Unidirectional spin-wave propagation and devices

The propagation of MSSWs is nonreciprocal due to the exponential decay of the SW amplitude from the two surfaces. In the MSSW mode configuration, the magnetization direction is in the film plane and perpendicular to the wavevector direction, as illustrated in figure 1(a). The MSSW is often known as Damon–Eshbach (DE) surface SWs [33] and they could travel from the left to right across the magnetic film while no equivalent waves can travel in the opposite direction [34]. The wavevector of a plane wave is given as $k = \hat x{k_x} + \hat y{k_y} + \hat z{k_z}$. The reflections from the top and bottom surfaces continuously reverse the sign of ${k_z}$ and a static condition can be fulfilled with the magnetostatic potential in the film as [35]:

$$\begin{equation}{{\Psi }} = {{{\Psi }}_0}{{\text{e}}^{ikr}}\frac{{({{\text{e}}^{i{k_z}z}} + {{\text{e}}^{ - i{k_z}z}})}}{2} = {{{\Psi }}_0}\cos \left( {{k_z}z} \right){{\text{e}}^{iv{k_y}y}},{ }\end{equation} \tag{ 1 }$$

Zoom In Zoom Out Reset image size

where ${{{\Psi }}_0}$ is the arbitrary amplitude and the parameter $v = \pm 1$ indicates the propagating direction (antiparallel (+1) and parallel (−1) to the y axis). The potential of the modes above and below the film can be assumed as:

$$\begin{equation}{{{\Psi }}_{{\text{up}}}} = C{{\text{e}}^{ - {k_y}z + iv{k_y}y}},{ }{{{\Psi }}_{{\text{down}}}} = D{{\text{e}}^{{k_y}z + iv{k_y}y}}.\end{equation} \tag{ 2 }$$

The Walker's equation for in the film can be written as

$$\begin{equation}\left( {1 + \chi } \right)\left( {k_z^2 + k_y^2} \right) = 0,\end{equation} \tag{ 3 }$$

which requires $k_z^2 = - k_y^2$. Here, $\chi = \frac{{{\omega _0}{\omega _M}}}{{\omega _0^2 - {\omega ^2}}},$ where ${\omega _0} = - \gamma {\mu _0}{H_0}$ and ${\omega _{\text{M}}} = - \gamma {\mu _0}{M_{\text{S}}}$. The SWs propagate in the $y$ direction, so ${k_y}$ is real and ${k_z}$ is imaginary. We could replace the potential in the film as:

$$\begin{equation}{{\Psi }} = ({{{\Psi }}_{0 + }}{{\text{e}}^{kz}} + {{{\Psi }}_{0 - }}{{\text{e}}^{ - kz}}){{\text{e}}^{ivky}}.\end{equation} \tag{ 4 }$$

To fulfil the boundary conditions at the film surface as well as the normal vector, the condition gives:

$$\begin{equation}\left[ {\begin{array}{*{20}{c}} {\left( {\chi + 2 - v\kappa } \right){{\text{e}}^{kd/2}}}&{ - \left( {\chi + v\kappa } \right){{\text{e}}^{ - kd/2}}} \\ { - \left( {\chi - v\kappa } \right){{\text{e}}^{ - kd/2}}}&{\left( {\chi + 2 + v\kappa } \right){{\text{e}}^{kd/2}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{{\Psi }}_{0 + }}} \\ {{{{\Psi }}_{0 - }}} \end{array}} \right] = 0,{ }\end{equation} \tag{ 5 }$$

where $\kappa = \frac{{\omega {\omega _{\text{M}}}}}{{\omega _0^2 - {\omega ^2}}}$ and $d$ is the film thickness. The dispersion relation (frequency vs wavevector) of MSSWs can then be obtained when the determinant of the matrix is vanishing:

$$\begin{equation}{\omega ^2} = {\omega _0}\left( {{\omega _0} + {\omega _{\text{M}}}} \right) + \frac{{\omega _{\text{M}}^2}}{4}\left[ {1 - {{\text{e}}^{ - 2kd}}} \right].\end{equation} \tag{ 6 }$$

The potential function of the MSSW can be expressed as:

$$\begin{equation}{{\Psi }} = {{{\Psi }}_0}\left( {{{\text{e}}^{kz}} + \frac{{\chi + 2 - v\kappa }}{{\chi + v\kappa }}{{\text{e}}^{kd - kz}}} \right){{\text{e}}^{ivky}}.\end{equation} \tag{ 7 }$$

