Surface acoustic waves interaction with confined acoustic phonons in a coupled nanoridges dimer set atop of a multilayer design

Abstract

We present a comprehensive study using finite element numerical analysis of the acoustic localized phonons supported by a gold nanoridges dimer-based multilayer design. The latter consists in a SiO₂-substrate over which a gold film covered with a thin polymer is deposited. We investigate first the mechanical eigen-modes analysis of a single monomer ridge, where we find flexural and compressional type modes in the sub-GHz frequency range. This is realized by either setting the ridge in a periodic structure, which enables to get the dispersion curve of the modes, or by considering an isolated system bounded by perfect matched layers, where we use the equivalent of the local density of states to track the modes. A good agreement is obtained between the two methods. Similarly, we find in case of the coupled dimer ridges hybridized modes, namely the flexural and compressional modes of a monomer split-up into in- and -out-of-phase type modes. We demonstrate efficient coupling between the monomer/dimer localized phonons with surface acoustic waves (SAWs) as the simulated transmission spectra show dips at the frequencies of the monomer/dimer eigenmodes. For symmetry reasons, some of the dimer modes are expected to be optomechanically active. The proposed SAW-based device is meant to help design acousto-optic modulators or ultrasensitive sensors.

Keywords:

• Dimer nanoridges
• localized acoustic phonons
• multilayer design
• surface acoustic waves

1. Introduction

Control and modulation of light properties in nanoscale photonic structures based on the design of acousto-optic nanodevices has concentrated a lot of attention lately (Laude et al., 2017; Gao et al., 2023). Such components can be useful in quantum information processing as well as to significantly enhance light-matter interaction at the nanoscale (Sansa et al., 2020; Barzanjeh et al., 2022). Physically speaking, the acousto-optical coupling (AOC) is described by two mechanisms namely, i) the photoelastic and ii) the moving boundary effects. In the latter, the boundaries deformations between two adjacent materials induce a local variation of the dielectric constant all along the interface (Royer & Dieulesaint, 2000) while in the former effect, the dielectric is modulated locally within the volume by the acoustic strain field (Balram, Davanço, Lim, Song, & Srinivasan, 2014). In order to reach a meaningful AOC magnitude, an appreciable overlap between the phonon field and the optical mode is essential, and this is often realized by colocalizing the latter in the same volume such as in an optomechanical cavity (Arregui et al., 2023). In particular, a very promising new kind of OM cavities has been proposed based on the so-called phoxonic crystals that exhibit band gaps for both photons and phonons, giving rise to highly localized light and sound waves, and subsequently to enhanced AOC rates (Aram & Khorasani, 2018).

Lately, nanoscale multilayered structured designs have attracted a great deal of interest due to their exquisite properties in terms of highly localized acoustic and optical modes they exhibit (Dobrzynski et al., 2017; Noual, Akiki, Pennec, El Boudouti, & Djafari-Rouhani, 2020). A special class of these structures is often referred to as film-coupled nanoparticles or particles on mirror due to the coupling of metallic nanoparticles to their image within a metal-film underneath (Li, Zhang, Maier, & Lei, 2018). One key feature of such designs is the strongly localized surface plasmons resonances modes they support, which have been harnessed in various applications (Saada et al, 2020; Oo, Silva et al, 2016; Cheng & Xu, 2021). In particular, localized surface plasmons have been shown useful in order to enhance inelastic Brillouin scattering (Noual, Kang, et al., 2021; Vasileiadis et al., 2022). Besides, the use of surface acoustic waves (SAWs) with high amplitudes ($\sim nm$), helps to increase vibration magnitudes of localized phonons in metallic nanoparticles, which enhances the AOC rate meaningfully (Lin, Lin, & Hsu, 2015; Noual, Akiki, Lévêque, Pennec, & Djafari-Rouhani, 2021).

