Controlling acoustic waves using magnetoelastic Fano resonances

TL;DR

This paper introduces a theoretical framework for magneto-elastic devices that manipulate acoustic waves via Fano resonances, enabling energy-efficient control of wave transmission for signal processing applications.

Contribution

It presents a novel theoretical analysis of magneto-elastic devices utilizing Fano resonances to control acoustic wave scattering, including methods to enhance coupling and mitigate losses.

Findings

Resonant control of acoustic wave transmission via magnetic field tuning.

Enhanced magnetoelastic coupling through oblique incidence geometry.

Potential for energy-efficient acoustic signal processing devices.

Abstract

We propose and analyze theoretically a class of energy-efficient magneto-elastic devices for analogue signal processing. The signals are carried by transverse acoustic waves while the bias magnetic field controls their scattering from a magneto-elastic slab. By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic waves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic waves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behaviour akin to inelastic scattering in atomic and nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magnetoelastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss potential benefits and issues in realistic systems and suggest further routes to enhance performance of the proposed devices.

Tables

Table 1. Table 1: Comparison of the figure of merit Υ Υ \Upsilon for different materials, assuming δ = 20 𝛿 20 \delta=20 nm, μ 0 ​ H B = 50 subscript 𝜇 0 subscript 𝐻 B 50 \mu_{0}H_{\mathrm{B}}=50 mT and C 0 = 298 subscript 𝐶 0 298 C_{0}=298 GPa.

Parameters	YIG	Co	Py
${\rm Y}\left(\theta =0^{\circ }\right)$	$4.3$x${10}^{-2}$	$1.7$x${10}^{-3}$	$2.7$x${10}^{-4}$
${\Gamma }_R$ (ns-1)	$1.9$x${10}^{-4}$	$7.5$x${10}^{-3}$	$2.0$x${10}^{-4}$
${\Gamma }_{\mathrm{FMR}}$ (ns-1)	$4.4$x${10}^{-3}$	4.3	0.74
${\rm Y}\left(\theta ={30}^{\circ }\right)$	$4.1$x${10}^{-2}$	$2.5$x${10}^{-3}$	$2.8$x${10}^{-4}$
${\Gamma }_R$ (ns-1)	$1.8$x${10}^{-4}$	$1.1$x${10}^{-2}$	$2.1$x${10}^{-4}$
${\Gamma }_{\mathrm{FMR}}$ (ns-1)	$4.4$x${10}^{-3}$	4.3	0.74
$f_{\mathrm{ME}}={\omega }_{\mathrm{ME}}/2\pi$ (GHz)	2.97	7.14	6.26
$B$ (MJm-3)	0.55	10	-0.9
$C$ (GPa)	74	80	50
$\rho$ (kgm-3)	5170	8900	8720
$\alpha$	$0.9$x${10}^{-4}$	$1.8$x${10}^{-2}$	$4.0$x${10}^{-3}$
$M_s$ (kAm-1)	140	1000	760

Equations

$$F_{\mathrm{ME}}=\frac{B}{M_{\mathrm{s}}^2}\sum _{i\ne j}{M}_i{M}_j{u}_{ij}+\frac{B^{\prime }}{M_{\mathrm{s}}^2}\sum _i{M}_i^2{u}_{ii},i,j=x,y,z,$$

$$F_{\mathrm{ME}}=\frac{B}{M_{\mathrm{s}}}(M_x{u}_{xz}+M_y{u}_{yz}).$$

$$-\frac{\partial m_x}{\partial t}$$

$$\frac{\partial m_y}{\partial t}$$

$$\rho \frac{{\partial }^{2}U}{\partial t^2}=\frac{\partial }{\partial x}\left(C\frac{\partial U}{\partial x}+\frac{B}{M_{\mathrm{s}}}{m}_x\right)+\frac{\partial }{\partial y}\left(C\frac{\partial U}{\partial y}+\frac{B}{M_{\mathrm{s}}}{m}_y\right),$$

