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We theoretically investigate the topological aspects of the magnon-phonon hybrid excitation in a simple two-dimensional (2D) square-lattice ferromagnet with perpendicular magnetic anisotropy [1]. Several distinguishing features of our model are as follows. Our model is optimized for atomically thin magnetic crystals, i.e., 2D magnets. The recent discovery of magnetism in 2D van der Waals materials opens huge opportunities for investigating unexplored rich physics and future spintronic devices in reduced dimensions [2,3]. In our 2D model, we ignore the nonlocal dipolar interaction, which is not a precondition for a finite Berry curvature in 2D magnets. Moreover, the Berry curvature we find requires neither a special spin asymmetry such as the Dzyaloshinskii-Moriya interaction (DMI) nor a special lattice symmetry: Our 2D model description is applicable for general thin-film ferromagnets. Therefore, we show that even without such long-range dipolar interaction, DMI, or special lattice symmetry, the nontrivial topology of a magnon-phonon hybrid can emerge by taking account of the well-known magnetoelastic interaction originates from the magnetocrystalline anisotropy. Because the magnetocrystalline anisotropy is ubiquitous in ferromagnetic thin-film structures, our result does not rely on specific preconditions and thus is quite generic. Furthermore, we show that the topological structures of the magnon-polaron bands can be manipulated by effective magnetic fields via topological phase transition. We uncover the origin of the nontrivial topological bands by mapping our model to the well-known two-band model for topological insulators [4], where the Chern numbers are read by counting the number of topological textures, called skyrmions, of a certain vector in momentum space. In this picture, the magnon-phonon hybridization induces the chiral texture of the momentum space vector. As an experimental probe for our theory, we propose the thermal Hall conductivity [Fig. 2].

**References:**

[1] G. Go et al., Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers, Phys. Rev. Lett. 123, 237207 (2019).&lt;br/&gt;[2] C. Gong et al., Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature (London) 546, 265 (2017).&lt;br/&gt;[3] B. Huang et al., Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature (London) 546, 270 (2017).&lt;br/&gt;[4] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/436ed988646e642bcb99ed25391293f6.jpg)
Fig. 1. (a) The schematic illustration of the magnon and phonon system. The ground state of the magnetization is given by the uniform spin state along the &lt;i&gt;z-&lt;/i&gt;axis (red arrow). (b) The Chern number of our magnon-phonon hybrid system. &lt;i&gt;H &lt;/i&gt;represents the effective magnetic field including the anisotropy field and the external magnetic field. Band dispersion (c) without the magnon-phonon interaction and (d) with the magnon-phonon interaction.

![](https://assets.underline.io/uploads/markdown_image/1/image/2bc284e0ccbc6784cc3e8d6d0786fed4.jpg)
FIG. 2. (a) Dependence of thermal Hall conductivity on the effective magnetic field &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;eff&lt;/sub&gt; and with different temperatures &lt;i&gt;T&lt;/i&gt; using the parameters in the main text. (b) Dependence of the Chern number on the effective magnetic field &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;eff &lt;/sub&gt;using the parameters in the main text.

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 Topological magnon-polaron in a two-dimensional ferromagnet

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We theoretically investigate the topological aspects of the magnon-phonon hybrid excitation in a simple two-dimensional (2D) square-lattice ferromagnet with perpendicular magnetic anisotropy [1]. Several distinguishing features of our model are as follows. Our model is optimized for atomically thin magnetic crystals, i.e., 2D magnets. The recent discovery of magnetism in 2D van der Waals materials opens huge opportunities for investigating unexplored rich physics and future spintronic devices in reduced dimensions [2,3]. In our 2D model, we ignore the nonlocal dipolar interaction, which is not a precondition for a finite Berry curvature in 2D magnets. Moreover, the Berry curvature we find requires neither a special spin asymmetry such as the Dzyaloshinskii-Moriya interaction (DMI) nor a special lattice symmetry: Our 2D model description is applicable for general thin-film ferromagnets. Therefore, we show that even without such long-range dipolar interaction, DMI, or special lattice symmetry, the nontrivial topology of a magnon-phonon hybrid can emerge by taking account of the well-known magnetoelastic interaction originates from the magnetocrystalline anisotropy. Because the magnetocrystalline anisotropy is ubiquitous in ferromagnetic thin-film structures, our result does not rely on specific preconditions and thus is quite generic. Furthermore, we show that the topological structures of the magnon-polaron bands can be manipulated by effective magnetic fields via topological phase transition. We uncover the origin of the nontrivial topological bands by mapping our model to the well-known two-band model for topological insulators [4], where the Chern numbers are read by counting the number of topological textures, called skyrmions, of a certain vector in momentum space. In this picture, the magnon-phonon hybridization induces the chiral texture of the momentum space vector. As an experimental probe for our theory, we propose the thermal Hall conductivity [Fig. 2].

**References:**

[1] G. Go et al., Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers, Phys. Rev. Lett. 123, 237207 (2019). [2] C. Gong et al., Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature (London) 546, 265 (2017). [3] B. Huang et al., Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature (London) 546, 270 (2017). [4] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/436ed988646e642bcb99ed25391293f6.jpg)
Fig. 1. (a) The schematic illustration of the magnon and phonon system. The ground state of the magnetization is given by the uniform spin state along the z-axis (red arrow). (b) The Chern number of our magnon-phonon hybrid system. H represents the effective magnetic field including the anisotropy field and the external magnetic field. Band dispersion (c) without the magnon-phonon interaction and (d) with the magnon-phonon interaction.

