The Extraction of Coupling-of-Modes Parameters in a Layered Piezoelectric Substrate and Its Application to a Double-Mode SAW Filter

This paper presents an advanced method that combines coupling-of-modes (COM) theory and the finite element method (FEM), which enables the quick extraction of COM parameters and the accurate prediction of the electroacoustic and temperature behavior of surface acoustic wave (SAW) devices. For validation, firstly, the proposed method is performed for a normal SAW resonator. Then, the validated method is applied to analysis of an I.H.P. SAW resonator based on a 29°YX−LT/SiO2/SiC structure. Via optimization, the electromechanical coupling coefficient (K2) is increased up to 13.92% and a high quality (Q) value of 1265 is obtained; meanwhile, the corresponding temperature coefficient of frequency (TCF) is −10.67 ppm/°C. Furthermore, a double-mode SAW (DMS) filter with low insertion loss and excellent temperature stability is also produced. It is demonstrated that the proposed method is effective even for SAW devices with complex structures, providing a useful tool for the design of SAW devices with improved performance.


Introduction
Surface acoustic wave (SAW) devices have been widely used in wireless communication systems; particularly, filters and duplexers based on SAW resonators are massproduced and applied in the radio frequency (RF) front-end due to their advantages of high isolation, low insertion loss, etc.With the advent of the 5G era, there is a growing demand for SAW devices in terms of their high frequency, large electromechanical coupling coefficient (K 2 ), high quality (Q), good temperature coefficient of frequency (TCF) etc.
The performance of SAW devices mainly depends on their piezoelectric substrates.A normal SAW devices on bulk piezoelectric single-crystal materials such as lithium tantalate (LiTaO 3 , LT) and lithium niobate (LiNbO 3 , LN) have a limitation to their high frequency and Q factor [1][2][3][4][5][6][7][8][9].To solve these challenges, T. Takai et al. proposed the I.H.P. SAW device based on a LiTaO 3 /SiO 2 /AlN/Si multi-layered structure and demonstrated that the resonators employing I.H.P. SAW device offers higher Bode-Q values of over 6000 at 0.9 GHz to1900 at 3.5 GHz and a very small TCF of −8 ppm/ • C [4].Because of increasingly stringent requirements for performance enhancement, techniques for the analysis and design of SAW devices on such layered piezoelectric substrates with more complex structures are called for [10,11].
In the available numerical simulation techniques, the coupling-of-modes (COM) model is commonly used and proven to be an efficient modeling approach to design SAW devices [10,[12][13][14][15].However, the COM model is a phenomenological model, and its COM parameters have to be first determined either via measurements or precise theoretical parameters have to be first determined either via measurements or precise theoreti merical methods, for example the finite element method (FEM), or the boundary e method combined with a finite element method (FEM/BEM) [16][17][18][19][20][21].Many SAW res ers have made efforts to precisely extract COM parameters.V. Plessky et al. prop two-parameter model that approximates the dispersion of leaky waves by characte the dispersion relationship of pure surface shear waves propagating in a Bragg g within the forbidden band [22].B. P. Abbott and K. Hashimoto et al. further combin V. Plessky dispersion relation with the COM theory, establishing the STW-COM and providing a method for extracting STW-COM parameters [23].The current models take into account phenomena including the excitation, propagation, and sca of surface acoustic waves, and simulate normal SAW devices well.Nevertheless, te ature-dependent COM parameter extraction is still absent.
Therefore, this work proposes a new method combining the COM model wit tiphysics quasi-3D periodic FEM model considering the temperature field, which e the quick and accurate prediction of electroacoustic properties and temperature be for both normal and layered SAW devices.For validation, the admittances of two r tors based on 36°YX−LT and 36°YX−LT/SiO2/SiC are calculated by using the P-matr the extracted COM parameters.Then, the proposed method is employed in the simu of a full-sized double-mode SAW (DMS) filter, and its temperature behavior is s and discussed.Furthermore, a design of a high-performance DMS filter with low in loss and improved temperature stability is achieved.