$$m_x$$

$$m_y$$

$$k_{\omega ,x}^2=\frac{\frac{\rho }{C}{\omega }^2\left({\omega }^2-{\tilde{\omega }}_x{\tilde{\omega }}_y\right)-k_{\omega ,y}^2\left({\omega }^2-{\tilde{\omega }}_x{\tilde{\omega }}_y+\frac{\gamma B^2}{M_{\mathrm{s}}C}{\tilde{\omega }}_x\right)}{\left[{\omega }^2-{\tilde{\omega }}_x{\tilde{\omega }}_y+\frac{\gamma B^2}{M_{\mathrm{s}}C}{\tilde{\omega }}_y\right]},$$

$$Z_{\omega ,\mathrm{ME}}^{(F/B)}=\frac{C{k}_{\omega ,x}}{\omega }\left(1+\frac{\gamma B^2}{C{M}_{\mathrm{s}}}\frac{{\tilde{\omega }}_y\mp i\omega \frac{k_{\omega ,y}}{k_{\omega ,x}}}{{\omega }^2-{\tilde{\omega }}_x{\tilde{\omega }}_y}\right).$$

$${\omega }_{\mathrm{ME}}=\sqrt{{\omega }_x{\omega }_y-\frac{\gamma B^2}{M_{\mathrm{s}}C}{\omega }_y}.$$

$$R_{\omega }$$

$$T_{\omega }$$

$$R_{\omega }$$

$$\varphi$$

$${\Gamma }_{\mathrm{R}}=\frac{\gamma B^2}{2M_{\mathrm{s}}{C}^2\cos \theta }\sqrt{{\rho }_0{C}_0}\left({\omega }_y{\cos }^2\theta +\frac{C^2}{C_0^2}{\omega }_x{\sin }^2\theta \right)\delta .$$

$${\rm Y}=\frac{{\Gamma }_{\mathrm{R}}}{{\Gamma }_{\mathrm{FMR}}}=\frac{\gamma \delta B^2}{2}\sqrt{{\rho }_0{C}_0}\frac{\left(H_{\mathrm{B}}{\cos }^2\theta +\frac{C^2}{C_0^2}{M}_{\mathrm{s}}{\sin }^2\theta \right)}{\alpha C^2{M}_{\mathrm{s}}^2\cos \theta },$$

Full text

††preprint: AIP/123-QED

Optical and, more generally, wave-based computing paradigms gain momentum on a promise to replace and complement the traditional semiconductor-based technology.Feitelson (1988) The energy savings inherent to non-volatile memory devices has spurred the rapid growth of research in magnonics, Kruglyak, Demokritov, and Grundler (2010); Nikitov et al. (2015) in which spin wavesAkhiezer, yakhtar, and Peletminskii (1968) are exploited as a signal or data carrier. Yet, the progress is hampered by the magnetic loss (damping).Krivoruchko (2015); Azzawi, Hindmarch, and Atkinson (2017) Indeed, the propagation distance of spin waves is rather short in ferromagnetic metals while low-damping magnetic insulators are more difficult to structure into nanoscale devices. In contrast, the propagation distance of acoustic waves is typically much longer than that of spin waves at the same frequencies.Collins (1984) Hence, their use as the signal or data carrier could reduce the propagation loss to a tolerable level. Notably, one could control the acoustic waves using a magnetic field by coupling them to spin waves within magnetostrictive materials.Kittel (1958); Bömmel and Dransfeld (1959); Dreher et al. (2012) To minimize the magnetic loss, the size of such magneto-acoustic functional elements should be kept minimal. This implies coupling propagating acoustic waves to confined spin wave modes of finite-sized magnetic elements. As we show below this design idea opens a route towards hybrid devices combining functional benefits of magnonicsKruglyak, Demokritov, and Grundler (2010); Nikitov et al. (2015) with the energy efficiency of phononics.Collins (1984); Li et al. (2012); Maldovan (2013)

