## Introduction

Magnonics is an extremely active research area which exploits the wave nature of magnons, the quanta of spin waves, in order to advance data storage, communication, and information processing technology. However, a current drawback in the excitation, manipulation, and detection of magnons exists due to relatively low conversion efficiencies1. Coupling to the phononic system is a less explored avenue for the manipulation of magnons, and has been shown to be a promising means of lowering the switching energy of nanoelements2,3. With this in mind, a more thorough understanding of the coupling between the spin and phonon systems in nanostructures is necessary.

Because of this, the magnon–phonon interaction has recently been the focus of increased research, specifically as it is applied to surface acoustic wave ferromagnetic resonance (SAW-FMR)4,5,6,7. Significant effects of optically excited SAWs on the magnetization dynamics of patterned nanomagnet arrays have been demonstrated8. In these systems, the SAW frequencies arise from the geometric arrangement of the elements9, with the SAW energy located in the substrate layer below the nanoelements. Due to the magneto-elastic coupling, the magnetic resonance is pinned at the SAW frequency. This effect was used to extract the Gilbert damping parameter in a nanomagnet10. In these structures, the dynamics are akin to a driven oscillation rather than to two coupled oscillators11. Another exciting development was the experimental demonstration of magnons transferring spin to the phonon system12. Despite all of this progress, observation, and quantification of the hybridized magnon–phonon modes itself remain challenging tasks13, and so far dynamics of the hybridized modes have not been resolved spectroscopically in relevant structures14.

In this article, we utilize the vibrational modes of a single, isolated Ni nanostructure to optically initiate phononic dynamics in the GHz frequency range (5 GHz–25 GHz) along with the intrinsic magnetic resonances of the magnet. Access to this higher frequency range has been a limiting factor in resolving the mode splitting in previous experiments using interdigital transducers14. The confined geometry of the magnetic nanostructure isolates the phononic modes much like semiconductor microcavities control excitonic states. Using an external magnetic field in the appropriate geometries, the magnonic mode is tuned through the phononic resonances and the avoided crossings characteristic of coupled systems is observed15,16.

## Results

### Hybridized magnon–phonon dynamics

The magnon–phonon dynamics of the nanomagnet (Fig. 1a) were measured using a two-color TR-MOKE setup using a balanced photodiode detection scheme which enables us to measure the magnetic system and the non-magnetic (phononic) system at the same time17 (see the Methods section). When the pump pulse hits the nanomagnet, the energy is absorbed by the electron system which then equilibrates with the phonon systems within a few picoseconds according to the two-temperature model18. This excites the spin and phonon systems concurrently within the nanomagnet (Fig. 1b).

In Fig. 1, colormaps from the magnetic (c) and non-magnetic (d) channels are displayed. Each field represents a different TR-MOKE scan that has been transformed into the frequency domain using an FFT algorithm. The colors represent the frequency components present at that field. In order to see the frequency variation more clearly, the frequency amplitude was normalized for each field bin. The magnetic channel displays a dominant frequency which changes as the external field changes (Fig. 1c). In addition, there are field independent modes which arise due to the coupling to the phononic system. Confirmation of this is seen in the non-magnetic spectra (Fig. 1d), where the frequencies are all H-field independent and match the positions of the magnetic channel’s field independent frequencies. These spectra look similar to those observed in patterned arrays8. However, in the single isolated nanomagnet, the coupling between the two systems is stronger and more direct. It arises within the element itself rather than with surface acoustic waves located in the substrate, leading to qualitatively different behavior around the crossovers between resonances.

The in-plane components of the phononic mode’s k-vector are $$k_{x,y} = \frac{{n_{x,y}\pi }}{{l_{x,y}}}$$, where lx,y is the dimension of the nanoelement along the x or y directions. Therefore, the phononic mode is characterized by the indices (nx,ny) and the in-plane angle from the x-axis, φk=tan−1(ny/nx) (Fig. 1e). Because the material is elastically isotropic, it can be characterized by the Lamé constants, λ=Ev/(1+v)(1−2v) and μ=E/(2(1+v)), where E is the Young’s Modulus and v is the Poisson ratio19. We assume the density of Ni, ρ = 8900 kg/m3 and v = 0.3120 and fit E to the experimentally measured phononic modes using the relation

$$\omega _{{\mathrm{ph}}}^2 = \frac{{\left( {2\lambda + 3\mu } \right)k^2}}{{2\rho }}$$
(1)

derived in Supplementary Note 2. The fitted value was $$209_{ - 29}^{ + 31}$$ GPa, in excellent agreement with the range of values reported in ref. 21.

