Chiral Phonon Transport Induced by Topological Magnons
CChiral Phonon Transport Induced by Topological Magnons
Even Thingstad, Akashdeep Kamra, Arne Brataas, and Asle Sudbø
Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
The plethora of recent discoveries in the field of topological electronic insulators has inspired asearch for boson systems with similar properties. There are predictions that ferromagnets on a two-dimensional honeycomb lattice may host chiral edge magnons. In such systems, we theoreticallystudy how magnons and phonons couple. We find topological magnon-polarons around the avoidedcrossings between phonons and topological magnons. Exploiting this feature along with our findingof Rayleigh-like edge phonons in armchair ribbons, we demonstrate the existence of chiral edgemodes with a phononic character. We predict that these modes mediate a chirality in the coherentphonon response and suggest measuring this effect via elastic transducers. These findings reveal apossible approach towards heat management in future devices.
Introduction .— Topological electronic insulators [1–5]are characterized by an insulating bulk with conduct-ing ‘chiral’ edge states. The unidirectional propagationof these chiral modes is ‘topologically protected’ againstdefects at low temperatures when we can disregard in-elastic scattering from phonons [5]. This has led to thedevelopment of a wide range of essential concepts, includ-ing Majorana modes [6–9], topological quantum compu-tation [10, 11], and chiral transport. Inspired by thesefindings, there has been an upsurge of efforts towardsfinding similar states in other systems [12] with an em-phasis on bosonic excitations [13–19]. There are predic-tions of topological magnons [15–17] in honeycomb ferro-magnets with an engineered Dzyaloshinskii-Moriya inter-action [20, 21] that induces the necessary band gap. Incontrast to fermionic systems with Fermi energy withinthis band gap, the bulk is not necessarily insulating inbosonic systems [22].The field of magnonics [23–26] focuses on pure spintransport mediated by magnons [27]. It is possible to ex-ploit the low-dissipation and wave-like nature of these ex-citations in information processing [28, 29]. The coherentpumping of chiral surface spin wave (Damon-Eshbach)modes induces cooling via incoherent magnon-phononscattering [30]. Besides application oriented properties,the bosonic nature of magnons, combined with spintronicmanipulation techniques [24, 31], allows for intriguingphysics [32–35]. The coupling [36] between magnons andphonons fundamentally differs from the electron-phononinteraction and results in a coherent hybridization of themodes [37], in addition to the temperature dependent in-coherent effects [30, 38] discussed above. The direct influ-ence of the hybridization between magnons and phonons,known as magnon-polarons [39, 40], has been observed inspin and energy transport in magnetic systems [41–46].In this Letter, we address the robustness of the topo-logical magnons in a honeycomb ferromagnet [15–17]against their coupling with the lattice vibrations. Incontrast to the case of electron-phonon coupling, wherephonons can be disregarded at low temperatures, themagnon dispersion may undergo significant changes with new states emerging in the band gap [45, 46]. We findthat in the honeycomb ferromagnet with spins orientedorthogonal to the lattice plane, only transverse phononmodes with out-of-plane displacement couple to spin. Tounderstand the eigenmodes, we evaluate and analyze thecoupled spin and out-of-plane phonon modes for an in-finitely large plane as well as for a finite ribbon geometry.We quantify the effect of the magneto-elastic coupling onthe magnon Hall conductivity and find a non-monotonicdependence on the coupling strength. Our analysis of thefinite ribbons shows that topological magnons hybridizewith bulk phonons around the avoided crossings in theircoupled dispersion, forming magnon-polarons with topo-logical chiral properties. Hence, while their edge local-ization is weakened, the magneto-elastic coupling doesnot completely remove the topological magnons. Fur-thermore, we find that armchair edges support Rayleigh-like edge phonon modes in sharp contrast to the zigzagedges. When topological magnons hybridize with theseedge phonons, edge magnon-polarons with almost undi-minished chirality are formed. We suggest a setup whichutilizes this induced chirality in coherent phonon trans-port. Such systems enable the observation of the topolog-ical physics and serve as a prototype for a unidirectionalheat pump. This offers a highly feasible alternative toproducing topological phonon diodes [47–49].
