Full-Wave Analysis of Ultrahigh Electromechanical Coupling Surface Acoustic Wave Propagating Properties in a Relaxor Based Ferroelectric Single Crystal/Cubic Silicon Carbide Layered Structure

This paper describes a full-wave analysis of ultrahigh electromechanical coupling surface acoustic wave (SAW) of Y-cut X propagating Pb(In1/2Nb1/2)O3-Pb(Mg1/3Nb2/3)O3-PbTiO3 (YX-PIMNT) single crystals on a cubic silicon carbide (3C-SiC) substrate. There are several eigenmodes including shear horizontal (SH) and Rayleigh SAWs. Based on the finite-element method (FEM), the phase velocity (Vp) and coupling factor (K ) of SAWs varying with the top electrode thickness, thickness, and Euler angle (θ) of the YX-PIMNT substrate have been investigated. K of SH SAW can reach an extremely high value of 75.9%. The proper control of structural parameters can suppress unwanted responses caused by othermodes without deteriorating the coupling factor. The large K value of SH SAW and suppression of unwanted responses have highly promising applications in developing ultrawideband and tunable SAW filters. Finally, the performance of 3C-SiC and 6H-SiC as substrates was investigated, and 3C-SiC was identified as a more attractive substrate candidate than 6H-SiC.

3C-SiC is an attractive candidate substrate given its high velocity and  2 [15,16].This paper reports a full-wave analysis of YX-PIMNT/3C-SiC substrate conducted via the finite-element method (FEM) with the commercial software package COMSOL Multiphysics.The variation in the phase velocity (V  ) and coupling factor ( 2 ) of SAWs with the top electrode thickness, thickness, and Euler angle () of the YX-PIMNT substrate was investigated.Results showed that  2 of SH SAW can reach a maximum value of 75.9% in this configuration.By properly controlling structural parameters, unwanted responses caused by other modes can be suppressed without deteriorating the coupling factor.The large  2 value of SH SAW and suppression of unwanted responses extremely have highly promising applications in developing ultrawideband and tunable SAW filters.Finally, The performance of 3C-SiC and 6H-SiC as substrates was investigated, and 3C-SiC was found to be a more attractive substrate candidate than 6H-SiC.

Modeling and Simulation
Figure 1 shows the modeled structure of an electrode layer/YX-PIMNT/3C-SiC substrate configuration.The thickness of each layer was denoted as ℎ  , ℎ PIMNT , and ℎ 3C-SiC .
In the model, the wavelength () was set to 2 m.Only a half period was considered, and the antisymmetric periodic boundary condition was applied to the field variables at the left (Γ  ) and right surfaces (Γ  ).The perfectly matched layer was applied to the bottom to decrease model size and suppress the spurious resonances caused by the reflection at the bottom.In this study, we assumed that waves propagate along -direction; then, waves that propagate along other directions could be ignored.Thus, the periodic boundary condition was applied to the variables between the + and − surfaces.A 3D periodical model of the layered structure was established in the commercial software package COMSOL Multiphysics.Au was chosen as the interdigital transducer (IDT) electrode material.The material constants of YX-PIMNT and 3C-SiC crystal were obtained from [17][18][19].
Harmonic analysis was conducted with 1 V loaded to the electrode.Figure 2 presents an image of the calculated acoustic wave radiation.It is shown that SH and Rayleigh SAWs mainly concentrate their energy near the metal electrode, whereas bulk waves radiate their energy to the interior of the model.
Figure 3 shows the calculated relative admittance in decibels, namely, 20 lg|()| of infinitely long IDTs on the layered structure as a function of frequency when ℎ PIMNT = 0.3.Five eigenmodes were observed, and the frequency ranged from 0.3 GHz to 3 GHz.The first eigenmode with   ≈ 650 MHz is the main mode that corresponds to SH SAW.The second eigenmode is associated with the traditional Rayleigh SAW.Higher order eigenmodes are also observed and identified as bulk waves as shown in Figure 3.