The potential function in the magnetic film decays exponentially along the film thickness direction, as shown in figure 1(a). When changing the propagation direction, the high potential mode shifts from one surface to the other, which is responsible for the SW nonreciprocity. In thick films with $d \gg \lambda$ ($\lambda$ is the SW wavelength) and $kd \gg 1$, the MSSW only propagates on the top or bottom film surface and unidirectional SWs can be detected either electrically with a microwave antenna patterned on top of the magnetic film [40, 41] or optically by Brillouin light scattering (BLS) spectroscopy [38, 39]. The two MSSWs waves have the same frequency and the propagation direction on the two surfaces is defined by the vectorial product $\vec k = \vec H \times \hat n$, where $\hat n$ is the unit vector normal to the film surface. This suggests that by changing the direction of the external magnetic field, the nonreciprocity can be reversed. We investigate the most relevant case for applications: the waves propagating in in-plane magnetized ferromagnetic films at a particular angle to the applied magnetic field. In addition, Kostylev theoretically demonstrated that the fundamental mode of the dipole-exchange spectrum in a thin ferromagnetic metallic film is localized at the film surface opposite the localization surface of the DE surface SWs [42], while such an effect cannot affect the SW nonreciprocity of the SW excitation by microwave antennas.

The localization of the modes on the two surfaces of the film is explained by considering the asymmetry of the dynamic dipolar fields, which create volume and surface magnetic charges which in turn generate dipolar magnetic fields. On the lower surface, the two fields add up while, on the upper one, they partially offset each other. This asymmetry of the dipolar field is at the origin of the localization asymmetry of the mode at the film surfaces. To compensate for this asymmetry, dynamic magnetization increases its amplitude by precession near the surface of the film, where the dynamic dipolar field is smaller [43].

Unidirectional MSSWs were detected in thick magnetic films by microwave-based technology for several decades in thick Yittrium Iron Garnet (YIG) slabs [40, 41]. In 2014, Wong et al experimentally obtained the SW amplitude ratios by measuring SWs propagating in opposite directions in magnetic films with different thicknesses, including a 20 nm permalloy film, a 160 nm YIG film and a 2 μm YIG film [36]. Two microwave antennas are fabricated on the top surface of the film with a separation distance of 8 μm. By directly comparing the SW amplitude extracted by scattering parameters, ${S_{12}}$ and ${S_{21}}$, from a vector network analyzer, the SW nonreciprocity is found to increase with an increasing sample thickness, as shown in figure 1(b). However, when the thickness of the YIG film is 254 μm, the nonreciprocity is greatly enhanced and SW unidirectionality is achieved. In 2009, Demidov et al found that, using a stripe antenna, the excitation of the MSSWs is nonreciprocal in amplitude, although the decay rate of the left and right of the antenna is the same [44]. In addition, it is found that when using a low-damping thick YIG film, the unidirectional SWs can convey heat and, at the sample, end up to 10 mm from the microwave source [45].

Madami et al [46] studied the focusing of magnetostatic backward volume waves (MSBVWs) excited by a curvilinear microwave coplanar waveguide in 5 mm-thick epitaxial YIG film within an in-plane bias field oriented along the symmetry axis of the transducer. It was shown by both the micro-BLS technique and micromagnetic simulation that two-dimensional maps of MSBVW beam intensity at distances larger than the focus position demonstrate nonreciprocity and absence of mirror symmetry with respect to the direction of the bias field. These effects result from the nonreciprocity of MSBVWs traveling at an angle to the bias field direction.

The unidirectional SW propagation in magnetic thin films is challenging to achieve in thin magnetic films, so engineering the nonreciprocity of MSSWs is very important. In 2016, Kwon et al found a way to enhance the nonreciprocity by putting a Ta layer on top of permalloy [47]. They found that the nonreciprocity depends on the Ta thickness, which induces a modification of the interfacial anisotropy. The nonreciprocity ratio was found to reach a value of 60 in the time and frequency domain.

The mode localization of MSSWs can also be detected by BLS spectroscopy, which is an optical technique based on the inelastic scattering of photons with thermal magnons [48, 49]. Sandercock et al studied surface magnons in thick Fe and Ni films and, by reversing the direction of the bias magnetic field, observed that the sharp peak associated with the DE wave moves from one side of the spectrum to the other, as the signature of the SW nonreciprocity, as shown in figure 1(c) [37]. Grünberg et al studied perpendicular standing SWs (PSSWs) and the DE surface wave in Fe films by BLS and found that the remarkable Stokes-anti-Stokes intensity asymmetry indicated the SW nonreciprocity [38]. For magnetic films thinner than the penetration depth of light in metallic films (typically of 15–20 nm), the DE mode can be detected on both the Stokes and anti-Stokes side of the BLS spectra while, for larger film thickness, it is observed on one side only. In a 1170 Å thick Fe film, Hicken et al found that the surface SW mode exhibited strong nonreciprocity, being observed on one anti-Stokes side of the spectrum, while the PSSW modes were reciprocal, as shown in figure 1(e) [39].