Model .—We consider a ferromagnetic material with lo-calized spins on a two-dimensional honeycomb lattice,allow for out-of-plane vibrations of the lattice sites, andassume there is magneto-elastic coupling. This systemcan be modelled by a Hamiltonian of the form H = H m + H ph + H me , where H m is the magnetic Hamilto-nian, H ph describes the phonons, and H me represents themagneto-elastic coupling.The Hamiltonian we consider is inspired by the Hal-dane model [1] given by [15–17] H m = − J ∑ ⟨ i,j ⟩ S i ⋅ S j + D ∑ ⟪ i,j ⟫ ν ij ˆ z ⋅ S i × S j − B ∑ i S zi . (1)The first term describes the ferromagnetic exchange cou-pling between nearest neighbour sites, while the sec- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r ond accounts for the Dzyaloshinskii-Moriya interaction[20, 21] between next-to-nearest neighbours [50]. TheHaldane sign ν ij = ± d and the next-to-nearest neighbour distance be a . Refs. [16, 17] discussthe dispersion relation and Berry curvature of this spinmodel in linear spin wave theory.For the phonon Hamiltonian, we consider only the out-of-plane degrees of freedom since only these modes coupleto the spin to lowest order in the linear spin wave ex-pansion. We assume nearest-neighbour interactions withelastic constant C , let the mass associated with the spinson the lattice sites be m , and disregard substrate cou-pling. Introducing S k = ∑ β cos ( k ⋅ βββ ) , where the sum isover the three next-to-nearest neighbour vectors βββ of Fig.1(a), we obtain the dispersion relation ω ph ± ( k ) = √ Cm √ ± √ + S k (2)for the free phonon modes.Motivated by the continuum limit description [36, 37],we write down the lattice magneto-elastic coupling tolinear order in the magnon amplitude, obtaining H me = κ ∑ D ∑ i ∈ D ∑ ααα D S i ⋅ ααα D ( u zi − u zi + ααα D ) (3)where κ parametrizes the strength of the magnon-phononcoupling, ∑ D denotes the sum over sublattices, ∑ i ∈ D isthe sum over the lattice sites on the D sublattice, and ααα D are the corresponding nearest neighbour vectors. Theout-of-plane deviation for lattice site i is denoted by u zi . Bulk spectrum .—We introduce the Holstein-Primakoffrepresentation of spins and use linear spin wave theoryin the spin- and magneto-elastic terms [27]. Within therotating wave approximation [51], the resulting Hamilto-nian describing the phonon and magnon modes of thesystem is obtained as H = ∑ k ψ † k M k ψ k , where ψ † k =( a † k , b † k , c † k − , c † k + ) . Here, a k and b k are annihilation op-erators for the sublattice magnon modes on the A and B sublattices, while c k ± are the annihilation operators forthe phonon modes. The matrix M k takes the form M k = ⎛⎜⎜⎜⎝ A + h z h − g A − g A + h + A − h z g B − g B + g ∗ A − g ∗ B − ω ph k − g ∗ A + g ∗ B + ω ph k + ⎞⎟⎟⎟⎠ , (4)where A = JS + B , h z ( k ) = D S ∑ β sin ( k ⋅ βββ ) , h − ( k ) = − JS ∑ α exp (− i k ⋅ ααα ) , and h + = ( h − ) ∗ . Thecoupling between the D -sublattice magnons and thephonon branch ± is captured by g D ± , which is pro-portional to the dimensionless coupling strength ˜ κ = KK' Γ M (a)(c) 𝛽 𝛽 𝛽 𝛼𝛼 𝛼 i j 𝜈 ij =1 𝜈 =-1 𝛽 𝛽 𝛽 𝛼 𝛼 𝛼 =-1 K M En e r g y [ J S ] PhM ji AB (b) FIG. 1. (a) Lattice geometry showing the nearest neighbourvectors ααα , next-to-nearest neighbour vectors βββ , and the Hal-dane sign ν ij = ±
1. (b) The first Brillouin zone in reciprocalspace, including the paths along which we plot the dispersionrelation in figure (c). The parameter values used are
D = . J , B = . JS , √ C / m = . JS , and rescaled coupling strength˜ κ = .
