br Numerical results and discussion

Numerical results and discussion
As an example, consider the AlN on Si filter with Mo electrodes in [17,18]. The thicknesses of the top electrode, the AlN film, the ground electrode, and the Si layer are Mo (100nm)/AlN (1μm)/Mo (100nm)/Si (5μm). The width of the input and output electrode fingers are a1=a2=a=10μm. The spacing between the electrode fingers is b=5μm. These are the same as those in [17,18]. However, in [17,18], the dimensions of the unelectroded parts at the left and right edges were not specified. We use c=30μm which is about five times the total plate thickness in order to exhibit the Sulfo-NHS-Biotin trapping behavior of the modes of interest. We consider the case when the input and output electrodes each has four fingers for our parametric study below. This corresponds to P=17. As to be seen below, P=17 is sufficient to show the basic behaviors of the filter while in real devices a larger P is sometimes used (P=41 in [17,18]).
Fig. 3 shows the seven trapped modes found in the frequency interval determined by (10) and (12) in the order of increasing frequency. The frequencies of thickness-extensional modes are mainly determined by the plate thickness [26,27]. Therefore they only increase slightly and slowly in Fig. 3 because n=4 is already fixed for the 4th-order thickness-extensional modes in [17,18]. The first mode in Fig. 3(a) does not have a nodal point (zero) along x1 and all of the top electrode fingers vibrate in phase, which is ideal as the operating mode of the filter. Notice that in the mode in Fig. 3(a) the vibration decays to zero exponentially near the plate edges, which is the desired energy trapping. The mode in Fig. 1(a) has small oscillations with peaks corresponding to the fingers of the top electrodes and valleys corresponding to the spacing among the electrode fingers. This is also because of the exponentially decaying behavior of the vibration away from the edges of the electrode fingers. As the frequency increases, from (b) to (g), the modes have one to six nodal points, respectively. When there are nodal points, the fingers of the input (or output) electrode may vibrate out of phase, causing cancellations of the charges on different fingers of the input (or output) electrode, which may be undesirable. The modes in Fig. 3 are either symmetric or antisymmetric about the middle of the plate alternatingly. This symmetry or antisymmetry of the modes about the plate center and the successive increase of nodal points are typical behaviors of plate thickness-extensional modes with in-plane variations and are as expected. They are long plate modes compared to the plate thickness. Because of the presence of the electrode fingers, these long plate modes are modulated by the electrode fingers which are responsible for the small oscillations of the modes.
When plotting Fig. 4, we increased the electrode finger width a, from the 10μm in  to 20μm and all other parameters were kept the same as those used for Fig. 3. In this case there are 11 trapped modes, with the first two shown in Fig. 4 which is sufficient for our purpose. Comparison of Fig. 4 with Fig. 3(a) and (b) shows that the frequencies have become slightly lower because wider electrode fingers have more inertia. The width of the small peaks are related to the electrode finger width and has become wider.
In Fig. 5 the electrode finger spacing b is changed from the 5μm in  to 10μm. All other parameters are the same as those used in Fig. 3. In this case there are eight trapped modes. Only the first two are shown in Fig. 5. Comparing Fig. 5 with Fig. 3(a) and (b), it can be seen that the frequencies have increased slightly when b increases. This is because larger electrode finger spacing effectively means less electrode inertia on the crystal plate and hence higher frequencies.
In Fig. 6 we increase c, the dimension of the two unelectroded edge parts, from 30μm to 50μm while keeping all other parameters the same as those in Fig. 3. Again only the first two trapped modes are shown. Within five significant figures the frequencies remain the same. This is reasonable because there is not much vibration in the edge parts. The main difference between Fig. 6 and Fig. 3(a) and (b) is that the near the plate edges the modes in Fig. 6 are more flat, showing stronger energy trapping.