Magnon-phonon coupling in mechanically strained ferrimagnetic membranes

Introduction

Magnons and phonons have been coupled since the early theory of ferromagnetic resonance and spin-wave interaction with ultrasonic strain fields [1,2]. In a magnetically ordered solid, strain changes magnetic anisotropy and therefore modulates the spin-precession frequency. Conversely, precessing magnetisation can drive elastic motion through magnetostriction. This reciprocal coupling is attractive for microwave signal processing, spin-mechanical transduction, strain sensing, and hybrid magnonic devices.

Surface acoustic waves and elastic resonators have already been used to drive ferromagnetic resonance in thin films, with quantitative agreement between microwave spectroscopy and magnetoelastic theory [3,4]. In yttrium iron garnet, direct observations of magnon-phonon coupling and magnon-polaron effects have shown that low magnetic damping can make the interaction spectroscopically visible [5,6,7]. Cavity magnomechanics extends the same physics to microwave cavities, phonons, and collective ferrimagnetic modes [8,9].

Recent work has broadened the materials landscape. Magnetoelastic coupling has been observed in ferrimagnetic oxides such as GdTiO3, and strong magnon-phonon coupling has been reported in two-dimensional and layered antiferromagnets [10,11,12,13]. Reviews now frame magnon-phonon coupling as a field that connects fundamental hybridisation, thermal transport, and device concepts [14]. Yet there remains a device-level gap between bulk crystals or spheres and planar integrated structures.

Mechanically strained membranes offer one route across that gap. A membrane localises acoustic energy, allows static strain to be imposed without large substrate dilution, and can be integrated with microwave conductors. The cost is that release processing introduces edge damage, strain gradients, and acoustic leakage. This article studies that trade-off in ferrimagnetic garnet membranes designed for GHz acoustic modes and microwave ferromagnetic resonance.

The central question is practical: can static mechanical strain tune both magnon frequency and magnon-phonon coupling in a reproducible membrane device? We answer with a fictional but physically grounded experiment combining vector-network-analyser spectroscopy, laser vibrometry, finite-element mode modelling, and a coupled-oscillator magnetoelastic analysis.

Membrane fabrication

The starting material was a 95 nm yttrium iron garnet film grown on a sacrificial garnet-compatible oxide stack. The film was capped with 35 nm low-stress silicon nitride, patterned into 80 um x 120 um windows, and bonded to a piezoelectric frame before release. The nitride layer provided mechanical support during release and reduced membrane wrinkling. After release, the garnet surface roughness measured by AFM was 0.42 nm over 5 um x 5 um in the central region and 0.9-1.4 nm within 5 um of the clamped edge.

Each device included a coplanar waveguide for microwave magnetic excitation and a pair of aluminium scandium nitride interdigital transducers for acoustic drive. The acoustic pitch selected Lamb-mode frequencies from 2.1 to 5.8 GHz. The membrane long axis was aligned within 4 deg of the in-plane magnetic easy axis. Static strain was applied by biasing the piezoelectric frame; the strain transfer to the membrane was calibrated with digital image correlation on gold fiducials and with shifts in the acoustic eigenfrequencies.

The devices are intentionally more complex than ordinary FMR test films. The point is to place a low-damping ferrimagnet inside a mechanical structure whose strain field is known. YIG and related garnets are widely used because they combine ferrimagnetic order with very low damping, which is why they appear in magnon-polaron and cavity-magnomechanics experiments [5,6,8].

The fabrication yield was 41% for fully released membranes with working microwave and acoustic ports. The most common failures were cracks at the waveguide ground-plane corners and delamination near the acoustic transducer. These failed devices are included in the supplementary process statistics because they constrain realistic scalability.

Measurement geometry

Microwave transmission was measured with a vector network analyser from 1.5 to 7.0 GHz. The external magnetic field was applied in the membrane plane unless otherwise stated. Field sweeps at fixed frequency identified the ferromagnetic resonance branch, while frequency sweeps at fixed field resolved avoided crossings with acoustic modes. Acoustic drive was measured through the interdigital transducers, and membrane displacement was mapped by heterodyne laser Doppler vibrometry.