03 (see main text). The magnon (yellow) and phonon(purple) character of the modes is indicated with colors. Themodes are significantly affected by the magneto-elastic cou-pling only close to avoided crossings. ( κd / JS ) √̵ h S / m ( C / m ) . The spectrum obtained bydiagonalizing this matrix is plotted in Fig. 1(c) along thepaths displayed in Fig. 1(b). Hall conductivity .—The topological nature of the spinmodel is manifested in the magnon Hall conductivity thatarises because of the time-reversal symmetry breakingcaused by the Dzyaloshinskii-Moriya interaction.The spin current operator J γ may be found from a con-tinuity equation or magnon group velocity approach [52],both yielding J γ = ∑ k ( a † k b † k ) ( ∂H m ( k ) ∂k γ ) ( a k b k ) (5)along the Cartesian direction γ . Here, H m ( k ) is the ma-trix representation of the magnon Hamiltonian. Assum-ing we apply a weak in-plane magnetic field gradient ∇B ,we are interested in the current j = σ ∇B , which is de-termined by the conductivity tensor σ [52]. The Hallconductivity can be calculated using the Kubo formula,giving σ xy = ∑ k ∑ α,β ≠ α n B ( E kα ) C αβ ( k ) , (6)where E kα is the energy eigenvalue of band α and n B ( E kα ) is the corresponding Bose factor. Thecurvature-tensor C αβ is given by C αβ ( k ) = i J αβy ( k ) J βαx ( k ) − J αβx ( k ) J βαy ( k )( E kα − E kβ ) , (7)where ( α, β ) are band-indices, and J αβγ ( k ) are the en-ergy eigenstate matrix elements of the current operatorat quasimomentum k . Disregarding the magneto-elasticcoupling, the band-curvature C α = ∑ β ≠ α C αβ can be iden-tified as the Berry curvature.Expressing the sublattice magnon operators in termsof the eigenmode operators, one may identify the currentmatrix elements J αβγ and integrate the curvature overthe Brillouin zone to obtain the Hall conductivity. Weare particularly interested in the effect of the magneto-elastic coupling, and therefore present the dependence ofthe Hall conductivity on the dimensionless coupling ˜ κ inFig 2.To understand this dependence, we consider thecurvature-tensor C αβ . When the bands α and β bothhave a predominant magnon content, the topologicalnature of the underlying magnons gives a finite curva-ture. This magnon curvature is largest close to theDirac points [16, 17]. Close to an avoided crossing,the magneto-elastic coupling changes the spectrum andcauses transfer of band-curvature between the relevantbands α and β . The latter can be seen by plotting thecurvature-tensor element C αβ for the band-pairs withavoided crossings, as shown in the insets of Fig 2. Theresulting change in Hall conductivity is given by thesecurvature-tensor elements weighted with the differencebetween the Bose factors of the relevant bands. This fol-lows from the anti-symmetry property of the curvature-tensor. The two band-pairs in the insets contributeoppositely to the Hall conductivity, and the competi-tion between their curvature transfer explains the non-monotonic behaviour of the Hall conductivity. Ribbon geometry and coherent transport .—Due to thetopological nature of the magnon model under consider-ation and the bulk-boundary correspondence, there aregapless magnon edge states in a finite sample [5, 15–17]. Considering an armchair ribbon with finite width,the one-dimensional projection of the energy spectrumis plotted in Fig. 3. The corresponding spectrum forthe zigzag edge ribbon is given in the Supplemental Ma-terial [53], where also Refs. [54–58] appear. Magnonand phonon modes hybridize in regions with an avoidedcrossing. When the upper phonon band lies within thebandgap of the pure magnon spectrum, there are modeswith a mixed content of chiral magnon edge states andphonons. Although the spectra look qualitatively simi-lar, there is a crucial distinction between the two cases.For the zigzag edge configuration, all the phonon modes x y / x y T = 0.10 T = 0.30 T = 0.60 a a a a a Pair (1,2) a a a Pair (2,3)
FIG. 2. Dependence of the Hall conductivity on the magneto-elastic coupling strength ˜ κ for parameter values D = . J , B = . JS , and √ C / m = . JS for different temperatures T in units of JS . The insets show the quasimomentum de-pendence of the curvature-tensor at ˜ κ = .
