Magnon polaron formed by selectively coupled coherent magnon and phonon modes of a surface patterned ferromagnet

Strong coupling between two quanta of different excitations leads to the formation of a hybridized state which paves a way for exploiting new degrees of freedom to control phenomena with high efficiency and precision. A magnon polaron is the hybridized state of a phonon and a magnon, the elementary quanta of lattice vibrations and spin waves in a magnetically-ordered material. A magnon polaron can be formed at the intersection of the magnon and phonon dispersions, where their frequencies coincide. The observation of magnon polarons in the time domain has remained extremely challenging because the weak interaction of magnons and phonons and their short lifetimes jeopardize the strong coupling required for the formation of a hybridized state. Here, we overcome these limitations by spatial matching of magnons and phonons in a metallic ferromagnet with a nanoscale periodic surface pattern. The spatial overlap of the selected phonon and magnon modes formed in the periodic ferromagnetic structure results in a high coupling strength which, in combination with their long lifetimes allows us to find clear evidence of an optically excited magnon polaron. We show that the symmetries of the localized magnon and phonon states play a crucial role in the magnon polaron formation and its manifestation in the optically excited magnetic transients.


INTRODUCTION
Magnons are collective spin excitations in magnetically-ordered materials. Nowadays the manipulation of coherent high-frequency magnons on the nanoscale is one of the most prospective concepts for information technologies, also in the quantum regime. In this respect, the hybridization of magnons with phonons has been considered as a powerful method for spin control [1][2][3][4][5][6]. The magnon-phonon hybridization phenomenon in bulk materials is well understood [7,8] and has been realized in conventional magneto-acoustical experiments in the MHz frequency range [9,10]. Within the last decade, the magnon-phonon interaction has been actively studied in experiments with high-frequency coherent magnons and phonons [11][12][13][14][15][16][17][18][19][20]. For instance, the excitation of coherent magnons has been realized by broadband coherent phonon wave packets [11,13,19,20], localized monochromatic phonons [14,15,17,18] and propagating surface acoustic waves [12,16]. However, these experiments demonstrate only the one-way process of transferring energy from phonons to magnons with no evidence of strong coupling, characterized by a reversible energy exchange between them, and formation of a hybridized state.
Direct evidence of the formation of a hybridized state, further referred as magnon polaron, would be the avoided crossing of the magnon and phonon dispersion curves at their intersection [7,8]. A spectral splitting between the two branches arising from hybridization clearly indicates strong coupling of two underlying excitations, like in the case of two strongly coupled harmonic oscillators [21,22]. The experimental observation of the avoided crossing for magnon polarons is possible when the energy splitting at the magnon-phonon resonance exceeds the spectral widths of the interacting modes. In this case the experimental methods with selective probing of magnons and phonons with a certain wave vector, e.g. Brillouin light scattering [3] or neutron scattering [23], indicate the avoided crossing and confirm formation of the hybridized state. However, in the time-resolved experiment with excitation of a bunch of the phonon and magnon modes within broad coherent wave packets, fast dephasing of both phonons and magnons suppresses formation of a magnon polaron. Only recently, experiments with Ni nanomagnets [24] have demonstrated that by means of quantization of the magnon and phonon spectra by a three-dimensional confinement, the avoided crossing becomes detectable in the magnon transients.
Quantitatively, the hybridization is characterized by the cooperativity = 2 ph M ⁄ where is the coupling strength, i.e. the energy exchange rate between phonons and magnons, and ph and M are their individual damping rates [25]. High cooperativity ≫ 1 ensures deterministic manipulation of the coupled states with periodic conversion of magnons into phonons and vice versa with high fidelity, but is extremely challenging to realize in practice due to the weak magnon-phonon interaction and broad magnon and phonon spectra with respective quick dephasing of these excitations. The latter factor is crucial for ferromagnetic metals.
Here, we demonstrate an original approach for achieving pronounced hybridization of magnons and phonons with high cooperativity in a metallic ferromagnet. By surface patterning of a ferromagnetic film, we achieve magnon and phonon modes with significantly long lifetimes. Spatial matching of their wave distributions determines the coupling strength and selects particular modes in the phonon and magnon spectra for hybridization. This allows us to solve the main problem of quick dephasing of the interacting excitations, and we achieve ≈ 8 sufficient for pronounced magnon polaron formation. The suggested approach is verified in time-resolved experiments revealing the avoided crossing between the magnetic polaron branches in the transient magnon evolution that is excited by a femtosecond optical pulse. We demonstrate and discuss the scenarios in which coherent magnon polarons are generated and detected depending on the spatial symmetry of the phonons and magnons in the nanostructure and on the initial and boundary conditions of the coupled system. Our theoretical analysis based on coupled oscillators illustrates the physics behind the experimental observations.