To design a SAW-device for enhanced AOC, we thoroughly investigate in this work localized acoustic phonons in a gold dimer ridges-based component using SAW. Specifically, numerical simulations using Comsol-Multiphysics are performed, so to find localized mechanical eigen-modes supported by the nanostructure, with a particular focus on the phonons considered potentially useful for AOC. The proposed device consists in SiO₂ substrate over which a gold film covered with a thin polymer is deposited. Atop the structure surface, a gold nano-ridge dimer is placed, where each ridge contains a thin polymer layer lying in the middle. This feature can be used for the excitation of localized plasmons within the polymer layer that could be strongly modulated by localized phonons in the ridge. Prior studies have only considered localized plasmons mostly under the particles, meaning that the current design offers a supplemental degree of freedom. In order to understand the physical origin of the dimer-based system eigenmodes, we analyze the modes of a single monomer first. We show efficient excitation of compressional-like and flexural type eigen-modes of the dimer using SAWs in the sub-GHz range, where the coupled ridges hybridize their modes, yielding in- and -out-of-phase type modes. The symmetry of the modes indicate that compressional modes and anti-phase flexural modes should be optomechanically active (Noual, Akiki, et al., 2021). The reported findings in this paper could help design SAW-based acousto-optic modulators or ultrasensitive nanosensors, due to the easily accessible fabrication technology of SAW-devices nowadays (Mei, Zhang, & Friend, 2020; Kelly, Northfield, Rashid, Bao, & Berini, 2022).

2. Single monomer ridge eigenmodes

In this section, we analyse the mechanical eigenmodes of a single monomer ridge. We shall do this using two complementary approaches, namely, by studying first a periodic structure and then by performing a thorough eigen-analysis of the isolated ridge.

2.1. Periodic structure

We consider a periodic structure in which the single ridge atop the multilayer is repeated periodically along the $x-$ direction as depicted in Figure 1(a). More precisely, the structure design is made of a gold ridge of height h and width w, with a thin layer of polyamide of height h_pol placed at the middle. The ridge monomer is deposited over a thin polyamide layer of thickness e_pol which coats a gold film measuring e_Au. We chose a SiO₂ semi-infinite substrate to support the multilayer (Figure 1(a)). It is worth indicating that the method followed here, enables us to compute the dispersion curve of the system which gives a global insight regarding the modes supported by the structure, including the localized type modes and their properties in terms of dispersion and symmetry. For that matter, we perform an eigenfrequency study of the system as the wave vector $k$ spans the first Brillouin zone, $[0-\pi /a],$ where $a$ is the period. Numerically, the periodicity is set using Bloch boundary conditions, and the eigenvalue equation is solved using structural mechanics module of Comsol 5.2a. At the bottom of the unit-cell of the lattice we use perfectly matched layer (PML) which absorb incident waves without reflections, in order to mimic a semi-infinite (SiO₂) substrate. The elastic properties of the materials are defined based on young’s modulus E, Poisson ratio $\mathrm{\nu }$ and the density $\mathrm{\rho }$ as follows: $E=3.6\mathrm{GPa},$ Poison ratio $\mathrm{\nu }=0.34$ and the density $\mathrm{\rho }=1420\mathrm{kg}/{\mathrm{m}}^3$ for the polyamide, and $E=72.2\mathrm{GPa},\mathrm{ }\mathrm{\nu }=0.168\text{ and ρ}=2200\mathrm{kg}/{\mathrm{m}}^3$ for the glass substrate SiO₂ (Royer & Dieulesaint, 2000). As for the gold layer which is an anisotropic material with face centered cubic crystal, its elastic constants ${\mathrm{C}}_{11},\mathrm{ }{\mathrm{C}}_{12}\text{ and }{\mathrm{C}}_{44}$ were taken from the same reference. Figure 1(b) shows the simulated dispersion curve for $a =2 \mu m,\mathrm{ }w=320\mathrm{nm},\mathrm{ }h=300\mathrm{nm},\mathrm{ }{\mathrm{h}}_{\mathrm{pol}}=50\mathrm{n}\mathrm{m},\mathrm{ }{\mathrm{e}}_{\mathrm{pol}}=50\mathrm{n}\mathrm{m}$ and ${\mathrm{e}}_{\mathrm{Au}}=100\mathrm{n}\mathrm{m},$ while $H\approx 2.5a$ was taken large enough to mimic a semi-infinite substrate. As mentioned above, we focus our analysis on the modes at the edge of the first Brillouin zone, i.e. $k=\pi /a,$ situated under the SiO₂-sound line labelled as A1, A2, A3, and A4, which do not propagate within the substrate. The field maps of the displacement field norm associated with such modes near the monomer are given in Figure 1(c). According to the field maps, the four branches modes (A1, A2, A3, and A4) are localized within the ridge with different rates, meaning that these are rather eigenmodes of the latter. As a matter of fact, the first branch A1 is a first order flexural mode at ${\mathrm{f}}_{\mathrm{A}1}\sim 0.227\mathrm{GHz},$ the mode A2 at ${\mathrm{f}}_{\mathrm{A}2}\sim 0.463\mathrm{GHz}$ is a compressional mode, whilst modes (A3, A4) at ${\mathrm{f}}_{\mathrm{A}4}\sim 0.555\mathrm{GHz}$ and ${\mathrm{f}}_{\mathrm{A}5}\sim 0.684\mathrm{GHz},$ correspond to the second order flexural mode which splits into two branches. This, happens due to its coupling with the gold/polymer films vibrations possibly induced by the periodicity effect, namely, the interaction with neighboring cells.

