Tripartite phonon–magnon–plasmon coupling, parametric amplification, and formation of a phonon–magnon–plasmon polariton in a two-dimensional periodic array of magnetostrictive/plasmonic bilayered nanodots Open Access

Plasmonics^1,2 and magneto-plasmonics^3,4 have been busy fields of study, spawning novel technologies such as magneto-optical nano-antennas^5,6 and circuits.⁷ In magneto-plasmonics, one usually examines the influence of magnetic fields on plasmons, while ignoring the reciprocal effect where surface plasmons, potentially hybridized with entities like phonons,^8–10 impact the dynamic magnetic properties of magnetic media, specifically SWs. Here, we report a study of this latter effect, which we have recently shown could reveal intriguing features such as the formation of acousto-plasmo-magnonic frequency combs.¹¹ This field is an extension of the well-explored and popular field of magnetoelastic coupling^12–23 where pure phonon modes interact with magnon modes (magnon–phonon coupling) in magnetostrictive nanomagnets to produce such entities as magnon–polarons.^20,21 The next frontier is obviously tripartite coupling where three different entities—phonons, magnons and plasmons—couple to reveal new physics and portend new applications.

Conventional wisdom will hold that phonon–magnon–plasmon coupling would be an elusive phenomenon since plasma wave frequencies far exceed those of SWs. Hence, any coupling between them will encounter a nearly insurmountable barrier because of the resulting large phase mismatch. Surface plasmons or surface plasmon polaritons in metals, however, can have much lower frequencies than bulk plasmons,²⁴ and, hence, they can couple with acoustic phonons in appropriate media to form hybrid phonon–plasmon modes^8–10 whose frequencies are close to those of spin waves in ferromagnetic media. These hybrid modes (phonons + plasmons) can couple strongly with SW modes (magnons) to produce tripartite coupling between phonons, magnons, and plasmons, giving rise to acousto-plasmo-magnonics.

In this paper, we have employed time-resolved magneto-optical Kerr effect (TR-MOKE) microscopy to showcase the intriguing coupling between SWs and hybridized phonon–plasmon waves. Our study employs a two-dimensional periodic array of $∼100$ nm (Co/Al) heterostructured nanodots deposited on an Si substrate. Acousto-plasmo-magnonic modes are excited in this system when it is exposed to ultrashort (⁠ $∼100$ fs) laser pulses. The periodic heating and cooling caused by the laser pulses give rise to acoustic waves (phonons) in the Si substrate owing to periodic stresses caused by the unequal thermal expansion/contraction coefficients of the Si substrate and the nanodots on top. These phonons mix with the surface plasmons or surface plasmon polaritons due to the Al layer to form the hybrid phonon–plasmon modes, which then couple with the SW modes generated in the nanomagnets, with or without an external magnetic field, to form the acousto-plasmo-magnonic modes whose dispersion relations may show no dependence on any magnetic field because they are not magnetic in origin.

The samples are fabricated with e-beam lithography and sequential e-beam evaporation of different metals. An e-beam resist is spun onto a Si substrate and patterned with e-beam in a Raith system to open windows for the nanodots, followed by the sequential evaporation of Ta (10 nm: for adhesion), Al (10 nm), Co (6 nm), and Au (2 nm), followed by lift-off. The Au layer is needed to prevent the oxidation of Co. Both Al and Au are plasmonic metals, but because the Au layer is $5×$ thinner than the Al layer and is much more distant from the Si substrate where the phonons reside, its contribution to plasmon–magnon hybrid modes, if any, is much lower than that of Al. We cannot separate out the contributions from Au and Al, but in our context, it is of no consequence. We fabricate two sets of samples: (i) a two-dimensional array of elliptical Co nanomagnets (NMs) on a Si substrate and (ii) an identical array of (Co/Al) bilayered nanodots of the same shape and dimensions as the Co NMs on a similar Si substrate. Only the latter hosts strong enough hybrid plasmon–phonon modes that can couple with magnon modes in the nanomagnets because of the 10 nm thick Al layer close to the Si substrate.

In order to study acousto-plasmo-magnonics and differentiate it from the more familiar acousto-magnonics (i.e., tripartite phonon–plasmon–magnon coupling vs bipartite phonon–magnon coupling), the investigation of ultrafast magnetization dynamics of the samples was conducted by employing a custom-built TR-MOKE microscope²⁵ based on a two-color collinear pump–probe technique under ambient conditions. A schematic representation of this experimental setup is shown in Fig. 1(c).

