Phononic Crystal Made of Multilayered Ridges on a Substrate for Rayleigh Waves Manipulation

We present a phononic crystal to achieve efficient manipulation of surface acoustic waves (SAW). The structure is made of finite phononic micro-ridges arranged periodically in a substrate surface. Each ridge is constructed by staking silicon and tungsten layers so that it behaves as one-dimensional phononic crystal which exhibits band gaps for elastic waves. The band gap allows the existence of resonance modes where the elastic energy is either confined within units in the free end of the ridge or the ones in contact with the substrate. We show that SAW interaction with localized modes in the free surface of the ridge gives rise to sharp attenuation in the SAW transmission while the modes confined within the ridge/ substrate interface cause broad band attenuations of SAW. Furthermore, we demonstrate that the coupling between the two kinds of modes within the band gap gives high SAW transmission amplitude in the form of Fano like peaks with high quality factor. The structure could provide an interesting solution for accurate SAW control for sensing application for instance. Introduction There has been numerous works and investigations upon phononic crystals (PCs) as they provide the solution for acoustic and elastic waves control and dispersion manipulation. PCs are constructed by periodic arrangement of solid materials or fluids having different elastic properties which allows the existence of phononic band gaps (BG) where wave propagation is prohibited [1,2]. This property paved the way to consider PC for many applications such as filtering and waveguiding [3-8], sensing [9,10], acoustic wave focusing and lensing [11-14], and topological phononics [15,16]. The rapid progress in phononics led to the appearance of acoustic metamaterials (AM) which demonstrate exceptional properties such as negative effective mass density and/or bulk modulus or compressibility [17-20]. Such behavior led to create new designs of artificial systems capable of remarkable acoustic wave manipulation such as sound insulation [21], sub-wavelength focusing [22,23], and acoustic cloaking [24].


Introduction
There has been numerous works and investigations upon phononic crystals (PCs) as they provide the solution for acoustic and elastic waves control and dispersion manipulation. PCs are constructed by periodic arrangement of solid materials or fluids having different elastic properties which allows the existence of phononic band gaps (BG) where wave propagation is prohibited [1,2]. This property paved the way to consider PC for many applications such as filtering and waveguiding [3][4][5][6][7][8], sensing [9,10], acoustic wave focusing and lensing [11][12][13][14], and topological phononics [15,16]. The rapid progress in phononics led to the appearance of acoustic metamaterials (AM) which demonstrate exceptional properties such as negative effective mass density and/or bulk modulus or compressibility [17][18][19][20]. Such behavior led to create new designs of artificial systems capable of remarkable acoustic wave manipulation such as sound insulation [21], sub-wavelength focusing [22,23], and acoustic cloaking [24].
Among the works on PC and AM, structures based on plates attracted more interest as they offer good platforms for elastic wave propagation and integration with low loss and weak mode conversion at wavelength larger than the plate thickness. The first PC systems were constructed by considering periodic distribution of holes or inclusions in the plate [25,26]. Then, the phononic community quickly turns their interest into elastic pillars deposited on thin plates as they showed promising dispersion properties for Lamb waves. Pennec et al. [7,27] and Wu et al [28,29] proposed a pillar based plate by which they show the arising of local resonance BG (LRBG) created by the resonances of the pillars which couple with the Lamb waves in the band structure. This behavior was experimentally demonstrated by Wu al [28] using a laser ultrasonic technique in an aluminum based pillar plate. In other studies, the acoustic properties of silicone rubber pillars deposited on a rigid plate were investigated [20,30,31]. The structure can produce tunable LRBG at the sonic regime [32], and showed also great performance as a thin sonic insulator with high sound transmission losses [33,34]. Meanwhile, the process of introducing a point or a line defect inside the PC lattice allows the creation of elastic modes in the BG with zero or slow group velocity.
Pennec et al. [7,27] performed elastic waveguiding using a line defect, either by creating a space between two rows of pillars, or changing the pillars height or their material in a row. In another study, the same group proposed a system with hollow pillars where whispering gallery modes can be used to achieve efficient elastic waveguiding [35]. The modes' frequency can be tuned by filling the hollow parts with liquid [36,37]. Wang et al. [38] have shown experimentally similar behavior in a phononic strip. In another study, Wu et al. [29] have demonstrated experimentally the waveguiding feature in a stubbed PC plate using a bent waveguide to change the wave propagation direction in the ultrasonic regime. Very recently, Jin et al. [39] studied the scattering of an incident Lamb wave with a single or a line of pillars and discussed the amplitudes and phases of the monopolar and dipolar waves emitted by these resonators.
The Lamb waves confinement and waveguiding were also demonstrated by the pillar based system in the sonic regime [40][41][42].