03 for band-pairs(1,2) and (2,3), where the bands are labelled according totheir energy and band 1 is the lowest band. The dominantcontribution in these band-pairs comes from the regions withavoided crossings of the respective bands. are delocalized throughout the sample, while the arm-chair edges host “Rayleigh-like” edge phonon modes. Indirect analogy with Rayleigh modes on the surface ofa three-dimensional material, the localization length ofthese modes is directly proportional to their wavelength,as shown in the Supplemental Material [53]. These edgephonon modes are supported by the half-hexagon pro-trusions of the armchair edge that can pivot around thebonds parallel to the edges connecting these protrusions,see Fig. 3. No such parallel bonds exist for the zigzagedge.The Hall conductivity is a hallmark of topological elec-tronic properties and motivates a similar role for the Hallconductivity mediated by topological magnons. How-ever, in contrast to electrons, the bosonic nature of themagnons results in the lack of a general proportionalitybetween the magnon Hall conductivity and the Chernnumber [52]. Furthermore, the observation of a magnonplanar Hall effect [59] in a cubic, non-topological mag-net suggests that this Hall conductivity may not be re-garded as a smoking-gun signature for topological prop-erties. Thus, we suggest a complementary approach toobserve the topological nature of the underlying magnonsby elastically probing the chirality of the magneto-elastichybrid modes.We propose to observe coherent chiral phonon propa-gation in the experimental setup of Fig. 4(b) by utiliz-ing the edge modes, as depicted schematically in Fig.4(a) [60], on the upper armchair edge of the sam-ple. Taking inspiration from previous related experi-ments [41, 61], we suggest to inject elastic energy intothe sample middle at the upper edge using a nano-scalevariant of the interdigital transducer design [62, 63], elab-orated further in the Supplemental Material [53]. For agiven transducer design, modes are excited with fixedwavevectors ± k x and a tunable frequency. Similar trans-ducers can be used to detect the elastic response p L / R on the left (L) and right (R) edges of the sample. Here, p L / R is the elastic power detected at the transducers.Figure 4(a) schematically depicts the dispersion forthe magnetoelastic modes localized on an armchair edge.Disregarding magnetoelastic coupling, the edge hosts twocounterpropagating Rayleigh-like edge phonons and asingle chiral edge magnon. There is thus no chiral-ity in the phononic response. Due to magnetoelasticcoupling, the Rayleigh-like phonon with wavevector − k x hybridizes with the chiral magnon to form a magnon-polaron while the other phonon remains unchanged. Thisbreaks the symmetry between the counterpropagatingphononic modes and the result is non-zero chirality χ =( p R − p L )/( p R + p L ) . Furthermore, as shown in Fig.4(a), the hybridization with the magnon mode reversesthe group velocity direction of the participating phononmode. In principle, this gives perfectly chiral phonontransport.The wavevector location of the avoided crossing canbe tuned via the Zeeman shift in the magnon disper-sion. Performing a frequency integrated measurementover an energy range of the same order as the magneto-elastic coupling, one obtains a peaked chirality when themagnetic field is such that the wavevector of the avoidedcrossing coincides with the wavevector of the transducer,obtaining a chirality as shown in Fig. 4(b). Performinga similar transport experiment on the zigzag edge doesnot give chiral phonon transport since the delocalizedphonons hybridize with counterpropagating magnons onboth the edges, thereby destroying the overall chirality.In addition, the size of the avoided crossing is smaller dueto the smaller overlap with the localized chiral magnon.The armchair edge is therefore crucial for obtaining thechirality. Summary .— We have examined the robustness of topo-logical magnons in a honeycomb ferromagnet againsttheir interaction with phonons. Their topological prop-erties, albeit weakened, survive the magneto-elastic cou-pling. The magnon Hall conductivity of the system isfound to depend on the magneto-elastic coupling strengthin a non-monotonic, temperature-sensitive manner. Ex-ploiting the Rayleigh-like edge phonons in armchair rib-bons, we predict the existence of topological magnon-polarons confined to the boundary. We have suggestedan experimental setup capable of probing the chiralnature of the topological magnon-polarons by elasticmeans, which thus serves as a platform for chiral coherentphononic transport.