Figures 1(a) and 1(b) show the scanning electron microscope (SEM) images of systems (i) and (ii), which are without and with the Al layer, which is the major source of surface plasmons and surface plasmon polaritons. The Co NMs are shaped like elliptical disks with major axis dimension $∼105$ nm, minor axis dimension $∼100$ nm, and thickness $∼6$ nm. Along the major axes (x axis), the edge-to-edge separation between the NMs is $∼45$ nm, and in the direction of the minor axes, it is $∼50$ nm. The lateral dimensions of the nanomagnets and the edge-to-edge separation between them exhibit a maximum deviation of $±5%$ and $±8%$⁠, respectively.

The fundamental laser beam (wavelength = 800 nm, fluence = 2 mJ cm $− 2$⁠, pulse width = 80 fs) from a mode-locked Ti-sapphire laser (Tsunami, Spectra-Physics) is used to probe the polar Kerr rotation (hereafter referred to as the probe beam) and its frequency-doubled counterpart (wavelength = 400 nm, fluence = 16 mJ cm $− 2$⁠, pulse width = 100 fs), hereafter referred to as the pump beam, is used to excite the magnetization dynamics of the samples. The latter has high enough intensity that it can penetrate through the Au and Co layers to reach the Al layer and excite plasma oscillations there (surface plasmons and surface plasmon polaritons). The Kerr signal is measured using an optical bridge detector (OBD) as a function of time delay between the pump and probe beams. Achieving the spatial overlap of these two beams is critical. The slightly defocused pump beam (of diameter $∼1$  $μ$m) at the focal plane of the tightly focused probe beam (of diameter $∼800$ nm) is made collinear and overlapping on the sample plane using a single microscope objective of numerical aperture 0.65. The OBD simultaneously measures both the reflectivity and Kerr rotation and separates these intertwined signals with the help of two lock-in amplifiers in a phase-sensitive manner to attain high sensitivity. To maintain temporal synchrony, the pump beam is subjected to periodic modulation at a frequency of about 2 kHz by a mechanical chopper. This modulated pump beam serves as a reference frequency and is conveyed to the lock-in amplifiers, anchoring the phase relationships within the system. Furthermore, the probe beam is systematically positioned at the precise center of the pump beam using an $x$– $y$– $z$ piezoelectric scanning controller, guided by a feedback loop and a white-light illumination system, enhancing the fidelity of the experiment. A static magnetic field with varying magnitude is applied with a small tilt (of $∼ 10 °$– $15 °$⁠) from the sample plane to introduce the necessary out-of-plane demagnetizing field along the direction of the pump beam, which helps to induce precessional magnetization dynamics (SWs) in the sample during pumping. The experimental time window of 2 ns was found to be sufficient to resolve the SW peaks with a temporal resolution of 10 ps from the fast Fourier transform (FFT) of the bi-exponential background-subtracted time-resolved traces.

We compared the experimental results of the samples without Al (no surface plasmons or surface plasmon polaritons present) with theoretical results obtained from micromagnetic simulations using Object Oriented Micromagnetic Framework (OOMMF) software²⁶ , employing a $4×4$ array of nanomagnets whose specifications are based on what we observed in the SEM images. The discretized array featured rectangular parallelepiped-shaped cells (⁠ $3×3×6$ nm $3$⁠) with a two-dimensional periodic boundary condition. The unit cell length was kept below the exchange length of Co (⁠ $∼4.93$ nm), to allow the incorporation of exchange interactions. Simulation parameters used included saturation magnetization, $M s=1400$ emu cm $− 3$⁠, anisotropy constant, $K=0$⁠, $γ=17.6$ MHz Oe $− 1$⁠, the exchange stiffness constant, $A e x=3.0× 10 − 6$ erg cm $− 1$⁠,²⁷ and Gilbert damping coefficient, $α=0.008$⁠.²⁸ 

To simulate magnetization dynamics, we first obtained the static magnetic state after system relaxation to equilibrium at a specific bias-field. Optical excitation was emulated with a square pulsed magnetic field (10 ps rise and fall time, 200 ps width, 20 Oe peak amplitude) perpendicular to the sample plane. The FFT of the simulated time-resolved out-of-plane magnetization (⁠ $m z$⁠) revealed the SW spectra, consistently showing three dominant peaks (M1, M2, and M3) across various bias magnetic fields. These theoretical results showed excellent agreement with the experimental observations.

We were not able to simulate the magnetization dynamics in the samples containing Al since no software is available to us that can incorporate the effect of surface plasmons or surface plasmon polaritons in magneto-dynamics. Therefore, we are unable to compare theory with experiments in the Al-containing samples. Our experiments, however, showed Stark differences between the plasmonic (Al-present) and non-plasmonic (Al-absent) samples, which can only accrue from coupling with surface plasmons and/or surface plasmon polaritons.