Modification of the MSSW dispersion (frequency vs wavevector) to achieve unidirectional SW propagation can also be achieved in exchange and dipolar-coupled ferromagnetic films. Grassi et al [50] showed careful engineering of the dispersion for MSSWs in a CoFeB/NiFe bilayer, where it is possible to reduce the group velocity of waves traveling in a particular direction to a very low value (slow waves) while maintaining a large value for those propagating the other way. Similar results were obtained by Mruczkiewicz et al [51] in a NiFe/Ni bilayer.

Gallardo et al [52] demonstrated that the dipolar interaction produced by the dynamic magnetizations between two ferromagnetic layers separated by a nonmagnetic spacer is a notable source of nonreciprocity in the SW frequency, with a remarkable property of reconfigurability that relies on control of the relative magnetic orientation of the interacting ferromagnetic layers.

The DMI is the asymmetric exchange interaction in magnetic systems, which could induce chiral magnetic textures such as skyrmions and chiral domain walls. The DMI between two neighbouring spins takes the form [53]:

$$\begin{equation}{H_{{\text{DMI}}}} = - {{{\boldsymbol{D}}}_{12}} \cdot \left( {{{{\boldsymbol{S}}}_1} \times {{{\boldsymbol{S}}}_2}} \right),{ }\end{equation} \tag{ 8 }$$

where ${{{\boldsymbol{S}}}_1}$ and ${{{\boldsymbol{S}}}_2}$ are neighbouring spins and ${{{\boldsymbol{D}}}_{12}}$ is the DM vector. Two types of DMI can be classified depending on the type of inversion symmetry breaking. The iDMI corresponds to the inversion symmetry breaking at the interface, which is inversely proportional to the film thickness. Typical magnetic systems which host iDMI are magnetic multilayers, where one layer should provide strong spin–orbit couplings, such as Pt, Ta, W and Ir. The bulk DMI is found mostly in B20 structures such as MnSi and FeGe. In this section, we mainly focus on the iDMI.

When SWs propagate in a magnetic system with iDMI, the dispersion is reformed due to the inversion symmetry breaking. Consider that SWs propagate in the film plane and are perpendicular to the external magnetic field in a magnetic film with iDMI, as shown in figure 2(a). The SW dynamics can be described by the Landau–Lifshitz–Gilbert (LLG) equation as:

$$\begin{equation}\frac{{{\text{d}}\widehat {{\boldsymbol{m}}}}}{{{\text{d}}t}} = - \gamma {\mu _0}\widehat {{\boldsymbol{m}}} \times {{{\boldsymbol{H}}}_{{\text{eff}}}} + \alpha \left( {\widehat {{\boldsymbol{m}}} \times \frac{{{\text{d}}\widehat {{\boldsymbol{m}}}}}{{{\text{d}}t}}} \right),\;\;\;\;\;\;\end{equation} \tag{ 9 }$$

Zoom In Zoom Out Reset image size

where $\alpha$ is the damping parameter. The effective field ${{{\boldsymbol{H}}}_{{\text{eff}}}}$ can be written as:

$$\begin{equation}{{{\boldsymbol{H}}}_{{\text{eff}}}} = vH{{\boldsymbol{x}}} + \frac{{2A}}{{{\mu _0}{M_{\text{S}}}}}{\nabla ^2}\widehat {{\boldsymbol{m}}} - \frac{{2D}}{{{\mu _0}{M_{\text{S}}}}}\left( {{{\boldsymbol{x}}} \times \frac{{{\text{d}}\widehat {{\boldsymbol{m}}}}}{{{\text{d}}t}}} \right) + {{{\boldsymbol{H}}}_{{\text{dip}}}}, \end{equation} \tag{ 10 }$$

where $H$ is the external field, $v = \pm 1$ indicates the propagating direction, $A$ is the exchange stiffness, $D$ is the iDMI constant and ${{{\boldsymbol{H}}}_{{\text{dip}}}}$ is the dipolar field depending on the film thickness and the wavevector. The dispersion, by neglecting the nonlocal magnetostatic contribution, can be expressed as:

$$\begin{align}\omega &= \gamma {\mu _0}{\left[ {\left( {H + \frac{{2A}}{{{\mu _0}{M_{\text{S}}}}}{k^2}} \right)\left( {H + {M_{\text{S}}} + \frac{{2A}}{{{\mu _0}{M_{\text{S}}}}}{k^2}} \right)} \right]^{\frac{1}{2}}} \nonumber\\ &\quad+ \frac{{2vD}}{{{\mu _0}{M_{\text{S}}}}}k.\end{align} \tag{ 11 }$$