Figure 1. (a) Schematic representation of the multilayer design unit-cell in case when a single monomer is set atop. Bloch periodic conditions are set along $x-$ axis. (b) Simulated dispersion curves of the geometry of (a), for the geometrical parameters: $a=2 \mu m, w=320 nm, h=300 nm, h_{\mathit{pol}}=50 nm, e_{\mathit{pol}}=50 nm, H\approx 2.5a$ and $e_{Au}=100 nm.$ (c) Displacement field norm maps near the monomer at the frequencies of the first Brillouin zone edge modes referred to as A1, A2, A3, and A4 in the dispersion plot, in (b).

Indeed, the ridge-films coupling is apparent in the field maps (Figure 1(c)) of modes A3 and A4. Note in particular the strong Sezawa type mode in the films interacting with the flexural mode A4 of the ridge, where the two modes vibrate in-phase. Conversely, for mode A3 the films and the ridge seem to vibrate out-of-phase, meaning that they tend to cancel-out. This might explain why the branch A3 is less dispersive comparing to A4, where the film and the ridge add-up their vibrations. In other words, there is more coupling with neighboring cells in this latter mode, hence the dispersion. The branch A1 is pretty much flat, due to a strong localization of such mode in the ridge as compared to the rest of the modes. In such mode, the ridge is clearly decoupled from its surrounding i.e. the films and substrate.

2.2. Isolated single ridge

Here, we study the eigen-vibrations of a single monomer ridge. In other words, the particle is completely isolated which enables to retrieve the modes of the ridge independently of any exterior perturbation by opposition to the periodic structure. This is performed by surrounding the ridge atop the multilayer with PMLs all along except the top surface which is set to vibrate freely; this enables to absorb incoming waves towards the boundaries (PMLs) with no reflections. Numerically speaking, we solve the same eigen-value problem where the Bloch periodic conditions are replaced by PMLs as boundary conditions. To track the modes, we compute the integral of the square modulus of the displacement field in the monomer normalized to its value in the whole system, namely: (1) $${\rho }_N=\mathrm{ }{\int }_{\mathit{Monomer}}^{}{\Vert \stackrel{\rightharpoonup }{U}\Vert }^{2}dV/{\int }_{\mathit{system}}^{}{\Vert \Vert \stackrel{\rightharpoonup }{U}\Vert \Vert }^{2}dV$$(1) where, $\overrightarrow{U}=\left(u, v, w\right)$ represents the displacement field with components $u,$ $v,$ $w$ along the spatial axes $x,$ $y,$ and $z,$ respectively (which is adopted throughout the paper). Noting that, ${\Vert \Vert \stackrel{\rightharpoonup }{\mathit{U}}\Vert \Vert }^2$ is the equivalent of the local density of states (LDOS) (Economou, 2006), the computed quantity ${\rho }_N$ is proportional to the normalized integral of LDOS within the ridge, which yields the localized eigenmodes of the latter. In Figure 2(c), we give the simulations results of ${\rho }_N$ versus the eigen-frequency. The plot shows the existence of three peaks at the frequencies ${\mathrm{f}}_{\mathrm{M}1}\sim 0.226\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{M}2}\sim 0.517\mathrm{GHz},\text{ and }{\mathrm{f}}_{\mathrm{M}3}\sim 0.602\mathrm{GHz}.$ Such peaks match respectively the modes (A1, A2, A3/A4) studied previously (Figure 1(b)). In particular, it is interesting to note that modes A3 and A4 now appear as one mode (at ${\mathrm{f}}_{\mathrm{M}3}\sim 0.602\mathrm{GHz}$) namely the second order flexural mode of the ridge, this is due to the non-coupling with neighboring cells as the ridge is now isolated.