GALFENOL NANOGRATING AND EXPERIMENTAL TECHNIQUES
To verify the suggested approach, we use a magnetostrictive alloy of iron and gallium (Fe0.81Ga0.19), known as Galfenol. This metal possesses both enhanced magnon-phonon interaction [26] and welldefined magnon resonances [27,28]. A film of Fe0.81Ga0.19 of 105-nm thickness was epitaxially grown on a (001)-GaAs substrate after 10 periods of a GaAs/AlAs (59 nm/71 nm) superlattice. The Galfenol film was capped by a 3-nm thick Cr layer to prevent oxidation. The film was patterned into a shallow onedimensional nanograting (NG) of 25×25-µm 2 size, which is illustrated in Fig. 1(a). Parallel grooves in the sample surface along the [010] crystallographic axis of the GaAs substrate were milled using a focused beam of Ga ions (Raith VELION FIB-SEM). The grooves have depth a=7 nm and width w=100 nm, which equals their separation; the respective NG lateral period is d=200 nm. Partial removing of the Cr cap layer did not affect the main properties of the studied sample and no signs of degradation due to oxidation of Galfenol were noticed in the experimental signals. The sample was fixed by a silver paste to a massive copper plate which served as a heat sink and was located between the poles of an electromagnetic coil. The measurements were performed at ambient conditions at room temperature.
The experimental technique shown in Fig. 1 nm (probe pulses). The energy density in the focused pump spot of 5-m diameter was 12 mJ/cm 2 . The energy density in the 1-m-diameter spot of the linearly polarized probe pulse at the NG surface was 1 mJ/cm 2 . The probe polarization and its relative orientation to NG were controlled by a Glan-Taylor prism with 1000:1 extinction ratio. The temporal resolution was achieved by means of an asynchronous optical sampling (ASOPS) technique [29]. The pump and probe oscillators were locked with a frequency offset of 800 Hz. In combination with the 80-MHz repetition rate, it allowed measurement of the timeresolved signals in a time window of 12.5 ns with time resolution limited by the probe pulse duration.
To measure the time evolution of the NG magnetization, we use a detection scheme that monitors the polar Kerr rotation (KR) of the probe's polarization plane, ( ), where is the time delay between the probe and pump pulses [30]. The KR signal was measured at parallel orientation of the probe polarization plane and the NG groves, when the photoelastic modulation did not affect the KR signal detected by means of a differential scheme based on a Wollaston prism and a balanced optical receiver with 10-MHz bandwidth. By measuring the intensity of the reflected probe pulse, Δ ( ), we monitor the coherent photoelastic response of the NG [31].