In the small $k$ limit, where the exchange interaction can be neglected, the dispersion is reformed as:

$$\begin{align}\omega = \gamma {\mu _0}{\left[ {H\left( {H + {M_{\text{S}}}} \right)} \right]^{\frac{1}{2}}} + \frac{{M_{\text{S}}^2\left| k \right|d}}{{4{{\left[ {H\left( {H + {M_{\text{S}}}} \right)} \right]}^{\frac{1}{2}}}}} + \frac{{2vD}}{{{\mu _0}{M_{\text{S}}}}}k.\end{align} \tag{ 12 }$$

The dispersion of SWs is then asymmetric, depending on the propagation direction, and shows a linear dependence of the $k$ when considering the iDMI. The calculated SW dispersion with $D = -$1.5, 0, 1.5 mJ m⁻² in the large $k$ limit is shown in figure 1(b), where the saturation magnetization ${M_{\text{S}}} = 800$ kA m⁻¹, the exchange constant $A$ = 1.3$\, \times {10^{ - 11}}$ J m⁻¹ and the thickness $d$ = 1 nm is used in the calculation.

The asymmetric dispersion induced by iDMI results in the nonreciprocal SWs in the frequency domain. BLS is an efficient method to detect the asymmetric SW dispersion with the wavevector resolution [57–59]. The propagating SWs with opposite directions give rise to the Stokes and anti-Stokes peaks, which can characterize frequency differences induced by the asymmetric dispersion. In 2015, Di et al directly observe the iDMI in a Pt/Co/Ni multilayer by BLS [60]. An iDMI constant of 0.44 mJ m⁻² is found and the linewidths of the counter-propagating SWs are different, which is in agreement with the theoretical simulations. Tacchi et al further investigate the effect of the heavy metal thickness of the iDMI in Pt/CoFeB films by BLS [54]. The authors found that by increasing the Pt thickness, the iDMI increases and saturates to a value around 0.45 mJ m⁻² for Pt thicknesses larger than 2 nm, as shown in figure 2(c).

IDMI and nonreciprocal SWs can also be detected and a chiral SW group velocity is found to be induced by iDMI in ultrathin magnetic garnet films, where SWs propagating in the chirally favoured direction travel faster than in the counter-direction, as shown in figure 2(d) [55, 61]. The chirality behaviour of SW group velocities is attributed to a DMI-induced drift group velocity whose direction follows a right-handed rule as:

$$\begin{equation}{v_{{\text{DMI}}}} = \left[ {\left( {\widehat {\mathbf{n}} \times \widehat {\mathbf{H}}} \right) \cdot \widehat {\mathbf{k}}} \right]\frac{{2\gamma }}{{{M_{\text{S}}}}}D.\end{equation} \tag{ 13 }$$

The orientation of the drift velocity can be controlled by the external magnetic field as well as the propagation direction. In the spectra extracted from the electrical propagating SW spectroscopy, the SW group velocity can be calculated as:

$$\begin{equation}{v_{\text{g}}} = \frac{{{\text{d}}\omega }}{{{\text{d}}k}} = {{\Delta }}f \cdot s,{ }\end{equation} \tag{ 14 }$$

where ${{\Delta }}f$ is the frequency span indicating an SW phase change of 2$\pi$, and $s$ is the SW propagation distance. One can observe in figure 2(d) that by reversing either the SW propagation direction or the external magnetic field, the SW velocity shifts due to the iDMI-induced drift velocity. The asymmetry of SW group velocity increases when decreasing the film thickness, indicating that the DMI is an interfacial effect in thin YIG films. An overview of the experimental techniques as well as their theoretical background and models for the quantification of the DMI constant $D$ on iDMI measurements can also be found in [62].

Taking advantage of the asymmetric SW dispersion induced by the iDMI, the lowest frequency point in the SW dispersion shifts from $k = 0$. Ma et al [63] found that when the excitation frequency is below the frequency of the mode with $k = 0$, SWs can only propagate in one direction. By using micromagnetic simulations, the authors present spatial maps of SWs with different frequencies. Due to different velocities and relaxation times, SWs with a frequency higher than Ferromagnetic Resonance (FMR) also propagate nonreciprocally in opposite directions with different decay lengths. Although unidirectional SWs are found in magnetic multilayers with iDMI, the experimental realization of the unidirectional SW propagation in such systems is still missing. Finding a magnetic thin film with both large iDMI and low damping is required for functional unidirectional SWbased devices.

Magnonic crystals (MCs) are periodic magnetic structures that can modify the band structure of SWs in GHz frequencies [64–70]. The asymmetric dispersion induced by iDMI could host nonreciprocal SW propagation. Gallardo et al investigated periodic iDMI-based magnonic crystals [56] using theoretical calculations and micromagnetic simulations. The periodic iDMI can be realized by patterning periodic heavy metal wires with strong spin–orbit coupling on top of a ferromagnetic thin film. Theoretical as well as numerical calculations are performed and the simulated magnon band structures are shown in figure 2(e). Due to the periodic iDMI, the SW dispersion is nonreciprocal and the magnonic band gap opens. By increasing the iDMI value, the bottom magnon bands of the dispersion shifts down and the first band gap is enhanced. The indirect band gaps are observed due to the fact that the maximum of the first band and the minimum of the second band shift in a different way when changing the iDMI value.

