# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 2 Lumped-Parameter Modeling of Piezoelectric Metamaterial
### Piezoelectric Metamaterial with Cavity  
First, the role of the electro-mechanical coupling coefficient in the context of piezoelectric metamaterial, warrants further discussion here. Apparently, according to Equation (16), this non-dimensional coupling coefficient is primarily related to the transducer material property, \(h_{31}\), i.e., the piezoelectric coupling constant at the material-level. It is also related to the piezoelectric capacitance, the Young's moduli, the geometry of the substrate and the transducers, and the wavenumber involved. It can be increased with the integration of a negative capacitance element or negative stiffness effect [28]. These indicate that there is potential to increase the system-level electro-mechanical coupling through, either unit-cell material/geometry tailoring and/or circuit design. In this research, the material  
Figure 2: Equivalent circuit model of a unit-cell. properties of the transducer and the host structure are kept unchanged. The circuit configuration also remains unchanged. In conventional piezoelectric metamaterial, a significant portion of the energy of the acoustic wave is stored in the mechanical stiffness of the structure, including both the host structure and the piezoelectric transducers, and is not converted into electrical energy due to the limited electro-mechanical coupling coefficient. Subsequently, only a small portion of acoustic wave energy can be transferred into electrical energy, which creates reactive force to affect the wave propagation. As a result, the wave manipulation capability of the system is very limited, reflected by the relatively narrow width of the bandgap. Based on the fundamental modeling and the abovementioned qualitative analysis, in this research, the design enhancement of piezoelectric metamaterial by geometry tailoring is explored. Specifically, it is proposed to introduce a cavity in the host structure substrate of each unit-cell, as shown in Figure 3a,b. Our hypothesis is that the cavity can effectively reduce the unit-cell stiffness, and hence increase the portion of energy that is converted into electrical energy through the piezoelectric effect when the substrate is subjected to wave motion. Consequently, the reactive force from LC resonance would be increased, and the wave attenuation effect and the bandgap width would be enhanced. In such a case, the lumped parameter mass and stiffness of the unit-cell system are modified as:  
\[m_{c}=\frac{2\sin(\frac{k_{x}l_{b}}{2})}{k_{x}l_{b}}\rho_{b}(h_{b}b_{b}l_{b}-h_ {c}b_{c}l_{c})+\frac{2\sin(\frac{k_{x}l_{b}}{2}\frac{l_{p}}{l_{b}})}{k_{x}l_{b }}2\rho_{p}h_{p}b_{p}l_{b} \tag{17a}\] \[k_{c}=2E_{b}(I_{b}-I_{cav}){k_{x}}^{3}\sin(\frac{k_{x}l_{b}}{2})+4E_{p}I_{p}{k_{x}} ^{3}\sin(\frac{k_{x}l_{b}}{2}\frac{l_{p}}{l_{b}}) \tag{17b}\]  
where \(h_{c},b_{c}\), and \(l_{c}\) are the height, width, and length of the cavity introduced, respectively, and \(I_{cav}\) represents the moment of inertia of the cavity volume. With the introduction of the cavity, the resonant frequency of the unit-cell (before the circuitry is integrated), then becomes  
\[\omega_{0c}^{2}=\frac{k_{c}}{m_{c}} \tag{18}\]  
Figure 3: (**a**) Unit-cell with cavity; (**b**) Configuration of the piezoelectric metamaterial and the coordinate system. The system-level electro-mechanical coupling with substrate cavity takes the form  
\[k_{cc}^{2}=\frac{k_{1}^{2}C}{k_{c}}=\frac{\left(-2\frac{F_{p}h_{31}k_{x}}{b_{p1p}} \sin(\frac{k_{x}l_{b}}{2}\frac{l_{p}}{l_{b}})\right)^{2}}{\left(2E_{b}(I_{b}-I_ {cav}){k_{x}}^{3}\sin(\frac{k_{x}l_{b}}{2})+4E_{p}I_{p}k_{x}{}^{3}\sin(\frac{k_ {x}l_{b}}{2}\frac{l_{p}}{l_{b}})\right)\frac{\beta_{33}h_{p}}{b_{p}l_{p}}} \tag{19}\]  
The dispersion equation can be re-written as  
\[\left(-\frac{\omega^{2}}{\omega_{0c}^{2}}+1\right)\left(-\frac{\omega^{2}}{ \omega_{LC}^{2}}+1\right)-2k_{cc}^{2}=0 \tag{20}\]  
Recall that the local LC resonant frequency (\(\omega_{LC}^{2}\)) is usually much smaller than the Bragg scattering frequencies.