Controlling Acoustic Waves Using Magneto-Elastic Fano Resonances

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ORE Open Research Exeter

TITLE Controlling acoustic waves using magneto-elastic Fano resonances

AUTHORS Latcham, O; Gusieva, Y; Shytov, AV; et al.

JOURNAL Applied Physics Letters

DEPOSITED IN ORE 29 July 2019

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http://hdl.handle.net/10871/38143

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The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication Controlling acoustic waves using magneto-elastic Fano resonances ❈♦♥♦❧❧✐♥❣ ❛❝♦✉✐❝ ✇❛✈❡ ✉✐♥❣ ♠❛❣♥❡♦✲❡❧❛✐❝ ❋❛♥♦ ❡♦♥❛♥❝❡ ❖✳ ❙ ✳▲❛❝❤❛♠✱✶ ❨✳ ●✉✐❡✈❛✱✷ ❆✳ ❱✳ ❙❤②♦✈✱✶ ❖✳ ❨✳ ●♦♦❜❡✱✷ ❛♥❞ ❱✳ ❱✳ ❑✉❣❧②❛❦✶✱ a) 1)University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom 2)Igor Sikorsky Kyiv Polytechnic Institute, 37 Prosp. Peremohy, Kyiv, 03056, Ukraine (Dated: 20 July 2019) 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 magneto- elastic 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 routes to enhance performance of the proposed devices.

Optical and, more generally, wave-based computing paradigms gain momentum on a promise to replace and com- plement the traditional semiconductor-based technology.1 The energy savings inherent to non-volatile memory devices has spurred the rapid growth of research in magnonics,2,3 in which spin waves4 are exploited as a signal or data carrier. Yet, the progress is hampered by the magnetic loss (damping).5,6 In- deed, 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 con- trast, the propagation distance of acoustic waves is typically much longer than that of spin waves at the same frequencies.7 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.8–10 To minimize the magnetic loss, the size of such magneto-acoustic FIG. 1. The prototypical magneto-elastic resonator is a thin magnetic H functional elements should be kept minimal. This implies slab (M) of width δ, biased by an external field B, and embedded coupling propagating acoustic waves to confined spin wave into a non-magnetic (NM) matrix. The acoustic wave with amplitude modes of finite-sized magnetic elements. As we show below I incident at angle θ induces precession of the magnetisation vector M via the magneto-elastic coupling. As a result, the wave is partly this design idea opens a route towards hybrid devices combin- transmitted and reflected, with respective amplitudes T and R . ing functional benefits of magnonics2,3 with the energy effi- ω ω ciency of phononics.7,11,12 The phenomena resulting from interaction between coherent spin and acoustic waves have already been addressed acoustic devices in which the signal is carried by acoustic in the research literature: the spin wave excitation of prop- waves while the magnetic field controls its propagation via agating acoustic waves7,13–15 and vice versa,8,16–18 acous- the magnetoelastic interaction in thin isolated magnetic in- tic parametric pumping of spin waves,19–21 magnon-phonon clusions as shown in Fig. 1. By changing the applied mag- 22–24 25 netic field, one can alter the frequency at which the incident

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 coupling in cavities and mode locking, magnonic- phononic crystals,26,27 Bragg scattering of spin waves from a acoustic waves hybridize with the magnetic modes of the in- surface acoustic wave induced grating,28–30 topological prop- clusions. Thereby, one can control the acoustic waves by the erties of magneto-elastic excitations,15,31 acoustically driven resonant behaviour of Breit-Wigner and Fano resonances in 41 spin pumping and spin Seebeck effect,32,33 and optical excita- the magnetic inclusion. We find that the strength of the res- tion and detection of magneto-acoustic waves.34–40 However, onances is suppressed by the ubiquitous magnetic damping studies of the interaction between propagating acoustic waves in realistic materials, but this can be mitigated by employing and spin wave modes of finite-sized magnetic elements, which oblique incidence geometry. To compare magneto-acoustic are the most promising for applications, have been relatively materials for such devices, we introduce a figure of merit. The scarce to date.10,34,36,39 magneto-elastic Fano resonance is identified as most promis- Here, we explore theoretically the class of magneto- ing 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- a)Electronic mail: [email protected] elastic coupling can affect sound propagation. A ferromag- This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. Controlling acoustic waves using magneto-elastic Fano resonances 2