Figure 2. (a) Sketch of the geometry model used to study the transmission of SAWs through one ridge. PMLs are set all around the structure to get Rid of undesired reflections. (b) Displacement field component $\mathit{v},$ normalized to the maximum displacement field norm in the system, in the absence of the monomer, at $\mathit{f}\mathit{ }=\mathit{ }0.444\mathit{ }\mathit{GHz}.$ (c) Integral of the square modulus of the displacement field in the monomer normalized to its integral in the whole system. (d) Transmission coefficient through a monomer normalized to that of a flat surface without the latter. (e) Zoom in of the displacement field around the monomer at the frequencies, ${\mathit{f}}_{\mathit{M}1}=\mathit{ }0.231\mathit{ }\mathit{GHz},\mathit{ }{\mathit{f}}_{\mathit{M}2}=0.518\mathit{ }\mathit{GHz}$ and ${\mathit{f}}_{\mathit{M}3}=0.604\mathit{ }\mathit{GHz}$ of the modes referred to as M1, M2, and M3 in (d).

One observes that the frequencies of the modes are slightly different comparing with the periodic structure except for the first flexural mode (${\mathbf{f}}_{\mathbf{M}1}\sim 0.226 \mathbf{GHz}$). As mentioned earlier, there is some coupling in modes A2 and A3/A4 with neighboring cells while for mode A1 the coupling is weak. As a result, for the isolated ridge, the frequency of the latter is almost the same whereas for the other modes there is a slight shift. Besides, the relative height of the peaks shows that the first-order flexural mode (P1) is the most localized mode (i.e. the elastic energy if localized solely in the ridge) followed by the higher-order flexural mode and then the compressional mode where part of the elastic energy is diffused in the surface layers and the substrate under the ridge.

Now, we use the SAW-platform in order to excite the ridge mechanical eigen-modes. The structure design is depicted in Figure 2(a). The geometrical parameters are the same as in Figure 1(a). It should be pointed-out that we perform a frequency response analysis of the structure, where the excitation of SAW is realized using a harmonic vertical line force $\overrightarrow{\mathit{F}}=\mathit{F}{\mathit{e}}^{\mathit{i}2\mathit{\pi }\mathit{ft}}\overrightarrow{\mathit{z}}$ at the system inlet on top (see Figure 2(a)); where $\mathit{f}$ is the frequency and $\mathit{t}$ is the time. Figure 2(b) shows an example of SAW excitation at around $0.444\mathit{ }\mathit{GHz},$ in which we have depicted $\mathbf{w}/\mathbf{max}\left(\sqrt{{\left|\mathbf{u}\right|}^2+{\left|\mathbf{w}\right|}^2}\right)$ map, where $\mathbf{max}\left(\sqrt{{\left|\mathbf{u}\right|}^2+{\left|\mathbf{w}\right|}^2}\right)$ represents the maximum value of the displacement norm in the whole system. We see the effective excitation of the SAWs and their complete absorption inside the PMLs regions preventing any undesired reflections. Let us indicate that the normalized transmission coefficient of the structure is computed by taking a line average of the registered displacement field magnitude $〈\sqrt{{\left|\mathit{u}\right|}^2+{\left|\mathit{w}\right|}^2}〉$ near the output. The normalization is performed with respect to the system in absence of the ridges. Figure 2(d) shows the simulated transmission spectrum computed in the frequency range $\left[0.2 \mathbf{GHz}-0.7 \mathbf{GHz}\right].$ The curve shows the excitation of three modes at the frequencies ${\mathbf{f}}_1 \approx 0.231 \mathbf{GHz}$ (M1), ${\mathbf{f}}_2 \approx 0.518 \mathbf{GHz}$ (M2) and ${\mathbf{f}}_3\approx 0.604 \mathbf{GHz}$ (M3). The frequencies match very well the position of the peaks shown in Figure 2(c). These modes correspond to the first-order flexural mode (M1), the compressional mode (M2), and the higher-order flexural mode (M3). These results show clearly, the efficient excitation of the localized phonons of the single ridge using SAWs.

