Amorphous SiC/c-ZnO-Based Quasi-Lamb Mode Sensor for Liquid Environments

The propagation of the quasi-Lamb modes along a-SiC/ZnO thin composite plates was modeled and analysed with the aim to design a sensor able to detect the changes in parameters of a liquid environment, such as added mass and viscosity changes. The modes propagation was modeled by numerically solving the system of coupled electro-mechanical field equations in three media. The mode shape, the power flow, the phase velocity, and the electroacoustic coupling efficiency (K2) of the modes were calculated, specifically addressing the design of enhanced-coupling, microwave frequency sensors for applications in probing the solid/liquid interface. Three modes were identified that have predominant longitudinal polarization, high phase velocity, and quite good K2: the fundamental quasi symmetric mode (qS0) and two higher order quasi-longitudinal modes (qL1 and qL2) with a dominantly longitudinal displacement component in one plate side. The velocity and attenuation of these modes were calculated for different liquid viscosities and added mass, and the gravimetric and viscosity sensitivities of both the phase velocity and attenuation were theoretically calculated. The present study highlights the feasibility of the a-SiC/ZnO acoustic waveguides for the development of high-frequency, integrated-circuit compatible electroacoustic devices suitable for working in a liquid environment.


Introduction
Lamb waves are acoustic modes that propagate along plates with a finite thickness: two sets of modes propagate, the symmetric and anti-symmetric modes (S i and A i , with i representing the mode order), whose velocities depend on the plate thickness-to-wavelength ratio [1]. Assuming as x 1 the wave propagation direction and as x 3 the plate normal, the symmetric modes are characterized by a longitudinal (U 1 ) and transverse (U 3 ) displacement components that are symmetric and anti-symmetric about the mid-plane of the plate; the anti-symmetric modes exhibit transverse (U 3 ) and longitudinal (U 1 ) components that are symmetric and anti-symmetric about the mid-plane of the plate. An up-to-date literature review and evaluation of ultrasonic guided waves can be found in Reference [2]. Acoustic plate modes with a displacement component normal to the plate surface are not suitable for sensing applications in liquids; when the plate contacts a liquid environment, the modes are converted into compressive waves in the surrounding liquid. The fundamental anti-symmetric Lamb mode, A 0 , while being elliptically polarized, can travel along thin plates that are in contact with a liquid, but only for a limited plate thickness range corresponding to mode velocities lower than the compressional velocity of the surrounding liquid medium. In Reference [3], the A 0 mode propagation along amorphous SiC/cAlN composite plates is theoretically investigated, specifically addressing the design of enhanced coupling, microwave devices for liquid sensing applications. In-plane polarized modes, such as longitudinal and shear horizontal modes, are suitable for liquid sensing applications since, being the normal component of the particle displacement zero, they propagate while radiating very little energy into the liquid phase. In References [4][5][6], sensors based on the propagation of shear horizontal acoustic plate modes (SHAPMs) are described for liquid viscosity and for liquid/solid phase transition measurement applications. The fundamental symmetric Lamb mode S 0 , as well as the higher order quasi-longitudinal modes, are also suitable for liquid environment applications at specific plate thickness-to-wavelength ratios corresponding to U 3 << U 1 . Reference [7] describes the test of a liquid sensor array based on the propagation of the fundamental and higher order Lamb modes travelling along a single crystal bulk piezoelectric material (ST-x quartz plate) and showing predominant longitudinal polarization. In Reference [8], the propagation of the fundamental S 0 Lamb mode along thin piezoelectric membranes (BN, ZnO, InN, AlN and GaN) is theoretically studied, aiming at the design of electroacoustic devices suitable for work in a liquid environment. This study demonstrates the suitability of the thin suspended membranes for micro electromechanical systems (MEMS) applications where a high electroacoustic coupling efficiency and a high wave velocity are desirable. Moreover, unlike the single crystal bulk piezoelectric plates, this type of device can be part of an integrated system chip where the sensor coexists with the surrounding electronic circuitry. In Reference [9], some of the important characteristics of the acoustic modes propagation in both simple and composite (or multilayered) membranes are reviewed, and the terms used for the different types of modes in both the two membranes are summarized. In References [10,11], the Lamb wave propagation along bi-layered AlN/3C-SiC composite membranes is theoretically and experimentally investigated. The results clearly show that 3C-SiC-based devices can provide advantages of high frequency and high quality factor for Lamb wave devices due to the high acoustic velocity and low propagation loss. In Reference [12], some high-order quasi-Lamb wave modes in an AlN/3C-SiC plate were studied that have larger electromechanical couplings than the corresponding Lamb wave modes in an AlN thin plate. It was found that the quality factor of the third quasi-symmetric Lamb wave mode can be as high as 5510 at an operation frequency of 2.92 GHz. In Reference [13], the theoretical analysis of Rayleigh wave propagation in a-SiC/AlN and 3C-SiC/AlN demonstrates that the sensors based on a-SiC/AlN show performances that are comparable with those based on 3C-SiC/AlN, and are competitive with those based on Si(001)/AlN. An example of an a-SiC-based surface acoustic wave (SAW) device is described in Reference [14]: SAW devices implemented on a-SiC films (197 and 422 nm thick) grown on GaAs by plasma-enhanced chemical-vapor-deposition technique showed a Q in the range from 3500 to 5000, at 1 GHz operating frequency. Since the a-SiC thin film can be a useful compound for high-frequency acoustic devices, the propagation of Lamb modes in ZnO/a-SiC-based thin plates has been studied with the aim to design enhanced-coupling, microwave frequency sensors for applications in probing the solid/liquid interface. Non piezoelectric amorphous SiC substrate offers interesting properties such as a high acoustic wave velocity, resistance to chemicals, and high hardness. The a-SiC films deposition onto Si(100) substrates from a sintered SiC target by a radio frequency magnetron sputtering system has been demonstrated [13,15,16] and has the advantage of being compatible with the integrated circuit technology [17]. Piezoelectric wurtzite ZnO thin film technology has been widely used for many years for the fabrication of surface acoustic wave (SAW) devices onto non piezoelectric substrates, such as silicon, glass, and sapphire, to name just a few. When the piezoelectric ZnO film is grown onto high-velocity materials, such as diamond or SiC, it is a promising candidate for high frequency, high sensitivity micro sensors [18]. The aim of the present theoretical calculations is to investigate the influence of the thickness of both ZnO and a-SiC layers on the performances of a Lamb wave device for liquid sensing applications.

