues of exploration for short-wavelength spin waves in ultrathin films. Moreover, such detection techniques could be implemented with a spatial array of sensors, which could pave the way toward information processing with caustics.

Focused beams can also occur for systems with fast spin dynamics. For example, spin waves in antiferromagnets coupled to electromagnetic fields are polariton excitations with frequencies in the THz range. Dispersion relations for these waves can be highly anisotropic and display fascinating features such as the Goos–Hänchen effect and negative refraction. Formation of tunable caustics, a three-dimensional analogy to the focusing described here, is predicted for beams transmitted through thin antiferro- and ferri-magnetic films. Moreover, some multiferroics with magneto-electric coupling display exotic spin orderings and chiral interactions that lead to highly anisotropic electromagnetic energy flows at short wavelengths. These materials may offer new possibilities for electric and magnetic field control of focusing.
<figure><img src="image_36.png" /><figcaption>**Figure 32.** Isofrequency curves and the resulting focusing pattern for a thin ferromagnetic film influenced by iDMI. There is a magnetic field (vertically) parallel to the *xy* plane. The figures in the top panel show the isofrequency curves and the bottom sets of panels present the focusing patterns calculated analytically and numerically. Reprinted (figure) with permission from [200], Copyright (2016) by the American Physical Society.</figcaption></figure>

Spin systems have recently been investigated with the idea that spin waves could encode and process information with significant reductions in energy consumption when compared to electronic circuits. The fact that spin waves of different frequencies will be sent in different directions from a point source offers multiple practical advantages. For example, one can: (1) envision a magnonic de-multiplexer [197, 198] where multiple frequencies enter an antenna and are automatically sent in different directions according to their individual frequencies; (2) create narrow spin wave beams without the need for physical waveguides, and (3) tune the direction of a spin wave beam by changing the magnitude or direction of a static magnetic field or electric field [199], thereby allowing logic devices to be developed.

Up to this point, we have discussed the focusing of spin waves in ferromagnetic films with thicknesses ranging from 10 nm to a few microns. For these materials, internal dipolar fields provide the anisotropy required to cause focusing and caustics. However, for ultrathin ferromagnetic films only 1–2 nm in thickness, the anisotropy caused by dipolar effects is minimal and the constant frequency curves are nearly circular. Interestingly, there are several methods to produce and control caustic beams in this limit. One example occurs in thin ferromagnetic films which lie on top of a substrate that induces an iDMI. In this case, the combination of the dipolar

effects with the DMI can not only produce caustic beams, but create nonreciprocal caustic beams [200] where beam in the *x* direction does not have an equivalent beam pointing in the −*x* direction.

An example of nonreciprocal caustics is shown in figure 32 for a 2 nm thick film in an external field of $\mu_0H_0 = 0.8$ T. The top panels show isofrequency curves for two different frequencies. The middle panels show the focusing pattern calculated analytically, and the bottom panels present micro-magnetic simulations illustrating the spin wave energy as a function of position, calculated from the time average of an oscillating component of the magnetization, $m_z^2(r, t)$, over two periods following 150 periods of oscillation. The results are dramatic. For example, at 4.2 GHz the isofrequency curve has a very flat region on the left side and a rounded region on the right side. This leads to a single beam propagating in the −*x* direction.

A second method to create and modify caustics in ultra-thin films has recently been investigated [201]. All of the previous examples dealt with wavelengths that are large compared to a lattice constant. When one looks as high wavevector spin waves, the lattice structure itself can create anisotropic isofrequency curves, with the resulting focusing of energy flows.

## 17.2. Current and future challenges

Large scale applications for ferromagnetic spin waves in magnonic computing may require creation of multiple beams and methods to tune the focusing on a small, local scale. It will be necessary to investigate a number of options, including tuning the focusing by local magnetic fields and/or by polarized spin currents.

Recent advances in microscopy techniques may allow focusing effects to be probed on extremely short length scales. This might require artificial point sources such as anti-dots [202]. Progress in spin wave imaging with NV-center magnetometry allows caustics to be imaged with high spatial resolution [203], which opens up new avenues of exploration for short-wavelength spin waves in ultrathin films. Moreover, such detection techniques could be implemented with a spatial array of sensors, which could pave the way toward information processing with caustics.

Focused beams can also occur for systems with fast spin dynamics. For example, spin waves in antiferromagnets coupled to electromagnetic fields are polariton excitations with frequencies in the THz range. Dispersion relations for these waves can be highly anisotropic and display fascinating features such as the Goos–Hänchen effect and negative refraction. Formation of tunable caustics, a three-dimensional analogy to the focusing described here, is predicted for beams transmitted through thin antiferro- and ferri-magnetic films. Moreover, some multiferroics with magneto-electric coupling display exotic spin orderings and chiral interactions that lead to highly anisotropic electromagnetic energy flows at short wavelengths. These materials may offer new possibilities for electric and magnetic field control of focusing.
<figure><img src="image_39.png" /><figcaption>**Figure 33.** (a) Schematic diagram shows the origin of PMA at FM/oxide interface due to hybridization of out-of-plane 2p orbitals of O and out-of-plane 3d orbitals of FM. (b) The schematic illustration shows the mechanism of VCMA. When dc gate voltage $V_G$ is applied, the charge density at the out-of-plane 3d orbitals of FM is changed with respect to the in-plane orbitals. This affects interfacial orbital hybridization and changes PMA through SOC of FM.</figcaption></figure>

### 17.3. Concluding remarks

Because, in part, of the potential applications, the field of focused power flows outlined above is continuing to expand and appear in a variety of different areas. In addition, focused beams may allow us to learn about issues such as tunable heat transport or tunable transport of angular momentum. Given the range of applications and fundamental issues, the field of focused power flows in magnetic materials is likely to flourish.

### Acknowledgments

RLS: acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG). REC thanks UCCS for Distinguished Professor research funding.

## 18. Voltage-controlled magnetic anisotropy induced spin waves

Bivas Rana<fnref n="1" /> and YoshiChika Otani<fnref n="1" />,<fnref n="2" />

### 18.1. Status

Apart from fundamental interests, the magnonics also aims to find out efficient ways to excite and manipulate spin waves (SWs) for the development of future magnonic devices, which will eventually surpass the operational speed, efficiency, functionality, integration density of current semiconductor-based electronic devices. So far, the SWs have been excited by conventional charge current, light, and thermally induced methods, such as radio-frequency current-induced Oersted

fields, spin-transfer-torques, femtosecond pulsed laser beams, and thermal energy. Despite having some advantages, all these methods also have several disadvantages such as high energy dissipation in the form of Joule heating, instability of the miniaturized electrical junctions due to high current density, the difficulty of integration in nanoscale devices, excitation of coherent SWs, i.e., excitation of magnons in the ballistic regime and so on.

As an alternative, the excitation of SWs by electric-field is highly desirable. Importantly, the electronic spins localized in partially filled orbitals govern the magnetic properties of ordered magnetic materials. Although external electric-field cannot directly couple with those localized spins, it can interact indirectly with the spins via several means. In ferromagnetic semiconductors, electric-field modulation of carrier density controls the spins. In multiferroic materials, the strong coupling between magnetic and electric polarization enables us to control the spins by electric-field. In piezoelectric materials, the electric-field induced strain can deform an adjacent magnetostrictive ferromagnetic film (e.g., Ni), where the lattice deformation couples with the spins through SOC, known as magnetostriction. In the quest of a new electric-field induced method, the VCMA has emerged as a novel means to control the spins by electric-field. The PMA occurs at the interfaces between ultrathin 3d transition ferromagnetic (FM) metals (e.g., Fe, CoFeB) and nonmagnetic insulators (e.g., MgO, Al₂O₃) due to the hybridization of out-of-plane 3d orbitals of FMs and out-of-plane 2p orbitals of O (figure 33(a)). The electric field applied at the FM/oxide interfaces changes the electronic population significantly in the out-of-plane 3d orbitals of FMs compared with in-plane orbitals (figure 33(b)), eventually modifying the interfacial orbital hybridization and controls the interfacial spins through relativistic SOC of FMs [204].

As VCMA relies upon the modification of electronic population in 3d orbitals of FM, it is suitable for microwave applications. The excitation of FMR [205] and SWs [206, 207]

<footnote n="1">¹RIKEN, Japan</footnote>

<footnote n="2">²University of Tokyo, Japan</footnote>
<figure><img src="image_37.png" /><figcaption>**Figure 33.** (a) Schematic diagram shows the origin of PMA at FM/oxide interface due to hybridization of out-of-plane 2p orbitals of O and out-of-plane 3d orbitals of FM. (b) The schematic illustration shows the mechanism of VCMA. When dc gate voltage $V_G$ is applied, the charge density at the out-of-plane 3d orbitals of FM is changed with respect to the in-plane orbitals. This affects interfacial orbital hybridization and changes PMA through SOC of FM.</figcaption></figure>

### 17.3. Concluding remarks

Because, in part, of the potential applications, the field of focused power flows outlined above is continuing to expand and appear in a variety of different areas. In addition, focused beams may allow us to learn about issues such as tunable heat transport or tunable transport of angular momentum. Given the range of applications and fundamental issues, the field of focused power flows in magnetic materials is likely to flourish.

### Acknowledgments

RLS: acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG). REC thanks UCCS for Distinguished Professor research funding.

## 18. Voltage-controlled magnetic anisotropy induced spin waves

Bivas Rana<fnref n="1" /> and YoshiChika Otani<fnref n="1" />,<fnref n="2" />

### 18.1. Status

Apart from fundamental interests, the magnonics also aims to find out efficient ways to excite and manipulate spin waves (SWs) for the development of future magnonic devices, which will eventually surpass the operational speed, efficiency, functionality, integration density of current semiconductor-based electronic devices. So far, the SWs have been excited by conventional charge current, light, and thermally induced methods, such as radio-frequency current-induced Oersted

fields, spin-transfer-torques, femtosecond pulsed laser beams, and thermal energy. Despite having some advantages, all these methods also have several disadvantages such as high energy dissipation in the form of Joule heating, instability of the miniaturized electrical junctions due to high current density, the difficulty of integration in nanoscale devices, excitation of coherent SWs, i.e., excitation of magnons in the ballistic regime and so on.

As an alternative, the excitation of SWs by electric-field is highly desirable. Importantly, the electronic spins localized in partially filled orbitals govern the magnetic properties of ordered magnetic materials. Although external electric-field cannot directly couple with those localized spins, it can interact indirectly with the spins via several means. In ferromagnetic semiconductors, electric-field modulation of carrier density controls the spins. In multiferroic materials, the strong coupling between magnetic and electric polarization enables us to control the spins by electric-field. In piezoelectric materials, the electric-field induced strain can deform an adjacent magnetostrictive ferromagnetic film (e.g., Ni), where the lattice deformation couples with the spins through SOC, known as magnetostriction. In the quest of a new electric-field induced method, the VCMA has emerged as a novel means to control the spins by electric-field. The PMA occurs at the interfaces between ultrathin 3d transition ferromagnetic (FM) metals (e.g., Fe, CoFeB) and nonmagnetic insulators (e.g., MgO, Al₂O₃) due to the hybridization of out-of-plane 3d orbitals of FMs and out-of-plane 2p orbitals of O (figure 33(a)). The electric field applied at the FM/oxide interfaces changes the electronic population significantly in the out-of-plane 3d orbitals of FMs compared with in-plane orbitals (figure 33(b)), eventually modifying the interfacial orbital hybridization and controls the interfacial spins through relativistic SOC of FMs [204].

As VCMA relies upon the modification of electronic population in 3d orbitals of FM, it is suitable for microwave applications. The excitation of FMR [205] and SWs [206, 207]

<footnote n="1">¹RIKEN, Japan</footnote>

<footnote n="2">²University of Tokyo, Japan</footnote>
<figure><img src="image_40.png" /><figcaption>**Figure 34.** (a) Schematic illustration shows the excitation of coherent propagating SWs by VCMA. (b) Schematic diagram illustrates the magnetization precession in an elliptic trajectory. The out-of-plane component ($m_z$) of magnetization precession oscillates at a frequency twice than the in-plane components ($m_x, m_y$). (c) Power spectral density of the microwave signal emitted by a MTJ under VCMA induced parametric excitation. Curves are vertically offset for clarity and are listed in order of drive frequency. The figure is reproduced from reference [208]. (d) Periodic nature of electric field applied at CoFeB/MgO interface through periodically arranged metal gate electrodes, giving rise to two periodic regions: region 1 (outside of top electrodes) and region 2 (underneath the top electrodes). (e) Experimentally measured frequency versus wavevector dispersion at $V_G = -4$ V under the application of 200 mT bias magnetic field. Symbols represent measured SW frequencies, while blue lines denote SW intensities as calculated by plane wave method. The corresponding color map is given inside. The dashed vertical line indicates the position of anticrossing, and the corresponding magnonic band gap is shown by the shaded region. (f) and (g) Calculated spatial profiles of the SW modes for wavevector $k = 7.1 \times 10^6$ rad m$^{-1}$ under the application of $V_G = -4$ V at $\mu_0H = 200$ mT. Low frequency mode, i.e., mode 1 is confined in region 1 and high frequency mode, i.e., mode 2 is confined in region 2.</figcaption></figure>

in the gigahertz regime have been demonstrated with ultralow power consumption, which can be at least two orders of magnitude lower than the current induced spin-transfer torques excitation. The coherent propagating SWs can be excited by placing a metal gate electrode on top of the oxide layer and applying RF voltage across the gate electrode and FM (figure 34(a)). The RF electric field at the FM/oxide interface modulates interfacial PMA and excites SWs at the resonance condition. The excited SW amplitude is linearly proportional to the RF voltage [206]. The maximum wavenumber of excited SWs is given by the inverse of the lateral width of the gate electrode. One of the key advantages is that the VCMA excitation is local, and the excitation area is right underneath the gate electrode. This enables to utilize VCMA in nanoscale devices with high integration density. Although VCMA cannot excite coherent SWs when the magnetization is aligned either in-plane or perpendicular to the film because of zero VCMA torque, these particular magnetization configurations are suitable for nonlinear parametric excitation of SWs. In this case, the time-varying VCMA couples to the out-of-plane component of dynamic magnetization with elliptical trajectory (figure 34(b)) [207]. Therefore, SW fre-

quency is half of the frequency of VCMA (figure 34(c)) [208]. The critical advantage of parametric excitation is that it can efficiently excite and selectively amplify even shorter wavelength SWs. Unlike linear excitation, the parametric excitation requires a threshold power which depends upon excitation frequency, SW wavevector, the width of waveguide and gate electrode.