Motivated by surface acoustic wave (SAW) manipulation in the ultrasonic regime for sensing and filtering applications, some works dealt with the SAW dispersion by PC. The first systems were constructed by tailoring holes or finite inclusions on a substrate surface [5,8,[41][42][43][44][45][46]. Later on, the interest was turned to PC made of micro-pillars deposited in the surface of semi-infinite material for higher dispersion of the SAW. Khelif et al. [47] proposed a two dimensional array of cylindrical pillars by which they demonstrate theoretically the BG opening for SAW by local resonance of the pillars. In the band structure, they show that the BG creation results from the coupling between the resonance modes and the SAW which occurs below the sound cone. This feature is very interesting compared to PC based on inclusions since one can avoid the SAW interaction with bulk acoustic waves (BAW) (occurring above the sound line) which inevitably increases the SAW energy leakage into the bulk substrate. The mechanical vibrations of pillars were experimentally demonstrated by Achaoui et al. [48] using optical measurements and the arising of LRBG was evidenced through the SAW transmission decay at the resonance frequency.
Other works have also detected experimentally the resonance of the pillars in different pillar based PC [49][50]. Yudistira et al. [51] showed the opening of both Bragg BG and LRBG in a hypersonic PC made of pillars, and very recently, Ash et al. [52] were able to detect high SAW attenuation caused by LRBG using a PC system made of pillars in holes (ring holes).
Meanwhile, few works have investigated the possibility of SAW waveguiding. Lethawe et al. [53] proposed a pillar based structure where waveguiding is achieved through a chain of pillars. Very recently, Benchabane et al. [54] demonstrated experimentally the possibility to manipulate the elastic waves in the surface of a substrate using micro-pillars where the elastic energy can be strongly confined at the subwavelength scale. For SAW waveguiding achievement by taking advantage of the BG, the classical approach is to introduce a line defect inside the PC to create confined modes within the defect at the BG frequency range. However, this approach makes the defect modes suffer from mode's conversion and become leaky toward the bulk substrate. In the transmission spectrum, the defect modes have large bandwidth to even close the BG, which makes the system not suitable for sensing applications for example.
In this paper, we introduce a PC made of a repetition of phononic ridges built by alternating silicon and tungsten layers (figs.1 (a) and 2(a)). Each ridge is a one-dimensional (1D) PC displaying band gaps where confined modes can be created both in the upper surface of the ridge and its interface with the substrate. We then took advantages of the (1D) PC properties to perform efficient coupling between the SAW and the localized modes in the ridge in order to get high SAW transmission in a very narrow bandwidth. The objective is to get and control high transmitted narrow bands with low radiation into the bulk substrate leading to high QF features. The paper is organized as follows: section II presents the phononic ridge and its dispersion property for elastic waves, and discusses the SAW interaction with the PC made of finite phononic ridges. In section III, we study the SAW transmission through one and three ridge(s) and discuss the coupling mechanisms between the modes of interest.

II. SAW dispersion in the PC system with phononic ridges
We first consider an infinite phononic ridge or plate composed of alternating silicon (Si) and tungsten (W) layers (a periodic beam) in z direction having the same thickness h while both layers have  Regarding the SAW dispersion, we built a PC structure using a periodic repetition of finite phononic ridges on the surface of a thick silicon substrate ( fig.2(a)). Each ridge is made of 3 Si/W cells ended with a last Si layer on top. The repetition period of the ridges is a=7 µm (the cell period), and the band structure calculation was performed using a unit cell with periodic boundary conditions (PBC) ( fig.2(b)). The X' point indicates the Brillouin zone limit π/a in the propagation direction x for SAW. The blue line corresponds to the Si sound line so that the modes located in the cyan shaded region are BAW or leaky surface modes that can radiate into the bulk silicon substrate. The modes located below the sound line are either confined within the surface of the Si substrate or inside the ridge. Some of the modes appear in the orange shaded region which corresponds to the phononic ridge band gap BG1. A mode located in this region cannot propagate into the Si substrate or in the whole ridge. They can only be confined in specific regions of the ridge as we will see below. The other SAW branches outside the BG1 correspond to modes where the elastic energy is spread all along the ridges.
x y z We now investigate the phononic ridge behavior when changing its geometry. We consider the finite phononic ridge where we change the thickness of the top Si layer denoted h T in this study ( fig.3(a)).