Phase Velocity Dispersion Curves
Lamb modes velocity dispersion curves are strongly dependent upon the material type, its crystallographic orientation, the wave propagation direction, and the plate composition (single or multilayered plates). As an example of wave propagation in homogeneous isotropic plates, Figure 1a shows the phase velocity dispersion curves of the fundamental and higher order Lamb modes propagating along the isotropic homogeneous a-SiC plate with a thickness of h = 5 µm in contact with air.

Phase Velocity Dispersion Curves
Lamb modes velocity dispersion curves are strongly dependent upon the material type, its crystallographic orientation, the wave propagation direction, and the plate composition (single or multilayered plates). As an example of wave propagation in homogeneous isotropic plates, Figure 1a shows the phase velocity dispersion curves of the fundamental and higher order Lamb modes propagating along the isotropic homogeneous a-SiC plate with a thickness of h = 5 µm in contact with air. These modes are represented by two sets of curves, the symmetric (Si) and anti-symmetric (Ai) modes, with i representing the mode order; the blue and red curves refer to the symmetric and antisymmetric modes. For very small plate thicknesses, the S0 mode velocity approaches the longitudinal bulk acoustic wave velocity in a-SiC. With increasing the plate thickness, the plate becomes a halfspace and the A0 and S0 mode velocities tend to approach the velocity of the Rayleigh wave in a-SiC, while the velocities of the non-fundamental modes asymptotically reach the a-SiC transverse bulk acoustic wave velocity.
Lamb-like waves propagate in a homogeneous anisotropic plate and can show three displacement components (longitudinal, shear horizontal, and shear vertical components), depending on the plate crystallographic cut and wave propagation direction. When considering a composite bi-layered plate, the symmetry of the particle displacement components with respect to the mid-plane of the plate is lost, unlike the homogeneous isotropic and anisotropic plate, and the shape of each mode changes with respect to the frequency. As mentioned in Reference [11], the fundamental modes can be considered as quasi-S0 and quasi-A0 (qS0 and qA0) for a limited plate thickness range, while all the other modes can be generically labelled as ith mode. Figure 1b shows the phase velocity dispersion curves of the fundamental and higher order modes propagating along a composite a-SiC/ZnO plate with a total thickness of H = 10 µm. This plate is no longer symmetric with respect to the middle plane, even if the two layers have equal thicknesses (5 µm). The two materials have different physical constants (mass density, elastic, piezoelectric and dielectric constants) and crystal symmetry, thus the mode displacement profiles are no longer simply anti-symmetric or symmetric with respect to the neutral axis. When a ZnO layer is attached to a thick a-SiC plate, the displacement profiles of the Lamb modes in the isotropic a-SiC plate are distorted due to the presence of the ZnO layer; the phase velocity dispersion curves of the composite plate, shown in Figure 1b, have equal color as the distinction between mode types is somewhat artificial. These modes are represented by two sets of curves, the symmetric (S i ) and anti-symmetric (A i ) modes, with i representing the mode order; the blue and red curves refer to the symmetric and anti-symmetric modes. For very small plate thicknesses, the S 0 mode velocity approaches the longitudinal bulk acoustic wave velocity in a-SiC. With increasing the plate thickness, the plate becomes a half-space and the A 0 and S 0 mode velocities tend to approach the velocity of the Rayleigh wave in a-SiC, while the velocities of the non-fundamental modes asymptotically reach the a-SiC transverse bulk acoustic wave velocity.