Apart from excitation, the modulation of SW properties (e.g. frequency) by VCMA have also been demonstrated [209]. The VCMA can also be utilized for the formation of recon-figurable nanochannels for SW propagation [210]. When a number of parallel nanochannels are formed on an ultrathin ferromagnetic film by placing parallel gate electrodes and applying gate voltage, the parallel nanochannels can be used as 1D MCs (figure 34(d)). In absence of nanochannels, i.e., gate voltage, only a single SW mode with trivial dispersion is observed. Choudhury et al have experimentally demonstrated the presence of two SW modes with nontrivial dispersion curves separated by a tunable band gap in VCMA induced parallel nanochannels (figure 34(e)) [211]. The higher (mode 2) and lower (mode 1) frequency modes are confined within channel 2 and channel 1, respectively (figures 34(f) and (g)).
<figure><img src="image_38.png" /><figcaption>**Figure 34.** (a) Schematic illustration shows the excitation of coherent propagating SWs by VCMA. (b) Schematic diagram illustrates the magnetization precession in an elliptic trajectory. The out-of-plane component ($m_z$) of magnetization precession oscillates at a frequency twice than the in-plane components ($m_x, m_y$). (c) Power spectral density of the microwave signal emitted by a MTJ under VCMA induced parametric excitation. Curves are vertically offset for clarity and are listed in order of drive frequency. The figure is reproduced from reference [208]. (d) Periodic nature of electric field applied at CoFeB/MgO interface through periodically arranged metal gate electrodes, giving rise to two periodic regions: region 1 (outside of top electrodes) and region 2 (underneath the top electrodes). (e) Experimentally measured frequency versus wavevector dispersion at $V_G = -4$ V under the application of 200 mT bias magnetic field. Symbols represent measured SW frequencies, while blue lines denote SW intensities as calculated by plane wave method. The corresponding color map is given inside. The dashed vertical line indicates the position of anticrossing, and the corresponding magnonic band gap is shown by the shaded region. (f) and (g) Calculated spatial profiles of the SW modes for wavevector $k = 7.1 \times 10^6$ rad m$^{-1}$ under the application of $V_G = -4$ V at $\mu_0H = 200$ mT. Low frequency mode, i.e., mode 1 is confined in region 1 and high frequency mode, i.e., mode 2 is confined in region 2.</figcaption></figure>

in the gigahertz regime have been demonstrated with ultralow power consumption, which can be at least two orders of magnitude lower than the current induced spin-transfer torques excitation. The coherent propagating SWs can be excited by placing a metal gate electrode on top of the oxide layer and applying RF voltage across the gate electrode and FM (figure 34(a)). The RF electric field at the FM/oxide interface modulates interfacial PMA and excites SWs at the resonance condition. The excited SW amplitude is linearly proportional to the RF voltage [206]. The maximum wavenumber of excited SWs is given by the inverse of the lateral width of the gate electrode. One of the key advantages is that the VCMA excitation is local, and the excitation area is right underneath the gate electrode. This enables to utilize VCMA in nanoscale devices with high integration density. Although VCMA cannot excite coherent SWs when the magnetization is aligned either in-plane or perpendicular to the film because of zero VCMA torque, these particular magnetization configurations are suitable for nonlinear parametric excitation of SWs. In this case, the time-varying VCMA couples to the out-of-plane component of dynamic magnetization with elliptical trajectory (figure 34(b)) [207]. Therefore, SW fre-

quency is half of the frequency of VCMA (figure 34(c)) [208]. The critical advantage of parametric excitation is that it can efficiently excite and selectively amplify even shorter wavelength SWs. Unlike linear excitation, the parametric excitation requires a threshold power which depends upon excitation frequency, SW wavevector, the width of waveguide and gate electrode.

Apart from excitation, the modulation of SW properties (e.g. frequency) by VCMA have also been demonstrated [209]. The VCMA can also be utilized for the formation of recon-figurable nanochannels for SW propagation [210]. When a number of parallel nanochannels are formed on an ultrathin ferromagnetic film by placing parallel gate electrodes and applying gate voltage, the parallel nanochannels can be used as 1D MCs (figure 34(d)). In absence of nanochannels, i.e., gate voltage, only a single SW mode with trivial dispersion is observed. Choudhury et al have experimentally demonstrated the presence of two SW modes with nontrivial dispersion curves separated by a tunable band gap in VCMA induced parallel nanochannels (figure 34(e)) [211]. The higher (mode 2) and lower (mode 1) frequency modes are confined within channel 2 and channel 1, respectively (figures 34(f) and (g)).
## 18.2. Current and future challenges

We have to sort out many challenges to fully utilize the advantages of VCMA in magnonic devices [212]. Although many experimental reports demonstrate the excitation and characterization of VCMA induced FMR [205, 213], only a couple of experimental reports, on the other hand, show the VCMA induced propagating SWs [206, 208], manipulation of SW properties by VCMA [209, 211] and control of damping constant by electric field [214]. Therefore, more experimental studies are necessary for understanding the properties and characters of VCMA excited linear and nonlinear SWs.

So far, it is observed that most of the applied microwave power is reflected back from the devices while exciting FMR, SWs by VCMA because of impedance mismatch [205]. In principle, impedance mismatch can be minimized by reducing the thickness of the oxide layer. However, it may significantly increase the junction current leading to many spurious effects such as spin-transfer-torque induced SWs, Joule heating, etc. Hence, the optimization of junction impedance to minimize electrical losses and impedance mismatch with an optimum value of the VCMA coefficient is a real technical challenge.

The VCMA coefficient for FM/oxide heterostructures can be up to 100 fJ V⁻¹ m⁻¹. One of the major challenges is to enhance the VCMA coefficient at least up to pJ V⁻¹ m⁻¹ to fulfill the minimum criteria for the potential application of VCMA in practical devices. This can be achieved by engineering FM/oxide interface. It has been observed that doping of HMs or insertion of an ultrathin heavy metallic layer such as Ta, Pt, W, Ir at FM/oxide interface can significantly enhance the value of the VCMA coefficient. The enhancement of the VCMA coefficient is also possible even by inserting the ultrathin layer of relatively lighter materials such as Hf, Mg. The physical reason behind this is the modification of orbital hybridization at FM/oxide interface and suppression of surface oxidization of the FM layer. However, the VCMA coefficient cannot be increased more than few hundreds of fJ V⁻¹ m⁻¹ by these methods. Alternatively, the VCMA coefficient can be further increased up to few pJ V⁻¹ m⁻¹ by utilizing voltage-controlled redox reactions, charge trapping, electromigration. As these methods rely upon the slow movement of ions, require thermal activation process and have very slow response time, they cannot be applied in microwave devices. However, they can be used for other purposes such as modulation of SW properties, formation of nanochannels and so on.

As VCMA is an interfacial effect, it is limited to ultrathin FM films. Growing high-quality thin films with a smooth and defect-free clean interface is therefore highly desirable. Although molecular beam epitaxy can grow high-quality ultrathin films, it is inapplicable for large scale production, i.e., for commercial purposes. Consequently, it is a real challenge to produce high quality thin films by dc or RF sputtering.

The overall damping constant of ultrathin FM films is generally higher than thicker films, which limits the SW propagation length down to micrometer distance. The SW amplifiers are essential to increase the propagation length by compensating the energy losses of SWs during propagation.

Conventionally, charge current based spin-orbit-torque and parametric pumping methods are used to amplify SWs effectively. Alternatively, VCMA torque could also replace the charge current-induced torque to build up an energy-efficient amplifier. However, experimental demonstrations are lacking. Therefore, in addition to increasing the VCMA coefficient, the searching for FM materials with ultralow damping parameter is also getting equal importance. One may think about ferrimagnetic insulators and Heusler alloys as the possible alternatives; unfortunately, VCMA phenomena have not been observed yet in these materials.

Another future challenge will be the efficient detection of VCMA induced SWs or magnons in ultrathin FM films by electrical means. Recently, the local, quantitative, and phase-sensitive detection of the SWs have also been demonstrated by single NV centers in diamond. But, this method can increase the complexity of device fabrication as FM films need to be deposited directly on a diamond containing NV centers. Alternatively, MTJs can be used as an efficient way to detect SWs in ultrathin films. However, experimental demonstrations are still lacking.

Ultimately, the interaction of VCMA induced SWs with various types of spin configurations such as magnetic skyrmions, and domain walls need to be studied experimentally for the development of magnonic devices. For instance, the driving skyrmions, domain walls by VCMA induced SWs can be considered. Moreover, properties of the induced SWs in VCMA controlled reconfigurable nanochannels, 2D MCs, and magnonic logic gates need to be studied experimentally. Recently, interfacial PMA has also been examined in ultrathin FM films hybridized with 2D materials, showing a significant increment of interfacial PMA in these hybrid structures. Therefore, the study of VCMA phenomena and VCMA induced SWs may also pave the way for developing various types of voltage-controlled 2D magnonic devices. Consequently new ideas such as inducing in-plane magnetic anisotropy at FM/oxide interface by controlled magnetic annealing and electric field control of induced anisotropy [215] and inverse VCMA effect for detecting SWs are also getting importance due to possible technological advantages.

## 18.3. Concluding remarks

VCMA has emerged as a potential technique to excite and manipulate SWs in ultrathin FM films for the development of all-electric-field controlled low power magnonic devices. However, the VCMA operated magnonic devices are still in its infant stage of development. Indeed, we have to experimentally demonstrate all the theoretical proposals, such as VCMA induced parametric amplification of SWs, dynamic interaction of SWs propagating through voltage-controlled nanochannels [210], voltage-controlled logic gates, 2D MCs [216], SW caustics. Above all, it is indispensable to improve the VCMA coefficient by the interface and material engineering.

## Acknowledgments

This review is supported by Grant-in-Aid for Scientific Research on Innovative Area, 'Nano Spin Conversion Science'
## 18.2. Current and future challenges

We have to sort out many challenges to fully utilize the advantages of VCMA in magnonic devices [212]. Although many experimental reports demonstrate the excitation and characterization of VCMA induced FMR [205, 213], only a couple of experimental reports, on the other hand, show the VCMA induced propagating SWs [206, 208], manipulation of SW properties by VCMA [209, 211] and control of damping constant by electric field [214]. Therefore, more experimental studies are necessary for understanding the properties and characters of VCMA excited linear and nonlinear SWs.

So far, it is observed that most of the applied microwave power is reflected back from the devices while exciting FMR, SWs by VCMA because of impedance mismatch [205]. In principle, impedance mismatch can be minimized by reducing the thickness of the oxide layer. However, it may significantly increase the junction current leading to many spurious effects such as spin-transfer-torque induced SWs, Joule heating, etc. Hence, the optimization of junction impedance to minimize electrical losses and impedance mismatch with an optimum value of the VCMA coefficient is a real technical challenge.

The VCMA coefficient for FM/oxide heterostructures can be up to 100 fJ V⁻¹ m⁻¹. One of the major challenges is to enhance the VCMA coefficient at least up to pJ V⁻¹ m⁻¹ to fulfill the minimum criteria for the potential application of VCMA in practical devices. This can be achieved by engineering FM/oxide interface. It has been observed that doping of HMs or insertion of an ultrathin heavy metallic layer such as Ta, Pt, W, Ir at FM/oxide interface can significantly enhance the value of the VCMA coefficient. The enhancement of the VCMA coefficient is also possible even by inserting the ultrathin layer of relatively lighter materials such as Hf, Mg. The physical reason behind this is the modification of orbital hybridization at FM/oxide interface and suppression of surface oxidization of the FM layer. However, the VCMA coefficient cannot be increased more than few hundreds of fJ V⁻¹ m⁻¹ by these methods. Alternatively, the VCMA coefficient can be further increased up to few pJ V⁻¹ m⁻¹ by utilizing voltage-controlled redox reactions, charge trapping, electromigration. As these methods rely upon the slow movement of ions, require thermal activation process and have very slow response time, they cannot be applied in microwave devices. However, they can be used for other purposes such as modulation of SW properties, formation of nanochannels and so on.

As VCMA is an interfacial effect, it is limited to ultrathin FM films. Growing high-quality thin films with a smooth and defect-free clean interface is therefore highly desirable. Although molecular beam epitaxy can grow high-quality ultrathin films, it is inapplicable for large scale production, i.e., for commercial purposes. Consequently, it is a real challenge to produce high quality thin films by dc or RF sputtering.

The overall damping constant of ultrathin FM films is generally higher than thicker films, which limits the SW propagation length down to micrometer distance. The SW amplifiers are essential to increase the propagation length by compensating the energy losses of SWs during propagation.

Conventionally, charge current based spin-orbit-torque and parametric pumping methods are used to amplify SWs effectively. Alternatively, VCMA torque could also replace the charge current-induced torque to build up an energy-efficient amplifier. However, experimental demonstrations are lacking. Therefore, in addition to increasing the VCMA coefficient, the searching for FM materials with ultralow damping parameter is also getting equal importance. One may think about ferrimagnetic insulators and Heusler alloys as the possible alternatives; unfortunately, VCMA phenomena have not been observed yet in these materials.

Another future challenge will be the efficient detection of VCMA induced SWs or magnons in ultrathin FM films by electrical means. Recently, the local, quantitative, and phase-sensitive detection of the SWs have also been demonstrated by single NV centers in diamond. But, this method can increase the complexity of device fabrication as FM films need to be deposited directly on a diamond containing NV centers. Alternatively, MTJs can be used as an efficient way to detect SWs in ultrathin films. However, experimental demonstrations are still lacking.

Ultimately, the interaction of VCMA induced SWs with various types of spin configurations such as magnetic skyrmions, and domain walls need to be studied experimentally for the development of magnonic devices. For instance, the driving skyrmions, domain walls by VCMA induced SWs can be considered. Moreover, properties of the induced SWs in VCMA controlled reconfigurable nanochannels, 2D MCs, and magnonic logic gates need to be studied experimentally. Recently, interfacial PMA has also been examined in ultrathin FM films hybridized with 2D materials, showing a significant increment of interfacial PMA in these hybrid structures. Therefore, the study of VCMA phenomena and VCMA induced SWs may also pave the way for developing various types of voltage-controlled 2D magnonic devices. Consequently new ideas such as inducing in-plane magnetic anisotropy at FM/oxide interface by controlled magnetic annealing and electric field control of induced anisotropy [215] and inverse VCMA effect for detecting SWs are also getting importance due to possible technological advantages.

## 18.3. Concluding remarks

VCMA has emerged as a potential technique to excite and manipulate SWs in ultrathin FM films for the development of all-electric-field controlled low power magnonic devices. However, the VCMA operated magnonic devices are still in its infant stage of development. Indeed, we have to experimentally demonstrate all the theoretical proposals, such as VCMA induced parametric amplification of SWs, dynamic interaction of SWs propagating through voltage-controlled nanochannels [210], voltage-controlled logic gates, 2D MCs [216], SW caustics. Above all, it is indispensable to improve the VCMA coefficient by the interface and material engineering.