COM Theory and P-Matrix Model
The materials in Figure 1 depict a basic SAW transducer structure.The COM is widely employed to investigate the coupling behavior of two counter-propa acoustic waves in a periodic grating structure.It is assumed that there exist two counter-propagating acoustic wave modes R S(x) in an infinitely long periodic grating array.These modes couple to each other th the interdigital reflection effect of the metal grating.Additionally, an external exc It is assumed that there exist two counter-propagating acoustic wave modes R(x) and S(x) in an infinitely long periodic grating array.These modes couple to each other through the interdigital reflection effect of the metal grating.Additionally, an external excitation source alternating the voltage, V, on the transducer stimulates acoustic waves, which perturbs the forementioned wave modes.The existence of electrodes on the surface affects the SAW velocity of the free surface and introduces mode coupling.Assuming linear coupling between amplitude, voltage, and current, the COM equations can be written as follow [10]: where κ represents the coupling coefficient, α signifies the excitation coefficient, C indicates the static capacitance, and ∆ denotes the detuning parameter, which is defined by the following: where ω is given by ω = 2πf, in which f is the frequency of the two waves.v denotes the effective SAW velocity, k 0 indicates propagation wave number, and γ represents attenuation.
In Equation ( 1), the five independent variables including v, κ, α, γ, and C are COM parameters that need to be determined.Under the linear assumption, the transducer can be equivalently represented by a three-port matrix, which consists of two acoustic ports and one electrical port.Therefore, the solutions to the equations of COM can be obtained using the P-matrix [10]: The matrix only considers the physical quantities at its boundaries, including the incident acoustic waves (a 1 and a 2 ), the outgoing acoustic waves (b 1 and b 2 ), the input voltage V, and the current I flowing into it.Assuming the COM parameters are known, the P-matrix unit can be calculated using the following equation [10]: where D = ∆ 2 − |κ| 2 , and L denotes the length of the device.Furthermore, since practical SAW devices are typically composed of multiple transducers and reflection gratings, it is necessary to sequentially cascade each P-matrix unit to calculate device response.Therefore, let P A and P B be the neighboring P-matrices cascaded from left to right; the cascading formula for the P-matrix is expressed as follows [10]: As for a single-port resonator without reflectors, P 33 represents the admittance of the interdigital transducers (IDT).For this case, the boundary conditions are given by a 1 = a 2 = 0, and P 33 represents the input admittance of the SAW resonator.

COM Parameter Extraction
A theoretical method to extract precise COM parameters based on FEM software COMSOL Multiphysics 5.5 is presented.Figure 2 shows the schematic of a quasi-threedimensional periodic FEM model and a structure used for simulation.As shown, one period with periodic metal electrodes alternately applying a positive or negative voltage is considered, and a perfectly matched layer (PML) is applied to the bottom to reduce the model's size and suppress the unwanted boundary reflection.Periodic boundary conditions were defined on the left (Γ L ) and right (Γ R ) of the model.In addition, the meshing grid elements in each domain of the model are applied, and the maximum size of the grid cells in each material is specified as one-sixteenth of the wavelength to accurately resolve the stress waves in the solid domain.Furthermore, each IDT requires at least three solid cells in the thickness direction to ensure the accuracy of the calculation.The details of the quasi-3D periodic FEM model are reported in our previous work [24].The piezoelectric module involving the physical field of solid mechanic and electrostatic coupling with the temperature field is performed so that the temperature-dependent COM parameters can be obtained.For a given temperature (T), the thermal effect can be computed by incorporating the linear thermal expansion coefficients and temperature coefficients of elasticity into the model.Figure 3 shows the procedure for the extraction of COM parameters.As is shown, an eigen frequency and frequency domain analysis of the multiphysics quasi-3D periodic FEM model for the SAW resonator is performed.As shown, the symmetric frequency (fsc−) and anti-symmetric frequency (fsc+) can be obtained by performing eigen frequency anal- Figure 3 shows the procedure for the extraction of COM parameters.As is shown, an eigen frequency and frequency domain analysis of the multiphysics quasi-3D periodic FEM model for the SAW resonator is performed.As shown, the symmetric frequency (f sc− ) and anti-symmetric frequency (f sc+ ) can be obtained by performing eigen frequency analysis at short-circuited grating, and then the center frequency ( f 0 ), v, and κ p can be calculated using the following: Figure 3 shows the procedure for the extraction of COM parameters.As is shown, an eigen frequency and frequency domain analysis of the multiphysics quasi-3D periodic FEM model for the SAW resonator is performed.As shown, the symmetric frequency (fsc−) and anti-symmetric frequency (fsc+) can be obtained by performing eigen frequency analysis at short-circuited grating, and then the center frequency ( ), v, and κp can be calculated using the following:  Frequency domain analysis of the proposed model allows for the determination of the admittance, Y, and Q factor of the SAW resonator, which can be calculated via harmonic analysis under the condition where a sinusoidal signal with voltage, V/2, is applied to the electrode.Then, the harmonic admittance, Y, per IDT period is estimated using Y = 2π f jQ c /V, where f is the driving frequency and Q c is the total charge induced on the electrode [24].Therefore, the remaining COM parameters, α p , γ p , and C p , can be obtained using the following formulas: where Y r is the conductance at the resonant frequency, ∆ f is the half-peak width of the conductance, f ar is the anti-resonant frequency, and the quality factor is given by Q = f sc− ∆ f .When the above five COM parameters are available, the device response and temperature behavior of the SAW device can be calculated.