Lamb-like waves propagate in a homogeneous anisotropic plate and can show three displacement components (longitudinal, shear horizontal, and shear vertical components), depending on the plate crystallographic cut and wave propagation direction. When considering a composite bi-layered plate, the symmetry of the particle displacement components with respect to the mid-plane of the plate is lost, unlike the homogeneous isotropic and anisotropic plate, and the shape of each mode changes with respect to the frequency. As mentioned in Reference [11], the fundamental modes can be considered as quasi-S 0 and quasi-A 0 (qS 0 and qA 0 ) for a limited plate thickness range, while all the other modes can be generically labelled as ith mode. Figure 1b shows the phase velocity dispersion curves of the fundamental and higher order modes propagating along a composite a-SiC/ZnO plate with a total thickness of H = 10 µm. This plate is no longer symmetric with respect to the middle plane, even if the two layers have equal thicknesses (5 µm). The two materials have different physical constants (mass density, elastic, piezoelectric and dielectric constants) and crystal symmetry, thus the mode displacement profiles are no longer simply anti-symmetric or symmetric with respect to the neutral axis. When a ZnO layer is attached to a thick a-SiC plate, the displacement profiles of the Lamb modes in the isotropic a-SiC plate are distorted due to the presence of the ZnO layer; the phase velocity dispersion curves of the composite plate, shown in Figure 1b, have equal color as the distinction between mode types is somewhat artificial. The theoretical phase velocity dispersion curves were Sensors 2017, 17, 1209 4 of 12 calculated utilizing the ZnO and a-SiC material constants available in the literature [15,16,19] and using the COMSOL, DISPERSE [20], and MATLAB software.

Displacement Profile
The shape of the modes travelling in a bi-layered plate are no longer simply anti-symmetric or symmetric with respect to the neutral axis, but are distorted. The mode shape at different points along the same dispersion curve evolves continuously from symmetric to anti-symmetric and vice versa, thus it would be incorrect to label the dispersion curves as quasi symmetric or quasi anti-symmetric. We decided to assign each dispersion curve a number in the order in which they appear along the frequency axis. We investigated the fundamental quasi symmetric mode and two higher order modes that show, for specific thickness-to-wavelength ratio values, a predominant longitudinal polarization (U 3 << U 1 ) on the a-SiC layer-free side of the plate. These modes were studied with the aim to design a high frequency electroacoustic device suitable to work in a liquid environment. The latter modes are hereafter named quasi longitudinal modes, qL 1 and qL 2 .

The qS 0 Mode
The investigation of the acoustic field profile across the thickness of the a-SiC bare plate revealed that, up to an a-SiC thickness-to-wavelength ratio equal to h aSiC /λ = 0.1, the S 0 mode has polarization predominantly oriented along the propagation direction. Figure 2 shows the U 1 , U 3 and power flow depth profile for an a-SiC plate with a thickness of 5 µm (h aSiC /λ = 0.1, λ = 50 µm). The wave travels at velocity v = 10,522 m/s. U 1 and U 3 have been normalized to the U 1 value at the a-SiC plate surface; the power flow has been normalized to its value at the a-SiC plate surface. The theoretical phase velocity dispersion curves were calculated utilizing the ZnO and a-SiC material constants available in the literature [15,16,19] and using the COMSOL, DISPERSE [20], and MATLAB software.