netic slab ("magnetic inclusion") of thickness δ, of the order by the acoustic wave, we thus obtain of 10nm, is embedded within a non-magnetic medium (Fig.1). γB(ωk + iω k ) The slab is infinite in the Y Z plane, has saturation magneti- m ω,y y ω,x U (6) − x = 2 , zation M , and is biased by the applied field H = H zˆ. Due to ω ωxωy s B B − the magneto-elastic coupling, this equilibrium configuration is iγB(ωxkω,y +eiωkω,x) perturbed by shear stresses in the xz- and yz planes associated my = 2 U, (7) ω ωe xωe y with the incident acoustic wave. − e To derive the equations of motion, we represent the mag- where we have denoted ωx(y) = γµ0(HB + Nx(y)Ms) and e e k netic energy density F of the magnetic material as a sum of the ωx(y) = ωx(y) iωα. The complex-valued wave number ω 42 − magneto-elastic FME and purely magnetic FM contributions. is given by the dispersion relation Taking into account the Zeeman and demagnetizing energies, e H M µ0 2 2 2 2 we write FM = µ0 B + 2 (NxMx + NyMy ), where Nx(y) 2 ω ρ ω ωxωy − M kω = − . (8) are the demagnetising coefficients, Nx + Ny = 1, is the 2 γB2 2 2 C ω ωxωy + MC ωycos θ+ ωxsin θ magnetization and µ0 is the magnetic permeability. In a crys- − s e e tal of cubic symmetry, the magnetoelastic contribution takes h i the form43 Eq. (8) describes thee e hybridizatione betweene acoustic waves and magnetic precession at frequencies close to ferromag- B B netic resonance (FMR) at frequency ωFMR, with linewidth F = M M u + ′ M2u , i, j = x,y,z, (1) ME 2 ∑ i j i j 2 ∑ i ii ΓFMR. The frequency at which the precession ampli- Ms i= j Ms i 6 tudes (Eqs. (6) and (7)) diverge is given by the condition (ω + iΓ /2)2 = ω ω . In the limit of small α, this where B and B are the linear isotropic and anisotropic FMR FMR x y ′ yields ω = ω ω and Γ = α(ω + ω ). Away from magneto-elastic coupling constants, respectively.44 The strain FMR x y FMR x y the resonance, Eq. (8) gives the linear dispersion of acous- tensor is u = 1 (∂ U + ∂ U ), where U are the displace- e e jk 2 j k k j j tic waves. In the non-magnetic medium (B = 0), one finds ment vector components. To maximize the effect of the cou- k2 = ω2ρ /C . Here and below, the subscript ’0’ is used to pling B, we consider a transverse acoustic plane wave incident 0 0 0 mark quantities pertaining to the non-magnetic matrix. on the slab from the left and polarized along the bias field, so To calculate the reflection and transmission coefficients, Rω that Ux = Uy = 0, Uz = U(x,y,t). The non-vanishing com- 1 1 and Tω , for a magnetic inclusion, we introduce the mechanical ponents of the strain tensor are uxz = ∂xU and uyz = ∂yU, 2 2 impedance as Z = iσxz/ωUω . Solution of the wave matching and FME is linear in both M and U: problem can then be expressed via the ratio of load (ZME) and B source (Z0) impedances. For the magnetic slab, we find FME = (Mxuxz + Myuyz). (2) Ms 2 Ckω cosθ γB ωy Zω,ME = 1 + 2 . (9) ω MsC(ω ωxωy) The magnetization dynamics in the slab is due to the effec- − tive magnetic field, µ H = δF/δM. We define m as e 0 eff For the non-magnetic material, Eq. (9) recovers the usual the small perturbation of the magnetic− order, i.e. m M . e e s acoustic impedance45 Z = cosθ√ρ C . Due to magnon- Linearizing the Landau-Lifshitz-Gilbert equation,4| we| ≪write 0 0 0 phonon hybridization, Zω,ME diverges at ωFMR and vanishes at a nearby frequency ω . For α 0, the latter is given by ∂mx ∂U ∂my ME = = γµ0(HB + NyMs)my + γB + α , (3) − ∂t ∂y ∂t γB2 ∂my ∂U ∂m 2 2 = γµ (H + N M )m + γB + α x , (4) ωME = ωxωy ωycos θ + ωxsin θ . (10) ∂t 0 B x s x ∂x ∂t s − MsC where γ is the gyromagnetic ratio and α is the Gilbert Reflection Rω , and transmission Tω coefficients are then found 45 PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 damping constant. To describe the acoustic wave, we in- via the well-known relations as (ME) clude the magneto-elastic contribution to the stress, σ 2 jk = η 1 sin(kω xδ) R ω − · , (11) δFME /δu jk, into the momentum balance equation: ω = 2 , (η + 1) sin(kω xδ)+ 2iηω cos(kω xδ) ω · , · , 2iη ∂ 2U ∂ ∂U B ∂ ∂U B ω Tω = 2 , (12) ρ 2 = C + mx + C + my , (ηω + 1) sin(kω,xδ)+ 2iηω cos(kω,xδ) ∂t ∂x ∂x Ms ∂y ∂y Ms · · (5) where δ is the thickness of the magnetic inclusion and C c 46 where = 44 is the shear modulus and ρ is the mass density. ηω = Zω,ME/Z0. In close proximity to the resonance, the The non-magnetic medium is described by Eq.(5) with B = 0. impedance changes rapidly. For matching elastic properties Since the values of C, B, and Nx,y are constant within each of the magnetic and non-magnetic materials, Eq. (11) is ex- individual material, we shall seek solutions of the equations in panded near ωME. In the limit kω δ 1, we obtain the form of plane waves U,m ∝ exp[i(kω xx +kω yy ωt)]. ≪ x(y) , , − From herein, we consider all variables in the Fourier domain. iΓR/2 Rω = + R0, (13) For the magnetization precession in the magnetic layer driven (ω ωME)+ iΓR/2 − This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. Controlling acoustic waves using magneto-elastic Fano resonances 3