In order to extract the relevant magnetic parameters, an identical magnetic film was grown next to the nanostructures. Its precession resonances were measured at various fields and angles, and γ and MS were fit to the Kittel formula22

$$\omega _{\mathrm{M}}^2 = \omega _1\omega _2$$
(2)

where $$\omega _1 = \gamma \left( { - 4\pi M_{\mathrm{S}}\left( {\cos 2\theta _{\mathrm{M}}} \right) + H\cos \left( {\theta _{\mathrm{H}} - \theta _{\mathrm{M}}} \right)} \right)$$, $$\omega _2 = \gamma \left( - 4\pi M_{\mathrm{S}}\left( {\cos 2\theta _{\mathrm{M}}} \right) + H\cos \left( {\theta _{\mathrm{H}} - \theta _{\mathrm{M}}} \right) \right)$$, H is the applied field magnitude, θM and θH are the angles of the magnetization vector and the applied field with respect to the surface normal, γ is the gyromagnetic ratio, and MS is the saturation magnetization. The fitted frequencies yielded values of $$\gamma = 1.98_{ - 0.01}^{ + 0.02} \times 10^7$$ rad∙ Oe−1∙s−1 and $$M_{\mathrm{S}} = 203_{ - 16}^{ + 8}$$ emu∙cm−3.

Changing the external field modifies the frequency of the magnons so they can be brought into resonance with the various phononic modes. The crossings are displayed in Fig. 2a, b for the (1,1) and (2,0) crossings, respectively. The Fourier spectra exhibit two clear peaks at each field value at and around the crossing field. These two frequencies are attributed to the hybridization of the magnon and phonon eigenstates. In this region, the modes do not have a specific magnon or phonon character, but rather exist in both states.

In order to quantitatively characterize the coupling evident in the avoided crossings, we derived a closed-form expression for the eigenfrequencies of the coupled magnon–phonon system. The full derivation is shown in Supplementary Note 2 and yields the characteristic Eq.

$$\left( {\omega ^2 - \omega _{\mathrm{M}}^2} \right)\left( {\omega ^2 - \omega _{{\mathrm{ph}}}^2} \right) - \omega _{\mathrm{C}}^4 = 0$$
(3)

Where

$$\omega _{\mathrm{C}}^4 = \left( {\frac{{\gamma M_{\mathrm{S}}}}{\rho }} \right)\left( {\omega _1C_2 + \omega _2C_1} \right)k^2$$
(4)

with

$$C_1 = \sin ^22\theta _{\mathrm{M}}\Big( {b_1^2\left( {\cos ^2\varphi _{\mathrm{k}}\cos ^2\varphi _{{\mathrm{mp}}} + \sin ^2\varphi _{\mathrm{k}}\sin ^2\varphi _{{\mathrm{mp}}}} \right)} \\ {+ \frac{3}{4}b_1b_2\sin 2\varphi _{\mathrm{k}}\sin 2\varphi _{{\mathrm{mp}}} + \frac{1}{2}b_2^2\sin ^22\varphi _{{\mathrm{mp}}}} \Big)$$
(5)
$${C}_2 = \sin ^2\theta _{\mathrm{M}}\left( {{\mathrm{b}}_1^2\sin ^22\varphi _{{\mathrm{mp}}} + 2{\mathrm{b}}_2^2\cos ^22\varphi _{{\mathrm{mp}}}} \right)$$
(6)

The frequency peaks were selected from the (1,1) and (2,0) modes and fit simultaneously using Eq. (3), which shows an excellent match between data and fits (Fig. 2c, d). The parameters γ, MS, and E from the elastic and magnetic fits were used as initial fitting parameters and allowed to vary within their respective errors. Because the nanomagnet is polycrystalline, b1 and b2 are equal. In addition, in order to take into consideration the in-plane magnetization distribution within the nanomagnet, as well as the inevitable slight experimental disorientation of the magnetic field, φmp was also allowed to vary. The fitted value of b1 was 40 ± 4 × 10−4 Oe. Using $$b_1 = - \frac{{3\lambda _Sc_{44}}}{{M_{\mathrm{S}}}}$$, a polycrystalline magnetostriction value of λS = −34 ± 4 × 10−6 was obtained23, which is in agreement with the bulk value24 of −32 × 10−6 and in reasonable agreement with the value measured in Ni thin films21. Figure 2e shows the mode splitting corresponding to the coupling strength of the hybridized modes. We find for the (1,1) mode $$\Delta f_{{\mathrm{min}}} = 0.76_{ - 0.08}^{ + 0.08}$$ GHz and for the (2,0) mode $$\Delta f_{{\mathrm{min}}} = 0.85_{ - 0.08}^{ + 0.08}$$ GHz. The error is the standard deviation found using a Monte Carlo error-propagation scheme by randomly varying the frequencies within the FFT resolution and then fitting them to Eq. (3). The anticrossing is empirical evidence of the coupling between the magnon and phonon systems, and has not been observed in other experiments utilizing acoustic waves as an excitation mechanism9,14. In order to determine the coupling regime of the magnon–phonon resonances, we analyze the loss rates of the different systems by employing a least- squares curve-fitting algorithm to the decaying sinusoids in the time-domain signals. The field-dependent loss rate for the magnetic signal can be approximated by κM = αef, where f is the frequency and αe is the effective damping25. $$\alpha _{\mathrm{e}} = 0.038_{ - 0.03}^{ + 0.04}$$, and was extracted at 4.6 kOe, away from the crossing point, so that the mode is predominantly magnetic in character. This value is consistent with previous measurements we have made on the damping in Ni9. The loss rate of the phononic system, κP, is extracted from the non-magnetic signal. Note that the loss rates cannot be taken directly from the linewidth of the FFT spectra since they are artificially broadened due to the finite duration of the signal in the time domain26.