Displacement Profile
The shape of the modes travelling in a bi-layered plate are no longer simply anti-symmetric or symmetric with respect to the neutral axis, but are distorted. The mode shape at different points along the same dispersion curve evolves continuously from symmetric to anti-symmetric and vice versa, thus it would be incorrect to label the dispersion curves as quasi symmetric or quasi anti-symmetric. We decided to assign each dispersion curve a number in the order in which they appear along the frequency axis. We investigated the fundamental quasi symmetric mode and two higher order modes that show, for specific thickness-to-wavelength ratio values, a predominant longitudinal polarization (U3 << U1) on the a-SiC layer-free side of the plate. These modes were studied with the aim to design a high frequency electroacoustic device suitable to work in a liquid environment. The latter modes are hereafter named quasi longitudinal modes, qL1 and qL2.

The qS0 Mode
The investigation of the acoustic field profile across the thickness of the a-SiC bare plate revealed that, up to an a-SiC thickness-to-wavelength ratio equal to haSiC/λ = 0.1, the S0 mode has polarization predominantly oriented along the propagation direction. Figure 2 shows the U1, U3 and power flow depth profile for an a-SiC plate with a thickness of 5 µm (haSiC/λ = 0.1, λ = 50 µm). The wave travels at velocity v = 10,522 m/s. U1 and U3 have been normalized to the U1 value at the a-SiC plate surface; the power flow has been normalized to its value at the a-SiC plate surface. When a thin ZnO layer is added to the a-SiC plate, the qS0 mode field profile results quite unperturbed for a very small ZnO thickness range. The power transported by the mode along the plate per unit length perpendicular to the propagation direction and per unit time, P = ρω v |U | + |U | dx , was calculated for waves propagating along the x1 direction of the plate, with U1 and U3 being the longitudinal and transverse particle displacement components normalized to the power flow evaluated at the x3 = h/2 surface; ω = 2π·f is the angular frequency, f = vph/λ, ρ is the plate mass density, n is the mode order, and vgr is the group velocity of the mode. The P distribution inside the plate gives information about the location of the peaks of acoustic energy transmission. Figure 3 shows the qS0 mode field profile and power flow distribution into the plate depth for a-SiC and ZnO normalized thicknesses equal to 0.1 and 0.05 (λ = 50 µm), respectively; the mode travels at velocity v = 8293 m/s. As can be seen, the longitudinal displacement component U1 is almost constant When a thin ZnO layer is added to the a-SiC plate, the qS 0 mode field profile results quite unperturbed for a very small ZnO thickness range. The power transported by the mode along the plate per unit length perpendicular to the propagation direction and per unit time, was calculated for waves propagating along the x 1 direction of the plate, with U 1 and U 3 being the longitudinal and transverse particle displacement components normalized to the power flow evaluated at the x 3 = h/2 surface; ω = 2π·f is the angular frequency, f = v ph /λ, ρ is the plate mass density, n is the mode order, and v gr is the group velocity of the mode. The P distribution inside the plate gives information about the location of the peaks of acoustic energy transmission. Figure 3 shows the qS 0 mode field profile and power flow distribution into the plate depth for a-SiC and ZnO normalized thicknesses equal to 0.1 and 0.05 (λ = 50 µm), respectively; the mode travels at velocity v = 8293 m/s. As can be seen, the longitudinal displacement component U 1 is almost constant across the composite