The phenomena resulting from interaction between coherent spin and acoustic waves have already been addressed in the research literature: the spin wave excitation of propagating acoustic wavesCollins (1984); Höllander et al. (2018); Streib, Keshtgar, and Bauer (2018); Thingstad et al. (2019) and vice versa,Kittel (1958); Li et al. (2017); Gowtham et al. (2015); Ulrichs et al. (2017) acoustic parametric pumping of spin waves,Gurevich (1965); Keshtgar, Zareyan, and Bauer (2014); Chowdhury, Dhagat, and Jander (2015) magnon-phonon coupling in cavitiesLitvinenko et al. (2015); Zhang et al. (2016); Kong et al. (2019) and mode locking,Wang and lin Hsu (1970) magnonic-phononic crystals,Nikitov et al. (2012); Graczyk, Kłos, and Krawczyk (2017) Bragg scattering of spin waves from a surface acoustic wave induced grating,Chumak et al. (2010); Kryshtal and Medved (2017a, b) topological properties of magneto-elastic excitations,Thingstad et al. (2019); Takahashi and Nagaosa (2016) acoustically driven spin pumping and spin Seebeck effect,Uchida et al. (2011); Polzikova et al. (2018) and optical excitation and detection of magneto-acoustic waves.Yahagi et al. (2014); Kats et al. (2016); Berk et al. (2017); Yang et al. (2018); Deb et al. (2018); Mondal et al. (2018); Hashimoto et al. (2018) However, studies of the interaction between propagating acoustic waves and spin wave modes of finite-sized magnetic elements, which are the most promising for applications, have been relatively scarce to date.Dreher et al. (2012); Yahagi et al. (2014); Berk et al. (2017); Mondal et al. (2018)

Here, we explore theoretically the class of magneto-acoustic devices in which the signal is carried by acoustic waves while the magnetic field controls its propagation via the magnetoelastic interaction in thin isolated magnetic inclusions as shown in Fig. 1. By changing the applied magnetic field, one can alter the frequency at which the incident acoustic waves hybridize with the magnetic modes of the inclusions. Thereby, one can control the acoustic waves by the resonant behaviour of Breit-Wigner and Fano resonances in the magnetic inclusion.Limonov et al. (2017) We find that the strength of the resonances is suppressed by the ubiquitous magnetic damping in realistic materials, but this can be mitigated by employing oblique incidence geometry. To compare magneto-acoustic materials for such devices, we introduce a figure of merit. The magneto-elastic Fano resonance is identified as most promising in terms of frequency and field tuneability. To enhance resonant behaviour, we explore the oblique incidence as a means by which to enhance the figure of merit.

We consider the simplest geometry in which magneto-elastic coupling can affect sound propagation. A ferromagnetic slab ("magnetic inclusion") of thickness $\delta$, of the order of 10 nm, is embedded within a non-magnetic medium (Fig.1). The slab is infinite in the $Y-Z$ plane, has saturation magnetization $M_{\mathrm{s}}$, and is biased by the applied field $\mathbf{H}_{\mathrm{B}}=H_{\mathrm{B}}\hat{\bf z}$. Due to the magneto-elastic coupling, this equilibrium configuration is perturbed by shear stresses in the $xz$- and $yz$ planes associated with the incident acoustic wave.