The magnetic resonance follows the thin-film Kittel form with strain-dependent anisotropy fields [2]. We extracted the effective magnetisation, in-plane anisotropy, and linewidth from field-swept spectra far from acoustic resonances. These parameters were then held fixed when fitting coupled magnon-phonon avoided crossings. This separation is important because otherwise acoustic hybridisation can be mistaken for a strain-induced change in damping.

Static strain ranged from -0.06% compressive to +0.18% tensile along the membrane long axis. Larger tensile strain caused irreversible shifts in the acoustic spectrum, likely from partial slip at the bonded frame. Temperature was varied from 20 K to 295 K. We did not attempt millikelvin operation because the devices were designed to test tunable coherent coupling in a classical spectroscopy regime rather than quantum ground-state mechanics.

Magnetic-field orientation was swept over 180 deg in the membrane plane for three devices. The coupling was strongest when the dynamic strain rotated the magnetisation relative to the static anisotropy axis, consistent with magnetoelastic selection rules derived from surface-acoustic-wave driven FMR experiments [3,4].

Model

The data were fitted with a two-mode coupled-oscillator model. The magnon mode has angular frequency omega_m(H, epsilon) and linewidth kappa_m. The acoustic mode has frequency omega_p(epsilon) and linewidth kappa_p. The complex transmission near an avoided crossing is described by two hybrid eigenfrequencies separated by 2g when omega_m = omega_p. Cooperativity is defined as C = 4g^2/(kappa_m kappa_p).

The magnetoelastic coupling rate was estimated from the volume integral of the product of dynamic strain and magnetic susceptibility. The magnetoelastic energy contains terms proportional to b1 epsilon_ii m_i^2 and b2 epsilon_ij m_i m_j, following the same physical coupling introduced in early magnetoelastic theory [1]. In a membrane, the overlap integral can be positive or negative depending on the Lamb-mode symmetry and where the magnetic film sits relative to the neutral plane.

Finite-element acoustic simulations were performed with the measured membrane dimensions, nitride stress, garnet elastic constants, and clamped-frame geometry. The simulations predicted symmetric and antisymmetric Lamb families. Laser vibrometry provided an experimental check on mode order and nodal structure, but the vibrometer measures surface displacement rather than the full internal strain tensor. The finite-element model therefore remained necessary for coupling estimates.

We also included a slowly varying background transmission term because microwave leakage between waveguide and transducer creates Fano-like line shapes. The background was fitted using off-resonant field windows. Ignoring this term biased the extracted g by up to 18% for weakly coupled modes.

Strain tuning of magnetic resonance

Static tensile strain shifted the uniform magnon frequency upward for fields along the membrane long axis and downward for fields along the short axis. At 0.18% tensile strain, the long-axis Kittel-mode shift was 214 +/- 12 MHz at 70 mT. The shift was reversible over 50 strain cycles below 0.15% strain and showed less than 6 MHz hysteresis after returning to zero bias.

The strain-induced anisotropy field extracted from the angular data was 2.8 +/- 0.3 mT per 0.1% strain. This value is lower than a fully clamped single-domain estimate because strain relaxes near the membrane edges and because the nitride support layer carries part of the elastic load. The result illustrates a recurring membrane issue: local magnetoelastic coupling may be strong, but the device-level tuning is diluted by mechanics.

The magnetic linewidth at 295 K was 7.6-11.4 MHz depending on device and field orientation. At 20 K, the best membrane had linewidth 3.1 MHz. Released membranes were consistently broader than unreleased witness films, which had linewidth below 2 MHz at 20 K. We attribute the broadening to strain inhomogeneity and edge damage rather than to the intrinsic garnet film.

The strain tunability is already useful for engineering. It allows the magnon branch to be swept through fixed acoustic resonances without changing field by tens of millitesla. This decouples magnon-phonon tuning from field-dependent microwave coupling and makes mode-by-mode comparison cleaner.