![](https://assets.underline.io/uploads/markdown_image/1/image/2bc284e0ccbc6784cc3e8d6d0786fed4.jpg)
FIG. 2. (a) Dependence of thermal Hall conductivity on the effective magnetic field Heff and with different temperatures T using the parameters in the main text. (b) Dependence of the Chern number on the effective magnetic field Heff using the parameters in the main text.

technical paper

Studying spin system dynamics concerns different rapidly growing fields of technology such as developing new types of information transfer systems [1], quantum computing [2, 3], new approaches for magnetic recording and reading [4]. The most common way to excite spin oscillations is by using the inverse Faraday effect (IFE). It is based on a Raman-like coherent optical scattering process and does not require absorption of light. This gives us two main advantages of this effect: it is instantaneous and non-thermal [5]. IFE results in magnetization induced by high-power laser radiation according to the following formula: Min=χ/16π [E×E*], (1) where χ is magneto-optical susceptibility, E is the electric field of the incident light, and E* is its complex conjugate. The cross product in formula (1) is zero for linearly polarized incident light and is maximum for circularly polarized light. The excitation of high order spin waves requires the nonuniform distribution of the electromagnetic field in the medium. It is commonly achieved by using the nanostructured materials in which the resonant phenomena can occur breaking the uniformity of the field. For example, in work [6] the all-dielectric 1D grating was used to generate spin waves due to the excitation of propagating waveguide modes. Another way to excite spin waves is the utilization of anisotropic materials in which the polarization conversion of the incident light occurs. Such a transformation leads to the spatial dependency of magnetization induced via IFE. This nonuniformity of the magnetic torque allows one to excite the spin waves of a high order. Let us consider the case of biaxial crystals the influence on the incident light polarization of which is described by the permittivity tensor εij. The diagonal components of this tensor are assumed not to be equal ε11≠ε22≠ε33 and connected with cartesian coordinates x, y, and z respectively. Let the light fall under the normal incidence to the plane of the material and propagate along the z-axis. Placing the material under study in an external magnetic field along the z-axis leads to the appearance of additional off-diagonal components which are ε12=-i×g and ε21=i×g. Polarization of the light propagating in such a medium transforms due to the anisotropy and Faraday effect. This process is described theoretically in [7]. To be more specific, consider the YFeO3 crystal with an orthorhombic cell. Its crystallographic axes a, b, and c are oriented along x, y, and z axes respectively. The main refractive indices of such material are n1=2.365, n2=2.4, n3=2.337 at the wavelength of 633 nm (ε'' is assumed to be zero). The transformation of the incident light with circular polarization when it passes through anisotropic medium results in oscillations of the magnetization induced via IFE (blue line on Fig. 1). It worth noting that oscillations of magnetization occur for linearly polarized light. The maximum value of cross-product in formula (1) can be achieved for the polarization of incident light at an angle of 45° to the a crystal axis (green line on Fig. 1). In this case, the material behaves as a quarter-wave plate fully transforming the initial linear polarization to circular. One can observe the oscillations even for light polarized linearly along the crystal axis but their maximum value is two times smaller compared to the previous scenarios (Fig. 2). In this situation, the polarization transformation due to birefringence must disappear but the Faraday effect slightly rotates the polarization off the axis allowing the anisotropy to take its place. The oscillations of IFE due to the birefringent properties of the material under study have been theoretically demonstrated. Such a behavior of the induced magnetization is important for generating spin waves of a high order allowing one to avoid additional nanostructuring of the material. This work was supported by the Ministry of Science and Higher Education of the Russian Federation (mega-grant no. 075-15-2019-1934).

**References:**

[1] S. Neusser and D. Grundler, Advanced Materials, Vol. 21 p. 2927 (2009) [2] K. Ganzhorn, S. Klingler, T. Wimmer, et al, Applied Physics Letters, Vol. 109, p. 022405 (2016) [3] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, et al, Applied Physics Express, Vol. 12, p. 070101 (2019) [4] D.O. Ignatyeva, C.S. Davies, D.A. Sylgachev, et al, Nature Communications, Vol. 10, p. 1 (2019) [5] A.V. Kimel, A. Kirilyuk, P.A. Usachev, et al, Nature, Vol. 435, p. 655 (2005) [6] A.I. Chernov, M.A. Kozhaev, D.O. Ignatyeva, et al, Nano Letters, Vol. 20, p. 5259 (2020) [7] A.K. Zvezdin, V.A. Kotov, Modern magnetooptics and magnetooptical materials, CRC Press (1997)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9fd7730e609d00e4f4b1fe8becda2c13.jpg)
Fig. 1. Cross product of electric field and its complex conjugate depending on the propagation length for circularly (blue line) and linearly polarized at an angle of 45° to the crystal axis (green line) incident light.

![](https://assets.underline.io/uploads/markdown_image/1/image/a63c9245721032602764f78c072ba99b.jpg)
Fig. 2. Cross product of electric field and its complex conjugate depending on the propagation length for incident light polarized along the a crystal axis.

 Oscillating Behavior of Inverse Faraday Effect in YFeO3

Cavity magnonics built on strongly coupled cavity photons and magnons has gradually demonstrated its unique advantages in fundamental and applied research in the past few years. It has several advantages, including flexible turnability, excellent controllability, and high compatibility. Furthermore, a versatile platform based on coherently coupled microwave photons, magnons, phonons, superconducting qubits, and optical photons has begun to take shape. The hybrid system is expected to be applied in quantum information processing, transfer, and storage. Before 2018, all experiments were carried out in the case of coherent coupling between cavity photons and magnons. In 2018, an intriguing dissipative coupling between cavity photon and magnon was observed for the first time, which originated from the cooperative dissipation of the cavity mode and magnon mode. Unlike the coherent coupling case, where the spectrum is shown as an anti-crossing between cavity mode and magnon mode, the spectrum is shown as a level attraction in the case of dissipative coupling. The revelation of dissipative coupling greatly enhances the richness and functionality of the cavity magnonics and stimulates some fundamental research. We will introduce this trend with two recent works from our group. After revealing the dissipative coupling mechanism, we first realized nonreciprocity and unidirectional invisibility in cavity magnonics by utilizing the interference effect between the coherent and dissipative couplings. The isolation ratio can be relatively large with optimized small insertion loss. With the dissipative coupling mechanism and magnon mode linewidth engineering technique, we construct an anti-parity-time (APT) symmetric cavity magnonics system. Besides the dissipative coupling, the APT symmetric system also needs (i) the cavity mode and magnon mode have the opposite frequency, which can be effectively realized with a detuned cavity magnonics system in the rotating frame, (ii) the same damping or gain rates of the cavity mode and magnon mode, which can be engineered by adjusting the damping rate of the magnon mode to match that of the cavity mode. Further, we observed the exceptional points in this APT symmetric cavity magnonics system, which are the transition points between the PT-symmetry-preserved and PT-symmetry-broken phases. Between two EPs, the maximal coherent superposition of photon and magnon states is robustly sustained by the preserved APT symmetry. Moreover, we observed another kind of singularity, arising from the dissipative coupling of two antiresonances, is an unconventional bound state in the continuum (BIC). At the settings of BICs, the coupled system exhibits infinite discontinuities in the group delay. We find that both singularities coexist at the equator of the eigenvector mapped Bloch sphere, revealing a unique hybrid state that simultaneously exhibits the maximal coherent superposition and slow light capability. We expect to explore more interesting physics and applications based on the dissipative coupling in cavity magnonics.