To derive the equations of motion, we represent the magnetic energy density $F$ of the magnetic material as a sum of the magneto-elastic $F_{\mathrm{ME}}$ and purely magnetic $F_{\mathrm{M}}$ contributions.Comstock and Auld (1963) Taking into account the Zeeman and demagnetizing energies, we write $F_{\mathrm{M}}=-\mu_{0}\bm{H}_{\mathrm{B}}\bm{M}+\frac{\mu_{0}}{2}(N_{x}M_{x}^{2}+N_{y}M_{y}^{2})$, where $N_{x(y)}$ are the demagnetising coefficients, $N_{x}+N_{y}=1$, $\bm{M}$ is the magnetization and $\mu_{0}$ is the magnetic permeability. In a crystal of cubic symmetry, the magnetoelastic contribution takes the formKamra et al. (2015)

[TABLE]

where $B^{\prime}$ and $B$ are the linear isotropic and anisotropic magneto-elastic coupling constants, respectively.Callen and Callen (1965) The strain tensor is $u_{jk}=\frac{1}{2}\left(\partial_{j}U_{k}+\partial_{k}U_{j}\right)$, where $U_{j}$ are the displacement vector components. To maximize the effect of the coupling $B$, we consider a transverse acoustic plane wave incident on the slab from the left and polarized along the bias field, so that $U_{x}=U_{y}=0$, $U_{z}=U(x,y,t)$. The non-vanishing components of the strain tensor are $u_{xz}=\frac{1}{2}\partial_{x}U$ and $u_{yz}=\frac{1}{2}\partial_{y}U$, and $F_{\mathrm{ME}}$ is linear in both ${\bm{M}}$ and $U$:

[TABLE]

The magnetization dynamics in the slab is due to the effective magnetic field, $\mu_{0}\bm{H}_{\mathrm{eff}}=-\delta F/\delta\bm{M}$. We define $\bm{m}$ as the small perturbation of the magnetic order, i.e. $|\bm{m}|\ll M_{\mathrm{s}}$. Linearizing the Landau-Lifshitz-Gilbert equation,Akhiezer, yakhtar, and Peletminskii (1968) we write

[TABLE]

where $\gamma$ is the gyromagnetic ratio and $\alpha$ is the Gilbert damping constant. To describe the acoustic wave, we include the magneto-elastic contribution to the stress, $\sigma^{\mathrm{(ME)}}_{jk}=\delta F_{ME}/\delta u_{jk}$, into the momentum balance equation:

[TABLE]

where $C=c_{44}$ is the shear modulus and $\rho$ is the mass density. The non-magnetic medium is described by Eq.(5) with $B=0$.

Since the values of $C$, $B$, and $N_{x,y}$ are constant within each individual material, we shall seek solutions of the equations in the form of plane waves $U,m_{x(y)}\propto\mathrm{exp}[i(k_{\omega,x}x+k_{\omega,y}y-\omega t)]$. From herein, we consider all variables in the Fourier domain. For the magnetization precession in the magnetic layer driven by the acoustic wave, we thus obtain

[TABLE]

where we have denoted $\omega_{x(y)}=\gamma\mu_{0}(H_{\mathrm{B}}+N_{x(y)}M_{\mathrm{s}})$ and $\widetilde{\omega}_{x(y)}=\omega_{x(y)}-i\omega\alpha$. The complex-valued wave number $\bm{k}_{\omega,x}$ is given by the dispersion relation

[TABLE]

where $k_{\omega,y}$ is equal to that of the incident wave, and the branch with $\mathop{\mathrm{Im}}\nolimits k_{\omega,x}>0$ describes a forward wave decaying into the slab. Eq. (8) describes the hybridization between acoustic waves and magnetic precession at frequencies close to ferromagnetic resonance (FMR) at frequency $\omega_{\mathrm{FMR}}$, with linewidth $\Gamma_{\mathrm{FMR}}$. The frequency at which the precession amplitudes (Eqs. (6) and (7)) diverge is given by the condition $\left(\omega_{\mathrm{FMR}}+i\Gamma_{\mathrm{FMR}}/2\right)^{2}=\widetilde{\omega}_{x}\widetilde{\omega}_{y}$. In the limit of small $\alpha$, this yields $\omega_{\mathrm{FMR}}=\omega_{x}\omega_{y}$ and $\Gamma_{\mathrm{FMR}}=\alpha(\omega_{x}+\omega_{y})$. Away from the resonance, Eq. (8) gives the linear dispersion of acoustic waves. In the non-magnetic medium ($B=0$), one finds $k_{0}^{2}=\omega^{2}\rho_{0}/C_{0}$. Here and below, the subscript ’0’ is used to mark quantities pertaining to the non-magnetic matrix.