## Acknowledgments

This review is supported by Grant-in-Aid for Scientific Research on Innovative Area, 'Nano Spin Conversion Science'
(Grant Number 26103002). BR acknowledges RIKEN Incen-
tive Research Project Grant No. FY2019, for partial financial
support.

19. Indirect interactions between magnets

Weichao Yu<fnref n="1" />, Tao Yu<fnref n="2" /> and Gerrit E W Bauer<fnref n="1" />,<fnref n="3" />

19.1. Status

The exchange interactions between electron spins in con-
densed matter generates a rich variety of magnetic order.
It becomes exponentially small at larger than atomic dis-
tances, so different magnets interact by exchange only when
in direct contact. The weaker dipole interaction generates
forces between macroscopic magnets that decay algebraically
with distance. Magnets can also interact indirectly through
a non-magnetic medium. Mobile electrons in metals medi-
ate an oscillatory RKKY non-local exchange interaction over
nanometers. The coupling by non-equilibrium spin currents
through metal spacers can reach over micrometers. The
exchange of spin waves synchronizes magnetic oscillators.
Microwave photons in high-quality cavities are supremely
suited to communicate spin information because of their coher-
ence over large distances and the strong interaction of spin
ensembles such as ferromagnets with the ac magnetic fields
[217, 218]. Other waveforms such as magnons and phonons
also generate indirect interactions between small magnets over
relative large distances [76, 219]. In this roadmap article, we
outline the interest of letting two or more ferromagnets interact
in cavities or wave-guides for different mediating waveforms
but with emphasis on microwaves.

In free space, magnets interact by the magneto-dipolar
interaction, which is very weak for typical sub-nm magnetic
spheres at cm distances. However, they may couple strongly
over large distances by exchanging virtual photons in cavity
modes. Interpreting the collective Kittel mode of a single mag-
net as a magnonic hydrogen atom, an interacting pair forms
a magnonic hydrogen molecule with hybridized orbitals of
even and odd symmetry (figure 35) [220]. When probed by
an even cavity mode, microwave spectroscopy detects the first
‘bright’ mode, but not the second ‘dark’ one [217]. The (sub-
radiant) magnons in the dark mode do not suffer from radiative
damping and have longer decay time, which might be useful
for quantum information storage [221]. The indirect interac-
tion between magnets provides a large playground since the
cavity as well as position, size and type of the magnets is
arbitrary. Adding non-magnetic, but optically active structures
and materials, is another important option. The task to sys-
temize various configurations, to find new hybridizing mech-
anisms, and search for applications is a challenge that has
only just begun.

<figure><img src="image_41.png" /><figcaption>**Figure 35.** The collective dynamics of a ferromagnetic particle (Kittel magnon) can be interpreted as ground state of a bosonic atom. The interaction between different atoms, enhanced by the exchange of photons in open or closed cavities, leads to collective delocalized states.</figcaption></figure>

19.2. Current and future challenges

Customizing cavities: cavities come in different shapes and types. Here we focus on microwave cavities formed by a closed box of high-quality metal that confines photons without significant losses. Small orifices do not disturb the system significantly and act as non-invasive input and output terminals that allow measuring the properties of the hybrid system in terms of the microwave transmission and reflection coefficients. When magnetization damping is small as well, the cavity + load system approximately conserves energy and the system Hamiltonian is (almost) Hermitian. The magnon–photon coupling is then coherent, and photon and magnon levels repel each other when tuned to degeneracy. CPWs are open cavities that offer greater flexibility, but are leaky, i.e. have a smaller quality factor. Waveguides are intrinsically open cavities that channel the free flow of microwaves with associated radiative damping. Dissipation may change the qualitative physics since a non-Hermitian Hamiltonian governs the non-local interaction between magnets [221].

The indirect coupling already modifies the ground state. When not at the minimum of the free energy, mechanical and magnetic torques and forces arise that can be modulated by the magnetic configuration and the frequency detuning. Heat current flow in the presence of temperature differences in the cavity. Feeding the cavity with microwaves from external sources, either via the input and output ports or local coils that address individual magnets [217], can drive the system into highly excited states [222].

Cavities or resonators can also confine optical photons, but the domination electric field coupling is relatively weak. An interesting challenge is the use of cavities in the intermediate THz radiation regime. Thin films confine phonons normal to thin films where they may form phononic spin valves [219], while magnon cavities can be induced by nanostructures on magnetic thin films [76].

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

<footnote n="2">²Max Planck Institute for the Structure and Dynamics of Matter, Germany</footnote>

<footnote n="3">³Groningen University, The Netherlands</footnote>
(Grant Number 26103002). BR acknowledges RIKEN Incen-
tive Research Project Grant No. FY2019, for partial financial
support.

19. Indirect interactions between magnets

Weichao Yu<fnref n="1" />, Tao Yu<fnref n="2" /> and Gerrit E W Bauer<fnref n="1" />,<fnref n="3" />

19.1. Status

The exchange interactions between electron spins in con-
densed matter generates a rich variety of magnetic order.
It becomes exponentially small at larger than atomic dis-
tances, so different magnets interact by exchange only when
in direct contact. The weaker dipole interaction generates
forces between macroscopic magnets that decay algebraically
with distance. Magnets can also interact indirectly through
a non-magnetic medium. Mobile electrons in metals medi-
ate an oscillatory RKKY non-local exchange interaction over
nanometers. The coupling by non-equilibrium spin currents
through metal spacers can reach over micrometers. The
exchange of spin waves synchronizes magnetic oscillators.
Microwave photons in high-quality cavities are supremely
suited to communicate spin information because of their coher-
ence over large distances and the strong interaction of spin
ensembles such as ferromagnets with the ac magnetic fields
[217, 218]. Other waveforms such as magnons and phonons
also generate indirect interactions between small magnets over
relative large distances [76, 219]. In this roadmap article, we
outline the interest of letting two or more ferromagnets interact
in cavities or wave-guides for different mediating waveforms
but with emphasis on microwaves.

In free space, magnets interact by the magneto-dipolar
interaction, which is very weak for typical sub-nm magnetic
spheres at cm distances. However, they may couple strongly
over large distances by exchanging virtual photons in cavity
modes. Interpreting the collective Kittel mode of a single mag-
net as a magnonic hydrogen atom, an interacting pair forms
a magnonic hydrogen molecule with hybridized orbitals of
even and odd symmetry (figure 35) [220]. When probed by
an even cavity mode, microwave spectroscopy detects the first
‘bright’ mode, but not the second ‘dark’ one [217]. The (sub-
radiant) magnons in the dark mode do not suffer from radiative
damping and have longer decay time, which might be useful
for quantum information storage [221]. The indirect interac-
tion between magnets provides a large playground since the
cavity as well as position, size and type of the magnets is
arbitrary. Adding non-magnetic, but optically active structures
and materials, is another important option. The task to sys-
temize various configurations, to find new hybridizing mech-
anisms, and search for applications is a challenge that has
only just begun.

<figure><img src="image_39.png" /><figcaption>**Figure 35.** The collective dynamics of a ferromagnetic particle (Kittel magnon) can be interpreted as ground state of a bosonic atom. The interaction between different atoms, enhanced by the exchange of photons in open or closed cavities, leads to collective delocalized states.</figcaption></figure>

19.2. Current and future challenges

Customizing cavities: cavities come in different shapes and types. Here we focus on microwave cavities formed by a closed box of high-quality metal that confines photons without significant losses. Small orifices do not disturb the system significantly and act as non-invasive input and output terminals that allow measuring the properties of the hybrid system in terms of the microwave transmission and reflection coefficients. When magnetization damping is small as well, the cavity + load system approximately conserves energy and the system Hamiltonian is (almost) Hermitian. The magnon–photon coupling is then coherent, and photon and magnon levels repel each other when tuned to degeneracy. CPWs are open cavities that offer greater flexibility, but are leaky, i.e. have a smaller quality factor. Waveguides are intrinsically open cavities that channel the free flow of microwaves with associated radiative damping. Dissipation may change the qualitative physics since a non-Hermitian Hamiltonian governs the non-local interaction between magnets [221].

The indirect coupling already modifies the ground state. When not at the minimum of the free energy, mechanical and magnetic torques and forces arise that can be modulated by the magnetic configuration and the frequency detuning. Heat current flow in the presence of temperature differences in the cavity. Feeding the cavity with microwaves from external sources, either via the input and output ports or local coils that address individual magnets [217], can drive the system into highly excited states [222].

Cavities or resonators can also confine optical photons, but the domination electric field coupling is relatively weak. An interesting challenge is the use of cavities in the intermediate THz radiation regime. Thin films confine phonons normal to thin films where they may form phononic spin valves [219], while magnon cavities can be induced by nanostructures on magnetic thin films [76].

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

<footnote n="2">²Max Planck Institute for the Structure and Dynamics of Matter, Germany</footnote>

<footnote n="3">³Groningen University, The Netherlands</footnote>
<figure><img src="image_42.png" /><figcaption>**Figure 36.** Numerical simulation of six magnetic spheres on a circle inside a disk-shaped microwave cavity fed by attached ports. At resonance, the magnetizations (black arrows) precess coherently in the presence of a unidirectional photon current as illustrated by the Poynting vector field (magenta arrows). The colored background is the electric field component in the perpendicular direction. We interpret this system as a chiral magnonic benzene molecule (Yu *et al* 2020, *Phys. Rev. B* **102**, 064416).</figcaption></figure>

*Loading cavities:* we may fill the cavities by magnets in different numbers and structural as well as magnetic configurations. Figure 36 shows as an example of a collinear magnonic benzene molecule driven by a microwave feed from the input port. We are not limited to ferro- or ferrimagnets, but may use antiferro- and paramagnets. The magnonic atoms may ‘react’ with other particles made from, e.g., ferroelectrics, superconductors, Josephson qubits [239], metamaterials, or devices such as MTJs.

*Chiral interaction:* in the absence of relativistic effects, the angular and linear momentum of photons, phonons, and magnons waves are in general independent. However, chirality emerges in DE surface spin waves or by spin waves excited by magnetodipolar stray fields [76]. Cavities or waveguides induce admixtures of TM and TE modes that generate local chirality at special lines. When the mediating waves are chiral, the indirect coupling between magnets is unidirectional. A consequence is the accumulation of magnons at the edge of a chain of magnets in a cavity [222]. It is then possible to design chiral magnonic molecules analogous to conventional aromatic molecules, but with non-reciprocal magnon couplings and persistent currents (figure 36), facilitating the design of novel on-chip microwave isolators and circulators. Since spin waves in thin film can be chirally excited by magnetic stray fields, similar games become possible on a much smaller scale. The acoustic Rayleigh surface waves display rotation–momentum coupling and generate novel indirect couplings [223].

*Toroidal moment:* the toroidal dipole moment or anapole is a parameter to describe the electromagnetic far field, which is independent from the magnetic and electric dipoles [224]. Spatially distributed magnets can contribute either static or dynamic toroidal multipoles, which have peculiar response and radiating properties. As discussed above, in a microwave cavity loaded by magnets, chiral spin currents may flow by means of cavity photon exchange. We anticipate that an exchange-strained circular spin texture generates equilibrium

spin currents and associated toroidal moments, while injected microwaves in the configuration of figure 36 generates a giant dynamic anapole.

*Toward quantum:* strongly coupled microwave photons can drive a weakly damped magnonic system easily into the non-linear regime, which is an important prerequisite for interesting quantum effects such as quantum entanglement and quantum squeezing. Non-linearities generate highly entangled magnon–photon states or generate tripartite entanglement between magnons, photons and phonons [225].

### 19.3. Concluding remarks

Magnons in spatially separated magnets may hybridize by the coherent exchange of (quasi)particles such as photons, phonons, and continuum magnons. Coupled magnet assemblies form very flexible devices on various length scales. AC and DC magnetic fields easily manipulate their collective bosonic excitations, offering a unique platform to study quantum effects and test new functionalities needed for next-generation information technologies.

## Acknowledgments

This work was supported by JSPS Kakenhi (Grant No. 19H00645 and No. 20K14369). TY acknowledges financial support by Deutsche Forschungsgemeinschaft through the Emmy Noether Program (SE 2558/2-1).

# 20. Topological magnonics

Christian Back<fnref n="1" /> and Götz S Uhrig<fnref n="2" />

## 20.1. Status

Generally, topological effects in condensed matter display a certain robustness against perturbations such as imperfections and disorder resulting from sample growth and patterning. This idea extends also to magnonics and motivates to consider and to realize topological effects. A second appealing feature of topological magnetic excitations is their generic chirality. Their edge modes travel only in one particular sense, see figure 37, and section 16. Recently, it has been proposed that magnonic topological edge modes in ferromagnetic domains can be used for advanced information processing [226]. This route is promising because it can be hoped that such systems are less dissipative. If the magnons can be manipulated coherently quantum information processing comes within reach. This perspective has triggered a surge of interest in topological magnonics in recent years which makes it impossible to review them here so that we refer to a longer survey for a more complete list of references [227].

Notwithstanding the obvious interest in topologically protected magnon modes, the experimental reports concerning topological magnonics are still scarce and can be grouped into two categories: detection of the magnon Hall effect, e.g.

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

<footnote n="2">²Technical University of Dortmund, Germany</footnote>
<figure><img src="image_40.png" /><figcaption>**Figure 36.** Numerical simulation of six magnetic spheres on a circle inside a disk-shaped microwave cavity fed by attached ports. At resonance, the magnetizations (black arrows) precess coherently in the presence of a unidirectional photon current as illustrated by the Poynting vector field (magenta arrows). The colored background is the electric field component in the perpendicular direction. We interpret this system as a chiral magnonic benzene molecule (Yu *et al* 2020, *Phys. Rev. B* **102**, 064416).</figcaption></figure>

*Loading cavities:* we may fill the cavities by magnets in different numbers and structural as well as magnetic configurations. Figure 36 shows as an example of a collinear magnonic benzene molecule driven by a microwave feed from the input port. We are not limited to ferro- or ferrimagnets, but may use antiferro- and paramagnets. The magnonic atoms may ‘react’ with other particles made from, e.g., ferroelectrics, superconductors, Josephson qubits [239], metamaterials, or devices such as MTJs.

*Chiral interaction:* in the absence of relativistic effects, the angular and linear momentum of photons, phonons, and magnons waves are in general independent. However, chirality emerges in DE surface spin waves or by spin waves excited by magnetodipolar stray fields [76]. Cavities or waveguides induce admixtures of TM and TE modes that generate local chirality at special lines. When the mediating waves are chiral, the indirect coupling between magnets is unidirectional. A consequence is the accumulation of magnons at the edge of a chain of magnets in a cavity [222]. It is then possible to design chiral magnonic molecules analogous to conventional aromatic molecules, but with non-reciprocal magnon couplings and persistent currents (figure 36), facilitating the design of novel on-chip microwave isolators and circulators. Since spin waves in thin film can be chirally excited by magnetic stray fields, similar games become possible on a much smaller scale. The acoustic Rayleigh surface waves display rotation–momentum coupling and generate novel indirect couplings [223].