In Figs. 2(a) and 2(b), we show the background-subtracted time-resolved data for reflectivity of the non-plasmonic (without aluminum) and plasmonic (with aluminum) samples as a function of delay between the pump and the probe, obtained at 16 mJ cm $− 2$ pump fluence. We will discuss the non-plasmonic results first.

The FFT of the reflectivity oscillations in the non-plasmonic sample [Fig. 2(c)] reveals the dominance of two peaks at 1.5 GHz (⁠ $f 1$⁠) and 2.7 GHz (⁠ $f 2$⁠) in the spectrum. Because the maximum delay between the pump and probe in our setup is slightly more than 3 ns, we can observe only $∼5$ periods of the oscillation in the reflectivity but that is enough to allow resolution of the two peaks in the spectrum. These modes (peaks) are absent in the FFT of time-resolved reflectivity oscillations from the bare Si substrate (without NM), which attests to the fact that the NM array plays a role in determining these peaks.

For the plasmonic sample, the FFT of the oscillations reveals the dominance of four (instead of two) peaks at 2 (⁠ $f 1 ′$⁠), 5.5 (⁠ $f 2 ′$⁠), 7.1 (⁠ $f 3 ′$⁠), and 8.6 GHz (⁠ $f 4 ′$⁠), where the intensities of the last three are much lower than that of the first but still well above our noise floor. Thus, the reflectivity signals of the plasmonic and non-plasmonic samples are both qualitatively and quantitatively different.

Before we proceed to delve into the origin of the reflectivity modes in the plasmonic sample, we point out that the observed frequencies $f n$ (⁠ $n=1,2$⁠) (non-plasmonic sample) and $f n ′$ (⁠ $n ′=1,2,3,4$⁠) (plasmonic sample) are insensitive to the presence or absence of any magnetic field at any given fluence and show exceptional stability over the broad range of magnetic field, confirming the non-magnetic origin of these oscillations.

There is a compelling reason to believe that the four modes $( f n ′ )$ in the plasmonic sample arise from mixing of the BWs caused by laser heating/cooling (phonons) with plasma waves (surface plasmons or surface plasmon polaritons) caused by the excitation of plasma oscillations in the Al layer by the laser beam. This is the hybridization of phonon–plasmon modes. Non-linear mixing will cause additional frequencies to appear, which is why we observe two additional modes (four instead of two) when the Al layer is present. There is also a blueshift, i.e., the mode frequencies are higher in the Al-containing sample. That is expected if energy is coupled from the plasma waves into the BWs to increase their frequencies. The phonons in the BW are absorbing plasmons to increase their energy and, hence, the frequency.

It may be argued that the plasmonic nanodots consist of Ta/Al/Co/Au layers, while the non-plasmonic nanodots consist of Ta/Co/Au layers so that there is a mass difference between the two, which could make a difference between the modes observed in the two types of samples. Further consideration rules out this possibility. These modes accrue from the unequal thermal expansion/contraction coefficients of the substrate and the layer that is in contact with the substrate, which is Ta in both cases. Hence, the compositional or mass difference between the two types of nanodots should not have a significant impact. Furthermore, the plasmonic sample, being heavier, should experience a redshift in the frequency of the modes, but we see a blueshift. Finally, the mass difference is not likely to spawn additional modes. We think it is unlikely that the compositional or mass difference plays a significant role.

We also notice that the peak linewidths are larger in the plasmonic sample spectra, indicating that the hybrid phonon–plasmon modes are more lossy than pure phonon (BW) modes. Plasmons couple very strongly with electromagnetic waves,³¹ and, hence, the hybrid phonon–plasmon wave may radiate some photons (electromagnetic waves) causing more rapid loss of energy. This is a speculation, and we intend to verify this in future by measuring any possible electromagnetic wave emission from these samples. To ensure that the features we have reported are not mere artifacts of structural, geometrical, or other trivial differences between the Al-containing and Al-lacking samples, we have moved the laser spots of the pump and probe beams in the TR-MOKE setup to different regions of the wafer. The laser spot size is $∼1$  $μ$m, the nanodot diameter is $∼100$ nm, and the edge-to-edge separation is $∼50$ nm. Hence, the spot covers an array of $6×6$ nanomagnets, which constitutes a measured “sample.” By simply moving the spot elsewhere on the wafer, we measure a different “sample.” We have done that and observe similar differences between the Al-containing and Al-lacking samples, no matter where we focus the laser spot, which confirms that what we report is not an artifact. This study is reported in the accompanying supplementary material.