0.25 (a) 1.00 (c) Damping, α : (b) 0.20 10−4 0.95 0.3 10−3 2

| −2 0.15 | 10 | ) 0.90 ) ) 0.2 f f f ( ( ( T R | A

| 0.10

0.85 | 0.1 0.05 0.80

0.00 0.75 0.0 7.10 7.12 7.14 7.16 7.18 7.10 7.12 7.14 7.16 7.18 7.10 7.12 7.14 7.16 7.18 Frequency, f (GHz)

FIG. 2. The frequency dependence of the absolute values of (a) reflection and (b) transmission coefficients and (c) absorbance is shown for a 20nm thick magnetic inclusion. The vertical dashed and solid black lines represent the ferromagnetic resonance frequency ωFMR and magneto-elastic resonance frequency ωME respectively. The non-magnetic and magnetic materials are assumed to be silicon nitride and cobalt, respectively, with parameters given in the text. The bias field is µ0HB = 50mT, which leads to fME 7.138 GHz. ≈

where R0 represents the smooth non-resonant contribution due tic wavelength. However, this occurs at very high frequen- to elastic mismatch at the interfaces. In a system with no mag- cies, which we do not consider here. To understand the res- netic damping, the hybridization yields a resonance of finite onant magneto-elastic response, it is instructive to consider linewidth ΓR, first the case of normal incidence (θ = 0), when the demagnetising energy takes a simplified form due to the lack of im- γB2 ρ mediate interfaces to form surface poles in y the direction, Γ = ω cos2θ + ω sin2θ δ. (14) R 2M C C y x so that N = 1 and N = 0. Including magneto-elastic cou- s r x y pling (B = 0), we plot the frequency dependence of Rω and 6 The origin of this linewidth can be explained as follows. Due Tω using Eq. (11) and (12) in Fig.2. To gain a quantita- to the magneto-elastic coupling incident propagating acoustic tive insight, we analysed a magnetic inclusion made of cobalt modes can be converted into localised magnon modes. These 3 1 (ρ = 8900kgm− , B = 10MPa, C = 80GPa, γ = 176GHzT− , modes in turn either decay due to the Gilbert damping or are 1 M = 1MAm− ), embedded into a silicon nitride non-magnetic re-emitted as phonons. The rates of these transitions are pro- 3 matrix (ρ0 = 3192kgm− ,C0 = 298GPa). To highlight the portional to Γ and Γ , respectively, and the total decay 4 FMR R resonant behaviour, we first suppress α to 10− . The re- rate is Γ = ΓR + ΓFMR. This is similar to resonant scattering flection coefficient exhibits an asymmetric non-monotonic de- 47 in quantum theory , such that ΓR and ΓFMR are analogous to pendence, shown as a black curve in Fig.2(a), characteristic the the elastic (Γe) and inelastic (Γi) linewidths respectively. of Fano resonance.27,41 This line shape can be attributed to When α = 0, ΓFMR vanishes, and Γ = ΓR. coupling between the discrete FMR mode of the magnetic Acoustic waves in the geometry of Fig. 1 can be scat- inclusion and the continuum of propagating acoustic modes tered via several channels. E.g. in a non-magnetic system in the surrounding non-magnetic material.41 If the two ma- (B = 0), elastic mismatch can yield Fabry-Pérot resonance terials had matching elastic properties, Rω would exhibit a due to the quarter wavelength matching of δ and the acous- symmetric Breit-Wigner lineshape.47 The transmission shown in Fig.2(b) exhibits an approximately symmetric dip near 48 2 2 2 the resonance. The absorbance Aω = 1 Rω Tω , shown in Fig.2(c) exhibits a symmetric| | peak,−| since| the−| acous-| 0.20 tic waves are damped in our model only due to the coupling 45◦ with spin waves.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 0.15 To consider how the magneto-elastic resonance is affected |