Acoustic modes

The released membranes supported multiple GHz Lamb modes. The lowest measured acoustic resonance was at 2.14 GHz, while the highest mode used for coupling analysis was at 5.72 GHz. Acoustic quality factors ranged from 220 to 1650 at 295 K and from 480 to 4200 at 20 K. Modes with displacement antinodes near the clamped edge had lower Q because of frame leakage.

Laser vibrometry confirmed the expected nodal patterns for the first six modes. Symmetric modes showed large in-plane strain in the central region, while antisymmetric modes concentrated shear strain near the garnet-nitride interface. The latter coupled more strongly to the magnon branch despite lower displacement amplitude because the relevant quantity is strain overlap, not surface motion.

Static tensile strain increased most acoustic frequencies, with slopes from 0.6 to 2.4 MHz per 0.01% strain depending on mode order. The slopes agreed with finite-element predictions within 12% for central modes and within 28% for edge-localised modes. The poorer agreement near edges is expected because release damage and local clamping stiffness are difficult to measure.

The acoustic spectrum was stable under repeated microwave drive. Heating from the acoustic transducers shifted resonances by less than 1.5 MHz at the powers used for spectroscopy. This is small compared with the magnon linewidth but included in the fitting uncertainty.

Avoided crossings

Avoided crossings were observed when the field-tuned magnon branch intersected selected Lamb modes. The strongest 20 K crossing occurred at 4.38 GHz and 92 mT, where the fitted coupling rate was g/2pi = 28.6 +/- 1.9 MHz. The magnon and phonon linewidths were kappa_m/2pi = 4.6 MHz and kappa_p/2pi = 52 MHz, giving cooperativity C = 3.8. At 295 K the same mode had g/2pi = 24.1 +/- 2.4 MHz and C = 1.1 because the acoustic Q and magnetic linewidth degraded.

We observed six reproducible crossings across four devices. Coupling rates ranged from 7.4 to 28.6 MHz. Modes whose finite-element strain was antisymmetric across the membrane width coupled weakly or not at all, even when their displacement amplitude was large. This selection rule confirms that the coupling is magnetoelastic rather than a spurious microwave standing wave.

The avoided crossing changed sign under strain for one mode family. Tensile strain increased the magnon frequency but decreased the acoustic overlap for an antisymmetric Lamb mode, reducing g by 32% between zero strain and +0.16%. For a symmetric mode at 3.12 GHz, tensile strain increased both frequency detuning and overlap, raising g by 21%. The model reproduced the opposite trends by including strain-dependent mode shape, not only frequency tuning.

The observed coupling rates are smaller than those reported for optimised cavity magnomechanical systems, where a ferrimagnetic sphere can concentrate both magnetic and acoustic energy [8]. They are nevertheless large for a planar membrane geometry and sufficient for coherent classical energy exchange over several linewidths.

Temperature dependence

Cooling improved both magnetic and acoustic linewidths. The acoustic Q increased by a factor of 1.8-2.7 from room temperature to 20 K, while magnon linewidth narrowed by a factor of 2-3. The coupling rate changed less than 15%, indicating that the magnetoelastic overlap is mostly structural rather than thermally activated.

The magnon frequency shifted with temperature because the saturation magnetisation and anisotropy changed. This shift had to be separated from strain tuning. We measured zero-strain temperature sweeps before and after each strain cycle and used them as baselines. Without this correction, thermal drift would imitate an apparent strain hysteresis of about 0.02%.

The low-temperature linewidth still did not reach that of bulk YIG. That matters because cooperativity scales inversely with linewidth product. The membrane geometry improves acoustic localisation but degrades magnetic homogeneity. This is the central engineering compromise of the platform.

The temperature dependence is also relevant for spin-thermal phenomena. Magnon-polaron studies in spin Seebeck measurements show that hybridisation can strongly affect transport when magnon and phonon dispersions touch [6,7,18]. Our devices probe the same class of hybridisation spectroscopically, but not through thermal spin-current detection.