In the calculations, we consider a single finite phononic ridge with 3.5 Si/W periods where the bottom Si surface is fixed and the upper surface is kept free. If we choose h T =5.5 µm for the top Si layer, we can depict three confined modes in the band gap BG1 ( fig.3). We plot in fig.3(b) the total displacement field amplitude of these modes to understand their mechanical vibration. The first one denoted T 1 located at 258 can be expected knowing that the cavity modes tend to adapt to the defect size so that their wavelength becomes smaller when decreasing h T . We then investigate the SAW dispersion with periodic repetition of finite phononic ridges over the Si substrate surface when changing their top Si layer thickness h T . Fig.4(a) shows the band structure calculation in the case of h T =5.5 µm. In the frequency range of the band gap BG1, we can depict three local resonance modes with zero group velocity. We also display in fig.4(b) the displacement field amplitude and the deformation profiles of these modes to identify their mechanical behavior. When analyzing the displacement fields, we can clearly identify the flexural resonance modes T 1 and T 3 and the compressional mode T 2 , which correspond to the same modes depicted in fig.3(b). Furthermore, the modes T 1 , T 2 and T 3 are located at 258, 337.5 and 373.4 MHz which are at the same frequencies as in the case of isolated ridges presented in fig.3. We can easily tune the frequency of these localized resonance modes inside the BG1 region by changing the upper Si layer thickness of the ridges.

III. SAW transmission through a phononic ridge
In this section, we study the interaction between the SAW and a single phononic ridge by transmission calculations as schematically described in fig.6(a). Later we shall discuss the effect of a series of parallel ridges. The objective is to understand the coupling mechanisms between the surface modes, the localized modes in the bottom of the ridge and the SAW inside the BG1 region. We consider the same finite phononic ridge as the one displayed in fig.4. Fig.6(a)  For h T =5.5 µm, we can depict the presence of three dips inside the BG1 located at frequencies 257.9, 337.5 and 373.6 MHz. At these frequencies, we plot in fig.7 the total displacement field in the phononic ridge as well as the u z component in the substrate. We can clearly see that the dips originate from the excitation of resonance modes where the mechanical vibration is localized in the top Si/W layers of the ridge. They correspond to the resonance modes T 1 , T 2 and T 3 observed in fig.4 at the same frequencies.
The dips are created by the excitation of the resonance modes which couples with Rayleigh waves causing a narrow band transmission loss and part of the elastic energy radiates into the bulk substrate. More precisely, when the SAW hits the ridge, an amount of the elastic energy is transmitted through the ridge inside the BG as we consider only 3 cells. The resonance modes will be then excited in the top of the ridge. One can consider the top Si/W layers as a cavity where the elastic energy is stored at the resonance frequency and scattered back along the ridge into the substrate. This mechanism results in SAW dips (loss) in the transmission spectrum in a narrow band corresponding to the resonance modes. Also, it seems that the coupling is stronger for the flexural surface modes T 1 and T 3 where higher SAW attenuation is depicted compared to the case of the compressional mode T 2 . Intuitively, the reason behind this behavior is that a ridge offers less mechanical resistance for bending than compression. This mechanical scheme makes the resonance modes T 1 and T 3 which have flexural mechanical vibration to be easily excited and coupled with SAW compared to the compressional modes T 2 . Furthermore, when looking closely to the mechanical vibration amplitudes, we can notice that at the modes resonance frequency, the maximum of the displacement amplitude in the ridge is at least one order of magnitude higher than the u z amplitude of SAW for the three dips. Additionally, an interesting feature displayed by this system is that we can easily tune the frequencies of the dips only by changing the thickness h T of the top Si layer. When lowering the value of h T from 5.5 to 4.7 µm ( fig.6(b) and (c)), the dips frequencies can be increased inside the band gap BG1. The three dips experience high mechanical QF which is evaluated at 2.8×10 4 , 1.85×10 4 and 7.2×10 3 for the three modes T 1 , T 2 and T 3 respectively in fig.6(c) for instance. Transmission h T =4.7µm In the following study, we calculate the transmission of SAW through one finite phononic ridge where we change the thickness of the bottom Si layer of the ridge in contact with the substrate. Fig.8(a) shows the results where h b is fixed to 8 µm and the other layers thicknesses are kept equal to 3.5 µm in the ridge. We first observe a large attenuation band (broad dip) in the region of the band gap BG1 where the SAW transmission is around 60% between 293 and 361 MHz. Actually, this decay in the transmission is caused by the SAW excitation of the flexural mode A and the compressional mode B observed in fig.5 where the mechanical vibration is confined within the bottom Si/W layers of the ridge in contact with the substrate. More precisely the broad dip in the BG1 results from the overlapping of the two wide band dips associated with the modes A et B as we shall see more clearly when considering the transmission through a series of three ridges. Furthermore, we also notice the presence of a sharp transmission peak/dip around 342.7 MHz (zoom of the peak/dip in the right panel) in the transmission spectrum. We plot in fig.9(a), (b) and (c) the total displacement field in the ridge and the out-of-plane displacement component u z in the substrate, respectively at 300 and 356 MHz indicated by red arrows in the attenuation band and at the peak frequency 342.7 MHz in fig.8(a). Form fig.9(a)  phononic ridge in contact with the substrate. Also, we can easily deduce from fig.9(c) that the peak/dip located at 342.7 MHz corresponds to the flexural resonance mode T 1 at the surface of the ridge which is at the same frequency as in the case of fig.2.