**References:**

1. H. Huebl, et al, Phys. Rev. Lett. 111, 127003 (2013). 2. Y. Tabuchi, wt al, Phys. Rev. Lett. 113, 083603 (2014). 3. X. Zhang, et al, Phys. Rev.Lett. 113, 156401 (2014). 4. M. Goryachev, et al, Phys. Rev. Applied 2, 054002 (2014). 5. L. Bai, et al, Phys. Rev. Lett. 114, 227201 (2015). 6. Y.-P. Wang, Phys. Rev. Lett. 123, 127202 (2019). 7. Y. Yang, Phys. Rev. Lett. 125, 147202 (2020).

**Images:**

 Nonreciprocity and Unconventional Singularities in Cavity Magnonics INVITED

The field of magnonics considers the operations with data carried by spin waves and their quanta, magnons [1]. Traditionally, magnonics has been a room-temperature research discipline. Nowadays, the new direction of cryogenic magnonics attracts increasing attention. The decrease in temperature, in particular, results in the reduction of thermal population n of magnons below 1 opening access to single-magnon quantum physics [2]. Moreover, cryogenic temperatures allow for combining magnonics with superconductivity [3-5]. Ferromagnetism and superconductivity are fundamental cooperative phenomena in condensed matter physics. Entailing opposite spin orders, they share an essential conceptual similarity: Disturbances in magnetic ordering in magnetic materials can propagate in the form of spin waves (magnons) while magnetic fields penetrate the majority of technologically relevant (type II) superconductors as a lattice of magnetic flux quanta (fluxons, Abrikosov vortices). Despite a rich choice of wave and quantum phenomena predicted, magnon-fluxon coupling has lacked experimental scrutiny so far. In this talk, a selection of our recent results [5] on the interaction of spin waves with the flux lattice in ferromagnet/superconductor Py/Nb bilayers will be presented - see Fig. 1. In this system we found the magnon frequency spectrum to exhibit a Bloch-like band structure (as shown in Fig. 1(b)) which can be tuned by the biasing magnetic field. Furthermore, the frequency spectra of spin waves scattered on the flux lattice moving under the action of a transport current in the superconductor exhibit Doppler shifts. The resonance absorption of spin waves is explained by their inelastic scattering on the vortex lattice constituting a reconfigurable magnonic crystal [6]. The observed Doppler shifts arise due to the modified dispersion relation for spin waves scattered on the moving vortex lattice constituting a moving Bragg grating. In addition, manipulation of spin waves and tailoring of their transmission spectra in Py will be exemplified by using adjacent Nb layers with nanofabricated pinning potential landscapes. A special attention will be devoted to direct-write hybrid superconductor-based systems with fast relaxation of non-equilibrium effects and supporting ultra-fast vortex motion at &gt;10km/s velocities [7]. Research leading to these results was done in collaboration with R. Sachser, T. Brächer, T. Böttcher, V. Kruglyak, R. Vovk, V. Shklovskij, M. Huth, B. Hillebrands and A. Chumak.

**References:**

[1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, et al., Nat. Phys. 11 (2015) 453 [2] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi,et al., Science 367 (2020) 425 [3] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, et al., Adv. Func. Mater. 28 (2018) 1802375 [4] A. A. Bespalov, A. S. Mel’nikov, and A.I. Buzdin, Phys. Rev. B 89 (2014) 054516 [5] O. V. Dobrovolskiy, R. Sachser, T. Brächer, et al., Nat. Phys. 15 (2019) 477 [6] A. V. Chumak, A. A. Serga, and B. Hillebrands, J. Phys. D: Appl. Phys. 50 (2017) 244001 [7] O. V. Dobrovolskiy, D. Yu. Vodolazov, F. Porrati, et al., Nat. Commun. 11 (2020) 3291

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/02511907edbb80ebd00a971a77b36b08.jpg)
(a) Sketch of the experimental system. (b) Fluxon-induced reconfigurable magnonic crystal: Bandgaps in the spin-wave transmission spectra in Py are observed when the Nb layer contains an array of Abrikosov vortices.

 Magnon Fluxonics INVITED

Acoustic wave has been widely integrated into modern electronics to enable various functions, including sensing, communication, signal filtering, etc. Conventionally, acoustic waves propagate equally forward and backward. Here, we employ a fundamentally new mechanism, magneto-rotation coupling[1], to break the symmetry of the propagation, and consequently induces an unprecedented rectification ratio to 100% at the optimized condition, in accordance with the theoretical prediction. As a unique feature of surface acoustic waves, individual lattice point undergoes a rotation motion, which changes its sign according to the wave propagation direction (see the blue and red oriented circles in figure 1(a)). More than 40 years ago, Maekawa and Tachiki developed a theoretical framework [1], proposing the coupling between the lattice rotation and the magnetization via magnetic anisotropy, namely magneto- rotation coupling. Thus, to break the symmetry of the propagation, here, we employ magneto-rotation coupling to capture the chirality of the acoustic waves in a perpendicularly anisotropic ultra-thin film Ta/CoFeB(1.6 nm)/MgO (see figure 1(b)), achieving a giant nonreciprocal acoustic wave attenuation (see figure 1(c)) with a rectification ratio up to 77% [2].