The qL1 and qL2 Modes
As an example, Figures 4 and 5 show the field profile and the in-plane power flow per unit area for the modes named quasi-longitudinal modes, qL1 and qL2, propagating at a velocity equal to 11,456 and 12,346 m/s, respectively, along the composite plate with a-SiC and ZnO fixed thicknesses (5 and 2.5 µm), for λ = 16.7 and 8.8 µm. The displacement components as well as the power flow have been normalized to their maximum values. As can be seen, these two modes have a very small U3 component at the a-SiC-free surface where U1, as well as the power flow, are at maximum values, thus confirming that these modes are suitable for the development of electroacoustic devices for applications to a liquid environment.
COMSOL FEM Multiphysics software was employed to simulate the resonance mode shape of the three modes, qS0, qL1, and qL2, propagating in the a-SiC/ZnO composite plate, with and without the liquid contacting the plate surface. 2D piezoelectric device simulation with solid mechanics and electrostatic modules was used for eigen-frequency analysis of the composite plate. The number of degrees of freedom to solve for the mode is minimized by providing periodic boundary conditions to the transmitting interdigital transducer which is a one-finger structure with a total width of one wavelength. Figure 6 shows the schematic of the COMSOL model with boundary conditions; as the liquid was modeled as a half space layer, a perfectly matched layer (PML) was added so that the wave propagating in the liquid was not reflected back to the plate. Figure 7 shows the U1 and U3 components of the three modes travelling along the composite plates in air; the a-SiC and ZnO thicknesses are fixed (5 and 2.5 µm), and the wavelength is λ = 50, 16.7 and 8.8 µm for the three modes.

The qL 1 and qL 2 Modes
As an example, Figures 4 and 5 show the field profile and the in-plane power flow per unit area for the modes named quasi-longitudinal modes, qL 1 and qL 2 , propagating at a velocity equal to 11,456 and 12,346 m/s, respectively, along the composite plate with a-SiC and ZnO fixed thicknesses (5 and 2.5 µm), for λ = 16.7 and 8.8 µm. The displacement components as well as the power flow have been normalized to their maximum values. As can be seen, these two modes have a very small U 3 component at the a-SiC-free surface where U 1 , as well as the power flow, are at maximum values, thus confirming that these modes are suitable for the development of electroacoustic devices for applications to a liquid environment.
COMSOL FEM Multiphysics software was employed to simulate the resonance mode shape of the three modes, qS 0 , qL 1 , and qL 2 , propagating in the a-SiC/ZnO composite plate, with and without the liquid contacting the plate surface. 2D piezoelectric device simulation with solid mechanics and electrostatic modules was used for eigen-frequency analysis of the composite plate. The number of degrees of freedom to solve for the mode is minimized by providing periodic boundary conditions to the transmitting interdigital transducer which is a one-finger structure with a total width of one wavelength. Figure 6 shows the schematic of the COMSOL model with boundary conditions; as the liquid was modeled as a half space layer, a perfectly matched layer (PML) was added so that the wave propagating in the liquid was not reflected back to the plate. Figure 7 shows the U 1 and U 3 components of the three modes travelling along the composite plates in air; the a-SiC and ZnO thicknesses are fixed (5 and 2.5 µm), and the wavelength is λ = 50, 16.7 and 8.8 µm for the three modes. across the composite plate depth, while U3 is negligible on the a-SiC side of the plate, and it increases slowly up to the ZnO side where it reaches a non-negligible value.