In contrast, modes “1” and “3” cannot be fitted with either the Kittel formula or the perpendicular standing SW (PSSW) mode relation. Mode “1” is likely an edge mode, while mode ”3” is identified as a hybrid magneto-dynamical mode, consistent with previous reports in these systems.³⁰ The corresponding SW mode analysis can be found in the supplementary material.

In stark contrast to the non-plasmonic sample with no Al present, the sample with Al (where plasmonic coupling is present) exhibits much richer spectra in a magnetic field as shown in Fig. 4(a). The corresponding FFT, showing five modes in the Kerr oscillations (M1, M2, M3, M4, *), is shown in Fig. 4(b). Note from Fig. 4(a) that the oscillations were measured up to a time delay (between the pump and probe beam) of 1.8 ns, which allows for a frequency resolution of 0.55 GHz in Fig. 4(b). This is at least four times smaller than the smallest separation between adjacent modes, which is why we can confidently claim that five modes are present in these data and that the mode frequencies have an error bar of no more than 0.55  GHz. Despite the modest signal-to-noise ratio available in these experiments where the probe beam can probe $(6×6)$ NMs, we still have enough measurement accuracy to justify the claims made in this paper.

One easily discernible high-frequency magnetic field-independent mode at $∼30.5$ GHz is always present, in addition to other field-dependent modes. The former has more power than all other modes. This field-independent mode is only present when the Al layer is there, confirming its plasmonic origin. Figure 4(b) illustrates this stable mode across four different bias magnetic fields (it is the mode marked with an asterisk). Further insight into this mode will require framing a rigorous theory of plasmon–phonon coupling in this complex system, which is outside the scope of this work.

Figure 5(a) reveals the presence of field-dependent modes in the plasmonic Al-containing sample in addition to the high-frequency field-independent mode. These are magnonic modes. The dispersions of modes M4 and M3 are fitted by the blue and red lines with arrowheads. These lines are merely guide to the eye. A distinct frequency gap of magnitude 2 $g ′$ opens up at a magnetic field of 820 Oe. This gap attests to the coupling between M4 and M3 (both magnonic modes), which is obviously mediated by the hybrid phonon–plasmon mode. This is magnon–magnon coupling mediated by the hybrid mode.

The gap 2 $g ′$ and the loss rates (half width at half maximum) of modes M4 and M3, denoted as $k 1$ and $k 2$⁠, are depicted in the inset in Fig. 5(a). To quantify these parameters, we applied a Lorentzian fit to the peaks corresponding to M4 and M3. The extracted values are, $g ′=0.78$ GHz, $k 1=0.36$ GHz, and $k 2=0.35$ GHz. The cooperativity factor for coupling is defined as $C= g ′ 2/ ( k 1 k 2 )$⁠, which, in our case, has a value of 4.8. In accordance with established criteria, strong coupling is delineated by $C>1$ and $g ′> k 1, k 2$⁠. Because we satisfy both conditions, we have strong coupling³³ between modes M4 and M3. The coupling agent is obviously the hybrid phonon–plasmon mode, which we observed in the reflectivity spectrum of Fig. 2(d) and it is coupling two magnon (SW) modes M4 and M3. This is tripartite coupling involving phonons, plasmons and magnons, and the coupling is strong.

We understand that M3 and M4 curves are fitted to dispersed data and that the dispersion in the data is considerable. Nonetheless, there is a clear gap opening up in Fig. 5(a) and even if we are inaccurate in estimating its magnitude, we do not overestimate it by as much as a factor of 4. Hence, irrespective of our inaccuracy, the value of the cooperativity factor surely exceeds unity, which attests to “strong” coupling.

In a regime of strong coupling, individual modes transform, giving rise to a novel quasi-particle, akin to the magnon–polaron observed in SW coupling with pure acoustic waves.^20,21 Our system exhibits robust tripartite coupling involving phonons, plasmons, and magnons, suggesting the emergence of a distinct phonon–plasmon–magnon polariton—a new quasi-particle. This phenomenon, reminiscent of strongly coupled magnon–plasmon polaritons reported in graphene-2D ferromagnet heterostructures,³⁴ underscores the significance of plasmons in our system.

Notably, the absence of coupling in the sample without aluminum (no plasmons) is evidenced by the lack of anti-crossing (gap opening) in the dispersion curves for field-dependent modes. This unequivocally demonstrates that plasmons introduce qualitative distinctions, as also seen in the reflectivity spectra, not merely quantitative ones, and coupling between SW modes occurs in their presence because the hybrid phonon–plasmon wave mediates the coupling.