The loss rates are $$\kappa _{\mathrm{M}} = 0.41_{ - 0.03}^{ + 0.05}$$ GHz for the magnetic system at the (1,1) crossover and $$\kappa _{\mathrm{M}} = 0.53_{ - 0.04}^{ + 0.05}$$ GHz for the magnetic system at the (2,0) crossover. $$\kappa _{\mathrm{P}} = 0.31_{ - 0.07}^{ + 0.12}$$ GHz for the (1,1) phononic mode and $$\kappa _{\mathrm{P}} = 0.47_{ - 0.07}^{ + 0.12}$$ GHz for the (2,0) phononic mode. The coupling rate is given by half of the mode splitting, $${\mathrm{\Gamma }}_{\mathrm{c}} = \Delta f_{{\mathrm{min}}}/2$$. This allows us to calculate the cooperativity $${C} = {\mathrm{\Gamma }}_{c}^2/\kappa _{\mathrm{M}}\kappa _{\mathrm{P}}$$, which for the (1,1) crossing is $${C} = 1.14_{ - 0.30}^{ + 0.48}$$ and for the (2,0) crossing is $${C} = 0.74_{ - 0.14}^{ + 0.21}$$. Keeping in mind the error, this places these two crossings in the intermediate coupling regime characterized by 1 > C > 0.127.

### Tuning the coupling regime

As evidenced through C1 and C2, the splitting energy depends on the type of strains present (normal, b1; shear, b2) and is a function of φmp and θM. In addition, these angular dependencies are weighted by the magnetic energy terms ω1 and ω2 which depend on MS and H and limit the range of experimentally accessible angles. As a demonstration of the dependence on the in-plane angle of the magnetization and the phononic k-vector, we focus on the (2,0) mode which is characterized by normal strains (b2 = 0). When both strains are present (depending on the respective amounts), the in-plane dependence disappears due to the orthogonallity of the b1 and b2 in-plane angular functions.

Taking this into consideration, we calculate the weighted coupling strength, (ω1C2+ω2C1) for our particular experimental configuration and plot it as a function of H and φmp (Fig. 3a). The angle of the magnetization, θM, from the surface normal is a function of the applied-field configuration. At an applied-field angle of θH=60°, Fig. 3b shows the angles of the magnetization θM for the range of applied fields used in the experiment. This H-φmp dependence is further modified by the geometric and material properties ρ, γ, and MS, which dictate the field at which the two resonances cross. In the original configuration (the applied field along the x direction), the (2,0) crossing is shown as a star on the H-φmp plot (Fig. 3a). By rotating the nanomagnet in-plane, we can change φmp, which changes the coupling strength (moving along the dotted line in Fig. 3a) and is maximum at an angle of 45°. The calculated and experimentally measured minimum frequency splitting versus φmp are shown in Fig. 3c. The data match the calculated results quite well. Figure 3d shows the configuration and obtained spectra with the magnetization oriented parallel to the (2,0) mode, and Fig. 3e shows the case where the magnetization is oriented 45° to the (2,0) mode. For the 45° spectrum, φmp was fit to Eq. (3) using the previously obtained material parameters as constants. The value obtained was φmp = 50°, a very close match. More importantly, from the spectrum in Fig. 3e, it is evident that the splitting increases for φmp = 45° the bandgap starts to emerge in the spectrum. The frequency splitting increases to $$\Delta f_{{\mathrm{min}}} = 1.41_{ - 0.16}^{ + 0.16}$$ GHz, a 66% increase from the φmp = 0° case. The damping in this configuration was also measured at 4.6 kOe, and is $$\alpha _{\mathrm{e}} = 0.047_{ - 0.03}^{ + 0.03}$$ which corresponds to $$\kappa _{\mathrm{M}} = 0.65_{ - 0.04}^{ + 0.05}$$. At this crossing $${\mathrm{\Gamma }}_{\mathrm{c}} > \kappa _{\mathrm{M}}$$,κP, which translates into a cooperativity of $${C} = 1.65_{ - 0.32}^{ + 0.48}$$, placing this crossing in the strong coupling regime within error (C > 1). Therefore, by reorienting the magnetization vector using an externally applied magnetic field, we were able to increase the coupling from the intermediate into the strong coupling regime. Table 1 shows the rate values and cooperativities for the three crossings analyzed.