PHONON AND MAGNON MODES IN THE NANOGRATING AND THE EXPERIMENTAL CONCEPT
The phonon spectrum generated by the pump pulse in the NG includes several phonon modes localized in the near-surface region. These modes are excited due to the strong absorption (20-nm penetration depth) of the pump pulse in Galfenol. The calculated spatial profiles of the atom displacements corresponding to the localized modes are shown in Fig. 1(c). Both modes are standing pseudo-surface acoustic waves [32], which possess two displacement components along the x-and z-axis. Therefore, they are characterized by three components of the dynamical strain: , , , the frequencies and phases of which remain preserved within the respective lifetimes of the modes. The mode shown in the lower panel of Fig. 1(b) is a Rayleigh-like standing wave with dominant displacement along the z-axis, i.e. perpendicular to the NG plane. We refer to this mode as a quasi-transverse acoustic (QTA) mode. It is important to mention that there is a counterpart of the QTA mode, which has different symmetry properties and cannot be excited optically. This localized mode is almost degenerate with the QTA mode, but as we show below, can play a crucial role in the interactions with magnons.
Another mode, shown in the top panel of Fig. 1(c) is the second-order Rayleigh-like mode often referred to as Sezawa-mode [33]. It has a predominant in-plane displacement and we refer to this mode as a quasi-longitudinal acoustic (QLA) mode. The calculated frequencies [34] of the QTA and QLA modes are 13.1 and 15.3 GHz, respectively. The QTA and QLA modes are excited simultaneously and are expected to have similar amplitudes and lifetimes. However, due to the specific polarization and spatial distribution, the QLA mode provides much less contribution to the reflectivity signal [34]. This can be seen in Fig. 1(d), which shows Δ ( ) and its fast Fourier transform (FFT) measured at B=250 mT applied along the NG grooves when the interaction of the magnon and phonon modes of the NG is fully suppressed [35]. Now we consider the magnon spectrum of the NG. Because the studied grating is formed by shallow grooves, the depths of which are much smaller than the grating period and the ferromagnetic film thickness the magnon spectrum is close to that of an unpatterned film [37]. The transient KR signal measured for a plain Galfenol film (outside the NG) is shown in Fig. 1(e). The fast decaying oscillations reflect the magnetization precession excited by the femtosecond optical pulse, which induces ultrafast changes of the magnetic anisotropy [28,38,39]. The magnon spectrum of a plain film consists of several magnon modes quantized along the z-axis [28,38]. The high-order modes possess short lifetimes, which manifests as fast decay of the precession within several hundred picoseconds after optical excitation.
However, at t>1 ns the fundamental magnon (FM) mode with the lowest frequency fFM=18.2 GHz at B=200 mT contributes solely to the KR signal. In the whole range of the magnetic field used in the experiment, it has a considerably long lifetime FM=0.95 ns with respective HWHM of the power density spectrum FM=0.17 GHz. In the NG, the magnon modes possess additional spatial modulation along the x-axis with the NG period d due to the periodically modulated demagnetizing field (shape anisotropy) given by the NG spatial profile [37,40]. Despite the modulation along the x-direction, the FM mode remains the most pronounced in the magnon spectrum of the NG and possesses the same spectral width and a similar dependence of the spectral position on B as in the plain Galfenol film [37]. The main object of our study is the interaction of this FM mode with the two localized phonon modes, QTA and QLA.
The idea of our experiments is demonstrated schematically in Fig. 1  The resonance of the FM mode with the QLA mode is evidently seen as the bright red spot in Fig.   2(a) at B=140 mT when fFM=fQLA. The transient KR signal and its FFT measured at this resonance are shown in Fig. 3(a). The strong increase of the spectral amplitude at the QLA mode frequency is explicitly seen from comparison of the FFT spectra obtained in and out of the FM-QLA resonance and shown in the left panel of Fig. 3(b). This increase is the result of resonant phonon driving of magnetization precession [12,[15][16][17][18]24], which clearly indicates magnon-phonon interaction. However, no avoided crossing is observed at the intersection of the FM and QLA modes.

AVOIDED CROSSING AT THE MAGNON-PHONON RESONANCE
Summarizing the main experimental results, we clearly observe magnon polaron formation for the FM-QTA resonance through the normal mode splitting, but no avoided crossing is detected for the FM-QLA resonance. Another important difference between the two resonances is the strong increase of the KR signal due to phonon driving, which is observed only for the FM-QLA resonance.