Our experiments have unveiled two other intriguing findings: First, the magneto-optical Kerr effect (MOKE) signal, specifically the average peak-to-peak Kerr rotation amplitude across all modes at an arbitrarily picked bias magnetic field of 1 kOe, exhibits a notable 30% increase in the plasmonic sample with aluminum (Al) compared to the non-plasmonic sample without Al. This enhancement persists across a broad range of magnetic fields. Second, the amplitude of power in the FFT spectra corresponding to the anti-crossed acousto-plasmo-magnon SW modes in the Al-containing sample is approximately four times greater than that in the field-dependent modes of the Al-lacking sample when averaged across all modes. Finally, the intensity of the field-independent mode is ten times higher than that of the field-dependent modes, as illustrated in Fig. 5(c).

The significant increase in the MOKE signal, specifically a 30% rise in the average peak-to-peak Kerr rotation amplitude across all modes in the sample with aluminum (Al) compared to the Al-lacking sample, suggests the presence of parametric amplification.³⁵ In this context, energy is transferred from the hybridized phonon–plasmon mode to the intrinsic SW modes in the Al-containing sample, thereby amplifying the amplitude and power of the resultant acousto-plasmo-SW mode. This observation serves as confirmation of the robust tripartite plasmon–phonon–magnon coupling, since such power transfer can only happen if there is strong coupling. Parametric amplification requires the presence of a non-linearity. Note from Fig. 5(c) that the presence of the Al has not only increased the power amplitudes of the modes but also shifted the frequencies. Only a non-linearity can change the frequencies and, hence, we must conclude that a non-linearity is present. Non-linear SW coupling in magnonic crystals (which is what we have) is well known.³⁶ Hence, we expect the nonlinearity to be present. Chasing down the exact nature of the non-linearity, however, is outside the scope of this work. One other point needs some discussion. In a lossless system, ideal parametric amplification would require that the idler frequency $ω 1$⁠, the signal frequency $ω 2$⁠, and the pump frequency $ω 3$ obey the relation $ω 3= ω 1+ ω 2$⁠, as mandated by energy conservation. We certainly do not have a lossless system, and, hence, we do not expect this relation to hold. We merely call the observed effect “parametric amplification” because it mimics parametric amplification.

In conclusion, we report (i) tripartite coupling between phonons, plasmons, and magnons in periodic arrays of bilayered nanodots of a magnetostrictive material and a plasmonic material, spawning a high-frequency magnetic-field-independent acousto-plasmo SW mode; (ii) parametric amplification of SW modes by hybrid phonon–plasmon modes, and (iii) strong coupling between two SW modes mediated by the hybrid phonon–plasmon wave, leading to the formation of the magnon–plasmon–phonon polariton. These findings not only contribute to our understanding of fundamental interactions in magnonic and plasmonic systems but also hold significant implications for practical applications. The observed parametric amplification can be leveraged for SW device applications. Furthermore, these results open avenues for controlling spin dynamics through the interplay of plasmon–phonon–photon dynamics. This approach has the potential to mitigate intrinsic limitations associated with ohmic and other losses, offering exciting prospects for plasmon-driven coherent spintronics or magneto-optical activities.

See the supplementary material for the (I) position dependence of the reflectivity spectra for plasmonic and non-plasmonic samples and (II) spin-wave mode analysis for co nanomagnets (non-plasmonic samples).

A.B. gratefully acknowledges the Department of Science and Technology, Govt. of India (Grant No. DST/NM/TUE/QM-3/2019-1C-SNB) for financial assistance. A.B. and S.B. acknowledge the support from the Indo-US Science and Technology Fund Center grant “Center for Nanomagnetics for Energy-Efficient Computing, Communications, and Data Storage” (No. IUSSTF/JC-030/2018). The work of R.F. and S.B. was supported by the US National Science Foundation under Grant No. ECCS-2235789. S.P. and P.K.P. acknowledge the Council of Scientific and Industrial Research (CSIR), Govt. of India, for the respective senior research fellowships.

The authors have no conflicts to disclose.

S. P. and P. K. P. performed the TR-MOKE measurements. S.P. analyzed the data and performed the micromagnetic simulations. R. F. fabricated the samples. S. B. and A. B. conceived the idea and supervised the project. All authors contributed to writing the paper.

S. Pal: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Writing – original draft (lead). P. K. Pal: Data curation (equal); Formal analysis (supporting); Investigation (equal); Methodology (equal). R. Fabiha: Methodology (equal). S. Bandyopadhyay: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). A. Barman: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of the study are available within the article.