*Toroidal moment:* the toroidal dipole moment or anapole is a parameter to describe the electromagnetic far field, which is independent from the magnetic and electric dipoles [224]. Spatially distributed magnets can contribute either static or dynamic toroidal multipoles, which have peculiar response and radiating properties. As discussed above, in a microwave cavity loaded by magnets, chiral spin currents may flow by means of cavity photon exchange. We anticipate that an exchange-strained circular spin texture generates equilibrium

spin currents and associated toroidal moments, while injected microwaves in the configuration of figure 36 generates a giant dynamic anapole.

*Toward quantum:* strongly coupled microwave photons can drive a weakly damped magnonic system easily into the non-linear regime, which is an important prerequisite for interesting quantum effects such as quantum entanglement and quantum squeezing. Non-linearities generate highly entangled magnon–photon states or generate tripartite entanglement between magnons, photons and phonons [225].

### 19.3. Concluding remarks

Magnons in spatially separated magnets may hybridize by the coherent exchange of (quasi)particles such as photons, phonons, and continuum magnons. Coupled magnet assemblies form very flexible devices on various length scales. AC and DC magnetic fields easily manipulate their collective bosonic excitations, offering a unique platform to study quantum effects and test new functionalities needed for next-generation information technologies.

## Acknowledgments

This work was supported by JSPS Kakenhi (Grant No. 19H00645 and No. 20K14369). TY acknowledges financial support by Deutsche Forschungsgemeinschaft through the Emmy Noether Program (SE 2558/2-1).

## 20. Topological magnonics

Christian Back<fnref n="1" /> and Götz S Uhrig<fnref n="2" />

### 20.1. Status

Generally, topological effects in condensed matter display a certain robustness against perturbations such as imperfections and disorder resulting from sample growth and patterning. This idea extends also to magnonics and motivates to consider and to realize topological effects. A second appealing feature of topological magnetic excitations is their generic chirality. Their edge modes travel only in one particular sense, see figure 37, and section 16. Recently, it has been proposed that magnonic topological edge modes in ferromagnetic domains can be used for advanced information processing [226]. This route is promising because it can be hoped that such systems are less dissipative. If the magnons can be manipulated coherently quantum information processing comes within reach. This perspective has triggered a surge of interest in topological magnonics in recent years which makes it impossible to review them here so that we refer to a longer survey for a more complete list of references [227].

Notwithstanding the obvious interest in topologically protected magnon modes, the experimental reports concerning topological magnonics are still scarce and can be grouped into two categories: detection of the magnon Hall effect, e.g.

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

<footnote n="2">²Technical University of Dortmund, Germany</footnote>
<figure><img src="image_43.png" /><figcaption>**Figure 37.** Schematic illustration of a propagating chiral magnon localised at the edge of a confined sample realizing a ferromagnetic Shastry–Sutherland lattice, for details see reference [233].</figcaption></figure>

[228–230], and observation of topological magnon bands e.g. [231, 232].

In 2010, Onose *et al* [228] reported a transverse magnon current in the insulating collinear ferromagnet (FM) $Lu_2V_2O_7$ with pyrochlore structure. In this system and in similar pyrochlore compounds ($Ho_2V_2O_7$, $In_2Mn_2O_7$, $Lu_2V_2O_7$, and the quantum spin ice system $Tb_2Ti_2O_7$) [229] as well as in the material YIG [230], a transverse magnon current has been observed. Notably, the detection of the magnon Hall effect has been realized in insulating materials only so far, where a temperature gradient drives the magnon current. Detection is realized by thermal Hall effect measurements [228, 229] or, in the case of YIG, by direct observation of a temperature difference [230]. In the classical pyrochlore materials [228] the rather strong DMI is at the origin of the generation of a Berry curvature of the magnon bands which leads to the transverse deflection of the exchange dominated magnon current, see also section 13. In YIG [230], DMI is weak and the explanation of the origin of the sideways deflection of long-wavelength magnons in the dipole-exchange regime should be revisited.

In the large class of antiferromagnets, in many materials with strong SOC the DMI is strong enough to have a significant impact on the magnon band structure and non-collinear order may be induced [232]. In particular, quasi 2D kagome or honeycomb materials are of interest, where flat, topological magnon bands have been predicted and recently detected using thermal Hall and inelastic neutron scattering experiments [231, 232].

## 20.2. Current and future challenges

So far, topological effects have been measured in a limited number of systems which are mostly magnetically ordered so that the excitations are magnons of which the total number is generically conserved except if certain anisotropies are present, e.g., dipolar interactions [233]. Thermal Hall measurements have confirmed non-trivial topology. Clearly, the measurements of further properties providing evidence for topological behavior are called for. In parallel, it is of great current interest to identify more magnetic systems with non-trivial

topology, for a class of potential candidates, see reference [234] and figure 38. In addition, completely novel ideas such as using Moiré bands of twisted ferromagnetic bilayers are proposed [235]. It would be highly advantageous to find systems with larger exchange coupling and concomitantly larger couplings such as DMIs inducing the required Berry curvatures. The advantage of larger couplings is twofold: the phenomenon can be detected at higher temperatures and they occur on shorter time scales. The latter point is essential if one wants to move from equilibrium properties to non-equilibrium physics which is obviously needed to tackle switching and information processing.

In addition, the extensions of the current scope on ferromagnetic magnons to antiferromagnetic (AFM) magnons as such [236] or on top of non-collinear spin textures [237] as well as to other magnetic excitations represent fascinating challenges, both in concepts and for applications. Fundamentally, AFM magnons are different from FM magnons in that they display linear dispersions and that their number is not conserved. The linear dispersion leads to larger frequencies and faster dynamics at given length scales. Note, however, that experimental detection of magnon modes in AFMs is much more challenging and suitable experimental methods are to be developed. Beyond magnons, there are further intriguing excitations for instance fractional spinons which may be relevant in quantum spin liquids [229] and triplons in valence bond solids [238]. For the latter, non-trivial topology is verified by inelastic neutron scattering and they are protected by an energy gap.

Looking further ahead, the control and manipulation of topological magnetic excitations must be targeted, requiring to enter the field of non-equilibrium physics. Certainly, this represents a formidable challenge. Is it possible to create, to manipulate and to detect wave packets of magnons or even single excitation? Can this be achieved coherently? Which techniques are promising? A possible candidate is optical control by laser pulses, e.g., exploiting the inverse Faraday effect [239].

Finally, one may ask if realizable device concepts utilizing topological magnonics can be developed. This would require first of all theoretical concepts, but also clear recipes toward experimental realization. Ideally this would encompass room temperature operation and a selection of materials that are easily fabricated by standard deposition tools.

One could expect, for example, that transverse magnon currents are observable e.g. in a micro-focus BLS, experiment by observing magnon deflection in a temperature gradient. Given the superior sensitivity of BLS such an experiment on bulk samples might be feasible. Furthermore, one can foresee the design of magnetically ordered nanostructures where chiral edge current may be detectable using bespoke spin-to-charge conversion designs.

## 20.3. Concluding remarks

In summary, the field of topological magnonics is still at its infancy, but it holds great promise. Experimentally it is obviously difficult to realize materials that clearly show topological magnon bands or that show clear manifestations of effects that
<figure><img src="image_41.png" /><figcaption>**Figure 37.** Schematic illustration of a propagating chiral magnon localised at the edge of a confined sample realizing a ferromagnetic Shastry–Sutherland lattice, for details see reference [233].</figcaption></figure>

[228–230], and observation of topological magnon bands e.g. [231, 232].

In 2010, Onose *et al* [228] reported a transverse magnon current in the insulating collinear ferromagnet (FM) $Lu_2V_2O_7$ with pyrochlore structure. In this system and in similar pyrochlore compounds ($Ho_2V_2O_7$, $In_2Mn_2O_7$, $Lu_2V_2O_7$, and the quantum spin ice system $Tb_2Ti_2O_7$) [229] as well as in the material YIG [230], a transverse magnon current has been observed. Notably, the detection of the magnon Hall effect has been realized in insulating materials only so far, where a temperature gradient drives the magnon current. Detection is realized by thermal Hall effect measurements [228, 229] or, in the case of YIG, by direct observation of a temperature difference [230]. In the classical pyrochlore materials [228] the rather strong DMI is at the origin of the generation of a Berry curvature of the magnon bands which leads to the transverse deflection of the exchange dominated magnon current, see also section 13. In YIG [230], DMI is weak and the explanation of the origin of the sideways deflection of long-wavelength magnons in the dipole-exchange regime should be revisited.

In the large class of antiferromagnets, in many materials with strong SOC the DMI is strong enough to have a significant impact on the magnon band structure and non-collinear order may be induced [232]. In particular, quasi 2D kagome or honeycomb materials are of interest, where flat, topological magnon bands have been predicted and recently detected using thermal Hall and inelastic neutron scattering experiments [231, 232].

## 20.2. Current and future challenges

So far, topological effects have been measured in a limited number of systems which are mostly magnetically ordered so that the excitations are magnons of which the total number is generically conserved except if certain anisotropies are present, e.g., dipolar interactions [233]. Thermal Hall measurements have confirmed non-trivial topology. Clearly, the measurements of further properties providing evidence for topological behavior are called for. In parallel, it is of great current interest to identify more magnetic systems with non-trivial

topology, for a class of potential candidates, see reference [234] and figure 38. In addition, completely novel ideas such as using Moiré bands of twisted ferromagnetic bilayers are proposed [235]. It would be highly advantageous to find systems with larger exchange coupling and concomitantly larger couplings such as DMIs inducing the required Berry curvatures. The advantage of larger couplings is twofold: the phenomena can be detected at higher temperatures and they occur on shorter time scales. The latter point is essential if one wants to move from equilibrium properties to non-equilibrium physics which is obviously needed to tackle switching and information processing.

In addition, the extensions of the current scope on ferromagnetic magnons to antiferromagnetic (AFM) magnons as such [236] or on top of non-collinear spin textures [237] as well as to other magnetic excitations represent fascinating challenges, both in concepts and for applications. Fundamentally, AFM magnons are different from FM magnons in that they display linear dispersions and that their number is not conserved. The linear dispersion leads to larger frequencies and faster dynamics at given length scales. Note, however, that experimental detection of magnon modes in AFMs is much more challenging and suitable experimental methods are to be developed. Beyond magnons, there are further intriguing excitations for instance fractional spinons which may be relevant in quantum spin liquids [229] and triplons in valence bond solids [238]. For the latter, non-trivial topology is verified by inelastic neutron scattering and they are protected by an energy gap.

Looking further ahead, the control and manipulation of topological magnetic excitations must be targeted, requiring to enter the field of non-equilibrium physics. Certainly, this represents a formidable challenge. Is it possible to create, to manipulate and to detect wave packets of magnons or even single excitation? Can this be achieved coherently? Which techniques are promising? A possible candidate is optical control by laser pulses, e.g., exploiting the inverse Faraday effect [239].

Finally, one may ask if realizable device concepts utilizing topological magnonics can be developed. This would require first of all theoretical concepts, but also clear recipes toward experimental realization. Ideally this would encompass room temperature operation and a selection of materials that are easily fabricated by standard deposition tools.

One could expect, for example, that transverse magnon currents are observable e.g. in a micro-focus BLS, experiment by observing magnon deflection in a temperature gradient. Given the superior sensitivity of BLS such an experiment on bulk samples might be feasible. Furthermore, one can foresee the design of magnetically ordered nanostructures where chiral edge current may be detectable using bespoke spin-to-charge conversion designs.

## 20.3. Concluding remarks

In summary, the field of topological magnonics is still at its infancy, but it holds great promise. Experimentally it is obviously difficult to realize materials that clearly show topological magnon bands or that show clear manifestations of effects that
<figure><img src="image_44.png" /><figcaption>**Figure 38.** Panel (a) Shastry–Sutherland lattice formed in a rare earth (RE) silicide (two-dimensional projection). Panel (b) magnonic dispersion in a strip of the lattice in (a) extended in *a*-direction, confined in *b*-direction, for spin *S*, exchange coupling *J*, single-ion anisotropy *A* = 0.2*J*, and DMI *D*<sub>*z*</sub> = 0.25*J* in *z*-direction. The dispersion of the right-moving chiral edge mode is shown in orange, the one of the left-moving in blue, for details see reference [234].</figcaption></figure>

are based on topology such as the magnon Hall effect. On the theory side, concepts for novel effects need to be developed which put the particular properties of topologically protected modes to use. The possibility to tailor the spin wave velocity of magnonic edge modes is an example for such a concept.

## Acknowledgments

CB acknowledges fruitful discussion with Takuya Taniguchi; GSU thanks Maik Malki for useful discussions and providing the figures.

# 21. Low- and room-temperature hybrid magnonic heterostructures

Oleksandr V Dobrovolskiy<fnref n="1" />, Barbora Budinska<fnref n="1" />, Huajun Qin<fnref n="2" /> and Sebastiaan van Dijken<fnref n="2" />

## 21.1. Status

Hybrid magnonic systems have recently attracted great attention because of emerging fundamental physical phenomena unseen in their elementary building blocks and due to their application potential, particularly, in coherent information processing [240]. They provide a modern paradigm for combining platforms and devices based on strong coupling between excitations in the charge, lattice and spin degrees of freedom and allow for tailoring the response of the system in one degree of freedom via tuning the excitations in the other one [241]. Magnons are highly tuneable excitations and can be engineered to couple with various dynamic media and excitations such as photons [240], phonons [241], fluxons [242] etc. This section deals with only two types of hybrid magnonic systems—ferromagnet/superconductor and

ferromagnetic/ferroelectric heterostructures. Recent reviews devoted to hybrid magnonic systems of various other types can be found, e.g. in references [240, 241, 243].