**References:**

[1] S. Maekawa and M. Tachiki, Surface acoustic attenuation due to surface spin wave in ferro and antiferromagnets, AIP Conf. Proc. 29, 542 (1976). [2] M. Xu, K. Yamamoto, J. Puebla, K. Baumgaertl, B. Rana, K. Miura, H. Takahashi, D. Grundler, S. Maekawa, and Y. Otani, Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling, Vol. 6, no. 32, eabb1724 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/404c74cc9b48109c9508f92526b2d977.jpg)
Fig. 1. (a) Schematic of the nonreciprocity mechanism. (b) Illustration of the experimental setup. (c) Attenuation ofacoustic waves, P±k, near a spin-wave resonance condition for surface acoustic wavenumbers +k and -k.

 Magneto-rotation coupling inducing nonreciprocal surface acoustic waves

Very recently magnetic stripe domains, characterized by alternating up and down out-of-plane orientation of the magnetization, have received great interest due to the possibility to use stripe patterns to manipulate spin-wave (SW) propagation as in artificial magnonic crystals [1]. In this work, we demonstrate the control of the SW propagation by using reconfigurable regular stripe-pattern domain structure in the hybrid system. The investigated system consists of 64-nm-thick NdCo layer and 10-nm-thick NiFe layer, coupled through an Al layer of different thicknesses. Magnetic force microscopy (MFM) measurements show that, due to the perpendicular magnetic anisotropy of the NdCo film, the system develops stripe domains aligned with the last in-plane saturation direction [2]. The domain pattern is found to have a period of about 140 nm, which is almost independent on the thickness of the Al layer. The magnetization reversal of the trilayer system was investigated by vibrating sample magnetometer, showing that the hysteresis loop is characterized by a two-step process, due to the different coercivity of the NiFe and NdCo films. Detailed analysis of the hysteresis loops along with micromagnetic simulations indicates that the stray magnetic field coming from the NdCo layer induces a regular domain structure also in the NiFe layer, which is tuned by the thickness of Al spacer. In addition, upon reversing the applied magnetic field, an antiparallel state, characterized by an antiparallel alignment of the magnetization component parallel to the domain axis in the NdCo and NiFe stripes, is formed. Then, Brillouin light scattering spectroscopy has been used to measure the spectra of the SWs propagating in the direction perpendicular to stripe domains for the parallel and the antiparallel state. For both configurations, the dispersion relation shows a strongly nonreciprocal mode (Figs. 1-2). However, in the parallel state SWs propagating with positive and negative wavevector are both characterized by a positive dispersion, while in the reversed state SWs propagating with negative wavevector show a negative dispersion. The above experimental results have been satisfactorily reproduced by numerical simulations. The latter show that the detected SW mode is mainly localized in the NiFe layer and its frequency nonreciprocity can be ascribed to the static magnetization configuration as well as to the interaction with the NdCo induced by the SWs via the dynamic stray field.

**References:**

[1] C. Banerjee et al., Phys. Rev. B 96, 024421 (2017) [2] D. Markó et al., Appl. Phys. Lett. 115, 082401 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f952ef7a0b2bb8cf6cafa5252632c82.jpg)
Fig. 1. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the parallel state. Empty and full dots mark the results of Brillouin light scattering measurements and simulations, respectively. Size and color intensity of the numerical results indicate the predicted intensity in the measurements.

![](https://assets.underline.io/uploads/markdown_image/1/image/dbd287cdf366b5549a79bdda3afc82b8.jpg)
Fig. 2. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the antiparallel state.

 Reconfigurable spin-wave propagation in magnetic stripe domains in hybrid system

Einstein-de Haas (EdH) effect is the fundamental phenomenon that the microscopic spin angular momentum is converted to the angular momentum of mechanical rotation. A mechanical torque due to the EdH effect, the magnitude of which is proportional to the time derivative of the magnetization, is difficult to be detected, which keeps its practical application away. However, recent development of microfabrication technology has enabled to sensitively detect the tiny EdH torque using micro-fabricated mechanical resonators [1,2]. These findings show that the EdH effect can be utilized for the new principle of the mechanical torque generation in sub-micron scale device. Moreover, it was demonstrated that the ultrafast demagnetization due to the laser heating excited the transverse strain wave in a Fe film due to the EdH effect, where the timescale and conversion efficiency of the angular momentum transfer were quantitatively estimated [3]. Rayleigh-type surface acoustic waves (RSAWs), the displacement of which oscillates elliptically, have been widely used to excite the spin waves in a ferromagnetic thin film via the magnetoelastic interaction [4]. Recently, as the phenomena based on the angular momentum conversion from the lattice rotation to the magnetization or electron spin, the gyromagnetic spin-wave excitation [5] and the spin current generation in a nonmagnetic metal with a weak spin-orbit coupling [6] were demonstrated. In this study, we demonstrate both acceleration and deceleration of RSAW velocity in the ferromagnetic (FM) film based on the EdH effect. When the spin-wave is gyromagnetically excited by RSAW propagation through the FM film, the EdH torque is generated in the FM lattice as the back action of the spin-wave resonance (SWR) (Fig. 1(a)). As shown in Fig. 1(a), the direction of EdH torque is opposite to the time derivative of the magnetization. We theoretically expect that the transverse velocity of RSAW can be modulated by the EdH torque, resulting in the phase change of RSAW propagation. In the experiment, we successfully observed the phase change of RSAW propagation, whose magnitude and variation were consistent with our analytical prediction. Figure 1(b) shows the schematical experimental setup for measuring the phase change of RSAW. A pair of inter-digital transduces (IDTs) consisting of Ti(3)/Au(30) is fabricated on a piezoelectric LiNbO3 substrate by the electron beam evaporation. The numbers in the parentheses indicate the layer thickness in nanometers. The width and space of the electrode of IDT are 550 nm. The Ni81Fe19 (NiFe) film with a lateral size of 400×400 μm2 is sputtered between a pair of IDTs. By applying a microwave with the fundamental frequency of RSAW excitation into one of IDTs, RSAWs are excited via the piezoelectric effect. The RSAWs propagating through the NiFe film are electrically detected by the other of IDTs using a vector network analyzer (VNA) as the inverse process of the RSAW excitation. Figures 2(a) and 2(b) show the reduced RSAW energy dissipation ΔPnorm and phase difference Δθ, respectively, as a function of external magnetic field swept in the direction parallel to the RSAW wave vector at the RSAW frequency of 1.75 GHz. Steep peaks appear at 4.6 and -4.0 mT in Fig. 2(a), showing that the spin-wave is excited because the values of the resonant field and resonant frequency are matched with the spin wave dispersion of a 20-nm-thick NiFe film. A bipolar change of Δθ is observed in the vicinity of the resonance field of spin-waves. From the angle between the applied external field and RSAW wave vector, it can be seen that the spin-wave is excited via gyromagnetic effect, not the magnetoelastic coupling [5]. The increase and decrease in Δθ can be explained by our analytical model consisting of the elastic motion equation with the EdH torque term. The maximum value of phase change Δθmax is -0.035 rad at 5.0 mT. This value is as large as that in case using the magnetoelastic coupling in a single Ni film [7]. In addition, the order of Δθmax is comparative to the value analytically obtained by assuming a loss-less conversion of the angular momentum. The consistency in Δθmax suggests that the EdH effect is robust in the angular translation from the magnetization to the lattice in the gigahertz regime.