Comparison with layered magnets

The ferrimagnetic membrane platform differs from two-dimensional magnetic crystals. Van der Waals magnets established that magnetism can persist in atomically thin layers when anisotropy is sufficient [15,16,17]. Layered antiferromagnets have recently shown strong magnon-phonon coupling and acoustically driven magnon dynamics [11,12,13]. Those systems are appealing because their thickness is naturally small and their lattice can couple strongly to spin order.

Garnet membranes offer different advantages: low damping, microwave compatibility, and established ferrimagnetic resonance. Their weakness is fabrication complexity. A released garnet membrane is not as clean as an exfoliated crystal, and its clamped boundary controls acoustic loss. The best choice depends on the intended function. For fundamental high-field hybridisation in atomically thin magnets, layered antiferromagnets are compelling. For microwave transduction and integrated resonators, ferrimagnetic membranes remain attractive.

The comparison also clarifies what "strong coupling" should mean. In some optical or neutron experiments, strong coupling is identified by anticrossing spectra. In an integrated resonator, cooperativity and linewidth matter just as much. Our room-temperature C > 1 result indicates resolvable coherent hybridisation, not quantum-state transfer. This language keeps the claim aligned with the experiment.

Future devices may combine both directions by integrating van der Waals magnets onto acoustic membranes or by placing 2D magnets near low-loss garnet resonators. The same overlap and linewidth accounting used here would be needed to evaluate such hybrids.

Strain sensing and transduction

The measured strain-to-frequency slope suggests a membrane strain sensitivity of 5.1 x 10^-7 per square-root hertz for a 1 s microwave measurement bandwidth, limited by magnon frequency noise. This is not competitive with the best optical nanomechanical strain sensors, but it operates electrically and inside a magnetic material. Magnetoelastic resonance sensors have used related principles for remote strain measurement [19].

For transduction, the most relevant figure is not strain sensitivity but electromechanical-to-magnetic conversion efficiency. Driving the acoustic transducer at the 4.38 GHz mode produced a microwave sideband at the magnon resonance with conversion efficiency -48 dB at 20 K and -56 dB at room temperature. Loss was dominated by acoustic leakage into the frame and impedance mismatch at the transducer. Related nanomagnet studies show that magnetoelastic coupling can strongly reshape magnetisation dynamics when strain is localised efficiently [20].

Micromagnetic simulations indicate that reducing membrane width from 80 um to 30 um would increase acoustic frequency spacing and improve mode isolation, but at the cost of stronger edge strain gradients. A phononic shield around the membrane frame produced the largest simulated gain, improving acoustic Q by a factor of 4 without increasing magnetic linewidth.

The devices therefore sit in a useful intermediate regime. They are already tunable classical magnon-phonon components, but they are not yet efficient microwave-acoustic converters. Improving them will require mechanical engineering as much as magnetic-material optimisation.

Limitations

The first limitation is strain inhomogeneity. Static strain was calibrated as an average over the central membrane, but the edges and transducer regions experience different strain. This broadens magnetic resonance and makes the extracted coupling an effective mode-overlap value rather than a local material constant.

The second limitation is acoustic loss. The membrane clamps were not phononically shielded, so GHz acoustic modes leaked into the piezoelectric frame. The acoustic Q improved at low temperature but remained below what is needed for quantum-coherent transduction. Cavity magnomechanics in carefully shaped ferrimagnetic resonators has a more favourable acoustic confinement geometry [8,9].

The third limitation is heating. Acoustic and microwave drives both deposit power in a small suspended structure. We monitored resonance shifts and used low powers for spectroscopy, but high-power transduction measurements may contain nonlinear thermal contributions.

Finally, the model treats the magnon branch as a single uniform Kittel mode. At large strain or near edge defects, nonuniform spin-wave modes can hybridise with the same acoustic resonance. Weak satellite lines in two devices suggest that this already occurs. Future work should combine Brillouin light scattering or spatially resolved Kerr microscopy with microwave spectroscopy to map the magnetic mode profiles directly.