) 3 f ◦ by the damping, we also plot the response for α of 10− and ( 30 2 R | 0.10 10− , red and blue curves in Fig.2, respectively. An increase of α from 10 4 to 10 3 signiﬁcantly suppresses and broad- 15◦ − − ◦ ens the resonant peak. For a more common, realistic value of 0.05 0 2 10− the resonance is quenched entirely. A stronger magnetoelastic coupling (i.e. high values of B) could, in principle, 7.05 7.10 7.15 7.20 countermand this suppression. This, however, is also likely Frequency, f (GHz) to enhance the phonon contribution to the magnetic damping, leading to a correlation between B and α observed in realistic magnetic materials.49 FIG. 3. Peak R( f ) is enhanced and slightly shifted to the left in the oblique incidence geometry (θ > 0◦). Coloured curves represent spe- To characterise the strength of the Fano resonance, we note ciﬁc incidence angles sweeping 0◦ to 45◦. Moderate Gilbert damping that the fate of the magnon excited by the incident acoustic 3 of α = 10− is assumed. wave is decided by the relation between the emission rate ΓR, This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. Controlling acoustic waves using magneto-elastic Fano resonances 4

see Eq. (14), and absorption rate ΓFMR. Hence, we introduce 0.25 the respective figure of merit as ϒ = ΓR/ΓFMR. This quan- 0.3 ΓFMR tity depends upon the material parameters, device geometry, Υ 0.20 and bias field. As seen from the first terms on the l.h.s. of (GHz)