**References:**

[1] K. Harii, et. al., Nat. Commun. 10, 2616 (2019). [2] K. Mori, et. al., Phys. Rev. B 102, 054415 (2020). [3] C. Dornes, et. al., Nature 565, 209 (2019). [4] M. Weiler, et. al., Phys. Rev. Lett. 106, 117601 (2011). [5] Y. Kurimune, et. al., Phys. Rev. Lett. 124, 217205 (2020). [6] D. Kobayashi, et. al., Phys. Rev. Lett. 119, 077202 (2017). [7] R. Sasaki, et. al., Phys. Rev. B 95, 020407 (R) (2017). [8] S. Tateno, et.al., submitted.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/95a80b47499a8efb662564ebc84f2325.jpg)
Fig. 1 (a) Schematic of the stress based on EdH effect σEdHzx and the EdH torque τEdH in the Ni-Fe lattice. The direction of the EdH torque is opposite to the time derivative of the magnetization. (b) Experimental setup for measuring the phase changeΔθ of RSAW using a VNA. The external field is applied along the RSAW propagation direction.

![](https://assets.underline.io/uploads/markdown_image/1/image/6d3a013484d0f414781f2f3ebf4436f3.jpg)
Fig. 2 The field dependence of (a) ΔPnorm and (b) Δθ at the fundamental frequency of RSAW excitation, respectively. The red solid line in (b) shows the result of the best fit using our analytical model.

 Velocity Modulation of Surface Acoustic Waves via Einstein-de Haas Effect

In recent years, the coupling between the elastic and magnetic domain in magnetostrictive materials has gained strong interest [1]. The magnetoelastic coupling in these materials results in elastic response originating from magnetic excitations and vice versa. At microwave frequencies, this mutual interaction leads to coupling between elastic and magnetic waves, forming magnetoelastic waves. Whereas plane magnetoelastic waves are already studied in bulk materials some decades ago [2-4], the magnetoelastic waves in thin magnetostrictive films gained attention only recently [5,6]. By contrast, many recent spin-wave-based information processing applications employ nanoscale waveguides for information transfer and computation [7]. At these nanoscales, confinement of both the elastic and magnetic waves leads to multiple modes each of which with a specific mode profile and consequently also to confined magnetoelastic waves. The shape of the magnetoelastic field depends on the displacement profile and the shape of the magnetoelastic body force depends on the magnetization profile. As a consequence, it is expected that every mode experiences different magnetoelastic coupling strength. This presents a rather complex multiphysics problem where analytical methods are hard or even impossible, and thus numerical approaches are preferred. Therefore, in this work, we present the development of a novel mumax3 extension that allows the numerical calculation of magnetoelastic waves in scaled waveguides [8]. Furthermore, we present results of magnetoelastic waves in a nanoscale waveguide with static magnetization along the wave propagation direction. The studied system consists of a thin CoFeB waveguide with width of 200 nm and thickness of 20 nm. A weak external field of 5 mT is applied in the longitudinal direction of the waveguide to initiate the static magnetization along this direction. The resonant modes in the structure are numerically found by applying a 5 ps excitation pulses and performing a two-dimensional fast Fourier transform. By comparing the numerical result with the analytical uncoupled dispersion relations of confined elastic and magnetic waves, all modes were identified. The uncoupled resonant waves of this structure consist of confined magnetic spin waves, antisymmetric and symmetric lamb waves and out-of-plane shear elastic waves. Fig. 1a shows the numerically obtained dispersion relations and Fig. 1b shows the overlay with the analytically calculated dispersion relations for uncoupled elastic and magnetic waves. The magnetoelastic coupling leads to the formation of anti-crossings near the reciprocal point where the uncoupled elastic and magnetic dispersion relations would intersect with each other. These anti-crossings identify regions of strong interaction between the elastic and magnetic domain and correspond to pure magnetoelastic waves. The magnetoelastic waves are shown to have different behavior as compared to the pure elastic or magnetic waves. The magnetoelastic wave group velocity is rather unaffected in the quasi-magnetic or quasi-elastic regime whereas it strongly differs close to the anti-crossing. Furthermore, the magnetization and displacement value are shown to take both high values close to the anti-crossing. However, when approaching the quasi-elastic regime there is a very strong decay of the magnetization component of the magnetoelastic wave and vice versa for the displacement component and the quasi-magnetic regime. Lastly, careful analysis of the dispersion relations also shows reciprocal points with weak or even without anti-crossing. The frequency gap formed by the anti-crossing is a good measure of the magnetoelastic coupling strength and thus provides much information. The variation in amplitude of this frequency gap is attributed to the wave confinement and their mode profiles. Every elastic and magnetic mode has its own mode profile and thus also its own magnetoelastic body force and field. Considering these mode dependent body forces and fields together with the mode’s overlap integral allows to identify the strength of the coupling between the different elastic and magnetic waves. The strongest coupling is shown to exist between the odd magnetic modes and symmetric lamb modes and even magnetic modes and anti-symmetric lamb modes. This work has been partially funded by the European Union’s Horizon 2020 research and innovation programme within the FET-OPEN project CHIRON under grant agreement No. 801055. FV acknowledge the financial support from Fonds voor Wetenschappelijk Onderzoek (FWO) under the grant No. 1S05719N.