## 21.2. Ferromagnet/superconductor hybrid structures for cryogenic magnonics

Traditionally, magnonics is a room-temperature research domain. However, recent studies are extending magnonics to cryogenic temperatures [221, 242, 244, 245]. This extension is driven by investigations of strongly coupled light-spin hybrid systems for quantum computing that approach the quantum limit of excitation [221]. In addition, thermal effects such as reduced saturation magnetization and thermally activated motion of topological defects are less pronounced at low temperatures, that leads to new phenomena in the spin-wave dynamics. Importantly, in cryogenic magnonics, microwave experiments are often performed in environments of superconducting quantum circuits and various hybrid devices based on Josephson junctions. Because of experimental constraints for light intensities in a purely optomagnonic system, quantum magnonics requires hybridized systems of magnon excitations and nonlinear macroscopic quantum systems such as superconducting qubits [221], see figure 39(a). At the same time, ferromagnetism (F) and superconductivity (S) entail opposite spin orders and thus their combination gives rise to numerous novel phenomena [242, 244, 245]. However, so far coexistence of F and S in bulk systems remains a rare circumstance peculiar to complex compounds in which either a strong ferromagnetic order coexists with unconventional spin-triplet superconductivity or orbital coupling stipulates coexistence of antiferromagnetic order (section 15) with conventional spin-singlet superconductivity.

The coexistence of S and F can be readily achieved in artificial S/F heterostructures which allow for numerous approaches for S/F hybridization. For instance, in proximity-coupled (i.e. electrically contacted) S/F/S three-layers in in-plane fields, a

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

<footnote n="2">²Aalto University, Finland</footnote>
<figure><img src="image_42.png" /><figcaption>**Figure 38.** Panel (a) Shastry–Sutherland lattice formed in a rare earth (RE) silicide (two-dimensional projection). Panel (b) magnonic dispersion in a strip of the lattice in (a) extended in *a*-direction, confined in *b*-direction, for spin *S*, exchange coupling *J*, single-ion anisotropy *A* = 0.2*J*, and DMI *D*<sub>*z*</sub> = 0.25*J* in *z*-direction. The dispersion of the right-moving chiral edge mode is shown in orange, the one of the left-moving in blue, for details see reference [234].</figcaption></figure>

are based on topology such as the magnon Hall effect. On the theory side, concepts for novel effects need to be developed which put the particular properties of topologically protected modes to use. The possibility to tailor the spin wave velocity of magnonic edge modes is an example for such a concept.

## Acknowledgments

CB acknowledges fruitful discussion with Takuya Taniguchi; GSU thanks Maik Malki for useful discussions and providing the figures.

# 21. Low- and room-temperature hybrid magnonic heterostructures

Oleksandr V Dobrovolskiy<fnref n="1" />, Barbora Budinska<fnref n="1" />, Huajun Qin<fnref n="2" /> and Sebastiaan van Dijken<fnref n="2" />

## 21.1. Status

Hybrid magnonic systems have recently attracted great attention because of emerging fundamental physical phenomena unseen in their elementary building blocks and due to their application potential, particularly, in coherent information processing [240]. They provide a modern paradigm for combining platforms and devices based on strong coupling between excitations in the charge, lattice and spin degrees of freedom and allow for tailoring the response of the system in one degree of freedom via tuning the excitations in the other one [241]. Magnons are highly tuneable excitations and can be engineered to couple with various dynamic media and excitations such as photons [240], phonons [241], fluxons [242] etc. This section deals with only two types of hybrid magnonic systems—ferromagnet/superconductor and

ferromagnetic/ferroelectric heterostructures. Recent reviews devoted to hybrid magnonic systems of various other types can be found, e.g. in references [240, 241, 243].

## 21.2. Ferromagnet/superconductor hybrid structures for cryogenic magnonics

Traditionally, magnonics is a room-temperature research domain. However, recent studies are extending magnonics to cryogenic temperatures [221, 242, 244, 245]. This extension is driven by investigations of strongly coupled light-spin hybrid systems for quantum computing that approach the quantum limit of excitation [221]. In addition, thermal effects such as reduced saturation magnetization and thermally activated motion of topological defects are less pronounced at low temperatures, that leads to new phenomena in the spin-wave dynamics. Importantly, in cryogenic magnonics, microwave experiments are often performed in environments of superconducting quantum circuits and various hybrid devices based on Josephson junctions. Because of experimental constraints for light intensities in a purely optomagnonic system, quantum magnonics requires hybridized systems of magnon excitations and nonlinear macroscopic quantum systems such as superconducting qubits [221], see figure 39(a). At the same time, ferromagnetism (F) and superconductivity (S) entail opposite spin orders and thus their combination gives rise to numerous novel phenomena [242, 244, 245]. However, so far coexistence of F and S in bulk systems remains a rare circumstance peculiar to complex compounds in which either a strong ferromagnetic order coexists with unconventional spin-triplet superconductivity or orbital coupling stipulates coexistence of antiferromagnetic order (section 15) with conventional spin-singlet superconductivity.

The coexistence of S and F can be readily achieved in artificial S/F heterostructures which allow for numerous approaches for S/F hybridization. For instance, in proximity-coupled (i.e. electrically contacted) S/F/S three-layers in in-plane fields, a

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

<footnote n="2">²Aalto University, Finland</footnote>
<figure><img src="image_45.png" /><figcaption>**Figure 39.** Examples of ferromagnet/superconductor hybrid structures for cryogenic magnonics. (a) Strong dispersive regime of quantum magnonics. Interaction of strength $g_{q-m}$ between the Kittel mode (with frequency $\omega_m$) of a spherical ferrimagnetic crystal of YIG and a superconducting qubit ($\omega_q$), engineered through magnetic- and electric-dipole couplings to a microwave cavity mode (top). Normalized qubit spectrum measured as a function of the coil current (bottom) and at $\omega_q \approx \omega_m$ (right). From [221]. Reprinted with permission from AAAS. (b) Magnonic metamaterial on the basis of S/F hybridization: patterned Py films are placed onto a CPW made of Nb. Black and green arrows show the direction of the microwave propagation and of the external magnetic field. [245] John Wiley & Sons. [Copyright © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]. (c) Bragg scattering of spin waves on the Abrikosov vortex lattice in a Py/Nb bilayer. Spin waves are excited by antenna 1, propagate through the Py waveguide and are detected by antenna 2. The vortex lattice induces a spatially periodic magnetic field $h(x, y)$ in Py, which becomes alternating in time when the vortices move under the action of a transport current. Bragg scattering of spin waves on the moving vortex lattice is accompanied by the Doppler effect. The tunability of the vortex lattice parameter via changing the out-of-plane component of the magnetic field $H_\perp$ allows for engineering of the spin-wave transmission spectrum. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Physics] [242] (2019). (d) Cherenkov excitation of magnons by fast-moving fluxons. Vortex lattice moving with the velocity $v$ induces a spatially periodic ac magnetic field $h(x, t)$ which excites the system of magnetic moments shown by purple arrows (top). This additional dissipation results in current peaks in the $I-V$ characteristics (bottom). An additional magnetic ($M$) viscous force adds up to the BS viscosity due to vortex core alone, resulting in a voltage drop for currents above $I_{on}$ corresponding to some threshold vortex velocity $v_c$. Reprinted (figure) with permission from [246], Copyright (2011) by the American Physical Society.</figcaption></figure>
<figure><img src="image_43.png" /><figcaption>**Figure 39.** Examples of ferromagnet/superconductor hybrid structures for cryogenic magnonics. (a) Strong dispersive regime of quantum magnonics. Interaction of strength $g_{q-m}$ between the Kittel mode (with frequency $\omega_m$) of a spherical ferrimagnetic crystal of YIG and a superconducting qubit ($\omega_q$), engineered through magnetic- and electric-dipole couplings to a microwave cavity mode (top). Normalized qubit spectrum measured as a function of the coil current (bottom) and at $\omega_q \approx \omega_m$ (right). From [221]. Reprinted with permission from AAAS. (b) Magnonic metamaterial on the basis of S/F hybridization: patterned Py films are placed onto a CPW made of Nb. Black and green arrows show the direction of the microwave propagation and of the external magnetic field. [245] John Wiley & Sons. [Copyright © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]. (c) Bragg scattering of spin waves on the Abrikosov vortex lattice in a Py/Nb bilayer. Spin waves are excited by antenna 1, propagate through the Py waveguide and are detected by antenna 2. The vortex lattice induces a spatially periodic magnetic field $h(x, y)$ in Py, which becomes alternating in time when the vortices move under the action of a transport current. Bragg scattering of spin waves on the moving vortex lattice is accompanied by the Doppler effect. The tunability of the vortex lattice parameter via changing the out-of-plane component of the magnetic field $H_\perp$ allows for engineering of the spin-wave transmission spectrum. Reprinted by permission from Springer Nature Customer Service Centre GmbH: [Springer Nature] [Nature Physics] [242] (2019). (d) Cherenkov excitation of magnons by fast-moving fluxons. Vortex lattice moving with the velocity $v$ induces a spatially periodic ac magnetic field $h(x, t)$ which excites the system of magnetic moments shown by purple arrows (top). This additional dissipation results in current peaks in the $I-V$ characteristics (bottom). An additional magnetic ($M$) viscous force adds up to the BS viscosity due to vortex core alone, resulting in a voltage drop for currents above $I_{on}$ corresponding to some threshold vortex velocity $v_c$. Reprinted (figure) with permission from [246], Copyright (2011) by the American Physical Society.</figcaption></figure>
substantial reduction of the FMR field [244] is attributed to the generation of unconventional spin-triplet superconductivity and to an interplay of the F layer with S-induced magnetic flux. In particular, it has been established that the presence of a spatially varying magnetization at an S/F interface can generate spin-polarized triplet supercurrents via spin mixing and spin rotation processes. In case of proximity-decoupled S/F bi-layers in in-plane fields, see figure 39(b), it has been demonstrated experimentally that coupling of spin waves in F with S results in an enhanced phase velocity of the spin waves [245]. This enhancement occurs due to the Meissner screening of ac magnetostatic stray fields by S and it opens access to the development of a new class of metamaterials on the basis of S/F hybridization [245].

A number of novel effects have been observed experimentally for proximity-decoupled S/F bi-layer systems in out-of-plane magnetic fields [242]. When the S layer is in the mixed state, an external magnetic field penetrates it in the form of a lattice of Abrikosov vortices (fluxons). In proximity-decoupled S/F bi-layers, the stray fields emanating from the vortex cores produce a periodic modulation of the magnetic order in F, such that the S/F bi-layer can be viewed as a fluxon-induced MC (section 2). In such a MC, see figure 39(c), forbidden-frequency gaps (bandgaps) are formed for spin waves with Brillouin wavenumbers that correspond to the period of the vortex lattice [242]. The tunability of the vortex lattice parameter via changing the out-of-plane component of the magnetic field allows for engineering of the spin-wave transmission spectrum. Remarkably, experiments on Bragg scattering of spin waves on a flux lattice moving under the action of a transport current in the S layer have revealed that it is accompanied by Doppler shifts [242]. One further promising research direction is related to the experimental examination of the radiation of magnons by moving fluxons via a Cherenkov-type mechanism. The Cherenkov radiation of magnons by fluxons is expected when the vortex velocity exceeds some threshold value and, as predicted theoretically [246], it should to lead to peaks in the current-voltage curve and an additional magnetic (M) contribution to the Bardeen-Stephen (BS) core viscosity, see figure 39(d). Given the vortex lattice parameter which in the presence of out-of-plane fields between 2 T and 10 mT varies from 35 nm to 500 nm, Cherenkov radiation of short-wavelength magnons (section 7) by fast-moving vortices could become a route to the excitation of exchange-dominated spin waves in S/F heterostructures and antiferromagnetic superconductors.

## 21.3. Ferromagnetic/ferroelectric heterostructures for programmable magnonics at room temperature

Studies of coupled magnetic and ferroelectric (FE) phases are motivated by fundamental questions about ferroic order coexistence and their potential for low-power nanoelectronic devices. Hybrid ferromagnetic (FM)/FE materials in which magnetoelectric interactions arise from charge modulation, exchange coupling, or strain transfer at composite interfaces are particularly promising because they allow for electric-field control of magnetism at room temperature [247]. In addition, FE domains can be imprinted into a FM film to

form a programmable metamaterial without physical patterning [248, 249]. Here, we review magnonic phenomena arising from strong local coupling of FM and FE order parameters. Sections 8 and 18 describe electric-field manipulation of spin waves using piezoelectrics and gate dielectrics.

Recent studies exploiting FM films grown onto FE $BaTiO_3$ substrates have highlighted the potential of such hybrids for active spin-wave control. In this material system, local strain transfer from 90° stripe domains in $BaTiO_3$ induces a regular modulation of magnetic anisotropy in the adjacent FM film via inverse magnetostriction. If the magnetoelastic anisotropy dominates other magnetic energies, the two ferroic domain patterns correlate (figures 40(a) and (b)). Because domain walls in FE materials are only a few nanometres wide, the induced changes of anisotropy in the FM film are nearly abrupt. This has two important consequences; sharp lateral variations of the effective magnetic field and strong pinning of magnetic domain walls onto their FE counterpart.

The modulation of magnetic anisotropy in strain-coupled FM/FE bilayers confines the excitation of spin-wave modes to single stripe domains, as demonstrated experimentally in reference [250]. At low frequency, standing spin waves form only in domains with small effective field (every second stripe), allowing spin-wave localization and guiding (figure 40(c)). Additionally, the narrow anisotropy boundaries act as local sources of propagating spin waves under uniform microwave excitation (figure 40(d)). Micromagnetic simulations indicate that the wavelength of the emitted spin waves scales down well into the exchange-dominated regime, enabling short-wavelength magnonics (section 7). Mode localization and short-wavelength spin-wave emission persist up to large magnetic field, facilitating active tuning of the spin-wave frequency and wavelength [250].

Strong pinning of straight magnetic domain walls in FM/FE heterostructures offers various other magnonic prospects. For instance, when driven into periodic oscillations by a spin-polarized ac electric current, the domain walls act as a local source of propagating spin waves [251]. Additionally, spin-wave transport through pinned magnetic domain walls can be actively turned on and off by reversible switching between two non-volatile domain-wall spin textures [132]. This programmable effect, illustrated in figures 40(e)–(g), is explained by a large difference in domain-wall width if the magnetization is set to a head-to-head/tail-to-tail or a head-to-tail configuration.

Electric-field control of spin waves in FM/FE bilayers with correlated domain patterns is a logical next step. Switching of the spontaneous polarization reorients the unit cell of the FE material (figures 40(a) and (b)) and, via strain transfer, this alters the direction or strength of the magnetoelastic anisotropy in the FM film. Local magnetic switching, the writing and erasure of magnetic domain patterns and reversible motion of magnetic domain walls in an electric field are demonstrated already [249]. For magnonics, the ability to move magnetic domain walls and switch the anisotropy by low-power electric fields offers an attractive new approach for active spin-wave manipulation. Reports along this research direction will likely appear in the coming years.
substantial reduction of the FMR field [244] is attributed to the generation of unconventional spin-triplet superconductivity and to an interplay of the F layer with S-induced magnetic flux. In particular, it has been established that the presence of a spatially varying magnetization at an S/F interface can generate spin-polarized triplet supercurrents via spin mixing and spin rotation processes. In case of proximity-decoupled S/F bi-layers in in-plane fields, see figure 39(b), it has been demonstrated experimentally that coupling of spin waves in F with S results in an enhanced phase velocity of the spin waves [245]. This enhancement occurs due to the Meissner screening of ac magnetostatic stray fields by S and it opens access to the development of a new class of metamaterials on the basis of S/F hybridization [245].