To calculate the reflection and transmission coefficients, $R_{\omega}$ and $T_{\omega}$, for a magnetic inclusion, we introduce the mechanical impedance as $Z=i\sigma_{xz}/\omega U_{\omega}$. Solution of the wave matching problem can then be expressed via the ratio of load ($Z_{\mathrm{ME}}$) and source ($Z_{0}$) impedances. For impedances in the forward (F) and backward (B) directions in the magnetic slab, we find

[TABLE]

Here, the ‘-’ and ‘+’ signs correspond to (F) and (B), respectively. For the non-magnetic material, Eq. (9) recovers the usual acoustic impedanceBrekhovskikh and Godin (1997) $Z_{0}=\mathrm{cos}\theta\sqrt{\rho_{0}C_{0}}$. Due to magnon-phonon hybridization, $\mathop{\mathrm{Re}}\nolimits Z^{(F/B)}_{\omega,\mathrm{ME}}$ diverges at $\omega_{\mathrm{FMR}}$ and vanishes at a nearby frequency $\omega_{\mathrm{ME}}$. For $\alpha=0$, the latter is given by

[TABLE]

Reflection $R_{\omega}$ and transmission $T_{\omega}$ coefficients are then found via the well-known relationsBrekhovskikh and Godin (1997) as

[TABLE]

where $\delta$ is the thickness of the magnetic inclusion, $\eta_{\omega}=Z_{\mathrm{ME}}^{\mathrm{(F)}}/Z_{0}$ and $\widetilde{\eta}_{\omega}={Z}_{\mathrm{ME}}^{\mathrm{(B)}}/Z_{0}$.Born and Wolf (1964) In close proximity to the resonance, the impedances changes rapidly. Expanding Eq. (11) near $\omega_{\mathrm{ME}}$ in the limit $k_{\omega}\delta\ll 1$, we obtain

[TABLE]

where $R_{0}$ represents a smooth non-resonant contribution due to elastic mismatch at the interfaces, while $\phi$ represents a resonant phase, which is non-zero for finite $\theta$ and approaches $\pi$ rapidly. In a system with no magnetic damping, the hybridization yields a resonance of finite linewidth $\Gamma_{\mathrm{R}}$,

[TABLE]

The origin of this linewidth can be explained as follows. Due to the magneto-elastic coupling incident propagating acoustic modes can be converted into localised magnon modes. These modes in turn either decay due to the Gilbert damping or are re-emitted as phonons. The rates of these transitions are proportional to $\Gamma_{\mathrm{FMR}}$ and $\Gamma_{\mathrm{R}}$, respectively, and the total decay rate is $\Gamma=\Gamma_{\mathrm{R}}+\Gamma_{\mathrm{FMR}}$. This is similar to resonant scattering in quantum theoryLandau and Lifshitz (1965), such that $\Gamma_{\mathrm{R}}$ and $\Gamma_{\mathrm{FMR}}$ are analogous to the the elastic $(\Gamma_{\mathrm{e}})$ and inelastic $(\Gamma_{\mathrm{i}})$ linewidths respectively. When $\alpha=0$, $\Gamma_{\mathrm{FMR}}$ vanishes, and $\Gamma=\Gamma_{\mathrm{R}}$.