## Discussion

Hybridized magnon–phonon dynamics were measured by tuning the magnonic resonance to that of the intrinsic phononic vibrations of an isolated square nanomagnet. The hybridized modes were clearly resolved in the Fourier transforms of the time-dependent magneto-optic signals at resonance. A two-dimensional equation describing the dynamics was derived which fit the data with high accuracy. In addition, the coupling was shown to depend on the orientation of the magnetization vector and the phononic vector as predicted by the theory. By tuning the direction of the applied field, we were able to enhance the coupling so that the system was unambiguously in the strong coupling regime. The ability to uniquely tune the energy splitting of the hybridized mode using various external degrees of freedom is attractive from the perspective of reconfigurable magnonic devices28,29. In addition, this magnetomechanical system provides a means of studying the dynamics of coupled quantum systems which may aid in the development of more efficient transducers between phononic and magnonic systems.

## Methods

### Sample fabrication

A polycrystalline Ni square with dimensions 330 x 330 x 30 nm was fabricated using electron beam lithography on a (100) Si substrate capped by a 110-nm-thick hafnium oxide antireflection coating for enhanced magneto-optic sensitivity (Supplementary Fig. 1)23.

### Experimental setup

The dynamics of this nanomagnet upon pulsed optical excitation were measured using a two-color TR-MOKE setup (Supplementary Fig. 1b). In all, 800 -nm 165-fs pulses exit the femtosecond laser cavity at a repetition rate of 76 MHz. They are split by a beam splitter (BS1), and part of the light is sent through a second harmonic generator (SHG) where the frequency is doubled to produce the pump pulse at a wavelength of 400 nm. The pump pulse is then sent through a mechanical chopper wheel (MCW), which serves as a reference to the lock-in amplifiers. The other component of light serves as the probe pulse and has a much lower power than the pump pulse so that it has a negligible effect on the magnetic system. After the beam splitter, the probe pulse goes through a delay stage to adjust the arrival time Δt of the probe pulse and then through a polarizer (L(αP)). The two beams are recombined using a dichroic filter (DF). Both beams are then focused onto the sample using a ×100 microscope objective controlled by a Witec Microscope (WM) which is able to position the objective and the sample with submicron resolution. Due to a slight chromatic aberration from the objective, the two beams are defocused from one another which provides a larger radius of the pump beam at the focal point of the probe beam and ensures a homogeneous excitation of the nanoelement.

The reflected beams are collected by the objective and sent back along their incoming paths. When the probe reaches BS2, it is diverted from the path. A color filter is in place so as to get rid of any residual blue pump energy. The red beam is then split by a polarizing beam splitter (PBS) and focused onto a pair of photodiode detectors (BPD). By adjusting L(αP), the signal can be balanced so the photodiodes’ voltages are equal. The detectors are connected to a circuit which outputs the difference (Δ) and sum (Σ) signals of the two photodiodes to two lock-in amplifiers which are then read by a LabView program. Due to the magneto-optic Kerr effect, the polarization of the light is rotated after reflection off the sample. This changes the relative magnitudes of the components of light split by the PBS which affects the difference (magnetic) signal. The sum (non-magnetic) signal is unaffected by changes in polarization, but is affected by changes in the reflectivity which occur due to elastic modulation of the dielectric tensor30. Furthermore, since changes in reflectivity affect both components of the light split by the PBS equally, the difference channel is unaffected by reflectivity changes. In this way, the magnetic and elastic signals can be separated and detected simultaneously17.

An external magnetic field was applied using a pair of Nd permanent magnets and the orientation and magnitude of the field was characterized using a gauss meter. The magnetic field, H, was applied at θH = 60° with respect to the surface normal. This angle was chosen so that the magnonic system could be tuned to multiple phononic resonances.