3. Dimer ridges eigenmodes

3.1. Periodic structure

We analyse in this paragraph the eigenmodes of a coupled dimer made of two identical ridges having the same dimensions as in Figure 1(a), and separated with the distance ${\mathrm{d}}_{\mathrm{s}}=50 nm.$ For that, we follow the same approaches as in the case of the monomer (Figures 1 and 2) namely, in a first step, the ridges are placed atop the system in a unit-cell repeated periodically along $-x$ axis, as shown in Figure 3(a). Figure 3(b) shows the simulated dispersion curves (of the system in Figure 3(a)). We notice the existence of six branches, labelled as B1 up to B6, underneath the SiO₂-line with the corresponding frequencies at the edge of the first Brillouin zone ($k=\pi /a$), respectively, $f_{B1}\simeq 0.221 \mathit{GHz}, f_{B2}\simeq 0.236 \mathit{GHz}, f_{B3}\simeq 0.427 \mathit{GHz}, f_{B4}\simeq 0.522 \mathit{GHz},$ $f_{B5}\simeq 0.556 \mathit{GHz}$ and $f_{B6}\simeq 0.618 \mathit{GHz}.$ One observes a split-up of the modes (Figure 3(b)) associated with a single monomer, yielding the modes (B1, B2), (B3, B4) and (B5, B6); these are the in- and out-of-phase modes of the first-order flexural mode (A1), the compressional mode (A2) and the second-order flexural mode (A3), respectively. This splitting occurs due to the strong coupling within the dimer, which results in the hybridization of the individual modes associated with each monomer. This can be seen in the displacement field norm maps near the dimer at the frequencies of the latter modes (B1-B6), as depicted in Figure 3(c). By opposition with the monomer, the periodicity effect (coupling with neighbouring cells) at the second order flexural modes (B5, B6) of the dimer is different as no splitting is observed in the frequency range of interest. Also, the out-of-phase compressional mode B4 is influenced by periodicity as it is clearly entrained in a flexion-like movement induced by the films vibrations, in a Sezawa type mode.

Figure 3. (a) Depiction of the geometry model unit-cell used for the study of the dimer eigenmodes. Bloch periodic conditions are set for periodicity. (b) Computed dispersion curve for the system in (a), for the following parameters values: $a=2 \mu m, w=320\mathrm{nm},\mathrm{ }h=300\mathrm{nm},\mathrm{ }{\mathrm{h}}_{\mathrm{pol}}=50\mathrm{nm},\mathrm{ }{\mathrm{e}}_{\mathrm{pol}}=50\mathrm{nm},\mathrm{ }{\mathrm{e}}_{\mathrm{Au}}=100\mathrm{nm}$ and ${\mathrm{d}}_{\mathrm{s}}=50\mathrm{nm}.$ (c) Displacement field norm maps near the dimer at the frequencies of the first Brillouin zone edge modes referred to as B1 up to B6 in (b).

3.2. Isolated dimer ridges eigenmodes

Similar to the previous case of a monomer, the eigen modes of the dimer are studied using the geometry model sketched in Figure 3(a), except that the Bloch periodic conditions are replaced by PMLs, and the top surface is set free. The determination of the eigenmodes is done as earlier, that is, by computing the normalized integral of the equivalent of LDOS namely: (2) $${\rho }_N=\mathrm{ }{\int }_{\mathit{Dimer}}^{}{\Vert \Vert \stackrel{\rightharpoonup }{U}\Vert \Vert }^{2}dV/{\int }_{\mathit{system}}^{}{\Vert \Vert \stackrel{\rightharpoonup }{U}\Vert \Vert }^{2}dV$$(2)