SYMMETRIES OF THE MODES AND COUPLING SELECTIVITY
To understand the differences in the manifestations of the two magnon-phonon resonances, we analyze the magnon-phonon interaction in the NG in more detail. Our analysis is based upon the approach developed in Ref. [41]. It has been shown that the coupling strength of interacting magnon and phonon modes can be determined by the spatial overlap of the dynamical magnetization, , of a magnon mode and the strain components of a phonon mode. In this case, the interacting magnon and phonon modes can be considered as two coupled oscillators [41]. Due to the in-plane orientation of the external magnetic field the -component of the steady-state magnetization can be assumed to be zero. In this case, for modeling the magnon-phonon interaction we may consider only two strain components: ηxx and ηxz [35]. The coupling strength, ≈ Δ/2, for the magnon and phonon modes at resonance is defined by two overlap integrals: where ̃, and ̃, are the projections of the dynamical magnetization of the magnon mode and the strain components of the phonon mode, respectively, normalized in such a way that ∫̃, 2 = ∫̃, 2 = 1 ( is a dimensionless unit volume element). The coefficients 1 and 2, which have dimension of frequency, are defined by the material parameters including the magneto-elastic coefficients, the saturation magnetization , the mass density of the media, as well as the external magnetic field orientation and the resonant frequency [42].
To evaluate Eq. (1) we consider the spatial distributions of the phonon modes localized in the NG.
As mentioned earlier, there is a counterpart of the QTA mode referred to as QTA*. The spatial distributions of the strain components are shown in the bottom of Fig. 4(a). The QTA and QTA* modes are split due to Bragg reflections and interferences in the spatially periodic NG [43], but their splitting does not exceed 0.1 GHz. Thus, the QTA and QTA* modes may be considered as degenerate. The calculated HWHM of both modes for an ideal grating, in which the mode decays are determined by leakage to the substrate [44], is ~0.001 GHz. This value is much smaller than the measured value of 0.03 GHz for the QTA mode. This indicates that the QTA mode decay and spectral broadening are mainly due to imperfections of the NG, which induces scattering and additional leakage [44]. Because there is no alternative decay channel for the QTA* mode, we assume that imperfections determine also the QTA* decay rate and, thus, QTA*=QTA. The only difference between the QTA and QTA* mode is the symmetry along the x-axis. We refer to the QTA mode as symmetric due to the symmetry of the strain components ̃ and ̃ relative to the center of the groove of the NG. The QTA* mode is referred to as antisymmetric. Only the symmetric QTA mode is excited by the pump pulse [34]. The QTA* mode cannot be excited by the pump pulse due to the antisymmetric nature of ̃, and its zero overlap integral with the symmetric distribution of the laser intensity. Thus, it also cannot be optically detected [34].
However, the shear strain component ̃, has the opposite symmetry which will be important for the interaction with magnons: it is antisymmetric for QTA and symmetric for QTA* [see Fig. 4(a)].
Next, we consider the magnon spatial distribution, assuming mixed boundary conditions for the magnetization in the studied structure along the z-axis: pinning at the (Fe,Ga)/GaAs interface and free precession at the patterned surface of the NG [45]. Then, the spatial distribution of the normalized dynamical magnetization, δ̃, for the FM mode can be written as where z=0 corresponds to the (Fe,Ga)/GaAs interface. The spatial distribution of ̃ is shown in the center of Fig. 4(a). The calculations based on Eqs. (1) and (2)  crystallographic direction at 45 degrees to the applied magnetic field [34].
A similar analysis for the QLA mode, for which the spatial distribution is demonstrated in the upper part of Fig. 4(a), shows that the overlap of the FM mode with the optically excited QLA mode is nonzero ( QLA ≈ 1 ) and driving of the FM mode by the QLA mode should take place. This agrees with the experimental results in Fig. 3(b) where the increase of the KR signal in resonance is clearly seen. The absence of a normal mode splitting in this case could be related to the smaller value of QLA , for which the normal mode splitting becomes affected by the energy transfer from the optically-excited coherent phonons to magnons [24].

MODEL OF COUPLED OSCILLATORS AND COOPERATIVITY OF THE MAGNON-PHONON COUPLING
For better understanding of the interaction of the phonon and magnon modes in the NG and its experimental manifestations, we consider a model of three interacting oscillators using the following system of equations: where aj are the complex amplitudes of the respective modes ( =QTA * , QLA, or FM) and Aj is the amplitude of their excitation. The coupling tensor K has the form where GHz is the spectral width of the line at the resonance = FM = QLA shown in Fig. 3(a). The spectral width of the QLA mode, QLA , was not measured directly due to its extremely weak contribution to I(t), but we may assume that it is determined also by the NG imperfections and QLA = QTA = 0.03 GHz.
Thus, the coupling strength of the FM and QLA modes is QLA = 0.07 GHz, which provides a cooperativity QLA = 1. For such "moderate coupling", the formation of the magnon polaron may still take place, but the avoided crossing would be masked by the energy transfer from the phonon mode [34].
The difference between manifestations of the experimentally observed regimes of the magnonphonon coupling is clearly seen in the color map shown in Figs. 4(b) and 4(c), which has been obtained by analytical solution of Eq. (3) with the parameters given above. The chosen excitation amplitudes are QLA = 10; FM = 1; QTA * = 0 corresponding to the experimentally realized situation when the QTA* mode is not excited and the energy transferred from the optical pump pulse to the QLA mode is 100 times higher than the energy injected into the FM mode. The calculated magnetic field dependence shows strong similarities with the experimental color map in Fig. 2, so that the model can be considered as prototypical for the involved physics. The main features of the magnon spectra in the calculated color map of Fig. 4(b), i.e. the avoided crossing and the driving, are in good agreement with the experimentally measured magnon spectra shown in Fig. 2(a). Indeed, the avoided crossing effect is observed for the lower FM-QTA* resonance mode while for the upper FM-QLA resonance the splitting of the hybridized state is masked by strong driving of the FM mode by resonant phonons. It is worth mentioning, that if the modes coupled with C = 1 are excited with equal initial amplitudes an avoided crossing is well resolved [34].