Results and Discussion
The   and   values for each eigenmode can be obtained when the relative admittance reaches peaks and troughs, respectively. 2 is the electroacoustic energy conversion efficiency of a resonator, which can be estimated using the following formula derived from the equivalent circuit analysis [20]: Hence, a large distance between   and   indicates a large  2 . 2 of SH SAW is 68.5%, whereas that of other eigenmodes is approximately zero for ℎ PIMNT = 0.3.This finding indicates that unwanted responses can be suppressed by choosing a proper thickness of YX-PIMNT, which is extremely attractive in developing wideband and tunable SAW filters.
The dispersion curves of phase velocity and  2 as a function of YX-PIMNT thickness were estimated with the calculated admittance and illustrated in Figures 5(a 8 when ℎ PIMNT ranges from 0.1 to 0.7.When ℎ PIMNT = 0.3, the maximum  2 value of the higher modes is 4.8%, indicating that these modes have little energy.A fairly large SH SAW  2 of 69.1% and a high SH SAW phase velocity of 1440 m/s are simultaneously achieved.Figures 6(a) and 6(b) show the variation in the calculated effective velocities and  2 of the SH and Rayleigh SAWs with Euler angle  ranging from −90 ∘ to 90 ∘ .The effective velocities are defined by V  =    and V  =    at the resonance and antiresonance frequencies, respectively.V  of the SH SAW gradually increases and reaches a value similar to that of the Rayleigh SAW, which is over 2600 m/s.The Rayleigh SAW exists within a wide range of Euler angles from −65 ∘ to 70 ∘ .A pure SH SAW is achieved only when  is in the ranges of −90 ∘ to −70 ∘ and 75 ∘ to 90 ∘ .Different  2 values from 59.7% to 68.9% are obtained for SH SAW, whereas those for Rayleigh SAW are approximately zero, which is beneficial for designing SAW filters with different bandwidth. 2 of the Rayleigh SAW gradually increases from  = −65 ∘ , reaches the largest value of 24.2% when  = 35 ∘ , and decreases to zero when  = 75 ∘ .
The effects of mass loading on the electrode layer should not be neglected because the thickness of the electrode layer is comparable with that of the thinned YX-PIMNT substrate.Therefore, the dependence of the  2 of the SH and Rayleigh SAWs on the thickness of the Au electrode was investigated and is shown in Figure 7.In this case, ℎ PIMNT = 0.3 and Euler angle  = 90 ∘ ; thus, unwanted waves are sufficiently suppressed. 2 of the SH SAW ranges from 62.4% to 69.8%, whereas that of Rayleigh SAW is approximately zero.When ℎ  = 0.05,  2 of the SH SAW reaches a maximum value of 69.8%.
Figure 8 presents the simulation results of the  2 curves as a function of electrode thickness for SH and Rayleigh SAWs that propagate along the YX-PIMNT/3C-SiC and YX-PIMNT/6H-SiC substrates.The ℎ PIMNT was fixed at 0.3 and the Euler angle at 90 ∘ . 2 of SH SAW in the YX-PIMNT/3C-SiC substrate is considerably higher than its value in the YX-PIMNT/6H-SiC substrate, whereas the Rayleigh SAW has low  2 values in both substrates.Hence, 3C-SiC is a more attractive candidate as the substrate in this model than 6H-SiC.However, the material losses were not taken into account in this work.These losses can be simulated in such model by giving an imaginary part to the elastic, piezoelectric, and dielectric constants to consider the mechanical, coupling, and dielectric losses; then, we can calculate the  factors.Currently, we have not yet added these losses mainly due to the lack of a full complex matrix for reference.In the following work we will consider testing these parameters with the material suppliers for accurate simulation of material losses and  factors.

Conclusion
This study presents a full-wave analysis of ultrahigh electromechanical coupling SAW of YX-PIMNT on 3C-SiC substrate.There are several eigenmodes including SH and Rayleigh SAWs.FEM results show that V  and  2 of SAWs vary with the top electrode thickness, thickness, and Euler angle of the YX-PIMNT substrate.
Results show that  2 of SH SAW can reach an extremely high value of 75.9%.Unwanted responses caused by other modes can be suppressed by properly controlling structural parameters without deteriorating the coupling factor.A relatively pure SH SAW is obtained when the thickness of YX-PIMNT is 0.3 and  has a wide range of −90 ∘ to −70 ∘ and 75 ∘ to 90 ∘ .The largest  2 achieved for the pure SH SAW is approximately 69.8% when the thickness of electrode is 0.05 and  is 90 ∘ .This extremely large  2 value of SH SAW and suppression of unwanted responses have an extremely promising application in developing ultrawideband and tunable SAW filters.Finally, the performance of 3C-SiC and 6H-SiC as a substrate was investigated, and 3C-SiC is found to be a more attractive substrate candidate than 6H-SiC.

Figure 1 :
Figure 1: Quasi-3D periodic FEM model used in the simulation (not to scale).

Figure 2 :Figure 3 :
Figure 2: Image of the calculated acoustic wave radiation.

Figures 4 (
a) and 4(b) show the normalized displacement field distribution for SH and Rayleigh SAWs on the layered structure.The dominant displacement components of SH and Rayleigh SAWs are SH and SV, respectively.The peak value of displacement is located in the electrode layer and decays exponentially in the interlayer of YX-PIMNT and 3C-SiC.This finding indicates that the vibration energy is mainly concentrated in the electrode and the YX-PIMNT thin layer, and only limited energy penetrates the 3C-SiC layer.

Figure 4 :Figure 5 :
Figure 4: Normalized displacement field distribution for SH and Rayleigh SAWs on the layered structure: (a) SH SAW and (b) Rayleigh SAW.

Figure 6 :
Figure 6: (a) Effective velocities and (b) estimated  2 (as a function of  from −90 ∘ to 90 ∘ ) of SH and Rayleigh SAWs.

Figure 7 :
Figure 7: Estimated  2 values (as a function of ℎ  ) of the SH and Rayleigh SAWs.