As the maximum of the first band reaches the second Brillouin zone, this branch becomes flat when the iDMI value is large. The larger periodicity favours the flat bands due to the reduction of the interlayer coupling of SWs underneath heavy metal wires. Below the heavy metal wires, the nonreciprocal SWs are also found to propagate, due to the non-zero phase velocities in the regions where the iDMI is non-zero.

Recently, Silvani et al [71] also presented the impact of the iDMI on the band structure of a one-dimensional magnonic crystal using micromagnetic simulations. The authors found flat bands at positive wavevectors when a large iDMI strength exists. Those flat modes are separated by forbidden gaps whose amplitudes depend on the value of $D$. By performing the time evaluation of the magnetization dynamics of the flat modes, the authors found that these modes are confined in specific regions and propagate only along the positive x-direction, indicating the SW unidirectionality. The formation of the indirect band gap as well as the low-frequency flat bands are demonstrated in the system of the magnonic crystal with periodic DMI, which encourages further studies in the investigation of chiral magnonic crystals and other related physical properties.

Asymmetric band structure has also been predicted and observed in magnonic crystals without iDMI. Mruczkiewicz et al theoretically investigated the impact of a perfect metal overlayer on the SW dispersion relation of 1D bi-component MCs [72] and in thin NiFe film dynamically coupled to an array of Ni stripes [51]. In both cases, the nonreciprocity is manifested in the shifting of the magnonic bandgap edges from the Brillouin zone border.

By placing an array of nanomagnets on top of a low damping magnetic insulator thin film, unidirectional SWs can be generated by the dynamics of the nanomagnets due to the chiral spin pumping effect [73]. Since the magnetic film is ultrathin (with a thickness of tens of nanometres), the DE-mode-induced SW nonreciprocity can be neglected. Typical heterostructures, where magnetic nanowires (Co) are placed on top of the low-damping magnetic thin film (YIG), are investigated, as shown in figure 3(a) [73]. Assuming the magnetic thin film has polycrystalline structures with isotropic in-plane magnetic properties and ignoring the PSSWs, the hybrid system can be treated as quasi-one-dimensional and the Hamiltonian can be written as [73]:

$$\begin{equation}\frac{\widehat{\mathcal{H}}}{\hbar} = \sum\limits_n {(\omega _n^ + {{\hat \beta }_{ + k}}{{\hat \alpha }^\dagger } + \omega _n^ - {{\hat \beta }_{ - k}}{{\hat \alpha }^\dagger })} ,\end{equation} \tag{ 15 }$$

Zoom In Zoom Out Reset image size

where ${\hat \alpha ^\dagger }$ is the creative operator of the magnon modes in the magnetic nanowires, and ${\hat \beta _{ + k}}$ and ${\hat \beta _{ - k}}$ are creative operators of propagating SWs in the magnetic film with opposite directions. The coupling strengths between magnon modes are:

$$\begin{align} &{\omega _n^ + = - \frac{{2\gamma }}{{n\pi }}{\sigma _n}\sqrt {\left( {{\mu _0}M_{\text{S}}^{\text{W}}} \right)\left( {{\mu _0}M_{\text{S}}^{\text{F}}} \right)} {\text{ }}\int {\widehat {\boldsymbol{m}}}_{\text{W}}^{\text{*}} {{{\boldsymbol{\tilde \Lambda }}}^{\text{*}}}{{\widehat {\boldsymbol{m}}}_{\text{F}}}\;{{\text{e}}^{kx}}{\text{d}}x} \nonumber\\ &{\omega _n^ - = - \frac{{2\gamma }}{{n\pi }}{\sigma _n}\sqrt {\left( {{\mu _0}M_{\text{S}}^{\text{W}}} \right)\left( {{\mu _0}M_{\text{S}}^{\text{F}}} \right)} {\text{ }}\int {\widehat {\boldsymbol{m}}}_{\text{W}}^{\text{*}} {\boldsymbol{\tilde \Lambda }}{\mkern 1mu} \widehat {\boldsymbol{m}}_{\mathbf{F}}\;{{\text{e}}^{kx}}{\text{d}}x} \end{align} \tag{ 16 }$$