A number of novel effects have been observed experimentally for proximity-decoupled S/F bi-layer systems in out-of-plane magnetic fields [242]. When the S layer is in the mixed state, an external magnetic field penetrates it in the form of a lattice of Abrikosov vortices (fluxons). In proximity-decoupled S/F bi-layers, the stray fields emanating from the vortex cores produce a periodic modulation of the magnetic order in F, such that the S/F bi-layer can be viewed as a fluxon-induced MC (section 2). In such a MC, see figure 39(c), forbidden-frequency gaps (bandgaps) are formed for spin waves with Brillouin wavenumbers that correspond to the period of the vortex lattice [242]. The tunability of the vortex lattice parameter via changing the out-of-plane component of the magnetic field allows for engineering of the spin-wave transmission spectrum. Remarkably, experiments on Bragg scattering of spin waves on a flux lattice moving under the action of a transport current in the S layer have revealed that it is accompanied by Doppler shifts [242]. One further promising research direction is related to the experimental examination of the radiation of magnons by moving fluxons via a Cherenkov-type mechanism. The Cherenkov radiation of magnons by fluxons is expected when the vortex velocity exceeds some threshold value and, as predicted theoretically [246], it should to lead to peaks in the current-voltage curve and an additional magnetic (M) contribution to the Bardeen-Stephen (BS) core viscosity, see figure 39(d). Given the vortex lattice parameter which in the presence of out-of-plane fields between 2 T and 10 mT varies from 35 nm to 500 nm, Cherenkov radiation of short-wavelength magnons (section 7) by fast-moving vortices could become a route to the excitation of exchange-dominated spin waves in S/F heterostructures and antiferromagnetic superconductors.

## 21.3. Ferromagnetic/ferroelectric heterostructures for programmable magnonics at room temperature

Studies of coupled magnetic and ferroelectric (FE) phases are motivated by fundamental questions about ferroic order coexistence and their potential for low-power nanoelectronic devices. Hybrid ferromagnetic (FM)/FE materials in which magnetoelectric interactions arise from charge modulation, exchange coupling, or strain transfer at composite interfaces are particularly promising because they allow for electric-field control of magnetism at room temperature [247]. In addition, FE domains can be imprinted into a FM film to

form a programmable metamaterial without physical patterning [248, 249]. Here, we review magnonic phenomena arising from strong local coupling of FM and FE order parameters. Sections 8 and 18 describe electric-field manipulation of spin waves using piezoelectrics and gate dielectrics.

Recent studies exploiting FM films grown onto FE $BaTiO_3$ substrates have highlighted the potential of such hybrids for active spin-wave control. In this material system, local strain transfer from 90° stripe domains in $BaTiO_3$ induces a regular modulation of magnetic anisotropy in the adjacent FM film via inverse magnetostriction. If the magnetoelastic anisotropy dominates other magnetic energies, the two ferroic domain patterns correlate (figures 40(a) and (b)). Because domain walls in FE materials are only a few nanometres wide, the induced changes of anisotropy in the FM film are nearly abrupt. This has two important consequences; sharp lateral variations of the effective magnetic field and strong pinning of magnetic domain walls onto their FE counterpart.

The modulation of magnetic anisotropy in strain-coupled FM/FE bilayers confines the excitation of spin-wave modes to single stripe domains, as demonstrated experimentally in reference [250]. At low frequency, standing spin waves form only in domains with small effective field (every second stripe), allowing spin-wave localization and guiding (figure 40(c)). Additionally, the narrow anisotropy boundaries act as local sources of propagating spin waves under uniform microwave excitation (figure 40(d)). Micromagnetic simulations indicate that the wavelength of the emitted spin waves scales down well into the exchange-dominated regime, enabling short-wavelength magnonics (section 7). Mode localization and short-wavelength spin-wave emission persist up to large magnetic field, facilitating active tuning of the spin-wave frequency and wavelength [250].

Strong pinning of straight magnetic domain walls in FM/FE heterostructures offers various other magnonic prospects. For instance, when driven into periodic oscillations by a spin-polarized ac electric current, the domain walls act as a local source of propagating spin waves [251]. Additionally, spin-wave transport through pinned magnetic domain walls can be actively turned on and off by reversible switching between two non-volatile domain-wall spin textures [132]. This programmable effect, illustrated in figures 40(e)–(g), is explained by a large difference in domain-wall width if the magnetization is set to a head-to-head/tail-to-tail or a head-to-tail configuration.

Electric-field control of spin waves in FM/FE bilayers with correlated domain patterns is a logical next step. Switching of the spontaneous polarization reorients the unit cell of the FE material (figures 40(a) and (b)) and, via strain transfer, this alters the direction or strength of the magnetoelastic anisotropy in the FM film. Local magnetic switching, the writing and erasure of magnetic domain patterns and reversible motion of magnetic domain walls in an electric field are demonstrated already [249]. For magnonics, the ability to move magnetic domain walls and switch the anisotropy by low-power electric fields offers an attractive new approach for active spin-wave manipulation. Reports along this research direction will likely appear in the coming years.
<figure><img src="image_46.png" /><figcaption>**Figure 40.** (a) Schematic of a FM/FE heterostructure exhibiting full domain-pattern transfer. The spontaneous polarization and lattice tetragonality of the FE layer rotate by 90° at the FE domain walls. Through strain transfer and inverse magnetostriction, this induces an alternating uniaxial magnetoelastic anisotropy in the FM layer. The magnetization aligns along the anisotropy axes in zero magnetic field (arrows). (b) Polarization and Kerr microscopy images of the FE and FM domain structures in a 50 nm CoFeB/BaTiO₃ bilayer. (c) Formation of fifth order standing spin waves in every second stripe domain of the same heterostructure during uniform excitation (simulation) [250]. (d) Emission of propagating spin waves from an anisotropy boundary in the FM/FE heterostructure and the simulated spin-wave dispersion curves for a 60 mT bias field along and perpendicular to the uniaxial magnetic anisotropy of the stripe domains [250]. (e) Kerr microscopy images demonstrating switching between magnetic stripe domains with alternating head-to-head/tail-to-tail domain walls (top) and head-to-tail domain walls (bottom). The blue scale bar corresponds to 10 µm. (f) Schematic of a micro-focused BLS experiment for the analysis of spin-wave transmission through a programmable pinned magnetic domain wall. The orange scale bar corresponds to 2 µm. (g) Demonstration of strong spin-wave reflection if the pinned domain wall is set to a narrow head-to-tail state and nearly full spin-wave transmission after its structure is switched to a broad head-to-head configuration [132]. The BLS scans are performed near the frequency of an antinode resonance mode in the narrow domain wall.</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>**Figure 40.** (a) Schematic of a FM/FE heterostructure exhibiting full domain-pattern transfer. The spontaneous polarization and lattice tetragonality of the FE layer rotate by 90° at the FE domain walls. Through strain transfer and inverse magnetostriction, this induces an alternating uniaxial magnetoelastic anisotropy in the FM layer. The magnetization aligns along the anisotropy axes in zero magnetic field (arrows). (b) Polarization and Kerr microscopy images of the FE and FM domain structures in a 50 nm CoFeB/BaTiO₃ bilayer. (c) Formation of fifth order standing spin waves in every second stripe domain of the same heterostructure during uniform excitation (simulation) [250]. (d) Emission of propagating spin waves from an anisotropy boundary in the FM/FE heterostructure and the simulated spin-wave dispersion curves for a 60 mT bias field along and perpendicular to the uniaxial magnetic anisotropy of the stripe domains [250]. (e) Kerr microscopy images demonstrating switching between magnetic stripe domains with alternating head-to-head/tail-to-tail domain walls (top) and head-to-tail domain walls (bottom). The blue scale bar corresponds to 10 µm. (f) Schematic of a micro-focused BLS experiment for the analysis of spin-wave transmission through a programmable pinned magnetic domain wall. The orange scale bar corresponds to 2 µm. (g) Demonstration of strong spin-wave reflection if the pinned domain wall is set to a narrow head-to-tail state and nearly full spin-wave transmission after its structure is switched to a broad head-to-head configuration [132]. The BLS scans are performed near the frequency of an antinode resonance mode in the narrow domain wall.</figcaption></figure>
## 21.4. Current and future challenges

Given that magnetic moment couples to the current oscillations in hybrid Josephson junctions, care should be taken in the analysis of their FMR response and current-voltage characteristics [244]. Bloch-like band structures in the spin-wave transmission spectra of S/F bilayers require long-range order of the vortex lattice [242] which is hard to achieve in experiments. For the Cherenkov-like generation of magnons by moving fluxons, vortex velocities exceeding $1-3 \text{ km s}^{-1}$ are required [246]. At such high velocities vortex cores may collapse because of the flux-flow instability. To prevent instability, one can use, e.g., vortex guiding in S/F bilayers or superconductors with fast relaxation of disequilibrium, for which vortex velocities of $5-15 \text{ km s}^{-1}$ have recently been demonstrated [252].

Electric-field control of magnetism in FM/FE heterostructures offers a low-power mechanism for spin-wave manipulation at room temperature. To date, proof-of-principle experiments utilize thick single-crystal FE substrates, necessitating the use of relatively large voltages [249]. For practical devices, the voltage pulses should be substantially smaller. Epitaxial FE films could satisfy this requirement [248]. Yet, the growth of thin films with regular FE stripe domains is challenging. Many attractive magnonic features of FM/FE bilayers originate from a regular modulation of magnetic anisotropy and strong pinning of straight magnetic domain walls. Similar effects could be realized also in other material systems, for instance via local ion-beam irradiation of FM films.

## 21.5. Concluding remarks

Investigations of S/F hybrid structures are gaining a momentum, opening access to exciting emerging phenomena in spin-wave physics and advancing novel magnonics and spintronics functionalities. The extension of magnonics to cryogenic temperatures has given birth to the new research direction of magnon fluxonics and metamaterials on the basis of F/S hybridization. Moreover, S/F heterostructures allow for highly efficient magnon-photon coupling and, thus, are of crucial importance for the further development of quantum magnonics.

Heterostructures with coupled FM and FE order parameters have emerged as an attractive material platform for room-temperature programmable magnonics. Imprinting of FE domains into a FM film has enabled new approaches for short-wavelength spin-wave emission, magnonic guiding, and spin-wave manipulation through magnetic- and electric-field actuation. Integration of these magnonic effects in magnetoelectric devices promises new functionalities and low-power operation.

## Acknowledgments

OVD acknowledges financial support by the European Cooperation in Science and Technology via COST Action CA16218 (NANOCOHYBRI) and ERC Starting Grant 678309 MagnonCircuits. HQ acknowledges financial support from the Academy of Finland (Grant Nos. 317918 and 325480). SvD acknowledges financial support by the

European Research Council (Grant Nos. ERC-2012-StG 307502-E-CONTROL, ERC-2014-PoC 665215-EMOTION and ERC-PoC-2018 812841-POWERSPIN) and the Academy of Finland (Grant No. 321983).

## 22. Biologically encoded magnonics

Benjamin W Zingsem¹ and Michael Winklhofer²

¹The University of Duisburg-Essen, Germany

²The Carl von Ossietzky University of Oldenburg, Germany

### 22.1. Status

Biologically encoded magnonics or biomagnonics, for short, is a recent approach to magnonics research, wherein biogenic magnetic material is harnessed for magnonics applications. Magnetotactic bacteria present an ideal model system for biomagnonics studies. These microorganisms have the genetic machinery to form dedicated ensembles of magnetic nanoparticles—magnetosomes—for orientation by the Earth's magnetic field. The magnetosomes consist of a strongly magnetic material (magnetite, $Fe_3O_4$), have magnetic-single domain character, and are arranged in the form of one or several linear chains [253]. Each magnetic particle is enclosed in a membrane vesicle, which not only controls the shape and size of the particle ($<100 \text{ nm}$) but also keeps adjacent particles separated by a clear distance of $8-10 \text{ nm}$ [253]. Thereby, the particles are not exchange-coupled among each other but interact solely by their stray fields. The spatial distribution of the stray fields, and thus the spin-wave spectrum, depends on the geometric arrangement of the particles, which, in turn, is biologically controlled. The mode profile, i.e., the spatial distribution of phase and amplitude, of spin-waves in magnetic nanoparticles exhibits distinct differences from larger systems: at gigahertz excitation frequencies ($\le 20 \text{ GHz}$ at ca 300–400 mT), each nanoparticle responds with a uniform precession [254]. Dipolar pinning, which is known to unfavourably distort the mode profiles near the edges of a large continuous sample [255], is suppressed in nanoparticles due to the strong, short-range exchange coupling. This entails a macrospin behavior where each individual nanoparticle acts as a coherent oscillator. At the same time, wavelengths that are shorter than typical low-frequency exchange- or dipolar spin-waves, can be accommodated in particle chains by forcing neighboring particles to oscillate with different phases. It was also observed in [254, 255] that nanoconfinement leads to coherent oscillation and a significant reduction in spectral linewidth, i.e., reduced damping, even compared to homogeneous bulk samples.

An instructive example of a suitable magnonics device based on dipolar interactions are three single-domain particles arranged around the corner. The spectral response of such a system to a spatially uniform excitation reveals an intriguing magnonic property at an applied field angle of $45^\circ$ (figure 41(a))—a spectral gap. The origin of the gap can be rationalized from the analysis of spatial mode profiles in the system. The relevant modes are the high energy
## 21.4. Current and future challenges

Given that magnetic moment couples to the current oscillations in hybrid Josephson junctions, care should be taken in the analysis of their FMR response and current-voltage characteristics [244]. Bloch-like band structures in the spin-wave transmission spectra of S/F bilayers require long-range order of the vortex lattice [242] which is hard to achieve in experiments. For the Cherenkov-like generation of magnons by moving fluxons, vortex velocities exceeding $1–3 \text{ km s}^{-1}$ are required [246]. At such high velocities vortex cores may collapse because of the flux-flow instability. To prevent instability, one can use, e.g., vortex guiding in S/F bilayers or superconductors with fast relaxation of disequilibrium, for which vortex velocities of $5–15 \text{ km s}^{-1}$ have recently been demonstrated [252].