The qL1 and qL2 Modes
As an example, Figures 4 and 5 show the field profile and the in-plane power flow per unit area for the modes named quasi-longitudinal modes, qL1 and qL2, propagating at a velocity equal to 11,456 and 12,346 m/s, respectively, along the composite plate with a-SiC and ZnO fixed thicknesses (5 and 2.5 µm), for λ = 16.7 and 8.8 µm. The displacement components as well as the power flow have been normalized to their maximum values. As can be seen, these two modes have a very small U3 component at the a-SiC-free surface where U1, as well as the power flow, are at maximum values, thus confirming that these modes are suitable for the development of electroacoustic devices for applications to a liquid environment.
COMSOL FEM Multiphysics software was employed to simulate the resonance mode shape of the three modes, qS0, qL1, and qL2, propagating in the a-SiC/ZnO composite plate, with and without the liquid contacting the plate surface. 2D piezoelectric device simulation with solid mechanics and electrostatic modules was used for eigen-frequency analysis of the composite plate. The number of degrees of freedom to solve for the mode is minimized by providing periodic boundary conditions to the transmitting interdigital transducer which is a one-finger structure with a total width of one wavelength. Figure 6 shows the schematic of the COMSOL model with boundary conditions; as the liquid was modeled as a half space layer, a perfectly matched layer (PML) was added so that the wave propagating in the liquid was not reflected back to the plate. Figure 7 shows the U1 and U3 components of the three modes travelling along the composite plates in air; the a-SiC and ZnO thicknesses are fixed (5 and 2.5 µm), and the wavelength is λ = 50, 16.7 and 8.8 µm for the three modes.     Figure 8 shows the displacement components U1 and U3 of the three modes propagating along the a-SiC/ZnO plate that contacts the liquid environment from the a-SiC side of the plate. As can be seen, the acoustic energy is confined inside the plate. The liquid was modelled as a linear isotropic viscoelastic material with independent elastic constants, the bulk (K) and shear (G) moduli, and two independent bulk (ηb) and shear viscosities (ηv) from Reference [21].    Figure 8 shows the displacement components U1 and U3 of the three modes propagating along the a-SiC/ZnO plate that contacts the liquid environment from the a-SiC side of the plate. As can be seen, the acoustic energy is confined inside the plate. The liquid was modelled as a linear isotropic viscoelastic material with independent elastic constants, the bulk (K) and shear (G) moduli, and two independent bulk (ηb) and shear viscosities (ηv) from Reference [21].    Figure 8 shows the displacement components U1 and U3 of the three modes propagating along the a-SiC/ZnO plate that contacts the liquid environment from the a-SiC side of the plate. As can be seen, the acoustic energy is confined inside the plate. The liquid was modelled as a linear isotropic viscoelastic material with independent elastic constants, the bulk (K) and shear (G) moduli, and two independent bulk (ηb) and shear viscosities (ηv) from Reference [21].  Figure 8 shows the displacement components U 1 and U 3 of the three modes propagating along the a-SiC/ZnO plate that contacts the liquid environment from the a-SiC side of the plate. As can be seen, the acoustic energy is confined inside the plate. The liquid was modelled as a linear isotropic viscoelastic material with independent elastic constants, the bulk (K) and shear (G) moduli, and two independent bulk (η b ) and shear viscosities (η v ) from Reference [21].

The Coupling Coefficient Dispersion Curves
Lamb mode devices based on piezoelectric layers use input and output interdigital transducers (IDTs) to launch and receive the acoustic waves. In the a-SiC/ZnO composite plate, four piezoelectric coupling configurations can be obtained by placing the IDTs at the a-SiC/ZnO interface (STF) or at the ZnO-free film surface (SFT), further including a floating electrode opposite the IDTs (STFM and SMFT). The four configurations are depicted in Figure 9. The electromechanical coupling coefficient K 2 is an important parameter used as a direct estimate of the electrical to acoustic energy conversion efficiency; it is frequency dispersive and strongly affected by the electrical boundary conditions that influence the potential depth profile. K 2 can be approximated as K = 2 • v − v v , where vf and vm are the velocities along the electrically open and shorted surfaces of the ZnO film. The K 2 of the qS0 mode was studied with respect to the thickness of the a-SiC and ZnO layers and the electrical boundary conditions. The K 2 dispersion curves of the four configurations are shown in Figure 10: the a-SiC normalized thickness is fixed (haSiC/λ = 0.1); the abscissa of the graph represents the ZnO normalized thickness. As can be seen, the K 2 curves exhibit a large dependence on the IDTs and floating electrode arrangement [22]. The K 2 of the qS0 mode in the STFM coupling configuration is larger than that of the other configurations; the

The Coupling Coefficient Dispersion Curves
Lamb mode devices based on piezoelectric layers use input and output interdigital transducers (IDTs) to launch and receive the acoustic waves. In the a-SiC/ZnO composite plate, four piezoelectric coupling configurations can be obtained by placing the IDTs at the a-SiC/ZnO interface (STF) or at the ZnO-free film surface (SFT), further including a floating electrode opposite the IDTs (STFM and SMFT). The four configurations are depicted in Figure 9.