Meanwhile, we plot in fig.8 respectively. When analyzing the displacement field shapes of these two peaks displayed in fig.9(d) and (e), one can easily conclude that they correspond to the resonance modes T 1 and T 2 which are excited at their natural frequency as we can deduce from figure 3(c) for the chosen h T =4.5 µm. From fig.8, we can conclude that upon increasing of the top Si layer thickness h T , the frequencies of modes T 1 and T 2 decrease so that T 2 falls into the BG1 region when h T =4.5µm.
In summary, we can derive an interesting feature regarding this system. First, the SAW interaction 14  fig.10(a), shows the excitation of the resonant mode T 1 in the three ridges by the SAW at the peak frequency 300 MHz along with the transmitted SAW. We can also have the same behavior in the attenuation band B by coupling this mode with the resonance mode T 2 knowing that they both have the same mechanical motion nature, i.e. compressional vibration. In fact, by choosing Si top layer thickness h T =4.9 µm, the frequency of the compressional mode T 2 can be tuned to fall into the attenuation band B where the coupling between the modes results in the creation of the peak T 2 depicted in fig.10(b). At the frequency 356.5 MHz of the peak T 2 , a maximum transmission of 66% is reached for SAW. When analyzing the total displacement field amplitude in the ridges and the u z component in the substrate plotted in the lower panel of fig.10(b) we can clearly see the excitation of the resonant mode T 2 in the three ridges by the SAW. The QF of mode T 1 is about 1200 in the case of h T =4.25µm ( fig.10(a)) while the QF of mode T2 is 2.89×10 4 in the case of h T =4.9µm ( fig.10(b)). Moreover, for the two cases where either the resonance mode T 1 falls into the dip band A or mode T 2 falls into the band B, we have a situation very close to an acoustic analogue of the electromagnetically induced transparency (EIT) [58,59]. In fact, in plasmonic crystals for instance, the coupling between localized surface states (known as Tamm states) and the defect plasmonic state give an EIT-like behavior [60].

Conclusion
The SAW dispersion over a periodic repetition of periodic multilayered ridges undergoes interesting properties for efficient SAW manipulation. The phononic BG of the ridge allows the existence of resonance modes localized within the first phononic layers at the free end of the ridge. The SAW interaction with these modes results in a sharp decay of the SAW amplitude in the form of high QF dips.
Moreover, another family of confined modes can be created inside the band gap where the elastic energy is confined within the first layers of the ridge in contact with the substrate surface. The excitation of these modes by SAW causes a broad decay of the SAW amplitude in a relatively wide band. The high QF confined modes in the upper free edge of the ridge can couple with the localized modes in ridge/substrate interface to give rise to Fano like resonance with high QF SAW transmission peaks. This result is a very interesting achievement in the sense that the phononic ridge based PC can be proposed for efficient manipulation of SAW with low leakage into the substrate. The structure can introduce a new approach for using PC in SAW based devices to increase the performance of sensing for instance. Meanwhile, the high Transmission |u| Transmission |u| transmission peak created by coupling the localized surface modes and the modes confined within the ridge/substrate interface tends towards the arising of an acoustic analogue of the electromagnetically induced transparency (EIT). Moreover, a phononic ridge can be used to excite high QF surface phonons which can then couple with external photons. Very recently, Anguiano et al. [61] and Lamberti et al. [62] demonstrated the confinement of both phonons and photons with a strong optomechanic coupling in a composite multilayered GaAs/AlAs micopillar. The composite pillar based system would be a promising structure to observe quantum optomechanic phenomena, and even to be integrated into a biosensing platform for instance to increase the device sensitivity. For a practical application in biosensing, the top free surface of the ridge can be functionalized with antigens where specific antibodies can be covalently grafted. The molecular binding event can then be monitored through a real time observation of the mechanical frequency shift of the localized modes in the SAW spectrum. Meanwhile, the multilayered ridge system can also be used for macroscale application in the frame of earthquake damage control for instance. One can imagine an appropriate design of periodically structured ridges so that their transmission spectrum can be tuned to scatter the earthquake surface wave into the bulk [63]. Finally, one can also tailor a defect inside the phononic ridge or pillar to create other cavity modes and achieve the same previous properties to obtain high QF SAW transmission peaks.