Acknowledgments .—We acknowledge support from theResearch Council of Norway Grant Nos. 262633 “Cen- - /3 /6 0 /6 /3 k x d En e r g y [ J S ] PhM-0.41 -0.4 -0.39 -0.382.432.442.452.46
FIG. 3. One-dimensional projection of the dispersion relationfor the magnetoelastic modes on a honeycomb ribbon witharmchair edges. In addition to the bulk bands, there are twotopological edge magnon states crossing the magnon bandgap, as well as Rayleigh-like edge phonons. The inset showsthe avoided crossing of a topological magnon edge mode withthe two quasi-degenerate edge phonon modes. The parametervalues are
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Topological in-sulators and topological superconductors (Princeton Uni-versity Press).[58] Pascal Ruello and Vitalyi E Gusev, “Physical mecha-nisms of coherent acoustic phonons generation by ultra-fast laser action,” Ultrasonics , 21–35 (2015).[59] J. Liu, L. J. Cornelissen, J. Shan, T. Kuschel, and B. J.van Wees, “Magnon planar hall effect and anisotropicmagnetoresistance in a magnetic insulator,” Phys. Rev.B , 140402 (2017).[60] The inset of Fig. 3(b) shows two phonon modes. One isirrelevant since it is localized on the opposite edge.[61] P. G. Gowtham, T. Moriyama, D. C. Ralph, and R. A.Buhrman, “Traveling surface spin-wave resonance spec-troscopy using surface acoustic waves,” Journal of Ap-plied Physics , 233910 (2015).[62] S. Datta, Surface acoustic wave devices (Prentice-Hall,1986).[63] Alexander V. Mamishev, Kishore Sundara-Rajan, FuminYang, Yanqing Du, and Markus Zahn, “Interdigital sen-sors and transducers,” Proceedings of the IEEE , 808(2004). Supplemental Material to the manuscript “Chiral Phonon Transport Induced by TopologicalMagnons” by
Even Thingstad, Akashdeep Kamra, Arne Brataas, Asle Sudbø
RAYLEIGH-LIKE PHONON EDGE MODES
To describe the phonons, as discussed in the main paper, we consider a force constant model for out-of-plane phononmodes on the honeycomb lattice with only nearest neighbour interaction. This is described by the Hamiltonian H = ∑ i p i m + ∑ ⟨ i,j ⟩ C ( u i − u j ) , (8)where i and j are lattice site indices running over both the A and B sublattices of the honeycomb lattice.To investigate edge modes in the system, we consider a finite ribbon geometry with periodic boundary conditions inone direction, and with a finite number of unit cells in the other direction. The edges of such ribbons can mainly beof two types: zigzag and armchair. The lattice geometries of these cases are shown in Fig. 5.To find the phonon energy spectrum for these lattice geometries, we introduce the partial Fourier transform of thelattice site deviations and momenta, which for the lattice site deviation takes the form u Dx,y = √ N x ∑ k u ky exp ( ik ˆ x ⋅ r Dn ) , (9)where u Dx,y is the lattice site deviation on sublattice D in unit cell ( x, y ) , r Dx,y is the corresponding equilibriumposition, and N x is the number of unit cells in the horizontal direction. The periodicity requirement u Dx,y = u Dx + N x ,y then gives k = πn / N x λ , where λ is the periodicity of the lattice in the horizontal direction and n is an integer. For thezigzag edge ribbon, λ = √ d , while λ = d for the armchair edge ribbon. This determines the size of the Brillouin zone.Introducing u † k = ( u A − k , u B − k , u A − k , u B − k , . . . , u A − kN y , u B − kN y ) (10)with similar notation for the momentum, the phonon Hamiltonian can be written on the form H = m ∑ k p † k p k + C ∑ k u † k M k u k , (11)with a matrix M k coupling the deviations on the various sublattices and neighbouring unit cell layers. ThisHamiltonian is diagonalized through a unitary transform of the deviations and momenta followed by introducingphonon creation and annihilation operators c † k and c k [54]. The excitation spectrum is then given by the phononfrequencies ω kn , where ω kn = ( C / m ) λ kn and { λ kn } are the eigenvalues of M k .Following this procedure for the ribbon geometry with zigzag edges, we obtain the spectrum in Fig. 6(a). The upperand lower branches of the phonon spectrum meet at k x d = π / √ k x d = π / √
