# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 3 Case Analyses and Discussions
### Dispersion Analysis of Unit-Cell with Cavity  
The dispersion relation is studied first, with emphasis on cavity effect. The cavity influences the unit-cell level stiffness and electro-mechanical coupling coefficient. Without loss of generality, a cavity with constant height of 1 mm and length of 25.38 mm is introduced into the center of the substrate. The width is considered as a variable. The theoretical stiffness ratio (the ratio of the stiffness of unit-cell with cavity to that of unit-cell without cavity) and the electro-mechanical coupling coefficient ratio (the ratio of the electro-mechanical coupling coefficient of unit-cell with cavity to that of unit-cell without cavity) with respect to the cavity width, are presented in Figure 4. It can be readily observed that increasing the width of the cavity can effectively reduce the stiffness of the unit-cell, which then yields a significant increase of the system level electro-mechanical coupling. As indicated in Section 2.3, one key parameter for piezoelectric metamaterial is the system-level electro-mechanical coupling coefficient, and the increase of this coupling would yield the improvement of bandgap behavior. It is still worth mentioning that the stiffness reduction implies a trade-off, with respect to load carrying capability of the substrate. Therefore, proper selection of the dimension of the cavity is required in practical applications.  
The dispersion curves of the unit-cells, with and without cavity, are then plotted for comparison. One representative example is shown in Figure 5, where a cavity of \(25.38\times 12\times 1\) mm\({}^{3}\) is incorporated. These curves are plotted within the first Brillouin zone (\(0-\pi\)), since all propagating wave modes in periodic structures can be captured by restricting the dimensionless wavenumber to this zone, due to the periodicity. Finite element analysis is performed for validation. It can be observed that the integration of the LC shunt circuit in the original, conventional unit-cell, creates a bandgap with a range between 3229 Hz and 3274 Hz. This yields the bandgap width of 45 Hz. In comparison, the one with cavity integration has a bandgap width of 67 Hz (from 3202 to 3269 Hz). The theoretical bandgap widths are 48 Hz and 61 Hz, respectively, for the conventional piezoelectric metamaterial beam and the metamaterial beam with cavity. The percentages of deviation are 6.7% and 9.8%, respectively. The deviation stems from the fact that the finite element simulation is performed in the usual three-dimensional space, and yields more detailed displacement distribution. Fundamentally, as wave  
Figure 4: Stiffness ratio and electro-mechanical coupling ratio _vs_ the width of the cavity.  
propagates through the piezoelectric metamaterial, each unit-cell experiences a local deformation, which then yields energy conversion from the mechanical domain to the electrical domain through the direct piezoelectric effect. Meanwhile, the charge flow in the LC shunt induces the reverse piezoelectric effect, which converts electrical energy back to mechanical energy, i.e., creating a reactive force to the unit-cell to affect the wave propagation characteristics. The two-way energy conversion efficiency is determined by the system-level electro-mechanical coefficient, which is exactly analogous to the mass ratio [18] in conventional metamaterial employing local mechanical resonator in the unit-cell. The reduction of the local stiffness of the unit-cell through cavity introduction, can effectively increase the electro-mechanical coupling coefficient and broaden the bandgap width.