# High-Frequency and Spectrum-Clean Shear-Horizontal Acoustic Wave Resonators with AlN Overlay
## 4 Propagation Characteristics of SH Wave in AlN/LiNbO\({}_{3}\)/Si  
### Cut Angle  
Figure 5a,b depict the open-surface phase velocities \(v_{\rm p,o}\) for varied normalized thicknesses of AlN overlay layer on top of the IDT transducers sitting on LiNbO\({}_{3}\)/Si and the piezo-layer is across all rotation angles from the \(YX\)-LiNbO\({}_{3}\) with \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.25 and \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.5, respectively. The \(c\)-axis-oriented AlN material constants are from literature [15] and listed in Table 2. The nonleaky SH wave propagating in the AlN/LiNbO\({}_{3}\)/Si stack shows a similar trend to that in the LiNbO\({}_{3}\)/Si layered stack. For both LiNbO\({}_{3}\) thickness cases, the phase velocity can be effectively enhanced by the AlN overlay, and the improvement converges when \(h_{\rm AlN}\) is larger than 0.2\(\lambda\). The phase velocities of the SH modes in Figure 5a with \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.25 are in general larger than the case in Figure 5b with \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.5. As can be observed in Figure 5a, with \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.25 and for \(\theta\) between 20\({}^{\circ}\) and 80\({}^{\circ}\), the \(v_{\rm p,o}\) can be as high as above 5000 m/s thanks to the AlN coating with \(h_{\rm AlN}>0.2\lambda\).  
Figure 4: FEM-simulated (**a**) narrow-band response compared to LiNbO\({}_{3}\) substrate, and (**b**) wide-band response of the SH resonators with periodic structure (1.5D model) based on LiNbO\({}_{3}\)/Si with 30\({}^{\circ}\)\(YX\) LiNbO\({}_{3}\) and \(h_{\rm LiNbO_{3}}\)/\(\lambda\) = 0.25.  
Furthermore, the Rayleigh mode in the case of \(h_{\text{LiNbO}3}/\lambda=0.25\) are less coupled with the SH mode at a high rotation angle of around \(128^{\circ}\), and also less perturbed by \(\theta\) than the case of \(h_{\text{LiNbO}3}/\lambda=0.5\). At low \(\theta<30^{\circ}\); however, the Rayleigh spurious mode is slightly closer to the SH mode in the case of \(h_{\text{LiNbO}3}/\lambda=0.25\) than in the thicker case, which is in a similar trend with Figure 3a.  
Figure 6a and b depict the effective coupling coefficient \(k_{\text{eff}}^{2}\) for varied thicknesses of AlN overlay layer on LiNbO\({}_{3}\)/Si across all rotation angles from the \(YX\) LiNbO\({}_{3}\) with \(h_{\text{LiNbO}3}/\lambda=0.25\) and \(h_{\text{LiNbO}3}/\lambda=0.5\), respectively. Although the AlN overlay apparently lowers the \(k_{\text{eff}}^{2}\), the degraded \(k_{\text{eff}}^{2}\) would still be more than enough and much larger than the current technologies. Moreover, the reduction in \(k_{\text{eff}}^{2}\) would converge when \(h_{\text{AlN}}\) is larger than \(0.2\lambda\). For both LiNbO\({}_{3}\) thickness cases, and for cut angles between \(0^{\circ}\) and \(60^{\circ}\), the \(k_{\text{eff}}^{2}\) can be as high as above \(11\%\). Comparing two LiNbO\({}_{3}\) thicknesses, the peak \(k_{\text{eff}}^{2}\) of the SH mode in AlN/LiNbO\({}_{3}\)/Si across a wide rotation-angle range are at similar level.  
It is also interesting to note that at \(\theta\)-\(10^{\circ}\)-\(30^{\circ}\), the \(k_{\text{eff}}^{2}\) of SH mode is maximized and at the same time the \(k_{\text{eff}}^{2}\) of Rayleigh mode is minimized, where the Rayleigh spurious mode can be suppressed in the nonleaky SH SAW resonator or filter. With AlN overlays, the optimized cut angle for Rayleigh mode elimination shifts down a bit. The optimal cut-angle design range would be of \(10^{\circ}\)-\(30^{\circ}\) for simultaneously achieving high \(v_{\text{p},\alpha}\) and large \(k_{\text{eff}}^{2}\) for the SH mode, as well as low- \(k_{\text{eff}}^{2}\) for Rayleigh mode.  
Figure 5: Calculated open phase velocities \(v_{\text{p},\alpha}\) for varied thicknesses of AlN overlay layer on LiNbO\({}_{3}\)/Si across all rotation angles from the \(YX\)-LiNbO\({}_{3}\) with (**a**) \(h_{\text{LiNbO}3}/\lambda=0.25\), and (**b**) \(h_{\text{LiNbO}3}/\lambda=0.5\).