MAGNON POLARON IN THE REFLECTIVITY SIGNAL
The magnon-phonon coupling and the formation of a magnon polaron should be manifested also in the field dependence of the transient reflectivity signal. The spectral amplitude of the QTA mode in ∆ ( ) does not depend strongly on B, which agrees with our model where the symmetric QTA mode does not interact with the FM mode. We might expect an effect of the interaction between the symmetric QTA mode with the optically inactive antisymmetric counterpart of the FM magnon mode, which is also present in the spatially periodic NG. Actually, the spatial distribution of the antisymmetric FM mode along the z-axis is altered by the demagnetizing field of the NG and differs significantly from the distribution of the symmetric FM mode resulting in a much smaller overlap integral and the absence of strong coupling with the QTA mode. That is why the avoided crossing for the QTA mode is not observed.
In contrast to the QTA mode, the contribution of the QLA mode to ∆ ( ) demonstrates a strong dependence on B. Out of resonance the contribution of the QLA mode to the transient reflectivity is extremely weak, which is apparently due to an inefficient photo-elastic effect. It increases significantly at the FM-QLA resonance at B=140 mT as demonstrated in the right panel in Fig. 3(b) and in Fig. 3(c).
Such an increase can be due to the magnon-phonon interaction, which distorts the polarization of the phonon mode similarly to how it happens in the bulk [47] and is predicted for surface acoustic waves [48]. The in-plane shear component, , makes the QLA mode visible in the photo-elastic effect [34].
In this case, the reflectivity signal, ΔI(t), for the QLA mode increases at the resonance field (B=140 mT), which is observed in our experiment. However, the influence of quadratic magneto-optical Kerr effects [49] on the reflectivity signal at the resonance conditions, when the precession amplitude is large, cannot be fully excluded. Moreover, the selectivity in the coupling of magnon and phonon modes allows us to isolate the resulting hybridized state from the optically active phonon mode of the same frequency. This selectivity, which is determined by the spatial overlap of the interacting modes, can be achieved also by alternative ways of spatial matching. For instance, one could imagine asymmetric periodic structures or two-dimensional patterns, the spatial profile of which provides efficient coupling of particular phonon and magnon modes, while other modes remain uncoupled.

CONCLUSIONS
The suggested approach is applicable for the manipulation of propagating magnon polarons. While the ability of surface acoustic waves to carry magnons across extremely long distances has been recently demonstrated in the weak coupling regime [50], the strong magnon-phonon coupling will enrich collective magnon-phonon transport by pronounced nonreciprocity and give access to the manipulation of the polarization of surface acoustic waves [48]. For instance, at certain direction of external magnetic field, the attenuation and propagation velocity are expected to be different for the magnon polarons propagating in opposite directions along the reciprocal lattice vector of the nanograting [51][52][53]. Another possibility is the localization of magnon polarons on a defect in a periodic structure due to the gap in the magnon spectrum. This is particularly important in the context of spin-spin interactions and the related phenomenon of magnon Bose-Einstein condensation [3]. An especially appealing prospect is to use the long coherence times and tunability of magnon polarons for creating states with well-defined magnon polaron numbers [54] or coherent magnon-photon-phonon states [55,56], thus providing new routes for quantum information and metrology.