with the SW wavenumbers $k = \pi n/a$ ${\text{with }}n = 2,4,6 \ldots { }$ propagating along ${\text{the }} + {{\boldsymbol{y}}}$ direction for $\omega _n^ +$ and the $- {{\boldsymbol{y}}}$ direction for $\omega _n^ -$ and ${\sigma _n} = \sin \left( {\frac{{kw}}{2}} \right)\left( {1 - {{\text{e}}^{ - kh}}} \right)$. $\mu {{\text{s}}_0}M_{\text{S}}^{\text{W}}$ and ${\mu _0}M_{\text{S}}^{\text{F}}$ are the saturation magnetization of magnetic nanowires and film, respectively. Here, ${\hat m_{\text{W}}} = \left( {{{\hat m}_x},{{\hat m}_y}} \right)$ describes the magnetization dynamics in nanowires and ${\hat m_x} = {\left( {\frac{a}{{4hw}}\sqrt {\frac{{{H_0} + M_{\text{S}}^{\text{W}}{N_{yy}}}}{{{H_0} + M_{\text{S}}^{\text{W}}{N_{xx}}}}} } \right)^{1/2}}$ and ${\hat m_y} = {\left( {\frac{a}{{4hw}}\sqrt {\frac{{{H_0} + M_{\text{S}}^{\text{W}}{N_{xx}}}}{{{H_0} + M_{\text{S}}^{\text{W}}{N_{yy}}}}} } \right)^{1/2}},$ where ${N_{xx}}$ and ${N_{yy}}$ are demagnetization factors; $\widetilde {{\boldsymbol{\Lambda }}} = \left( {\begin{array}{*{20}{c}} 1& i \\ i& { - 1} \end{array}} \right)\,$ and ${\widehat {{\boldsymbol{m}}}_{\text{F}}} = \left( {\hat m_x^k,\hat m_y^k} \right)$ describes the magnetization dynamics in the magnetic film and $i\hat m_x^k = \hat m_y^k = i{\left( {\frac{1}{{4t}}} \right)^{1/2}}$. When $\left| {\omega _n^ + \left| \ne \right|\omega _n^ - } \right|$, the interlayer dipolar coupling is chiral and the chirality ratio of the counter-propagating SWs can be defined as:

$$\begin{equation}\eta = \left| {\frac{{{{\left( {\frac{{\omega _n^ + }}{{\omega _n^ - }}} \right)}^2} - 1}}{{{{\left( {\frac{{\omega _n^ + }}{{\omega _n^ - }}} \right)}^2} + 1}}{ }} \right|.\end{equation} \tag{ 17 }$$

In the parallel state, where the magnetization in nanowires is parallel to that of the film, $\omega _n^ - = 0$ $\omega _n^ + \ne 0$ and the SWs only propagate in one direction. In the antiparallel state, ${\widehat {{\boldsymbol{m}}}_{\text{F}}} = \left( {\hat m_x^k, - \hat m_y^k} \right)$ and $\omega _n^ + = 0$ $\omega _n^ - \ne 0.$ The chirality of SWs is reversed and SWs only propagate in the opposite directions. In addition, the unidirectional SWs emitted by a nanomagnet are found to be perfectly trapped by a second initially passive magnet by a dynamical interference effect, as shown in figure 3(b) [74]. Moreover, due to the chiral spin pumping effect, the line width broadening in a magnetic heterostructure consisting of a Co nanowire grating dipolar-coupled to a YIG film is probed by BLS microscopy, indicating tunable magnetic damping [76]. Unidirectional SWs are found in this system and oscillating behaviour of the magnon population in Co nanowire grating is observed due to the magnon trap effect.

An experimental investigation is also conducted using a nanoscale magnetic array to excite unidirectional SWs [75, 77]. A Co nanowire array is patterned on top of an ultra-low-damping YIG thin film with a period of 600 nm. On top of the nanowire array, two identical coplanar waveguides (CPW) are fabricated to excite and detect propagating SWs. Due to the large demagnetization field, the magnetization of Co nanowires cannot be switched when the field is swept from positive to negative values, forming the parallel and antiparallel states, respectively. Due to the magnon–magnon coupling effect [78–82], the high-order exchange SWs with the wavelength of 60 nm can be excited and figures 3(d) and (e) show the transmission spectra ${S_{21}}$ and ${S_{12}}$, respectively. Unidirectional SWs can be observed and, by switching the parallel and antiparallel states, the propagation SW direction is reversed, which is in line with the theoretical calculations.

Since the demagnetization field in YIG film is very small, we could rotate the external field angle to change the angle between the magnetization of the nanowires and YIG film. When the external field is tilted away from the parallel direction to the nanowire, the chirality is broken and the SWs are not unidirectionally propagating anymore. We find that at the angle around 60°, the SW nonreciprocity is totally suppressed. The material engineering of such devices could further enhance device functionalities. It is promising to use antiferromagnetic materials for exciting, even shorter wavelength unidirectional SWs due to the sub-THz resonance frequencies.