Note that the integration is obviously naturally performed within the coupled dimer, that is, the two ridges at the same time. In Figure 4(a), we present the simulated results, where we notice the peaks labelled as P1 up to P6 at the frequencies, ${\mathrm{f}}_{\mathrm{P}1}\simeq 0.218\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{P}2}\simeq 0.236\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{P}3}\simeq 0.454\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{P}4}\simeq 0.549\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{P}5}\simeq 0.577\mathrm{GHz},\text{ and }{\mathrm{f}}_{\mathrm{P}6}\simeq 0.622\mathrm{GHz},$ respectively. In the inset of the figure (near the top), we plot the spectrum of the average value of the square modulus of the displacement field in the dimer $〈{\Vert \Vert \overrightarrow{U}\Vert \Vert }^2〉$ in the frequency range, $0.4\mathrm{GHz}-0.7\mathrm{GHz}.$ This slightly different way of finding localized modes, enabled us to highlight the in-phase compressional mode P3, which is otherwise almost missing in the plot of ${\rho }_N.$ The reason for this is that such a mode strongly radiates and couples with the surrounding films and substrate, making it a less localized mode to the point that its peak height is nearly not visible in Figure 4(a). Overall, one notes a good match of modes P1-P6 with the modes obtained with the periodic lattice, B1-B6. There is slight shift in frequency depending on the mode which can be attributed as before to the periodicity effect or to the coupling with neighboring cells.

Figure 4. (a) Plot of the integral of the square modulus of the displacement field in the dimer normalized to its integral in the whole system. The inset shows the average of the square modulus of the displacement field in the dimer normalized to the maximum value in the whole system. The dimer-based structure configuration used for computations is similar to the one shown in Figure 3(a), except that PMLs are replaced by Bloch boundary conditions. (b) The dimer-based SAWs structure design. This geometry is similar to the one in Figure 2(a), except that a dimer is set atop. (c) Simulated normalized transmission coefficient of the design in (b). the transmission of the monomer in the blue curve is overlaid with the dimer transmission plotted in red. (d) Zoom in of the displacement field around the dimer at the frequencies associated with the position of the dips in the transmission labelled D1 up to D6, in (c).

At this stage, we would like to show the possibility of exciting the localized acoustic phonons of the dimer using SAWs. The SAWs-based platform, where a dimer is atop the multilayer is shown in Figure 4(b). The frequency response analysis of the system is performed as previously using vertical line force on top near the inlet (see Figure 4(b)), to excite SAWs. The normalized transmission spectrum of the dimer-based SAWs design is computed is Figure 3(c) using the same procedure as mentioned earlier. For a matter of comparison, we overlay in Figure 4(c) the transmission of the monomer (blue curve) and that of the dimer (red curve).

The dips observed in case of the dimer (red curve) are labelled D1 up to D6; they fall at the frequencies, ${\mathrm{f}}_{\mathrm{D}1}\simeq 0.22\mathrm{GHz},\mathrm{ }{\text{ f}}_{\mathrm{D}2}\simeq 0.238\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{D}3}\simeq 0.456\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{D}4}\simeq 0.542\mathrm{GHz},\mathrm{ }{\mathrm{f}}_{\mathrm{D}5}\simeq \mathrm{ }0.576\mathrm{GHz}$ and ${\mathrm{f}}_{\mathrm{D}6}\simeq 0.619\mathrm{GHz}$ respectively. According to the displacement field norm maps at these frequencies, the dips correspond, respectively, to the in-and out-of-phase first flexural mode, compressional mode and second order flexural mode. The modes’ frequencies match well the frequencies positions of the peaks in Figure 4(a). The comparison with the transmission of the monomer shows that each dip associated with an eigenmode of the monomer is split-up into two dips. Indeed, the elastic coupling in the dimer induces a split of the modes M1, M2 and M3 of the monomer, respectively, into the modes, (D1, D2), (D3, D4), and (D5, D6), of the dimer. These results show clearly the efficient excitation of the dimer mechanical eigen-modes using SAWs.