The Coupling Coefficient Dispersion Curves
Lamb mode devices based on piezoelectric layers use input and output interdigital transducers (IDTs) to launch and receive the acoustic waves. In the a-SiC/ZnO composite plate, four piezoelectric coupling configurations can be obtained by placing the IDTs at the a-SiC/ZnO interface (STF) or at the ZnO-free film surface (SFT), further including a floating electrode opposite the IDTs (STFM and SMFT). The four configurations are depicted in Figure 9. The electromechanical coupling coefficient K 2 is an important parameter used as a direct estimate of the electrical to acoustic energy conversion efficiency; it is frequency dispersive and strongly affected by the electrical boundary conditions that influence the potential depth profile. K 2 can be approximated as K = 2 • v − v v , where vf and vm are the velocities along the electrically open and shorted surfaces of the ZnO film. The K 2 of the qS0 mode was studied with respect to the thickness of the a-SiC and ZnO layers and the electrical boundary conditions. The K 2 dispersion curves of the four configurations are shown in Figure 10: the a-SiC normalized thickness is fixed (haSiC/λ = 0.1); the abscissa of the graph represents the ZnO normalized thickness. As can be seen, the K 2 curves exhibit a large dependence on the IDTs and floating electrode arrangement [22]. The K 2 of the qS0 mode in the STFM coupling configuration is larger than that of the other configurations; the The electromechanical coupling coefficient K 2 is an important parameter used as a direct estimate of the electrical to acoustic energy conversion efficiency; it is frequency dispersive and strongly affected by the electrical boundary conditions that influence the potential depth profile. K 2 can be approximated as , where v f and v m are the velocities along the electrically open and shorted surfaces of the ZnO film. The K 2 of the qS 0 mode was studied with respect to the thickness of the a-SiC and ZnO layers and the electrical boundary conditions. The K 2 dispersion curves of the four configurations are shown in Figure 10: the a-SiC normalized thickness is fixed (h aSiC /λ = 0.1); the abscissa of the graph represents the ZnO normalized thickness. As can be seen, the K 2 curves exhibit a large dependence on the IDTs and floating electrode arrangement [22]. The K 2 of the qS 0 mode in the STFM coupling configuration is larger than that of the other configurations; the qS 0 mode-based devices can reach remarkable K 2 values, from 3.3% for the SFT configuration up to about 6.5% for the STFM configuration. qS0 mode-based devices can reach remarkable K 2 values, from 3.3% for the SFT configuration up to about 6.5% for the STFM configuration. While designing a qS0 mode-based device suitable for contact with a liquid environment, the choice of the ZnO layer normalized thickness is restricted to the small range shown in the inset of Figure 10: it follows that the SMFT and STFM coupling configurations must be preferred to the SFT and STF configurations for their higher K 2 . Table 1 lists the K 2 of the four coupling configurations based on the propagation of the three modes, qS0, qL1, and qL2. The STFM configuration is the most efficient for the qS0 and qL1 modes, while the SMFT is the most efficient for the qL2 mode. The K 2 of the selected configuration is a critical factor to determine the optimum number of electrode finger pairs, N = π 4K , of both the input and output IDTs, required to obtain the minimum insertion loss and the maximum frequency bandwidth of the filter or delay line to be implemented on the a-SiC/ZnO plate.

Gravimetric Sensor
The most common sensing application for electroacoustic devices is based on the gravimetric principle for mass detection. A mass accumulation on the device surface changes the surface density of the propagating medium, resulting in a mode velocity shift. If the added mass consists of an ideal thin elastic film that moves synchronously with the oscillating surface, the fractional velocity change to the added mass ratio defines the gravimetric sensitivity of the phase velocity While designing a qS 0 mode-based device suitable for contact with a liquid environment, the choice of the ZnO layer normalized thickness is restricted to the small range shown in the inset of Figure 10: it follows that the SMFT and STFM coupling configurations must be preferred to the SFT and STF configurations for their higher K 2 . Table 1 lists the K 2 of the four coupling configurations based on the propagation of the three modes, qS 0 , qL 1 , and qL 2 . The STFM configuration is the most efficient for the qS 0 and qL 1 modes, while the SMFT is the most efficient for the qL 2 mode. The K 2 of the selected configuration is a critical factor to determine the optimum number of electrode finger pairs, N opt = π 4K 2 , of both the input and output IDTs, required to obtain the minimum insertion loss and the maximum frequency bandwidth of the filter or delay line to be implemented on the a-SiC/ZnO plate.