3, consistent with the resultobtained by taking the 1-dimensional projection of the bulk bands. In Fig. 7, we plot the spatial profile of someselected eigenstates at quasimomentum k x d = π / √
3. All the modes are delocalized.Examining the armchair ribbon spectrum in Fig. 6(b), one may notice that the modes marked with green arrowsstand out from the rest. If we were to compute the bulk spectrum and then perform a 1D projection, the two modesmarked in green would not be found. We therefore conclude that they must originate from an edge effect.This is confirmed by examining the spatial profile of the modes, as shown in Fig. 8 for the modes between the upperand lower bulk phonon branches. The deviation amplitudes are finite on the outer armchair edges of the sample,and exponentially decaying into the interior of the sample. The inset shows the decay length as function of theinverse quasimomentum, and demonstrates that ξ ∝ / k x . This is perfectly analogous to the behaviour of so-called y x x=1y=1 x=2y=1 x=3y=1x=1y=2 x=1y=3 x=1y=4 x=1y=5 (a) Zigzag edge ribbon. x=1y=1 x=1y=2 x=1y=3 x=1y=4x=2y=1 x=3y=1x=1y=5 x=1y=63dxy (b) Armchair edge ribbon FIG. 5.
Lattice geometries for the zigzag- and armchair edge ribbons, including unit cell labelling. We assume periodic boundaryconditions in the horizontal direction and a finite number of hexagon layers in the vertical direction.
23 3 43 3 23 k x d E ( J S ) (a) Zigzag edge ribbon /3 /6 0 /6 /3 k x d E ( J S ) (b) Armchair edge ribbon FIG. 6.
One-dimensional projection of the energy spectrum for phonon modes on the honeycomb ribbon geometry with zigzag andarmchair edges for N y = unit cells in the vertical direction (see Fig. 5). For the armchair ribbon, in addition to the bulk modes,there are edge modes marked with green arrows. Rayleigh modes on the surface of a three-dimensional material [56]. Our modes can therefore be characterized asone-dimensional analogs of Rayleigh modes.From the above discussion, it follows that that while the armchair edges support edge modes, the zigzag edge does not.This is rooted in the fact that on the edge unit cells of the armchair ribbon, both atoms have 2 nearest neighbours.For the zigzag ribbon, one atom has 2 nearest neighbours, but the other has 3. Vibrations are therefore easier toexcite on the edges of the armchair ribbon.
COUPLED MAGNETOELASTIC MODES IN ZIGZAG RIBBON
To compute the excitation spectrum for the model with coupled magnon and phonon modes, we first calculate thephonon and magnon edge modes for the uncoupled model. The phonon modes were discussed in the previous section,and we refer to the literature for the magnon spectrum [57]. Expressing the magneto-elastic coupling in terms ofthese eigenmodes and diagonalizing the resulting matrix, we obtain the excitation spectra.For the zigzag edge ribbon, the spectrum is shown in Fig. 9. All phonon modes are delocalized. In the inset, weshow the hybridization of the chiral edge magnon mode with some of these delocalized modes. The armchair ribbon y A m p li t u d e Mode 0Mode 1Mode 200Mode 201Mode 202
FIG. 7.
Deviation amplitudes for selected zigzag ribbon phonon modes at k x d = π / √ for N y = unit cells in the verticaldirection. Fast oscillations have been averaged out. All modes are delocalized.