Spin textures are non-collinear magnetic orders in the magnetic ground state, which are due to the interplay between dipolar interaction, symmetric Heisenberg exchange interaction and DMI [83–97]. By combining with spin textures such as domain walls, vortices and skyrmions, multiple functionalities can be realized in SW-based devices. Magnetic domain walls are boundaries between different magnetic domains and SWs can be channelled inside the domain wall. In 2015, Garcia-Sanchez et al demonstrated that SWs can propagate in narrow magnonic waveguides based on domain walls [98]. They found that SWs can travel in both Bloch- and Néel-type domain walls. In Néel-type domain walls, SWs are strongly nonreciprocal due to the iDMI. In 2016, Wagner et al experimentally demonstrated that magnetic domain walls can serve as a reconfigurable SW nanochannel [99]. SWs propagating inside magnetic-vortex-stabilized domain walls are also imaged using BLS and scanning transmission x-ray microscopy (STXM), respectively [100, 101].

Unidirectional SWs can also propagate in a Bloch-type domain wall (see figure 4(a)), which has recently been demonstrated by Henry et al [102]. In an array of parallel domain walls in the stripe domain configuration, SW dispersion is modified due to the sequence of up/down domain walls with opposite nonreciprocity. If SWs are excited within the gap formed between two domain wall channelled branches in its dispersion, unidirectional SWs can be launched, as shown in figure 4(b). Furthermore, if a nanomagnet array is placed on top of a Néel-type domain wall, unidirectional SWs can also be emitted due to the chiral spin pumping effect [103].

Zoom In Zoom Out Reset image size

The interplay between SWs and skyrmions has attracted considerable attention recently. When extended to a three-dimensional system, a skyrmion can form a string-like structure consisting of stacks of two-dimensional skyrmions, as shown in figure 4(c) [104]. Materials like Cu₂OSeO₃ can host those skyrmion strings in cryogenic temperatures [105]. Using all-electrical SW characterizations, Seki et al investigated the propagating SWs inside skyrmion strings in Cu₂OSeO₃ lamellas [104]. The authors find that the propagating SWs are reciprocal in frequencies and decay lengths as well as group velocities, which strongly depend on the excitation modes, namely, clockwise, counter-clockwise and breathing modes, as shown in figure 4(d). The nonreciprocal SWs in skyrmion strings show properties for unidirectional information transfer in a topological object. Theoretical studies have also demonstrated the elementary process for the skyrmion motion driven by SWs [106]. It is found that propagating SWs can be scattered by the skyrmion by solving the LLG equation, which is determined by the skyrmion Hall angle. The Gilbert damping as well as the SW amplitude could influence the speed of the skyrmion motion. Due to topological protection, the skyrmion cannot easily be deteriorated by SWs. Beyond magnetic domain walls and skyrmions, the investigation of unidirectional SWs in other magnetic textures, such as vortices and helimagnetic spirals, is required.

Magnetic textures can also be used to excite SWs with their high-frequency dynamic properties. In 2016, Van de Wiele et al proposed a method using pinned magnetic domain walls to excite SWs with short-wavelengths [107]. Then, Holländer et al demonstrated that magnetic domain walls can be an effective antenna for generating broadband SWs [108]. In 2020, Albisetti et al found that by using patterned shaped micro-antennas such as domain walls in synthetic antiferromagnetic thin films, spatially shaped SW wavefronts can be fully controlled, as shown in figure 5(a) [109]. By sweeping a heated scanning probe, the exchange bias direction can be modified as well as the underlayer CoFeB magnetization, and domain walls can be patterned in a reconfigurable manner [110]. Using STXM, the focusing of SWs emitted by a curved domain wall can be imaged, as shown in figure 5(b). The spatial profile extracted at the focal point shows the full width at half maximum of the beam amplitude is around 340 nm. This type of curved domain walls emitting SWs shows a strong nonreciprocity due to the particular spin textures.

Zoom In Zoom Out Reset image size

SWs can also be emitted by magnetic skyrmions. In 2020, Díaz et al found that skyrmion–antiskyrmion bilayers can form topological charge dipoles and can efficiently emit SWs with the wavelength down to 100 nm, as shown in figure 5(c) [111]. The different SW patterns are found to depend on the spiral or antispiral spatial profiles. Recently, Chen et al proved that chiral SWs can be generated by a magnetic skyrmion on top of a low-damping magnetic thin film, as shown in figure 5(d) [112]. The chirality of the SWs results from the dynamical dipolar coupling between the magnetic bilayer and is determined by the magnetization directions of the magnetic film as well as the film normal direction. Unidirectional SW propagation is observed in a certain direction.