**References:**

[1] A. D. Kent and D. C. Worledge, Nature Nanotechnol. Vol. 10, p. 187 (2015) [2] C. Kittel, Phys. Rev. Vol. 110, p. 836 (1958). [3] E. Schlomann and R. I. Joseph, J. Appl. Phys. Vol. 35, p. 2382 (1964). [4] P. A. Fedders, Phys. Rev. B. Vol. 9, p. 3835 (1974). [5] Y. Nozaki and S. Yanagisawa, Electr. Eng. Jpn. Vol. 204, p. 3 (2018). [6] S. Bhuktare, A. Bose, H. Singh, and A. A. Tulapurkar, Sci. Rep. Vol. 7, p. 840 (2017). [7] A. Khitun and K. L. Wang, J. Appl. Phys. Vol. 110, p. 34306 (2011). [8] F. Vanderveken, et al., https://github.com/Fredericvdv/Magnetoelasticity_MuMax3.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c4a42e298a703a137c4d16ff97d65c5e.jpg)
Figure 1: Dispersion relations of magnetoelastic waves in a 200 nm wide and 20 nm thick CoFeB waveguide. (a) Numerically calculated dispersion relations. The inset shows a magnification of a region near a magnetoelastic gap. (b) Analytically calculated dispersion relations of uncoupled elastic and spin waves superimposed to the numerically obtained results.

 Confined magnetoelastic waves in thin waveguides

The interaction between different types of wave excitation in hybrid systems is usually anisotropic. Magnetoelastic coupling between surface acoustic waves and spin waves strongly depends on the direction of the external magnetic field. However, in the present study we observe that even if the orientation of the field is supportive for the coupling, the magnetoelastic interaction can be significantly reduced for surface acoustic waves with a particular profile in the dirR/S-SAWection normal to the surface at distances much smaller than the wavelength. We use Brillouin light scattering to investigate thermally excited phonons and magnons in a magnetostrictive CoFeB/Au multilayer deposited on a Si substrate – see Fig.1. The experimental data interpretation is based on a linearized model of interaction and finite element method computations[1]. For propagating waves, the coupling requires matching both the frequencies and the wavelengths[2]. This condition can be fulfilled for spin waves (SWs) and surface acoustic waves (SAWs) existing in the same range of frequencies and wave vectors. However, the strength of the magnetoelastic interaction depends on the orientation of the wave vector with respect to the direction of the static magnetic field[3]. Moreover, this interaction is different for different types of SAWs, specifically, Rayleigh-SAWs (R-SAW) and Love-SAWs (L-SAW)[1,4] – see Fig.2. Thus, the coupling is strongly anisotropic and cannot be observed for arbitrary SAWs and SWs, even if their frequencies and wave vectors match. Our study reveals an additional factor limiting the interaction between SAWs and SWs. The SAW/SW coupling proves to require an appropriate profile of the elastic wave near the surface of the magnetostrictve structure, at distances much smaller than the wavelength. For R-SAWs the tangential component of displacement ux can have nodes within the magnetic layer, resulting in a reduction of the net strength of magnetoelastic interaction even if the strain εxx is locally significant. In an L-SAW the displacement uy does not have any nodes (uy changes monotonously in the normal direction). The location of R-SAW nodes depends on elastic properties of the system, specifically, the elastic material parameters of the multilayer and the substrate, and the thickness of the multilayer. We have shown that this additional factor plays a role for some types of surface acoustic waves (R-SAW), while other types (L-SAW) are insensitive to it. We believe that the studies on magnon-phonon interaction in confined geometries (surfaces, cavities) are very promising and can reveal unusual interaction mechanisms. Sample We used naturally oxidized (001) silicon as a substrate supporting the studied [Co20Fe60B20/Au]20 multilayers deposited on top of a 4 nm titanium (Ti) and a 15 nm gold (Au) buffer layers. The multilayers were deposited by magnetron sputtering in argon atmosphere. Method We studied the dispersion relations of thermally excited SAWs and magnetostatic SWs using a six-pass tandem Brillouin spectrometer (Scientific Instrumentsc©TFP2-HC),which ensures a contrast of 1015. A frequency-stabilized diode-pumped solid-state laser (Coherentc©VERDI V5) operating at λ0= 532 nm was used as a source of incident light. The measurements were performed in the 180o backscattering geometry with crossed (p-s)polarization of incident and scattered light for SWs and non-crossed (p-p) polarization for SAWs. Using the finite element method in COMSOL Multiphysics©, we solve numerically the coupled equations of motion for mechanic displacement and magnetization. The CoFeB/Au multilayer is treated as a 60-nm-thick effective magnetic layer, and the CoFeB/Au multilayer together with the Ti/Au buffer as an 80-nm-thick effective acoustic layer on a Si substrate. Acknowledgement The research leading to these results has received funding, hereby gratefully acknowledged, from the National Science Centre, Poland, project No. UMO-2016/21/B/ST3/00452. S. Mies. would like to additionally acknowledge the financial support from the National Science Centre, Poland, project No. UMO-2020/36/T/ST3/00542.