Model Verification
For validation, first, the COM parameters extracted using the proposed method are verified for the normal SAW resonator based on bulk 128 • YX−LiNbO 3 substrate.Due to the almost negligible attenuation of Rayleigh waves excited on LiNbO 3 substrates, the attenuation coefficient was not extracted.Table 1 presents a comparison between the extracted COM parameters and those of the previously reported results taken from Refs.[25,26] in our previous work [27].As can be seen, these extracted COM parameters at a temperature (T) of 25 • C are compared well.Additionally, the COM parameters at a temperature (T) of 100 • C are also calculated via the proposed method.It is obvious that the temperature-dependent COM parameters vary with the temperature change.Compared to the traditional COM and pure FEM methods, the proposed method takes into account the temperature effects, allowing for a combination of faster and more accurate consideration of practical influences when designing SAW devices in the manufacturing process.This enhances the alignment between the expected and achieved outcomes.Then, the extracted COM parameters are employed to simulate the admittance of the SAW device with the P-matrix model.Figure 4 presents the calculated admittance of the multi-layered SH-SAW resonator on a 36 • YX−LT/SiO 2 /SiC layered substrate.For comparison, the measurement taken from Ref. [28] is also presented.As shown, the admittance curves basically fit well, and there exists a small difference at the anti-frequency, which is mainly caused by a bulk wave.Nevertheless, it is noted that the proposed COM method accurately reproduces fluctuations at low frequencies and a spurious response at high frequencies.
Micromachines 2023, 14, x FOR PEER REVIEW 7 of 14 accurately reproduces fluctuations at low frequencies and a spurious response at high frequencies.
Figure 4.An admittance comparison between the proposed COM method and measured results [28] for the SAW resonator on a 36°YX−LT/SiO2/SiC multi-layered substrate.

Simulation of Resonators
Based on the advantages of fast and accurate simulations offered by the COM method, this work employs the COM method for the simulation and optimization of SAW