Acoustic waves in the geometry of Fig. 1 can be scattered via several channels. E.g. in a non-magnetic system ($B=0$), elastic mismatch can yield Fabry-Pérot resonance due to the quarter wavelength matching of $\delta$ and the acoustic wavelength. However, this occurs at very high frequencies, which we do not consider here. To understand the resonant magneto-elastic response, it is instructive to consider first the case of normal incidence ($\theta=0$), when the demagnetising energy takes a simplified form due to the lack of immediate interfaces to form surface poles in $y$ the direction, so that $N_{x}=1$ and $N_{y}=0$. Including magneto-elastic coupling ($B\neq 0$), we plot the frequency dependence of $R_{\omega}$ and $T_{\omega}$ using Eq. (11) and  (12) in Fig.2. To gain a quantitative insight, we analysed a magnetic inclusion made of cobalt ($\rho=8900\mathrm{kgm}^{-3}$, $B=10\mathrm{MPa}$, $C=80\mathrm{GPa}$, $\gamma=176\mathrm{GHzT^{-1}}$, $M=1\mathrm{MAm^{-1}}$), embedded into a non-magnetic matrix ($\rho_{0}=3192\mathrm{kgm^{-3}},C_{0}=298\mathrm{GPa}$). To highlight the resonant behaviour, we first suppress $\alpha$ to $10^{-4}$. The reflection coefficient exhibits an asymmetric non-monotonic dependence, shown as a black curve in Fig.2(a), characteristic of Fano resonance.Graczyk, Kłos, and Krawczyk (2017); Limonov et al. (2017) This line shape can be attributed to coupling between the discrete FMR mode of the magnetic inclusion and the continuum of propagating acoustic modes in the surrounding non-magnetic material.Limonov et al. (2017) If the two materials had matching elastic properties, $R_{\omega}$ would exhibit a symmetric Breit-Wigner lineshape.Landau and Lifshitz (1965) The transmission shown in Fig.2(b) exhibits an approximately symmetric dip near the resonance.Klaiman (2017) The absorbance $|A_{\omega}|^{2}=1-|R_{\omega}|^{2}-|T_{\omega}|^{2}$, shown in Fig.2(c) exhibits a symmetric peak, since the acoustic waves are damped in our model only due to the coupling with spin waves.

To consider how the magneto-elastic resonance is affected by the damping, we also plot the response for $\alpha$ of $10^{-3}$ and $10^{-2}$, red and blue curves in Fig.2, respectively. An increase of $\alpha$ from $10^{-4}$ to $10^{-3}$ significantly suppresses and broadens the resonant peak. For a more common, realistic value of $10^{-2}$ the resonance is quenched entirely. A stronger magnetoelastic coupling (i.e. high values of $B$) could, in principle, countermand this suppression. This, however, is also likely to enhance the phonon contribution to the magnetic damping, leading to a correlation between $B$ and $\alpha$ observed in realistic magnetic materials.Emori et al. (2017)

To characterise the strength of the Fano resonance, we note that the fate of the magnon excited by the incident acoustic wave is decided by the relation between the emission rate $\Gamma_{\mathrm{R}}$, see Eq. (14), and absorption rate $\Gamma_{\mathrm{FMR}}$. Hence, we introduce the respective figure of merit as $\Upsilon=\Gamma_{\mathrm{R}}/\Gamma_{\mathrm{FMR}}$. This quantity depends upon the material parameters, device geometry, and bias field. As seen from the first terms on the l.h.s. of Eqs. (6) and (7), the relation between the dynamic magnetisation components $m_{x,y}$ are determined by the quantities $\omega_{x}$ and $\omega_{y}$. Equating these terms, one finds $m_{x}\propto m_{y}\sqrt{\omega_{y}/\omega_{x}}$, i.e. the precession of $\bm{m}$ is highly elliptical,Kim (2012) due to the demagnetising field along $x$. This negatively affects the phonon-magnon coupling for normal incidence ($k_{y}=0$): the acoustic wave couples only to $m_{x}$, as given by the second term in Eqs. (6) and (7). One way to mitigate this is to increase $H_{\mathrm{B}}$, moving the ratio $\omega_{y}/\omega_{x}$ closer to 1 and thus improving the figure of merit. To compare different magneto-elastic materials, the dependence on the layer thickness $\delta$ and elastic properties of the non-magnetic matrix (i.e. $\rho_{0}$ and $C_{0}$) can be eliminated by calculating a ratio of the figures of merit for the compared materials. The comparison can be performed either at the same value of the bias field, or at the same operating frequency. The latter situation is more appropriate for a device application, but to avoid unphysical parameters, we present our results for the same $\mu_{0}H_{\mathrm{B}}$. An example of such comparisons for yttrium iron garnet (YIG), cobalt (Co) and permalloy (Py) is offered in Table 1.