Conclusion

Mechanically strained ferrimagnetic garnet membranes exhibit tunable magnon-phonon coupling between microwave ferromagnetic resonance and GHz Lamb modes. Static strain shifts the magnon frequency, changes acoustic mode shapes, and can either enhance or suppress the coupling depending on mode symmetry. The strongest device reached g/2pi = 28.6 MHz and cooperativity above unity at room temperature, demonstrating coherent classical hybridisation in a planar released structure.

The work identifies both opportunity and constraint. Strained membranes provide an integrated geometry for magnon-phonon transduction and strain sensing, but their performance is limited by edge-induced magnetic linewidth and acoustic leakage. The next step is not simply stronger magnetostriction; it is co-design of ferrimagnetic film quality, release process, phononic shielding, and strain delivery.

Data and code availability

The supplementary archive contains microwave S-parameters, acoustic transmission spectra, laser vibrometry maps, strain-calibration images, finite-element meshes, coupled-mode fit outputs, and Python scripts used to generate all figures. Failed-release process statistics are included to support reproducibility of the membrane fabrication window.

References

1. Kittel, C. Interaction of spin waves and ultrasonic waves in ferromagnetic crystals. Phys. Rev. 110, 836-841 (1958).
2. Kittel, C. On the theory of ferromagnetic resonance absorption. Phys. Rev. 73, 155-161 (1948).
3. Weiler, M. et al. Elastically driven ferromagnetic resonance in nickel thin films. Phys. Rev. Lett. 106, 117601 (2011).
4. Dreher, L. et al. Surface acoustic wave driven ferromagnetic resonance in nickel thin films: theory and experiment. Phys. Rev. B 86, 134415 (2012).
5. Man, H. et al. Direct observation of magnon-phonon coupling in yttrium iron garnet. Phys. Rev. B 96, 100406 (2017).
6. Kikkawa, T. et al. Magnon polarons in the spin Seebeck effect. Phys. Rev. Lett. 117, 207203 (2016).
7. Flebus, B. et al. Magnon-polaron transport in magnetic insulators. Phys. Rev. B 95, 144420 (2017).
8. Zhang, X., Zou, C.-L., Jiang, L. & Tang, H. X. Cavity magnomechanics. Sci. Adv. 2, e1501286 (2016).
9. Li, J., Zhu, S.-Y. & Agarwal, G. S. Magnon-photon-phonon entanglement in cavity magnomechanics. Phys. Rev. Lett. 121, 203601 (2018).
10. Lovinger, D. J. et al. Magnetoelastic coupling to coherent acoustic phonon modes in the ferrimagnetic insulator GdTiO3. Phys. Rev. B 102, 085138 (2020).
11. Liu, S. et al. Direct observation of magnon-phonon strong coupling in two-dimensional antiferromagnet at high magnetic fields. Phys. Rev. Lett. 127, 097401 (2021).
12. Lyons, T. P. et al. Acoustically driven magnon-phonon coupling in a layered antiferromagnet. Phys. Rev. Lett. 131, 196701 (2023).
13. Hwang, Y. et al. Strongly coupled spin waves and surface acoustic waves at room temperature. Phys. Rev. Lett. 132, 056704 (2024).
14. Wang, K., Ren, K., Hou, Y., Cheng, Y. & Zhang, G. Magnon-phonon coupling: from fundamental physics to applications. Phys. Chem. Chem. Phys. 25, 21802-21815 (2023).
15. Burch, K. S., Mandrus, D. & Park, J.-G. Magnetism in two-dimensional van der Waals materials. Nature 563, 47-52 (2018).
16. Gong, C. et al. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265-269 (2017).
17. Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270-273 (2017).
18. Ramos, R. et al. Room temperature and low-field resonant enhancement of spin Seebeck effect in partially compensated magnets. Nat. Commun. 10, 5162 (2019).
19. Huber, T. et al. Magnetoelastic resonance sensor for remote strain measurements. Appl. Phys. Lett. 101, 042402 (2012).
20. Yahagi, Y., Harteneck, B., Cabrini, S. & Schmidt, H. Controlling nanomagnet magnetization dynamics via magnetoelastic coupling. Phys. Rev. B 90, 140405 (2014).