Simulation of Resonators
Based on the advantages of fast and accurate simulations offered by the COM method, this work employs the COM method for the simulation and optimization of SAW resonators and DMS filters.
Figure 5 shows the calculated admittance of resonators for both bulk 36 • YX−LT substrate and the 36 • YX−LT/SiO 2 /SiC multi-layered structure.The structural parameters of the resonator are shown in Table 2.It can be observed that the multi-layered structure exhibits significantly higher K 2 and a larger oscillation amplitude.This can be attributed to the SiC substrate in the multi-layered structure, which has a higher velocity and thus offers an SAW energy confinement effect.This characteristic allows for the efficient conversion of electrical energy into mechanical vibration, enabling the high performance of SAW devices.Compared to the bulk LT substrate, the LT/SiO 2 /SiC multi-layered substrate has advantages in terms of a higher Q and improved temperature stability.From this point on, the 36 • YX−LT/SiO 2 /SiC multi-layered structure was further optimized in terms of different Euler angles and geometric structural parameters.The advantages of the multi-layered structure over those of a bulk LT structure, in terms of SAW resonator fabrication, typically result in a higher Q and improved temper ature stability.Therefore, in this study, the multi-layered structure is optimized and em ployed for the simulation of SAW devices.
Figure 6a presents the calculated velocities and electromechanical coupling coeffi cients of an SH wave mode on the LT/SiO2/SiC structure versus different β rotation angles of the LT layer.The maximum K 2 value of the device, which was approximately 13.88% was achieved by setting the Euler angle of the LT layer in the COMSOL as (0°, 151°, 0° when the LT layer thickness at 800 nm.In this configuration, the phase velocity reached  The advantages of the multi-layered structure over those of a bulk LT structure, in terms of SAW resonator fabrication, typically result in a higher Q and improved temperature stability.Therefore, in this study, the multi-layered structure is optimized and employed for the simulation of SAW devices. Figure 6a presents the calculated velocities and electromechanical coupling coefficients of an SH wave mode on the LT/SiO 2 /SiC structure versus different β rotation angles of the LT layer.The maximum K 2 value of the device, which was approximately 13.88%, was achieved by setting the Euler angle of the LT layer in the COMSOL as (0 • , 151 • , 0 • ) when the LT layer thickness at 800 nm.In this configuration, the phase velocity reached around 4064 m/s.
The advantages of the multi-layered structure over those of a bulk LT structure, in terms of SAW resonator fabrication, typically result in a higher Q and improved temperature stability.Therefore, in this study, the multi-layered structure is optimized and employed for the simulation of SAW devices.
Figure 6a presents the calculated velocities and electromechanical coupling coefficients of an SH wave mode on the LT/SiO2/SiC structure versus different β rotation angles of the LT layer.The maximum K 2 value of the device, which was approximately 13.88%, was achieved by setting the Euler angle of the LT layer in the COMSOL as (0°, 151°, 0°) when the LT layer thickness at 800 nm.In this configuration, the phase velocity reached around 4064 m/s.Additionally, Figure 6b illustrates the calculated velocities and electromechanical coupling coefficients versus different piezoelectric layer thicknesses on a 29°YX−LT/SiO2/SiC structure.A K 2 value of 13.89% is obtained when the piezoelectric layer thickness is 0.2λ, with a corresponding phase velocity of 4060 m/s.This reduction in velocity is deemed acceptable compared to the case for a piezoelectric layer thickness of 0.1λ.In the case of a 0.2λ thick LT layer, a K 2 value of 13.91% is obtained when the electrode thickness is 0.06λ, as shown in Figure 6c.It is worth noting that as the electrode thickness increases, the velocity of the SH wave mode gradually decreases due to the influence of mass loading.However, the velocity of 4085 m/s at the electrode thickness of 0.06λ is an acceptable value.Under the condition of a 0.2λ thick LT layer and 0.06λ thick Al electrodes, an optimized structure with a maximum K 2 of 13.92% and suitable phase velocity of 4122 m/s obtained simultaneously results when the metallization ratio is 0.4.
Figure 7a presents the simulation results of the optimized resonator compared to those of a resonator on a 36°YX−LT substrate.The optimized resonator on the 29°YX−LT/SiO2/SiC structure exhibits significantly larger K 2 values and higher amplitudes compared to the resonator on a 36°YX−LT substrate.Figure 7b illustrates the temperature characteristics of the optimized resonator on the 29°YX−LT/SiO2/SiC structure.It is evident that as the temperature increases, the overall admittance gradually shifts to the left.The temperature coefficient of the resonance frequency (TCFr) is −10.67 ppm/°C, while the temperature coefficient of the anti-resonance frequency (TCFa) is −40.36 ppm/°C [29][30][31].Table 3 presents a comparison for the extracted COM parameters at different temperatures.As can be seen, these extracted temperature-dependent COM parameters vary with the temperature change.The performance comparison between the resonators on a 36°YX−LT substrate and a 29°YX−LT/SiO2/SiC substrate is shown in Table 4.The multi-layered structure demonstrates a significant improvement in temperature stability, which effectively mitigates the impact of temperature variations on SAW resonators and meets the requirements for precise and stable performance in practical applications.Additionally, Figure 6b illustrates the calculated velocities and electromechanical coupling coefficients versus different piezoelectric layer thicknesses on a 29 • YX−LT/SiO 2 /SiC structure.A K 2 value of 13.89% is obtained when the piezoelectric layer thickness is 0.2λ, with a corresponding phase velocity of 4060 m/s.This reduction in velocity is deemed acceptable compared to the case for a piezoelectric layer thickness of 0.1λ.In the case of a 0.2λ thick LT layer, a K 2 value of 13.91% is obtained when the electrode thickness is 0.06λ, as shown in Figure 6c.It is worth noting that as the electrode thickness increases, the velocity of the SH wave mode gradually decreases due to the influence of mass loading.However, the velocity of 4085 m/s at the electrode thickness of 0.06λ is an acceptable value.Under the condition of a 0.2λ thick LT layer and 0.06λ thick Al electrodes, an optimized structure with a maximum K 2 of 13.92% and suitable phase velocity of 4122 m/s obtained simultaneously results when the metallization ratio is 0.4.
Figure 7a presents the simulation results of the optimized resonator compared to those of a resonator on a 36 • YX−LT substrate.The optimized resonator on the 29 • YX−LT/SiO 2 / SiC structure exhibits significantly larger K 2 values and higher amplitudes compared to the resonator on a 36 • YX−LT substrate.Figure 7b 4. The multi-layered structure demonstrates a significant improvement in temperature stability, which effectively mitigates the impact of temperature variations on SAW resonators and meets the requirements for precise and stable performance in practical applications.

