# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 3 Case Analyses and Discussions
### Effect of Cavity on Bandgap Characteristics  
In this section, the parametric influence of the size of the cavity to bandgap width is investigated. Here cavities are incorporated with constant heights of 1 mm and 2 mm, respectively, and with constant length of 25.38 mm. The boundaries and the width of the bandgap under cavities with various widths, are then plotted in Figures 6 and 7. These results are obtained through finite element analysis.  
Figure 5: Dispersion curves. (**a**) Unit-cell without cavity; (**b**) Unit-cell with cavity of 25.38 \(\times\) 12 \(\times\) 1 mm\({}^{3}\); (**c**) Zoom in curves for unit-cell without cavity; (**d**) Zoom in curves for unit-cell with cavity.  
Figure 6 shows the frequency boundaries and the width of the bandgap. All the cavities have constant height of 1 mm and constant length of 25.38 mm. It can be observed that in these results all bandgaps are created in the vicinity of 3200 Hz, although the stiffness of the unit-cell has been significantly reduced. This is because the location of the bandgap is primarily determined by the frequency of the LC resonance. With the increase of the width of the cavity, the location of the bandgap shifts to a lower frequency range. This can be attributed to the fact that local stiffness of the unit-cell also affects the local resonant bandgap, as indicated in Equation (21). The relation between the bandgap width and the width of the cavity, is illustrated in Figure 6b. It can be observed that, by increasing the width of the cavity, the bandgap width of the piezoelectric metamaterial can be continuously increased. The reason is that, increasing the size of cavity increases the system-level electro-mechanical coupling coefficient. It is worth noting that the piezoelectric metamaterial with a cavity width of 24 mm, has bandgap expanded from 45 Hz to 126.7 Hz. Moreover, the proposed mechanism for broadening bandgap width of piezoelectric metamaterial does not preclude the utilization of piezoelectric transducers with larger coupling constant or negative capacitance circuitry integration. The bandgap width of the piezoelectric metamaterial can be further expanded by methods proposed previously.  
Similarly, Figure 7 illustrates the frequency boundaries and the bandgap width of the unit-cell, where the cavity introduced has height of 2 mm. The results generally resemble those shown in Figure 6, i.e., the introduction of cavity underneath the substrate, can effectively increase the bandgap width. However, it is worth mentioning that in this case, the bandgap width decreases dramatically  
Figure 6: Unit-cell with cavity with height of 1 mm. (**a**) Boundaries of the bandgap; (**b**) Bandgap width.  
Figure 7: Unit-cell with cavity with height of 2 mm. (**a**) Boundaries of the bandgap; (**b**) Bandgap width.  
when the width of the cavity is increased to 22.8 mm. The shrinking of bandgap width is attributed to the fact that the mode of the unit-cell becomes uneven, i.e., the assumptions of displacements in Equation (10a,b) no longer hold. The non-sinusoidal mode of the unit-cell reduces the system level electro-mechanical coupling coefficient, and therefore reduces bandgap width.