Electric-field control of magnetism in FM/FE heterostructures offers a low-power mechanism for spin-wave manipulation at room temperature. To date, proof-of-principle experiments utilize thick single-crystal FE substrates, necessitating the use of relatively large voltages [249]. For practical devices, the voltage pulses should be substantially smaller. Epitaxial FE films could satisfy this requirement [248]. Yet, the growth of thin films with regular FE stripe domains is challenging. Many attractive magnonic features of FM/FE bilayers originate from a regular modulation of magnetic anisotropy and strong pinning of straight magnetic domain walls. Similar effects could be realized also in other material systems, for instance via local ion-beam irradiation of FM films.

## 21.5. Concluding remarks

Investigations of S/F hybrid structures are gaining a momentum, opening access to exciting emerging phenomena in spin-wave physics and advancing novel magnonics and spintronics functionalities. The extension of magnonics to cryogenic temperatures has given birth to the new research direction of magnon fluxonics and metamaterials on the basis of F/S hybridization. Moreover, S/F heterostructures allow for highly efficient magnon-photon coupling and, thus, are of crucial importance for the further development of quantum magnonics.

Heterostructures with coupled FM and FE order parameters have emerged as an attractive material platform for room-temperature programmable magnonics. Imprinting of FE domains into a FM film has enabled new approaches for short-wavelength spin-wave emission, magnonic guiding, and spin-wave manipulation through magnetic- and electric-field actuation. Integration of these magnonic effects in magnetoelectric devices promises new functionalities and low-power operation.

## Acknowledgments

OVD acknowledges financial support by the European Cooperation in Science and Technology via COST Action CA16218 (NANOCOHYBRI) and ERC Starting Grant 678309 MagnonCircuits. HQ acknowledges financial support from the Academy of Finland (Grant Nos. 317918 and 325480). SvD acknowledges financial support by the

European Research Council (Grant Nos. ERC-2012-StG 307502-E-CONTROL, ERC-2014-PoC 665215-EMOTION and ERC-PoC-2018 812841-POWERSPIN) and the Academy of Finland (Grant No. 321983).

## 22. Biologically encoded magnonics

Benjamin W Zingsem¹ and Michael Winklhofer²

¹The University of Duisburg-Essen, Germany

²The Carl von Ossietzky University of Oldenburg, Germany

### 22.1. Status

Biologically encoded magnonics or biomagnonics, for short, is a recent approach to magnonics research, wherein biogenic magnetic material is harnessed for magnonics applications. Magnetotactic bacteria present an ideal model system for biomagnonics studies. These microorganisms have the genetic machinery to form dedicated ensembles of magnetic nanoparticles—magnetosomes—for orientation by the Earth's magnetic field. The magnetosomes consist of a strongly magnetic material (magnetite, $Fe_3O_4$), have magnetic-single domain character, and are arranged in the form of one or several linear chains [253]. Each magnetic particle is enclosed in a membrane vesicle, which not only controls the shape and size of the particle ($<100 \text{ nm}$) but also keeps adjacent particles separated by a clear distance of $8–10 \text{ nm}$ [253]. Thereby, the particles are not exchange-coupled among each other but interact solely by their stray fields. The spatial distribution of the stray fields, and thus the spin-wave spectrum, depends on the geometric arrangement of the particles, which, in turn, is biologically controlled. The mode profile, i.e., the spatial distribution of phase and amplitude, of spin-waves in magnetic nanoparticles exhibits distinct differences from larger systems: at gigahertz excitation frequencies ($\le 20 \text{ GHz}$ at ca 300–400 mT), each nanoparticle responds with a uniform precession [254]. Dipolar pinning, which is known to unfavourably distort the mode profiles near the edges of a large continuous sample [255], is suppressed in nanoparticles due to the strong, short-range exchange coupling. This entails a macrospin behavior where each individual nanoparticle acts as a coherent oscillator. At the same time, wavelengths that are shorter than typical low-frequency exchange- or dipolar spin-waves, can be accommodated in particle chains by forcing neighboring particles to oscillate with different phases. It was also observed in [254, 255] that nanoconfinement leads to coherent oscillation and a significant reduction in spectral linewidth, i.e., reduced damping, even compared to homogeneous bulk samples.

An instructive example of a suitable magnonics device based on dipolar interactions are three single-domain particles arranged around the corner. The spectral response of such a system to a spatially uniform excitation reveals an intriguing magnonic property at an applied field angle of $45°$ (figure 41(a))—a spectral gap. The origin of the gap can be rationalized from the analysis of spatial mode profiles in the system. The relevant modes are the high energy
<figure><img src="image_47.png" /><figcaption>**Figure 41.** (a) Simulation (mumax3 [262]) of the spectral response of an arrangement of three magnetite (Fe₃O₄) particles to a spatially uniform microwave excitation in a static magnetic field of 360 mT, applied over a sequence of in-plane directions from 0° to 90° (inset). If the horizontal and vertical segment of the arrangement (encircled in pink and green in the inset) were spatially isolated from one another, their spectral responses (dashed lines in respective colors) would cross at 45°. Coupling the segments through the shared corner particle, however, causes a spectral gap to emerge at 45°. (b) Spatial mode-profile associated with the normal-modes at 45° in (a), for high-energy (i) and low-energy state (ii), respectively. (c) Examples of excitation under non-uniform driving forces. Left: antiphase driving (180° phase shifted) of end particles (P1 and P2, see scheme in (d) at point (i) of spectrum does not excite corner particle ‘out’. Right: driving just one end particle at point (ii) of spectrum produces in-phase oscillation in the opposite end particle, conveyed by anti-phase oscillation of corner particle, similar to mode profile (ii) depicted in (b). (d) Pictographic sketch of the three-particle magnonic logic-gate setup. (e) State table of the response of the corner-particle (green) to individual excitations of the end-particles (blue and red), with ‘1’ and ‘-1’ signifying phases π/2 and 3π/2, respectively. The box enclosing (a) and (b) indicates a spatial uniform excitation as opposed to the excitations of individual particles (c)-(f).</figcaption></figure>

mode (figure 41(b-i)), where the particles precess all with
the same phase, and the low energy mode (figure 41(b-ii),
where the corner-particle is locked to an anti-phase oscilla-
tion with respect to the end particles. Having incommensurate
spatial mode profiles (figure 41(b)), these two normal modes
(eigenstates) repel each other when approaching 45°, hence
the spectral gap. Importantly, an eigenstate can also be excited
by driving just a single particle at the eigenfrequency of the
corresponding eigenstate; the energy from the local driving
force is transmitted into collective oscillations through dipolar
coupling among the particles. By driving a second particle at
the same frequency, the eigenmode can be sustained or anni-
hilated, depending on the phase difference between the two
driving forces. For example, in the high-energy eigenmode,
the spin-wave amplitude in the corner particle will be anni-
hilated by anti-phase driving forces (figure 41(c-i)), but not by
in-phase driving forces. For this system, a logic gate can be

realized most conveniently when locally exciting the end par-
ticles such that the phase information of both end particles is
mixed, i.e., is processed, in the corner particle (figures 41(c)
and (d)). The logic states of this device for the simplest types
of phase relationships between the end particles (in-phase
and anti-phase) are shown in the logic tables (figure 41(e).
When selecting the low energy mode, the resulting gate is
inverse to that of the high energy mode (figure 41(f)). These
results demonstrate that several logical operations can be per-
formed in simple geometric formations of dipolar coupled
nanoparticles. In the three-particle structure modeled here, the
individual particles have no intrinsic anisotropy, so that the
angular dependence of the excitation spectrum is solely deter-
mined by the geometric arrangement of the particles. This sim-
ple system can also be tuned in terms of the spectral gaps,
which can be widened by increasing the dipolar coupling
strength, for example, by selecting a magnetic material with
<figure><img src="image_45.png" /><figcaption>**Figure 41.** (a) Simulation (mumax3 [262]) of the spectral response of an arrangement of three magnetite (Fe₃O₄) particles to a spatially uniform microwave excitation in a static magnetic field of 360 mT, applied over a sequence of in-plane directions from 0° to 90° (inset). If the horizontal and vertical segment of the arrangement (encircled in pink and green in the inset) were spatially isolated from one another, their spectral responses (dashed lines in respective colors) would cross at 45°. Coupling the segments through the shared corner particle, however, causes a spectral gap to emerge at 45°. (b) Spatial mode-profile associated with the normal-modes at 45° in (a), for high-energy (i) and low-energy state (ii), respectively. (c) Examples of excitation under non-uniform driving forces. Left: antiphase driving (180° phase shifted) of end particles (P1 and P2, see scheme in (d) at point (i) of spectrum does not excite corner particle ‘out’. Right: driving just one end particle at point (ii) of spectrum produces in-phase oscillation in the opposite end particle, conveyed by anti-phase oscillation of corner particle, similar to mode profile (ii) depicted in (b). (d) Pictographic sketch of the three-particle magnonic logic-gate setup. (e) State table of the response of the corner-particle (green) to individual excitations of the end-particles (blue and red), with ‘1’ and ‘-1’ signifying phases π/2 and 3π/2, respectively. The box enclosing (a) and (b) indicates a spatial uniform excitation as opposed to the excitations of individual particles (c)-(f).</figcaption></figure>

mode (figure 41(b-i)), where the particles precess all with
the same phase, and the low energy mode (figure 41(b-ii),
where the corner-particle is locked to an anti-phase oscilla-
tion with respect to the end particles. Having incommensurate
spatial mode profiles (figure 41(b)), these two normal modes
(eigenstates) repel each other when approaching 45°, hence
the spectral gap. Importantly, an eigenstate can also be excited
by driving just a single particle at the eigenfrequency of the
corresponding eigenstate; the energy from the local driving
force is transmitted into collective oscillations through dipolar
coupling among the particles. By driving a second particle at
the same frequency, the eigenmode can be sustained or anni-
hilated, depending on the phase difference between the two
driving forces. For example, in the high-energy eigenmode,
the spin-wave amplitude in the corner particle will be anni-
hilated by anti-phase driving forces (figure 41(c-i)), but not by
in-phase driving forces. For this system, a logic gate can be

realized most conveniently when locally exciting the end par-
ticles such that the phase information of both end particles is
mixed, i.e., is processed, in the corner particle (figures 41(c)
and (d)). The logic states of this device for the simplest types
of phase relationships between the end particles (in-phase
and anti-phase) are shown in the logic tables (figure 41(e).
When selecting the low energy mode, the resulting gate is
inverse to that of the high energy mode (figure 41(f)). These
results demonstrate that several logical operations can be per-
formed in simple geometric formations of dipolar coupled
nanoparticles. In the three-particle structure modeled here, the
individual particles have no intrinsic anisotropy, so that the
angular dependence of the excitation spectrum is solely deter-
mined by the geometric arrangement of the particles. This sim-
ple system can also be tuned in terms of the spectral gaps,
which can be widened by increasing the dipolar coupling
strength, for example, by selecting a magnetic material with
<figure><img src="image_48.png" /><figcaption>**Figure 42.** (a) and (b) Transmission electron micrographs of air-dried cells of wild type *Magnetospirillum gryphiswaldense* (a) versus Δ*mamK* mutant (b). The proportion of curved or dendritic particle arrangements (red highlights) is higher in the mutant compared to the wildtype in which mostly straight (blue highlights) chains occur. (c) Pictographic representation of desired development of suitable magnonic networks, from simple magnetosome (left) via suitable gate structures (center) to computational magnonic networks. (d) Moore's law for semiconductor electronics along with the transistor count in various CPU models. The blue marking (top right) indicates the estimated number of magnonic logic gates that could be accommodated in the volume of an 8-core Core i7 Haswell-E CPU, due to the obsolescence of heat sink material in magnonics computing. (a), (b) and (d) Reproduced from [254]. CC BY 4.0.</figcaption></figure>
<figure><img src="image_46.png" /><figcaption>**Figure 42.** (a) and (b) Transmission electron micrographs of air-dried cells of wild type *Magnetospirillum gryphiswaldense* (a) versus Δ*mamK* mutant (b). The proportion of curved or dendritic particle arrangements (red highlights) is higher in the mutant compared to the wildtype in which mostly straight (blue highlights) chains occur. (c) Pictographic representation of desired development of suitable magnonic networks, from simple magnetosome (left) via suitable gate structures (center) to computational magnonic networks. (d) Moore's law for semiconductor electronics along with the transistor count in various CPU models. The blue marking (top right) indicates the estimated number of magnonic logic gates that could be accommodated in the volume of an 8-core Core i7 Haswell-E CPU, due to the obsolescence of heat sink material in magnonics computing. (a), (b) and (d) Reproduced from [254]. CC BY 4.0.</figcaption></figure>
larger saturation magnetization or by decreasing the distance between particles.

As demonstrated recently on the basis of microcavity FMR measurements and micromagnetic simulations [254], bacterial magnetosome chains exhibit magnonic properties that make them interesting for applications; most notably, chains with a sharp kink were identified to have FMR spectra similar to the three-particle magnon logic gate. Since the geometrical configuration of magnetosomes determines the overall topology of its magnonic band structure, the key to controlling biologically encoded magnonics is primarily through the structural determinants of the magnetosome chain. In magnetotactic spirilla, which are amenable to genetic manipulation, two dozens of magnetosome-associated proteins have been identified, with some of them necessary for proper assembly, alignment, and positioning of chains (see reference [256] for the most recent work). In our recent study [254], we have focused on cells lacking the gene (*mamK*) that encodes a filament-forming protein (*MamK*), which in wild-type cells acts as a dynamical scaffold for magnetosome chains. Indeed, in the *ΔmamK* deletion mutant of *Magnetospirillum gryphiswaldense*, considerably more cells with kinked, closed, or other magnonically interesting chain structures can be obtained compared to wild type (compare figures 42(a) and (b)). Another crucial protein in magnetotactic spirilla is the cell-curvature sensing protein *MamY*, which is necessary for positioning the magnetosome chain along the geodetic cell axis of the helically wound cell body so that cells of the *ΔmamY* deletion mutant fail to produce straight chains [256]. Although *Magnetospirillum sp.* are genetic models of magnetotactic bacteria, other types of magnetotactic bacteria with non-helical cell morphology have great potential for biomagnonics, too. Particularly interesting here are magnetotactic cocci (round cells), whose magnetosome chains tend to form sharp kinks in response to the compressive forces during desiccation-induced cell shrinkage [257], offering the prospect of obtaining magnonic gates even without direct genetic manipulation. Besides structural determinants of the magnetosome chain geometry, secondary mechanisms for tuning the magnonic fine-structure and damping processes are available, too. When fed with Mn²⁺ and Fe³⁺ in equal proportions, magnetotactic spirilla incorporate 1% Mn²⁺ in their magnetite crystals, which somewhat reduces the particle size, saturation magnetization, and magnetocrystalline anisotropy [258] (isotropic Mn²⁺ replacing anisotropic high-spin Fe²⁺ at octahedral sites in the inverse spinel lattice); we note that similar effects may be achieved more cheaply by doping with cation vacancies during post-growth oxidation of magnetite to maghemite (γ-Fe₂O₃). Magnetite and maghemite can also be post-growth modified by impregnation with a Co²⁺-containing solution, resulting in an anisotropy gradient toward the particle surface and thus modifies the effective magnetic material parameters, such as damping, anisotropy, magnetization, and g-factor. Last, the dipolar coupling strength among the magnetosome particles may be reduced by preventing them from reaching their full size, e.g., under suitably chosen growth conditions, or in a mutant strain, such as the ΔGFDC deletion mutant of MSR-1 [259].