Simulation of DMS Filters
The advantages of the optimized 29°YX−LT/SiO2/SiC structure over a 36°YX−LT structure typically imply a larger bandwidth and lower insertion loss for filters.Therefore, this work extends the optimization results of multi-layered SAW resonators and applies the optimized structure for a DMS filter simulation.
In order to obtain a wider bandwidth, the test DMS structure of three IDTs in which the lateral IDTs are symmetrically arranged with respect to the center IDT is used [32].Figure 8 presents the electrode configuration.The corresponding parameters of DMS filters are shown in Table 5.

Simulation of DMS Filters
The advantages of the optimized 29 • YX−LT/SiO 2 /SiC structure over a 36 • YX−LT structure typically imply a larger bandwidth and lower insertion loss for filters.Therefore, this work extends the optimization results of multi-layered SAW resonators and applies the optimized structure for a DMS filter simulation.
In order to obtain a wider bandwidth, the test DMS structure of three IDTs in which the lateral IDTs are symmetrically arranged with respect to the center IDT is used [32].Figure 8 presents the electrode configuration.The corresponding parameters of DMS filters are shown in Table 5.     Figure 9 illustrates a comparison of the transmission spectrums between the filters on the 36°YX−LT and 29°YX−LT/SiO2/SiC substrates.The DMS filter on the 29°YX−LT/SiO2/SiC substrate exhibits a significantly wider bandwidth compared to that of the common filter.This expanded bandwidth enables the filter to transmit a broader range of frequencies, meeting the requirements for processing various frequency signals.Additionally, the 29°YX−LT/SiO2/SiC structure demonstrates a flatter passband, ensuring uniform signal transmission and avoiding an uneven frequency response and distortion.This is crucial for applications that demand good signal quality and accuracy.Therefore, the DMS filter on the 29°YX−LT/SiO2/SiC substrate provides more flexible and high-performance signal processing capabilities.6 presents the specific performance of the two DMS filters.In addition to the improvement in TCF, in-band insertion loss is optimized from −3 dB to −2 dB, effectively enhancing the application potential of the filter at high temperatures and enabling the better performance of signal transmission.It can be inferred that by employing the improved method, it is possible in the early stages of designing SAW devices to simulate the propagation characteristics and temperature properties of DMS filters rapidly and accurately in the manufacturing process.This facilitates the identification of filters that meet the expected requirements, enabling mass production and reducing material wastage caused by inaccurate or incomplete simulations.6 presents the specific performance of the two DMS filters.In addition to the improvement in TCF, in-band insertion loss is optimized from −3 dB to −2 dB, effectively enhancing the application potential of the filter at high temperatures and enabling the better performance of signal transmission.It can be inferred that by employing the improved method, it is possible in the early stages of designing SAW devices to simulate the propagation characteristics and temperature properties of DMS filters rapidly and accurately in the manufacturing process.This facilitates the identification of filters that meet the expected requirements, enabling mass production and reducing material wastage caused by inaccurate or incomplete simulations.