**References:**

 [1] N. K. P. Babu, A. Trzaskowska, P. Graczyk, et al., Nano Lett. (2020) https://doi.org/10.1021/acs.nanolett.0c03692 [2] Y. Hashimoto, D. Bossini, T. H. Johansen, et al., Phys. Rev., Vol. 97, 14040 (2016) [3] L. Dreher, M. Weiler, M. Pernpeintner, et al., Phys. Rev. B 86, 134415 (2012) [4] M. Geilen, F. Kohl, A. Nicoloiu, et al., Appl. Phys. Lett. Vol. 117, 213501 (2020) [5] N.E. Khokhlov, P.I. Gerevenkov, L.A. Shelukhin, et al., Phys. Rev. Applied Vol. 12, 044044 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/052c823eb22ae084daf2c19fe75deaf0.jpg)
Fig.1 In the studied system acoustic waves propagate along the surface of the silicon substrate and are mostly concentrated in the CoFeB/Au multilayer cover. The multilayer can be regarded as an effective magnetostrictive medium where surface acoustic waves can interact with spin waves. The anisotropy of magnetoelastic interaction (i.e. the dependence on the orientation f of magnetic field) is different for Love and Rayleigh/Sezawa acoustic waves (L-SAW, R/S-SAW); presented on the left are the respective polar plots of the in-plane component of magneto elastic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/bc1badb1dc20cc3a670ea3ed7a09b463.jpg)
Fig.2 The BLS experimental magnetoelastic dispersion relation measured for magnons (redlines) and phonons (blue lines) with an in-plane applied magnetic field of (a) 30 mT and (b) 50 mT for Φ=0 deg . The dashed shows the regions where the frequencies of fundamental spin wave fundamental mode (F-SW) and L-SAW or R-SAW are agreed. The interactions is seen for L-SAW only. (c) For Φ=45 deg observe the lack of R-SAW/F-SW interaction, which cannot be explained based on the interaction anisotropy (see hme,‖(Φ) for R-SAW in Fig.1). The R-SAW/F-SW interaction is suppressed to zero due to the presence of a nodal line for the εxx component of the strain tensor, which averages to zero the magnetoelastic coupling.

 The interaction between surface acoustic waves and spin waves: the role of anisotropy and spatial profiles of the modes

Modern quantum computing architectures are built to provide low-error environments within which to perform the unitary operations that form the bases of quantum computations. These low-error environments necessary for successful quantum computation require use of complex structures that make large-scale implementation difficult, leading to the investigation of coupled magnon-photon systems [1-6] that can achieve strong coupling rates (&gt; 100 MHz) as a potential platform for hybrid quantum computing architectures. Such coupled magnon-photon systems benefit from the magnetic field tunability of the resonant magnon frequency, where dynamic tuning of resonant magnon frequencies with a pulsed magnetic field at rates comparable to or less than the order of the magnon-photon energy exchange time can realize unique operations not possible with static tuning [7-8]. Previously, we analyzed this problem numerically and found that a parabolically shaped pulsed magnetic field profile can realize the coherent exchange of quantum data and the generation of entanglement by changing the curvature λ and maximum frequency Δ of the parabolic profile [8]. That is, in the same physical structure, depending on the shape of the pulsed magnetic field, different unitary magnon-mediated quantum gates can be realized [8]. Here, we consider an analytical approach to study this system. Namely, we consider a hybrid magnon-photon system, consisting of two photonic resonators having degenerate frequencies ωp and one magnonic resonator with a dynamically-tuned resonance frequency ωm(t). The photonic resonators are not directly coupled with each other but are both coupled with the magnonic mode with the coupling rate κ. We consider a simple analytically-solvable interaction protocol in this hybrid system, which can be experimentally realized with rectangular magnetic field pulses of sufficiently large magnitude. We assume that initially (t&lt;0) the frequency difference │ωm–ωp│&gt;&gt;κ and interaction between the subsystems can be ignored. At t=0 the magnonic mode frequency ωm(t) is brought within the efficient interaction interval Δ=ωm–ωp∼κ and is held constant for the duration T. After t=T, the magnonic frequency is once again moved out of the photonic resonance region and any interaction between the subsystems stops. Depending on the control pulse parameters Δ and T, different behaviors of the hybrid system can be realized. Of particular interest is the situation, in which the magnonic and photonic subsystems stay completely decoupled after the action of the control pulse. Such a case corresponds to a unitary operation on the photonic modes, realized through the indirect magnon-mediated coupling, and is of great importance for practical applications of magnonic resonators in quantum computing systems. Using our model, we derived a simple analytical condition of realizing unitary magnon-mediated operations, namely, for each detuning Δ there exist an infinite series of pulse durations Tn(Δ)=2nπ/(8κ2+Δ2)½ providing unitary conditions (here, n is a positive integer). The resulting effect of the magnon interaction on the photonic subsystem, in this case, can be described by 2×2 unitary matrix Un=Uei(nπ/2–Tn(Δ)Δ/4) relating photonic states before and after the pulse, which has components: U11=ei2πcos(θn), U12=eiπ/2sin(θn), U21=–e–iπ/2sin(θn), U22=e–i2πcos(θn) Here θn=nπ/2–Tn(Δ)Δ/4 has the meaning of inter-mode energy exchange angle. For example, for θn=π/2+mπ (m is an integer number) a complete exchange of two photonic states is achieved, while θn=π/4+mπ describes 50% mixing behavior that allows one to entangle the two photonic modes. The obtained results allow one to select pulse parameters realizing the desired unitary operation and, also, to analyze the limitations on the performance of the proposed magnon-mediated gates. Thus, it is clear that the simple rectangular pulse cannot realize all possible unitary gates since the matrix Un depends only on one continuous parameter (θn), while the general 2×2 unitary matrix has 4 independent parameters. On the other hand, this pulse profile is sufficient to achieve an arbitrary exchange angle θn. The shortest pulse duration for which a complete exchange (θn=π/2) is possible is Tmin=π/(κ√2)and is realized at exact resonance Δ=0. Our results, also, show a possibility to realize magnon-mediated unitary gates working in off-resonance conditions │Δ│≥κ. For example, θn=π/2 gate can be realized with Δ=18(2/19)½κ and T10=(19/2)½π/κ. While such off-resonance operations require longer interaction times, their advantage in comparison to resonance-type operations is that the magnonic and photonic subsystems remain only weakly coupled throughout the whole process. This may lead to weaker decoherence of the quantum photonic subsystem caused by spin-lattice relaxation processes in the magnonic resonator. In general, n-th unitary branch Tn(Δ) has n different detuning values Δ, for which a given exchange angle θn (up to multiplies of π) is achieved, which leaves a lot of space for optimization of practical magnon-mediated unitary gates. In summary, we proposed a simple model of magnon-mediated interaction in dynamically-tuned hybrid magnon-photon quantum systems. We derived analytically the conditions under which such interaction results in a unitary operation on the photonic subsystem and obtained expressions for the parameters of the control pulse necessary for achieving the desired operation. Our results can be used for estimation and optimization of performance parameters of practical hybrid magnon-photon systems.