Gravimetric Sensor
The most common sensing application for electroacoustic devices is based on the gravimetric principle for mass detection. A mass accumulation on the device surface changes the surface density of the propagating medium, resulting in a mode velocity shift. If the added mass consists of an ideal thin elastic film that moves synchronously with the oscillating surface, the fractional velocity unloaded and mass-loaded plate's phase velocity. If the gravimetric detection occurs inside a liquid environment, then the wave attenuation is supposed to also be affected by the liquid environment's characteristics. The gravimetric sensitivity of the Insertion Loss (IL), S IL m = ∆IL m , was calculated by IL = −40π log 10 e v i v r = −54.6 v i v r , where v r and v i are the real and imaginary parts of the wave velocity, ∆IL = IL 0 − IL m , and IL 0 and IL m are the insertion loss without and with the added mass attached to the sensor surface. The sensitivity calculation was performed for two different liquid environments, water and a 53% glycerol/water mixture, to estimate the effect of the liquid environment's viscosity on the sensor performance. It was assumed that the added mass is anchored onto the a-SiC side of the composite plate. The velocity and loss sensitivities to the added mass in liquid were calculated for the qS 0 , qL 1 , and qL 2 modes travelling in the a-SiC(5 µm)/ZnO(2.5 µm) plate, with λ equal to 50, 16.7, and 8.8 µm, respectively. The relative velocity changes per unit added mass in water are equal to −109, −488, and −628 cm 2 /g, for the qS 0 , qL 1 , and qL 2 modes, respectively. In the 53% glycerol/water mixture, the calculated sensitivities were equal to −110, −511, and −685 cm 2 /g for the qS 0 , qL 1 , and qL 2 modes, respectively. The velocity gravimetric sensitivity of the qS 0 mode was scarcely affected (about 1% enhancement) by the viscosity of the liquid environment, while for the qL 1 and qL 2 modes an increase of about 5% and 9% was observed. The IL changes per unit added mass in water were equal to 0.54 × 10 −3 , 13.7 × 10 −3 , and 38.4 × 10 −3 dB/λ mm 2 /µg for the qS 0 , qL 1 , and qL 2 modes. The IL changes per unit added mass in the 53% glycerol/water mixture resulted in about 16 times that observed in water: it was equal to 10 × 10 −3 , 176 × 10 −3 , and 553 × 10 −3 dB/λ mm 2 /µgr for the qS 0 , qL 1 , and qL 2 modes.
A Lamb wave delay line consisting of two IDTs with fingers width and spacing equal to λ/4, with a number of finger pairs N opt = π 4K 2 max equal to 5, 10, and 20 (being K 2 max the highest K 2 value between those listed in Table 1 for each mode), and with a center-to-center IDT spacing of 3N opt ·λ could experience an IL equal to about 8 × 10 −3 , 0.4, and 2.3 dB mm 2 /µgr in water, and 0.15, 5.3, and 33 dB mm 2 /µgr in the mixture of 53% glycerol in water. For many liquid media sensing applications, a mass anchored to the sensor surface has to be detected that is not necessarily well-represented by a continuum layer rigidly coupled to the device surface, as assumed here. Thus, the calculated mass sensitivity in liquid media should be understood as a purely qualitative tool for comparisons between acoustic wave sensors, and yields only limited information regarding the response that could be observed in a real experiment [23].

Viscosity Sensor
When a liquid contacts the acoustic waveguide, the in-plane particle displacement component of the acoustic mode couples to a very thin viscous boundary layer of thickness δ = 2η ωρ l , where η and ρ l are the liquid viscosity and mass density. The viscous liquid was supposed to be a mixture of water and glycerol; the fraction of glycerol by volume ranged from 0 (only water) to 0.6, and the √ ηρ l ranged from 0.95 to about 15 kg·m −2 ·s −0.5 [21]. The effects of both the viscosity η and the mass density ρ l of the water/glycerol mixture on the wave velocity and IL was analyzed numerically. The real and imaginary parts of the phase velocity (v i and v r ) of the three modes were calculated for different concentrations of the water/glycerol mixtures. The relative changes of the phase velocity ∆v/v 0 and the IL as a function of √ ηρ l are shown in Figure 11a,b. It was assumed that the examined glycerol-water mixture was in contact with the a-SiC surface of the composite plate.
ranged from 0.95 to about 15 kg·m ·s [21]. The effects of both the viscosity η and the mass density ρ of the water/glycerol mixture on the wave velocity and IL was analyzed numerically. The real and imaginary parts of the phase velocity (vi and vr) of the three modes were calculated for different concentrations of the water/glycerol mixtures. The relative changes of the phase velocity Δv/v0 and the IL as a function of ρ are shown in Figures 11a,b. It was assumed that the examined glycerolwater mixture was in contact with the a-SiC surface of the composite plate. As shown in Figure 11, at low values of viscosity, the fluid behaves as a Newtonian liquid with Δv/v0 and IL proportional to ρ η. This is observed for the qS0 and qL1 modes, whose time scale