Unidirectional information transfer is a basic element for modern logic architectures. A diode is a two-terminal electronic device that conducts an electrical current in one direction. The excitation and propagation of unidirectional SWs could result in a crucial functionality in magnonic logic and computing devices, for the ability to build SW diodes, SW insulators, SW interferometers, etc. An SW diode can be designed in imitation of manganic electronic diodes. In 2015, Lan et al proposed a method for a reconfigurable SW diode based on chiral bound states in magnetic domain walls with iDMI [113]. In a magnetic film with a Bloch-type domain wall separating neighbouring domains, the bound SWs can propagate identically in both directions in the domain wall without iDMI. However, in the presence of iDMI, SWs propagating in opposite directions are spatially separated to different edges of the domain wall. If two-terminal conduits are placed at the left of the domain wall, a prototypical function of the SW diode can be realized, as shown in figure 6(a). If SWs are excited from the top terminal, they can pass the left part of the domain wall and reach the bottom terminal, while the opposite direction is forbidden due to the fact that SWs can only propagate through the right part of the domain wall. Figure 6(b) shows the micromagnetic simulation results of the SW diode in the forward and reverse directions.

Zoom In Zoom Out Reset image size

The concept of an SW diode based on the chiral dipolar coupling in two exchange-coupled ferromagnetic layers was proposed by Grassi et al [50] They showed that the diode had a wide operation frequency window in the GHz range that can be adjusted by tuning the amplitude of the applied magnetic field, and its forward and reverse directions can be interchanged by switching the polarity of the field.

By using unidirectional magnetostatic coupling induced by the iDMI, SW diodes and circulators can also be designed in a magnetic multilayer system [114]. The concept of the SW diode is shown in figure 6(c). A Py (3 nm)/NM (5 nm)/Co (2 nm)/Pt multilayer is investigated and, due to the unidirectional coupling direction, propagating SWs can only transport in a single direction in Py thin film. A circulator design is also proposed in a similar multilayer structure, as shown in figure 6(d). SWs in Co film can only propagate from port 1 to port 2 due to the weak coupling with the Py. However, if SWs travel from port 2 to port 3, the coupling with Py is strong and the energy can only be transferred between two Co layers. The SW diode and circulator show promising potential as logic elements in energy-efficient magnonic circuits. The unidirectional SW diodes discussed above are mostly investigated using theoretical calculations and numerical simulations. Practical unidirectional SW diodes with more functionalities and sizes in the submicrometric range are required to buildmagnonic computing elements and circuits.

Topologically protected chiral edge SW modes have been predicted in both magnonic crystals and two-dimensional honeycomb lattices [115, 116]. These SWs are topologically nontrivial and propagate in a unidirectional manner without backward scattering. Using these unique features of chiral edge SWs, wave splitter and interferometer functions can be realized. A 1:4 SW beam splitter using three-domain walls to conduct the edge SWs is shown in figure 7(a). SWs are split at the edge of the structure and four output beams are observed with the same amplitude. A design of the chiral edge SW interferometer is also proposed in figure 7(d). The process can be represented in the diagram in figure 7(d), which is the same as the concept of the Mach-Zehnder interferometer in optics. Topologically protected chiral edge SWs are robust against defects and also possess a unidirectional nature, which is promising for topological magnonic devices and circuits.

Zoom In Zoom Out Reset image size

A prototype of a magnonic interferometer based on unidirectional SWs generated by nanomagnets has been proposed recently [117]. Two identical Co nanowires are patterned on top of a low-damping YIG thin film. Two beams of unidirectional exchange SWs can be excited respectively from two nanowires. Because SWs excited by two nanowires are emitted from a different position, a phase shift of ${{\Delta }}\phi = kd$ exists between two beams of SWs. When the phase shift ${{\Delta }}\phi = kd = n\pi$, where $n$ is an odd number, two beams of SWs interfere destructively, as shown in figure 7(c). In contrast, when the phase shift ${{\Delta }}\phi = kd = n\pi$, and $n$ is an even number, two beams of SWs interfere constructively and the SW amplitude is enhanced. The interference conditions depend on the separation distance $d$ as well as the wavevector $k$. By varying these two parameters, the SW interference patterns in the frequency domain can be observed, as shown in figure 7(d). The unidirectional SW devices discussed above are mostly in the microwave regime of coherent SWs. In a very recent paper by Han et al nonreciprocal transmission of incoherent magnons with asymmetric diffusion lengths are found in a magnetic bilayer structure by the spin Hall effect [118]. Incoherent magnons cover the spectrum from gigahertz up to terahertz and the SW nonreciprocity can be controlled in a non-volatile manner, which could lead to the passive directional signal isolation device in the diffusive regime.

Beyond the methods discussed above for the realization of unidirectional SW propagation, it is found that a saturation magnetization gradient of a perpendicular magnetized ferromagnetic film raises an asymmetric dispersion for nonreciprocal SW propagation [119]. Under certain circumstances, unidirectional SW propagation can also be achieved. In addition, very recently, it has been demonstrated that a simple design of a vortex core in a teardrop-shaped nano patch can serve as a unidirectional SW emitter [120].