300 320 340 360 380 400 y/a A m p li t u d e k_x=0.300k_x=0.079k_x=0.045k_x=0.032k_x=0.025k_x=0.02010 20 30 40 50 k x d / d ξ -k x +k x FIG. 8. (a) Rayleigh mode schematic. At a given armchair edge, two Rayleigh-like modes propagate along the edge with quasimomenta ± k x . The modes are localized within a distance ξ ( k x ) from the edge. (b) Deviation amplitudes for the Rayleigh-like edge modes asfunction of the vertical position y in units of the Bravais lattice constant a for different momenta k x . The amplitudes are normalized tothe value on the edge for easier comparison. The inset shows the localization length ξ ( k x ) as function of the inverse quasimomentum,demonstrating that ξ ∝ / k x , consistent with the behaviour expected from ordinary Rayleigh modes. spectrum has already been discussed in the main text. INTERDIGITAL ELASTIC TRANSDUCERSGeneral principles and qualitative description
The interdigital transducer [55, 63] (IDT) consists of two metallic electrodes with a series of sections, called fingers,which are patterned into a comb-like structure on top of a piezoelectric material (Fig. 10). When a voltage is appliedacross the two electrodes, it creates a pattern of alternating charges on adjacent fingers via the capacitive effect. Viathe constitutive properties of the piezoelectric material, this results in a pattern of alternating strains. An applied acvoltage with angular frequency ω thus excites acoustic waves at the same frequency in the piezoelectric material. Thewavelength is determined by the corresponding dispersion relation ω = ck [ ? ], where c is the speed of sound in thematerial and k is the wavenumber. If the ensuing wavelength λ = π / k is equal to the spacing between the adjacentfingers belonging to the same electrode, the acoustic signal interferes constructively and the excitation efficiency0 k x d En e r g y [ J S ] PhM1.4 1.5 1.6 1.72.62.72.82.9
FIG. 9.
Energy spectrum for the coupled magnon and phonon modes of the ribbon geometry with zigzag edge as function ofquasimomentum k x . The magnon (yellow) and phonon (purple) content of each mode is indicated with color. OutputInput
Acoustic wave
FIG. 10. Schematic depiction of an interdigital transducer employed to excite Rayleigh waves via an ac voltage on the left handside. The same structure converts the acoustic waves back to ac voltage on the right and enables their detection. The metallicelectrodes are lithographically patterned on top of a piezoelectric material, such as Lithium Niobate, into the depicted combstructure. is high. If there is a mismatch between the finger spacing and the wavelength excited at the applied ac voltagefrequency, the acoustic waves tend to cancel each other and excitation efficiency is low. With an increasing number N of fingers, the reinforcement or cancellation effect is stronger and the excitation resonances become sharper. Thus,the operation principle of an IDT is similar to that of a Bragg grating. Then, it is easy to understand that peaks inexcitation are observed at multiple frequencies (and wavelengths) corresponding to the finger spacing being multiplesof the acoustic wavelength. The fundamental peak is the strongest and subsequent overtones are progressively weakeras demonstrated by the frequency transfer characteristics discussed below.Conventionally, IDTs have been employed in applications such as analog filters, and their desired operation frequencyrange has been from MHz to several tens of GHz [55]. A typical piezoelectric material employed is Lithium Niobatewith a Rayleigh wave speed of 3.3 km / s. Thus, the fundamental peak corresponding to a center frequency of 1GHz requires finger spacing of around 1 µ m, which could easily be achieved via photolithography techniques. Withcontemporary electron-beam lithography techniques, a finger spacing of several tens of nanometers is readily possible,thus allowing a fundamental frequency of tens of GHz. Employing higher overtones allows pushing the operationfrequency to several tens of GHz, and is predominantly limited by the driving electronics [63]. Due to the purviewof their conventional applications, attempts to achieve higher frequencies have been limited. With recent advancesin ultrafast lasers, several conventional methods have been adapted to achieve coherent phonon generation in theTHz regime [58]. Thus, the operation range of the proposed method is estimated to be rather wide with up tohundreds of GHz in frequency and tens of nanometers in wavelength. The wavenumber selectivity can also beincreased, in principle to arbitrary values, by using a large number of fingers. Combined with the tunability of1 z FIG. 11. Modeling the acoustic output of an interdigital transducer within the delta function model. An applied voltagegenerates charges on the metallic electrodes via the capacitive effect. The accumulated charge (or equivalently electric field) isconverted into strain via the piezoelectric effect. the exact magnon-phonon anticrossing point (via an applied field, for example) across a broad range of frequenciesand wavevectors, the proposed experimental method is well within the range of the contemporary state-of-the-arttechnology.
Frequency resolved acoustic output