Conclusions
In this paper, an advanced method combining the COM theory with a quasi-3D periodic FEM model coupled with a thermal field is proposed.Accurate simulations and analyses of the electroacoustic and temperature behavior of SAW resonators and filters are performed.This is of great assistance in the manufacturing process of SAW devices, effectively preventing production wastage caused by inaccurate simulations or the neglect of temperature stability.On this basis, a comparison is made between resonators on a 36°YX−LT structure and those on a 29°YX−LT/SiO2/SiC multi-layered structure.This comparison demonstrates the advantages of the multi-layered structure with a larger K 2 of 13.92% and a lower TCF of −10.67 ppm/°C.Frequency response calculations are performed for both the SAW resonators and filters based on this optimized structure, revealing

Conclusions
In this paper, an advanced method combining the COM theory with a quasi-3D periodic FEM model coupled with a thermal field is proposed.Accurate simulations and analyses of the electroacoustic and temperature behavior of SAW resonators and filters are performed.This is of great assistance in the manufacturing process of SAW devices, effectively preventing production wastage caused by inaccurate simulations or the neglect of temperature stability.On this basis, a comparison is made between resonators on a 36 • YX−LT structure and those on a 29 • YX−LT/SiO 2 /SiC multi-layered structure.This comparison demonstrates the advantages of the multi-layered structure with a larger K 2 of 13.92% and a lower TCF of −10.67 ppm/ • C. Frequency response calculations are

Figure 1 .
Figure 1.Propagation of SAW in periodic structure.

Figure 1 .
Figure 1.Propagation of SAW in periodic structure.

Figure 2 .
Figure 2. The schematic of the FEM model and structure used for simulation.(a) A SAW resonator on a 36°YX−LT substrate, and (b) a SAW resonator on a 36°YX−LT/SiO2/SiC substrate.

Figure 2 .
Figure 2. The schematic of the FEM model and structure used for simulation.(a) A SAW resonator on a 36 • YX−LT substrate, and (b) a SAW resonator on a 36 • YX−LT/SiO 2 /SiC substrate.

Figure 2 .
Figure 2. The schematic of the FEM model and structure used for simulation.(a) A SAW resonator on a 36°YX−LT substrate, and (b) a SAW resonator on a 36°YX−LT/SiO2/SiC substrate.

Figure 3 .
Figure 3.The FEM analysis with quasi-3D periodic structures: (a) the procedure of COM parameter extraction; (b) modal displacements at the stopband edges.Figure 3. The FEM analysis with quasi-3D periodic structures: (a) the procedure of COM parameter extraction; (b) modal displacements at the stopband edges.

Figure 3 .
Figure 3.The FEM analysis with quasi-3D periodic structures: (a) the procedure of COM parameter extraction; (b) modal displacements at the stopband edges.Figure 3. The FEM analysis with quasi-3D periodic structures: (a) the procedure of COM parameter extraction; (b) modal displacements at the stopband edges.

Figure 4 .
Figure 4.An admittance comparison between the proposed COM method and measured results[28] for the SAW resonator on a 36 • YX−LT/SiO 2 /SiC multi-layered substrate.

Figure 5 .
Figure 5.The admittance comparison between the SH-SAW resonator with bulk 36 • YX−LT substrate and the SH-SAW resonator with the 36 • YX−LT/SiO 2 /SiC multi-layered substrate.
illustrates the temperature characteristics of the optimized resonator on the 29 • YX−LT/SiO 2 /SiC structure.It is evident that as the temperature increases, the overall admittance gradually shifts to the left.The temperature coefficient of the resonance frequency (TCF r ) is −10.67 ppm/ • C, while the temperature coefficient of the anti-resonance frequency (TCF a ) is −40.36 ppm/ • C [29-31].Table 3 presents a comparison for the extracted COM parameters at different temperatures.As can be seen, these extracted temperature-dependent COM parameters vary with the temperature change.The performance comparison between the resonators on a 36 • YX−LT substrate and a 29 • YX−LT/SiO 2 /SiC substrate is shown in Table

Figure 7 .
Figure 7. (a) The admittance comparison between the SH-SAW resonator on the 36°YX−LT substrate and the SH-SAW resonator on the optimized 29°YX−LT/SiO2/SiC structure.(b) Admittances of the optimized 29°YX−LT/SiO2/SiC structure at different temperatures.

Micromachines 2023 ,Figure 8 .Table 5 .Figure 9
Figure 8. Configuration of a test DMS structure using the first mode and third mode.