Another way to improve $\Upsilon$ is to employ the oblique incidence ($\theta\neq 0$), in which the acoustic mode is also coupled to the magnetisation component $m_{y}$. The latter is not suppressed by the demagnetisation effects if $N_{y}\ll 1$.

The resulting enhancement in $\Upsilon$ is reflected in the full equation by the inclusion of $\omega_{x}$ and $\omega_{y}$ from $\Gamma_{\mathrm{R}}$,

[TABLE]

where $\omega_{x}\gg\omega_{y}$ and $H_{\mathrm{B}}\ll M_{\mathrm{s}}$ is assumed. For small $\theta$, the approximation $N_{x}\simeq 1$ and $N_{y}\simeq 0$ still holds. As a result, non-zero $\theta$ increases peak reflectivity, as seen in Fig.3. The evolution of the curves in Fig.3 with $\theta$ is explained by the variation of the phase $\phi$ of the resonant scattering relative to that of the non-resonant contribution $R_{0}$. The latter changes its sign at incidence angle of about $30^{\circ}$, which yields a nearly symmetric curve (blue), and an inverted Fano resonance at larger angles (green). Although larger incidence angles may be hard to implement in a practical device, the resonant scattering is still enhanced at smaller angles.

Above, we have focused on the simplest geometry that admits full analytic treatment. To implement our idea experimentally, particular care should be taken about the acoustic waves polarization and propagation direction relative to the direction of the magnetization. Indeed, our choice maximises magnetoelastic response. If however, the polarization is orthogonal to the bias field $H_{\mathrm{B}}$, i.e. $U_{z}=0$, the coupling would be second-order in magnetization components $m_{x,y}$, and would not contribute to the linearized LLG equation. Furthermore, we have neglected the exchange and magneto-dipolar fields that could arise due to the non-uniformity of the magnetization. To assess the accuracy of this approximation, we note that the length scale of this non-uniformity is set by the acoustic wavelength $\lambda$, of about 420nm for our parameters rather than by the magnetic slab thickness $\delta$. The associated exchange field is $\mu_{0}M_{\mathrm{s}}(kl_{ex})^{2}\simeq 9$mT. The $k$-dependent contributions to the magneto-dipole field vanish at normal incidence but may become significant at oblique incidence, giving $\mu_{0}M_{\mathrm{s}}k_{y}\delta\simeq 98$mT at $\theta=15^{\circ}$. In principle, these could increase the resonant frequency of the slab by a few GHz but would complicate the theory significantly. The detailed analysis of the associated effects is beyond the scope of this report.

In summary, we have demonstrated that the coupling between the magnetisation and strain fields can be used to control acoustic waves by magnetic inclusions. We show that the frequency dependence of the waves’ reflection coefficient from the inclusions has a Fano-like lineshape, which is particularly sensitive to the magnetic damping. Figure of merit is introduced to compare magnetoelastic materials and to characterize device performance. In particular, the figure of merit is significantly enhanced for oblique incidence of acoustic waves, which enhances their coupling to the magnetic modes. We envision that further routes may be taken to transform our prototype designs into working devices, such as forming a magneto-acoustic metamaterial to take advantage of spatial resonance.

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