**References:**

[1] H. Huebl, C. W. Zollitsch, J. Lotze, et al., Phys. Rev. Lett., vol. 111, no. 127003, 2013. [2] Y. Tabuchi, S. lshino, T. Ishikawa, et al., Phys. Rev. Lett, vol. 113, no. 083603, 2014. [3] X. Zhang, C.-L. Zou, L. Jiang, et al., Phys. Rev. Lett., vol. 113, no. 156401, 2014. [4] Y. Tabuchi, S. lshino, A. Noguchi, et al., Science, vol. 349, no. 6246, pp. 405-408, 2015. [5] Y. Li, T. Polakovic, Y.-L. Wang, et al., Phys. Rev. Lett., vol. 123, no. 107701, 2019. [6] J. Hou and L. Liu, Phys. Rev. Lett., vol. 123, no. 107702, 2019. [7] C. Trevillian and V. Tyberkevych, FE-09, Digests of the Intermag Conference, 2020. [8] C. Trevillian and V. Tyberkevych, Q2-07, Abstracts of the 65th Annual Conference on Magnetism and Magnetic Materials, 2020.

**Images:**

 Analytical Model for Unitary Magnon-Mediated Quantum Gates in Hybrid Magnon-Photon Systems

Acoustically-driven ferromagnetic resonance (ADFMR) gained interest in 2012 as Dreher et al. (1) developed it as a powerful and versatile method of examining complex magnetoacoustic interactions. While initial studies established the technique, the physical mechanisms behind the coupling must be thoroughly understood if ADFMR is to be exploited in next-generation microwave communications devices. In particular, the nonlinearity of ADFMR driven at high powers (2) is not well characterized, which may result in unwanted or unpredicted behavior when devices become optimized. To address this aspect, we produce ADFMR devices using curved interdigitated transducers (IDTs), in contrast to the straight IDTs that are typically used. Like an optical lens, these IDTs are engineered to focus acoustic energy towards a region smaller than the IDTs themselves. In this way, acoustic and magnetic amplitudes in a small region in the center of the device may be driven much higher than was previously possible, allowing examination of nonlinear processes such as Suhl instabilities. This may also produce an avenue for improved magnetoacoustic coupling, as well as further device miniaturization. Our devices are produced in much the same way as previously described (2), using aluminum split-finger IDTs and a 20-nm thin-film nickel layer grown on a y-cut lithium niobate (LiNbO3) substrate. We examine a device using a scanning acoustic-wave imaging interferometer similar to that used in Ref. (3). A 785-nm laser is split into signal and reference paths using an acousto-optic modulator operated at 40 MHz. The signal component is focused onto the device with a 10X objective lens. The beams are then recombined and the interference between them is recorded using a lock-in amplifier. Meanwhile, the ADFMR device is driven at its fundamental frequency 284 MHz or a higher odd harmonic. The DC photodiode signal is simultaneously recorded as a reference image (Fig. 1a), and follows the reflectivity of the scanned spot, showing the exact positions of the metallic nickel and aluminum surfaces. The vibration of the probed spot, downmixed to the range of the lock-in, is shown in Fig. 1b. Here, the gaussian-like curve of the focused SAWs is clearly visible as they converge to a high-amplitude region near the center of the device. Extracting a longitudinal segment of this signal allows us to see the oscillatory behavior in Fig. 1c. These oscillations indicate a standing wave with wavelength v/f = 12 µm (v = 3488 m/s is the Rayleigh velocity of SAWs in the LiNbO3 and f = 284 MHz is the driving frequency), as the output of the device is reflected back towards the input. To quantify the power convergence, transverse slices were taken near the center and fitted with the lorentzians shown in Fig. 1d, using the fitting parameters in Fig. 1e. Near the center (X ≈ 20.7 mm), the amplitude reaches a value of nearly 5 times greater than it is near the IDTs, as the width shrinks to a minimum of 22 µm, demonstrating that the focused IDTs significantly increase the energy density in a small region. ADFMR measurements on a focused IDT device show extremely power-dependent behavior. The linear regime is accessed with a -15 dBm driving field (Fig. 2a), where the shape is independent of power. However, with only +15 dBm of input power (Fig. 2b) the absorption lobes begin to shift towards lower field, and with a weaker maximum absorption. At +25 dBm of input power (Fig. 2c) the lobes shift even further and are much weaker in comparison. These results are a clear indication that a nonlinear power regime is accessed with modest input powers with the assistance of the curved IDTs, dramatically concentrating acoustic energy density. *This material is based on work supported by the Air Force Office of Scientific Research under project number FA9550-20RXCOR074

**References:**

1. L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, “Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment,” Physical Review B, vol. 86, no. 13, p. 134415, 2012. 2. D. Bas, P. Shah, M. McConney and M. Page, “Optimization of acoustically-driven ferromagnetic resonance devices,” Journal of Applied Physics, vol. 126, no. 11, p. 114501, 2019. 3. Z. Shen, X. Han, C. L. Zou, and H. X. Tang “Phase sensitive imaging of 10 GHz vibrations in an AlN microdisk resonator,” Review of Scientific Instruments, vol. 88, No. 12, p. 123709, 2017.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cf30be12a3755d154f36adeba3fe0211.jpg)
Figure 1. Interferometric imaging of microwave surface acoustic waves on a device. (a) Reflectivity map of the device in arbitrary units, scanned with 5-µm steps, highlighting the curved interdigitated transducers (IDTs), thin film of nickel (Ni), and substrate of lithium niobate (LiNbO3). (b) Interferometric acoustic wave imaging showing the amplitude of surface vibrations in arbitrary units, demodulated at the fundamental frequency of the IDTs (284 MHz) after downmixing. (c) Line scan indicated by the blue dashed line in (b), fitted with a sinusoid of period 12 µm, which corresponds to the period of standing waves in LiNbO3 at 284 MHz. (d) A series of lorentzian fits along the Y-direction for the data indicated by the dashed orange box in (b). (e) Fitting parameters width and height for the curves in (d).

![](https://assets.underline.io/uploads/markdown_image/1/image/43305569f2e8ce46063c1c55933a4975.jpg)
Figure 2. Normalized ADFMR signal in a focused IDT device. Driving field is (a) -15 dBm, (b) +15 dBm, and (c) +25 dBm at 864 MHz.

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