Figure 8 .
Figure 8. Configuration of a test DMS structure using the first mode and third mode.

Figure 9 14 Figure 8 .
Figure 9 illustrates a comparison of the transmission spectrums between the filters on the 36 • YX−LT and 29 • YX−LT/SiO 2 /SiC substrates.The DMS filter on the 29• YX−LT/ SiO 2 /SiC substrate exhibits a significantly wider bandwidth compared to that of the common filter.This expanded bandwidth enables the filter to transmit a broader range of frequencies, meeting the requirements for processing various frequency signals.Additionally, the 29 • YX−LT/SiO 2 /SiC structure demonstrates a flatter passband, ensuring uniform signal transmission and avoiding an uneven frequency response and distortion.This is crucial for applications that demand good signal quality and accuracy.Therefore, the DMS filter on the 29 • YX−LT/SiO 2 /SiC substrate provides more flexible and high-performance signal processing capabilities.

Figure 9 .
Figure 9.The transmission spectrum comparison between the DMS filters on the bulk 36°YX−LT substrate and the optimized 29°YX−LT/SiO2/SiC structure.

Figure 9 .
Figure 9.The transmission spectrum comparison between the DMS filters on the bulk 36 • YX−LT substrate and the optimized 29 • YX−LT/SiO 2 /SiC structure.

Figure 10
Figure 10 illustrates the computed temperature characteristics of the two DMS filters, indicating that the optimized 29 • YX−LT/SiO 2 /SiC structure exhibits superior temperature stability compared to that of the 36 • YX−LT structure.Particularly, TCF suppression is evident at lower frequencies in the 29 • YX−LT/SiO 2 /SiC structure.TCF values at the lower edge (TCF l ) are reduced to −14.35 ppm/ • C, while TCF values at the higher edge (TCF h ) are reduced to −22.02 ppm/ • C compared to those of the DMS filter on a 36 • YX−LT structure (TCF l = −32.87ppm/ • C, TCF h = −37.55ppm/ • C).The 29 • YX−LT/SiO 2 /SiC structure effectively suppresses the TCF to a satisfactory level, emphasizing its contribution to improved temperature stability.Table6presents the specific performance of the two DMS filters.In addition to the improvement in TCF, in-band insertion loss is optimized from −3 dB to −2 dB, effectively enhancing the application potential of the filter at high temperatures and enabling the better performance of signal transmission.It can be inferred that by employing the improved method, it is possible in the early stages of designing SAW devices to simulate the propagation characteristics and temperature properties of DMS filters rapidly and accurately in the manufacturing process.This facilitates the identification of filters that meet the expected requirements, enabling mass production and reducing material wastage caused by inaccurate or incomplete simulations.

Figure 10
Figure 10 illustrates the computed temperature characteristics of the two DMS filters, indicating that the optimized 29°YX−LT/SiO2/SiC structure exhibits superior temperature stability compared to that of the 36°YX−LT structure.Particularly, TCF suppression is evident at lower frequencies in the 29°YX−LT/SiO2/SiC structure.TCF values at the lower edge (TCFl) are reduced to −14.35 ppm/°C, while TCF values at the higher edge (TCFh) are reduced to −22.02 ppm/°C compared to those of the DMS filter on a 36°YX−LT structure (TCFl = −32.87ppm/°C, TCFh = −37.55ppm/°C).The 29°YX−LT/SiO2/SiC structure effectively suppresses the TCF to a satisfactory level, emphasizing its contribution to improved temperature stability.Table6presents the specific performance of the two DMS filters.In addition to the improvement in TCF, in-band insertion loss is optimized from −3 dB to −2 dB, effectively enhancing the application potential of the filter at high temperatures and enabling the better performance of signal transmission.It can be inferred that by employing the improved method, it is possible in the early stages of designing SAW devices to simulate the propagation characteristics and temperature properties of DMS filters rapidly and accurately in the manufacturing process.This facilitates the identification of filters that meet the expected requirements, enabling mass production and reducing material wastage caused by inaccurate or incomplete simulations.

Table 2 .
The geometric structure parameters of the SAW resonator.

Table 2 .
The geometric structure parameters of the SAW resonator.
• C

Table 5 .
The geometric structure parameters of the DMS filter.

Table 5 .
The geometric structure parameters of the DMS filter.