**2. Materials and Methods**

In the present study, the SAW propagation characteristics of proposed multi-layered structure are calculated using the SAW Analysis software (MSDOS (version 2 or later), IEEE, Montreal, QC, Canada) developed by Farnell and Adler [13,14]. The structure consists of BeO thin film integrated over 128◦ YX LiNbO<sup>3</sup> single crystal and TeO<sup>3</sup> over layer placed on BeO thin film. The multi-layered structure and the coordinate system are presented in Figure 1.

**Figure 1.** Illustration of TeO3/Beryllium Oxide (BeO)/128◦ YX LiNbO<sup>3</sup> multilayer surface acoustic wave (SAW) structure and the coordinate system.

The Cartesian coordinate system is chosen in such a way that Rayleigh wave propagates along *x*1—axis in which its amplitude vanishes as *x*<sup>3</sup> tends to negative infinity, and *x*2—axis is parallel to the direction of particle polarization.

The electric potential *φ* and particle displacements *U<sup>k</sup>* (*k* = 1, 2, 3) in a piezoelectric medium are governed by the following elastic wave equations [1,13]:

$$
\rho\_{kij}\frac{\partial^2 \phi}{\partial \mathbf{x}\_k \partial \mathbf{x}\_l} = \rho \frac{\partial^2 \mathcal{U}\_j}{\partial t^2} - \mathbb{C}\_{ijkl} \frac{\partial^2 \mathcal{U}\_k}{\partial \mathbf{x}\_l \partial \mathbf{x}\_l} \,' \tag{1}
$$

$$
\varepsilon\_{jkl}\frac{\partial^2 \mathcal{U}\_j}{\partial \mathbf{x}\_l \partial \mathbf{x}\_l} - \varepsilon\_{jk}\frac{\partial^2 \phi}{\partial \mathbf{x}\_l \partial \mathbf{x}\_k} = 0; \qquad \text{i, j, k, l} = 1, 2, 3 \dots \dots \dots \tag{2}
$$

where *Cijkl* is the mechanical stiffness tensor, *εjk* is the dielectric permittivity tensor, *ekij* is the piezoelectric tensor, and *ρ* is the density of the medium. The material parameters, like density, elastic constant, piezoelectric constant, and dielectric constants at a given temperature of TeO3, Wurtzite BeO thin films, and 128◦ YX LiNbO<sup>3</sup> single crystal used in present study to estimate the SAW phase velocity of the layered structure, are taken from earlier reported data by Dewan et al., Duman et al., Cline et al., and Kovacs et al., respectively [12,18,22,23], and are presented in Table 1.

**Table 1.** Material constants and temperature coefficients used in simulations.


*2.1. Electromechanical Coupling Coefficient K* 2 

The effective coupling of inter digital transducer to the surface-wave is measured in terms of the electromechanical coupling coefficient, *K* <sup>2</sup> given by [1,26,27]

$$K^2(\%) = \frac{200 \ (v - v')}{v} \,\text{,}\tag{3}$$

where *v* and *v*0 are the SAW phase velocities for electric free and short circuit conditions, respectively.