4. Phonon modes selection for acousto-plasmonic coupling purposes

In this last section, we want to provide insights regarding the dimer-structure localized acoustic phonons modes that may be beneficial for acousto-plasmonic coupling. It is important to note that dimers-based designs have been shown to have superior optical (Li, Zhang, & Lei, 2016), elastic and optomechanical properties (Matheny, 2018) comparing with monomers, making them the best choice for enhanced phonon-plasmon interaction. As a matter of fact, we have demonstrated in this work that dimers display a richer spectrum of localized phonons in contrast to monomers, due to modes splitting of the latters. Besides, it has been found that the out-of-phase flexural modes of the dimer turned out to be optomechanically active while they become inactive in case of monomers, and the in-phase flexural and out-of-phase compressional modes are proven inactive (Noual, Akiki, et al., 2021). In the light of this, we shall analyse the acousto-plasmonic properties of the following localized phonons of the proposed dimer-system namely: the out-of-phase flexural modes (B2 and B6) and the in-phase compressional mode (mode B3).

In that regard, we would like to mention that the polymer material underneath the dimer has been shown to act as an effective optical cavity (see the encircled region in Figure 5(a)) where localized surface plasmons could be excited (Mrabti et al., 2016). Thus, we examine the deformation induced by the dimer phonons modes on this specific region. The modes that deform this area volume shall cause important local strain, which by virtue of the photoelastic and/or moving boundary effects should cause a shift in the plasmon frequency. In other words, such phonons can be considered as optomechanically active. In Figures 5(b)-5(d), we plot $u$ and $w$ components of the displacement field normalized to the max value of the latter in the system, and associated with modes, B2, B3 and B6, respectively. In both first order (Figure 5(b)) and second order out-of-phase flexural (Figure 5(d)) modes (B2 and B6), the cavity undergoes a symmetrical elongation with respect to the symmetry plane at the middle, for both $w$ and $v,$ resulting into an important deformation of its overall volume; hence such modes should be optomechanically active. In the in-phase compressional B3 (Figure 5(d)), the cavity is compressed according to $w,$ and symmetrically elongated based on the $u$-field, which indicates that this mode should also be active, in-line with earlier works (Noual et al., 2020). We have omitted to discuss the dimer phonons effect on probable localized surface plasmons within the polymer layer in the middle of each ridge. The reason is that except for the in-phase compressional mode which we believe should deform such a polymer layer where the plasmon would be localized, the other modes require further investigations, including the optical characterization of such plasmons. We intend to do this in a forthcoming work.

Figure 5. (a) Sketch of the dimer deposited on the polymer layer Covering the gold film, with a circular highlight of the polymer area underneath the dimer, shown to act as an effective plasmonic cavity. (b) Plot of $w$ and $u$ components of the displacement field, normalized with respect to the maximum value of the latter in the system, and associated with the first flexural mode of the dimer. Each field map is tagged with the corresponding $w$ or $v$ component. Arrows highlight the displacement field direction. (c), (d) Same as in (b), but for the in-phase compressional and second order flexural mode.

5. Conclusion

In conclusion, we have performed numerical simulations to investigate localized acoustic phonons supported by a gold dimer nanoridges-based multilayer design using finite element method. The structure design consists of a coupled gold ridges dimer set atop a thin polymer layer coating a thick gold film that is deposited over a semi-infinite SiO₂-substrate. We have performed eigen-frequency analysis of the system where the ridges are either isolated in a finite system bounded by perfect matched layers or set to form a periodic lattice which enables to get the dispersion curve. In both cases, we have analysed the dimer eigenmodes by comparison with a single monomer. In the latter case, we found flexural and compressional type modes in the sub-GHz region. The elastic coupling in the dimer results in the split of the monomer modes, yielding in- and out-of-phase flexural and compressional modes. We have demonstrated their efficient excitation using a SAW-based device as important dips associated with the dimer modes show in the transmission coefficient of the system. Performing a symmetry analysis of the modes based on their $w$ and $v$ components, we can deduce that out-of-phase flexural and in-phase compressional modes should be optomechanically active. The proposed dimer-based SAW-device can be used for designing acousto-optic modulators or ultrasensitive sensors.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.