## 22.2. Current and future challenges

Thus far, we have studied single cells of just two genotypes. Still, there are many more species of magnetotactic bacteria to be explored in terms of their intrinsic magnonic properties and tailorability. Single-cell measurements will have to be guided and underpinned by micromagnetic simulations (e.g., figure 41) to identify candidate geometries for magnonic applications. The geometries may be produced lithographically and biologically, with biogenic production having two distinct advantages: first, the biological growth mechanisms ensure excellent crystallinity, magnetic single-domain character, consistent shape, and narrow particle-size spectrum [253]. Secondly, bacterial cells can be produced in large batches from which candidate cells with potential logic gates can be isolated, amplified, and perhaps even further optimized by directed evolution. Once a bacterium has been selected for desired magnonic properties, self-reproduction yields exponential production scalability. At the same time, it is essential to further advance our understanding of the molecular cell biology of magnetotactic bacteria to allow for targeted genetic engineering of desirable magnonic logic gates and even computational magnonic networks (as illustrated in figure 42(c)). Damping and other transport properties can be tailored through particle size and material choice. Although the composition of the magnetic material in magnetotactic bacteria is under genetic control and thus only partially influenceable during active growth [258], it is nevertheless amenable to post-growth modification. Thereby, logic gates can be tuned toward fast operation (large damping) and long transport distances (small damping) simultaneously.

Another challenge to nanomagnonics is the input/output periphery for spin-wave excitations on the nanoscale. One option to write individual inputs and read individual outputs is to drive the magnons through spin-torque excitations. This technique can be used for magnetic switching [260], but also for dynamic excitation of spin-wave normal modes [261]. The drawback of involving electrical currents can be mitigated by using a minimal amount of electrical inputs and outputs to an, otherwise, all-magnonic network.

## 22.3. Concluding remarks

The reachable density of logic gates when using networks of nano-sized magnon logic elements (e.g., figure 41) has the potential to outperform conventional charge-based silicon transistors (figure 42(d)). Self-organized magnonic gates can be harvested from magnetotactic bacteria (figure 42(b)), allowing for sustainable production of magnonic logic gates with exponential scalability. Future attempts will focus on engineering biological cells to produce magnetic nanoparticle configurations that perform desired magnonic tasks.

## Acknowledgments

The authors thank Michael Farle, Ralf Meckenstock, and Thomas Feggeler who significantly contributed in acquiring the first ferromagnetic resonance spectra of individual
larger saturation magnetization or by decreasing the distance between particles.

As demonstrated recently on the basis of microcavity FMR measurements and micromagnetic simulations [254], bacterial magnetosome chains exhibit magnonic properties that make them interesting for applications; most notably, chains with a sharp kink were identified to have FMR spectra similar to the three-particle magnon logic gate. Since the geometrical configuration of magnetosomes determines the overall topology of its magnonic band structure, the key to controlling biologically encoded magnonics is primarily through the structural determinants of the magnetosome chain. In magnetotactic spirilla, which are amenable to genetic manipulation, two dozens of magnetosome-associated proteins have been identified, with some of them necessary for proper assembly, alignment, and positioning of chains (see reference [256] for the most recent work). In our recent study [254], we have focused on cells lacking the gene (*mamK*) that encodes a filament-forming protein (*MamK*), which in wild-type cells acts as a dynamical scaffold for magnetosome chains. Indeed, in the *ΔmamK* deletion mutant of *Magnetospirillum gryphiswaldense*, considerably more cells with kinked, closed, or other magnonically interesting chain structures can be obtained compared to wild type (compare figures 42(a) and (b)). Another crucial protein in magnetotactic spirilla is the cell-curvature sensing protein *MamY*, which is necessary for positioning the magnetosome chain along the geodetic cell axis of the helically wound cell body so that cells of the *ΔmamY* deletion mutant fail to produce straight chains [256]. Although *Magnetospirillum sp.* are genetic models of magnetotactic bacteria, other types of magnetotactic bacteria with non-helical cell morphology have great potential for biomagnonics, too. Particularly interesting here are magnetotactic cocci (round cells), whose magnetosome chains tend to form sharp kinks in response to the compressive forces during desiccation-induced cell shrinkage [257], offering the prospect of obtaining magnonic gates even without direct genetic manipulation. Besides structural determinants of the magnetosome chain geometry, secondary mechanisms for tuning the magnonic fine-structure and damping processes are available, too. When fed with Mn²⁺ and Fe³⁺ in equal proportions, magnetotactic spirilla incorporate 1% Mn²⁺ in their magnetite crystals, which somewhat reduces the particle size, saturation magnetization, and magnetocrystalline anisotropy [258] (isotropic Mn²⁺ replacing anisotropic high-spin Fe²⁺ at octahedral sites in the inverse spinel lattice); we note that similar effects may be achieved more cheaply by doping with cation vacancies during post-growth oxidation of magnetite to maghemite (γ-Fe₂O₃). Magnetite and maghemite can also be post-growth modified by impregnation with a Co²⁺-containing solution, resulting in an anisotropy gradient toward the particle surface and thus modifies the effective magnetic material parameters, such as damping, anisotropy, magnetization, and g-factor. Last, the dipolar coupling strength among the magnetosome particles may be reduced by preventing them from reaching their full size, e.g., under suitably chosen growth conditions, or in a mutant strain, such as the ΔGFDC deletion mutant of MSR-1 [259].

## 22.2. Current and future challenges

Thus far, we have studied single cells of just two genotypes. Still, there are many more species of magnetotactic bacteria to be explored in terms of their intrinsic magnonic properties and tailorability. Single-cell measurements will have to be guided and underpinned by micromagnetic simulations (e.g., figure 41) to identify candidate geometries for magnonic applications. The geometries may be produced lithographically and biologically, with biogenic production having two distinct advantages: first, the biological growth mechanisms ensure excellent crystallinity, magnetic single-domain character, consistent shape, and narrow particle-size spectrum [253]. Secondly, bacterial cells can be produced in large batches from which candidate cells with potential logic gates can be isolated, amplified, and perhaps even further optimized by directed evolution. Once a bacterium has been selected for desired magnonic properties, self-reproduction yields exponential production scalability. At the same time, it is essential to further advance our understanding of the molecular cell biology of magnetotactic bacteria to allow for targeted genetic engineering of desirable magnonic logic gates and even computational magnonic networks (as illustrated in figure 42(c)). Damping and other transport properties can be tailored through particle size and material choice. Although the composition of the magnetic material in magnetotactic bacteria is under genetic control and thus only partially influenceable during active growth [258], it is nevertheless amenable to post-growth modification. Thereby, logic gates can be tuned toward fast operation (large damping) and long transport distances (small damping) simultaneously.

Another challenge to nanomagnonics is the input/output periphery for spin-wave excitations on the nanoscale. One option to write individual inputs and read individual outputs is to drive the magnons through spin-torque excitations. This technique can be used for magnetic switching [260], but also for dynamic excitation of spin-wave normal modes [261]. The drawback of involving electrical currents can be mitigated by using a minimal amount of electrical inputs and outputs to an, otherwise, all-magnonic network.

## 22.3. Concluding remarks

The reachable density of logic gates when using networks of nano-sized magnon logic elements (e.g., figure 41) has the potential to outperform conventional charge-based silicon transistors (figure 42(d)). Self-organized magnonic gates can be harvested from magnetotactic bacteria (figure 42(b)), allowing for sustainable production of magnonic logic gates with exponential scalability. Future attempts will focus on engineering biological cells to produce magnetic nanoparticle configurations that perform desired magnonic tasks.

## Acknowledgments

The authors thank Michael Farle, Ralf Meckenstock, and Thomas Feggeler who significantly contributed in acquiring the first ferromagnetic resonance spectra of individual
magnetosomes. Both authors acknowledge financial fund-
ing by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), BZ: Project-ID 405553726—TRR
270, MW: Project-ID 395940726—SFB 1372.

Data availability statement

All data is made available by the authors upon reasonable
request.

ORCID iDs

Anjan Barman https://orcid.org/0000-0002-4106-5658

Gianluca Gubbiotti https://orcid.org/0000-0002-7006-0370

S Ladak https://orcid.org/0000-0002-0275-0927

A O Adeyeye https://orcid.org/0000-0001-9724-2468

M Krawczyk https://orcid.org/0000-0002-0870-717X

J Gräfe https://orcid.org/0000-0002-4597-5923

C Adelmann https://orcid.org/0000-0002-4831-3159

S Cotofana https://orcid.org/0000-0001-7132-2291

A Naeemi https://orcid.org/0000-0003-4774-9046

V I Vasyuchka https://orcid.org/0000-0002-5077-7059

B Hillebrands https://orcid.org/0000-0001-8910-0355

S A Nikitov https://orcid.org/0000-0002-2413-7218

H Yu https://orcid.org/0000-0002-3291-6713

D Grundler https://orcid.org/0000-0002-4966-9712

A V Sadovnikov https://orcid.org/0000-0002-8847-2621

A A Grachev https://orcid.org/0000-0003-2023-910X

S E Sheshukova https://orcid.org/0000-0001-5732-0038

J-Y Duquesne https://orcid.org/0000-0001-9156-8758

M Marangolo https://orcid.org/0000-0001-6211-8168

G Csaba https://orcid.org/0000-0001-7592-0256

W Porod https://orcid.org/0000-0002-9854-6540

V E Demidov https://orcid.org/0000-0002-7840-5679

S Urazhdin https://orcid.org/0000-0002-6972-0300

S O Demokritov https://orcid.org/0000-0003-4422-4201

E Albisetti https://orcid.org/0000-0002-8134-0482

D Petti https://orcid.org/0000-0002-9273-1884

R Bertacco https://orcid.org/0000-0002-8109-9166

H Schultheiss https://orcid.org/0000-0002-6727-5098

V V Kruglyak https://orcid.org/0000-0001-6607-0886

V D Poimanov https://orcid.org/0000-0003-3078-2480

S Sahoo https://orcid.org/0000-0003-0207-5682

J Sinha https://orcid.org/0000-0003-3208-0570

H Yang https://orcid.org/0000-0003-0907-2898

M Münzenberg https://orcid.org/0000-0002-1332-5678

T Moriyama https://orcid.org/0000-0001-7071-0823

S Mizukami https://orcid.org/0000-0001-9913-1833

P Landeros https://orcid.org/0000-0002-0927-1419

R A Gallardo https://orcid.org/0000-0002-4495-5592

G Carlotti https://orcid.org/0000-0002-1511-7987

J-V Kim https://orcid.org/0000-0002-3849-649X

R L Stamps https://orcid.org/0000-0003-0713-4864

B Rana https://orcid.org/0000-0002-1076-4165

Y Otani https://orcid.org/0000-0001-8008-1493

W Yu https://orcid.org/0000-0001-6675-5535

T Yu https://orcid.org/0000-0001-7020-2204

G E W Bauer https://orcid.org/0000-0002-3615-8673

C Back https://orcid.org/0000-0003-3840-0993

G S Uhrig https://orcid.org/0000-0003-1961-0346

O V Dobrovolskiy https://orcid.org/0000-0002-7895-8265

B Budinska https://orcid.org/0000-0001-7640-5213

H Qin https://orcid.org/0000-0002-6894-3967

S van Dijken https://orcid.org/0000-0001-6372-2252

A V Chumak https://orcid.org/0000-0001-5515-0848

A Khitun https://orcid.org/0000-0003-4501-4687

D E Nikonov https://orcid.org/0000-0002-1436-1267

I A Young https://orcid.org/0000-0002-4017-5265

B W Zingsem https://orcid.org/0000-0002-9899-2700

M Winklhofer https://orcid.org/0000-0003-1352-9723

References

[1] Bloch F 1932 Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika Z. Phys. **74** 295

[2] Holstein T and Primakoff H 1940 Field dependence of the intrinsic domain magnetization of a ferromagnet Phys. Rev. **58** 1098

[3] Dyson F J 1956 General theory of spin-wave interactions Phys. Rev. **102** 1217

[4] Brillouin L 1953 *Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices* 2nd edn (New York: Dover)

[5] Sykes C G, Adam J D and Collins J H 1976 Magnetostatic wave propagation in a periodic structure *Appl. Phys. Lett.* **29** 388

Parekh J P and Tuan H S 1977 Magnetostatic surface wave reflectivity of a shallow groove on a YIG film *Appl. Phys. Lett.* **30** 667

[6] John S 1987 Strong localization of photons in certain disordered dielectric superlattices *Phys. Rev. Lett.* **58** 2486

Yablonovitch E 1987 Inhibited spontaneous emission in solid-state physics and electronics *Phys. Rev. Lett.* **58** 2059

[7] Vasseur J O, Dobrzynski L, Djafari-Rouhani B and Puszkarski H 1996 Magnon band structure of periodic composites *Phys. Rev. B* **54** 1043

[8] Gulyaev Y V and Nikitov A A 2001 Magnonic crystals and spin waves in periodic structures *Dokl. Phys.* **46** 687

[9] Puszkarski H and Krawczyk M 2003 Magnonic crystals—the magnetic counterpart of photonic crystals *Solid State Phenom.* **94** 125

[10] Demokritov S O and Slavin A N (ed) 2013 *Magnonics from Fundamentals to Applications* (Berlin: Springer)

[11] Rezende S M 2020 *Fundamentals of Magnonics* (Berlin: Springer)

[12] Gubbiotti G (ed) 2019 *Three Dimensional Magnonics: Layered Micro- and Nanostructures* (New York: Jenny Stanford Publishing)

[13] Barman A and Sinha J 2018 *Spin Dynamics and Damping in Ferromagnetic Thin Films and Nanostructures* (Gewerbestrasse: Springer International Publishing)

[14] Cenker J, Huang B, Suri N *et al* 2020 Direct observation of two-dimensional magnons in atomically thin CrI₃ *Nat. Phys.* **17** 20

[15] Holanda J, Maior D S, Azevedo A and Rezende S M Detecting the phonon spin in magnon-phonon conversion experiments 2018 *Nat. Phys.* **14** 500

[16] Chen T *et al* 2016 Spin-torque and spin-Hall nano-oscillators *Proc. IEEE* **104** 1919

[17] Romera M *et al* 2018 *Nature* **563** 230