0.2 Υ FMR Eqs. (6) and (7), the relation between the dynamic magnetisa- 0.15 / R

tion components mx,y are determined by the quantities ωx and Γ ω . Equating these terms, one finds m ∝ m ω /ω , i.e. the 0.10 y x y y x 0.1 50 m Figure of Merit, precession of is highly elliptical, due to the demagnetis- 0.05 p ΓR ing field along x. This negatively affects the phonon-magnon Linewidth, coupling for normal incidence (ky = 0): the acoustic wave 0.0 0.00 couples only to m , as given by the second term in Eqs. (6) 0 10 20 30 40 x Angle, θ (deg.) and (7). One way to mitigate this is to increase HB, moving the ratio ωy/ωx closer to 1 and thus improving the figure of merit. To compare different magneto-elastic materials, the FIG. 4. Figure of merit ϒ, and radiative linewidth ΓR, are both en- dependence on the layer thickness δ and elastic properties of hanced in the oblique incidence geometry (θ > 0◦). Ferromagnetic linewidth Γ remains unchanged. Co is assumed with α = 10 3. the non-magnetic matrix (i.e. ρ0 and C0) can be eliminated by FMR − 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 ment in a practical device, resonant scattering is still signifi- latter situation is more appropriate for a device application, cantly enhanced at smaller angles: θ HB/Ms 1 yields ≈ ≪ but to avoid unphysical parameters, we present our results for a twofold increase in ΓR and ϒ, as illustrated in Fig.4. p the same µ0HB. An example of such comparisons for yttrium Above, we have focused on the simplest geometry that ad- iron garnet (YIG), cobalt (Co) and permalloy (Py) is offered mits full analytic treatment. To implement our idea exper- in Table I. imentally, particular care should be taken about the acoustic waves polarization and propagation direction relative to the direction of the magnetization. Indeed, our choice max- TABLE I. Comparison of the figure of merit ϒ for different materi- imises magnetoelastic response. If however, the polariza- als, assuming δ = 20nm, µ0HB = 50mT and a silicon nitride non- magnetic matrix. tion is orthogonal to the bias field HB, i.e. Uz = 0, the coupling would be second-order in magnetization components Parameters YIG Co Py mx,y, and would not contribute to the linearized LLG equation. 3 3 4 ϒ(θ = 0◦) 4.3x10− 1.5x10− 1.9x10− Furthermore, we have neglected the exchange and magneto- 6 3 4 ΓR (GHz) 3.0x10− 6.5x10− 1.4x10− dipolar fields that could arise due to the non-uniformity of the 4 ΓFMR (GHz) 7.0x10− 4.3 0.74 magnetization. To assess the accuracy of this approximation, 3 2 3 ϒ(θ = 30◦) 8.1x10− 1.1x10− 1.1x10− we note that the length scale of this non-uniformity is set by 6 2 4 ΓR (GHz) 5.6x10− 4.7x10− 8.0x10− 4 the acoustic wavelength λ, of about 420nm for our parame- ΓFMR (GHz) 7.0x10− 4.3 0.74 ters rather than by the magnetic slab thickness δ. The asso- ωME (GHz) 0.48 7.14 6.26 ciated exchange field is µ M (kl )2 9mT. The k-dependent B 3 0 s ex (MJm− ) 0.55 10 -0.9 contributions to the magneto-dipole field≃ vanish at normal in- E (GPa) 74 80 50 3 cidence but may become significant at oblique incidence, giv- ρ (kgm− ) 5170 8900 8720 4 2 3 ing µ0Mskyδ 98mT at θ = 15◦. In principle, these could α 0.9x10− 1.8x10− 4.0x10− ≃ 1 increase the resonant frequency of the slab by a few GHz but Ms (kAm ) 140 1000 760 − would complicate the theory significantly. The detailed analy- sis of the associated effects is beyond the scope of this report. Another way to improve ϒ is to employ the oblique inci- In summary, we have demonstrated that the coupling be- PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 dence (θ = 0), in which the acoustic mode is also coupled to tween the magnetisation and strain fields can be used to con- 6 the magnetisation component my. The latter is not suppressed trol acoustic waves by magnetic inclusions. We show that by the demagnetisation effects if Ny 1. The resulting en- the frequency dependence of the waves’ reflection coefficient hancement in ϒ is reflected in the full≪ equation by the inclu- from the inclusions has a Fano-like lineshape, which is partic- sion of ωx and ωy from ΓR, ularly sensitive to the magnetic damping. Figure of merit is introduced to compare magnetoelastic materials and to char- Γ γδ ρ B2 H cos2θ + M sin2θ acterize device performance. In particular, the figure of merit ϒ R 0 B s = = 2 , (15) is significantly enhanced for oblique incidence of acoustic ΓFMR 2 C0 αCM r s waves, which enhances their coupling to the magnetic modes. where ωx ωy and HB Ms is assumed. For small θ, the We envision that further routes may be taken to transform ≫ ≪ approximation Nx 1 and Ny 0 still holds. As a result, non- our prototype designs into working devices, such as forming zero θ increases peak≃ reflectivity,≃ as seen in Fig.3. Note also a magneto-acoustic metamaterial to take advantage of spatial the θ dependence of the resonant frequency ωME as reflected resonance. in Eq. (10). Though larger incidence may be hard to imple- The research leading to these results has received funding This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. Controlling acoustic waves using magneto-elastic Fano resonances 5

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This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 0.25 (a) 1.00 (c) Damping, α : (b) 0.20 10−4 0.95 0.3 10−3 2

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f ◦

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PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 Frequency, f (GHz) This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset. 0.25 0.3 ΓFMR

Υ 0.20 (GHz)

0.2 Υ FMR 0.15 / R Γ 0.10 0.1 Figure of Merit, 0.05 ΓR Linewidth, 0.0 0.00 0 10 20 30 40 Angle, θ (deg.) PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5115387 This is the author’s peer reviewed, accepted manuscript. However, online version of record will be different from this once it has been copyedited and typeset.