## 3 Bandgap Algorithm of the PMPIC by Extended Plane Wave Expansion Method
### Bandgap Structures of the PMPIC by an Extended Plane Wave Expansion Method
Based on the Bloch theorem, the displacement and output voltage in Equation (10) can be defined as  
\[w(\mathbf{r},t) =\mathrm{e}^{i(\mathbf{k}\mathbf{r}-\omega t)}w_{\mathbf{k}}( \mathbf{r}) \tag{21}\] \[V(\mathbf{r},t) =\mathrm{e}^{i(\mathbf{k}\mathbf{r}-\omega t)}V_{\mathbf{k}}( \mathbf{r})\]  
where \(\mathbf{k}\) is the Bloch wave vector which is restricted to the first irreducible Brillouin zone. \(w_{k}(\mathbf{r})\) is also a periodic function with the same period as that of material properties, so it can be expanded as  
\[w_{\mathbf{k}}(\mathbf{r}) =\sum_{\mathbf{G}^{\prime}}w_{\mathbf{k}}\big{(}\mathbf{G}^{ \prime}\big{)}e^{i\mathbf{G}^{\prime}\mathbf{r}} \tag{22}\] \[V_{\mathbf{k}}(\mathbf{r}) =\sum_{\mathbf{G}^{\prime}}V\big{(}\mathbf{G}^{\prime}\big{)}e^{i \mathbf{G}^{\prime}\mathbf{r}}\]Based on Equations (21) and (22), we will have  
\[ \begin{array}{l}w(\mathbf{r},t)=e^{-i\omega t}\underset{\mathbf{G}^{\prime}}{ \sum}w_{\mathbf{k}}\big{(}\mathbf{G}^{\prime}\big{)}e^{j(\mathbf{k}+\mathbf{G} ^{\prime})\mathbf{r}}\\ V(\mathbf{r},t)=e^{-i\omega t}\underset{\mathbf{G}^{\prime}}{\sum}V\big{(} \mathbf{G}^{\prime}\big{)}e^{j(\mathbf{k}+\mathbf{G}^{\prime})\mathbf{r}}\\ \end{array}  \tag{23}\]  
Substituting Equations (15) and (23) into Equation (10), we will have  
\[ \begin{array}{l}\omega^{2}\underset{\mathbf{G}^{\prime}}{\sum}\phi_{*} \big{(}\mathbf{G}^{\prime\prime}-\mathbf{G}^{\prime}\big{)}w_{\mathbf{k}}( \mathbf{G}^{\prime})=\underset{\mathbf{G}^{\prime}}{\sum}D_{*}\big{(}\mathbf{ G}^{\prime\prime}-\mathbf{G}^{\prime}\big{)}\Big{[}\big{(}\mathbf{k}+\mathbf{G}^{ \prime}\big{)}_{x}^{2}(\mathbf{k}+\mathbf{G}^{\prime\prime})_{x}^{2}+\big{(} \mathbf{k}+\mathbf{G}^{\prime}\big{)}_{y}^{2}(\mathbf{k}+\mathbf{G}^{\prime \prime})_{y}^{2}\big{]}w_{\mathbf{k}}(\mathbf{G}^{\prime})+\\ \underset{\mathbf{G}^{\prime}}{\sum}v_{*}\big{(}\mathbf{G}^{\prime\prime}- \mathbf{G}^{\prime}\big{)}\Big{[}\big{(}\mathbf{k}+\mathbf{G}^{\prime}\big{)} _{y}^{2}(\mathbf{k}+\mathbf{G}^{\prime\prime})_{x}^{2}+\big{(}\mathbf{k}+ \mathbf{G}^{\prime}\big{)}_{x}^{2}(\mathbf{k}+\mathbf{G}^{\prime\prime})_{y}^{ 2}\big{]}w_{\mathbf{k}}(\mathbf{G}^{\prime})+\\ \underset{\mathbf{G}^{\prime}}{\sum}\lambda_{*}\big{(}\mathbf{G}^{\prime\prime}- \mathbf{G}^{\prime}\big{)}\Big{[}\big{(}\mathbf{k}+\mathbf{G}^{\prime\prime} \big{)}_{x}^{2}+\big{(}\mathbf{k}+\mathbf{G}^{\prime\prime}\big{)}_{y}^{2} \big{]}V\big{(}\mathbf{G}^{\prime}\big{)}+\\ 2\underset{\mathbf{G}^{\prime}}{\sum}\chi_{*}\big{(}\mathbf{G}^{\prime\prime}- \mathbf{G}^{\prime}\big{)}\big{(}\mathbf{k}+\mathbf{G}^{\prime}\big{)}_{x} \big{(}\mathbf{k}+\mathbf{G}^{\prime}\big{)}_{y}(\mathbf{k}+\mathbf{G}^{ \prime\prime}\big{)}_{x}(\mathbf{k}+\mathbf{G}^{\prime\prime})_{y}w_{\mathbf{ k}}(\mathbf{G}^{\prime})\\ \end{array}  \tag{24}\]  
Similarly, Equation (14) can also be derived as Equation (25). (25)  
Finally, Equations (24) and (25) can be rearranged as the following standard eigenvalue equation. \[\bigg{\{}\bigg{[} \begin{array}{cc}\mathbf{E}&\mathbf{F}\\ 0&\mathbf{R}\end{array} \bigg{]}-\omega^{2}\bigg{[} \begin{array}{cc}\mathbf{ A}&0\\ \mathbf{B}&-\mathbf{C}\end{array} \bigg{]}\bigg{\}}\bigg{[} \begin{array}{c}w_{k} \big{(}\mathbf{G}^{\prime}\big{)}\\ V\big{(}\mathbf{G}^{\prime}\big{)}\end{array} \bigg{]}=0 \tag{26}\]  
where \(\begin{array}{c}\mathbf{E}-\underset{\mathbf{G}^{\prime}}{\sum}D_{*}( \mathbf{G}^{\prime\prime}-\mathbf{C})\big{[}\mathbf{k}+\mathbf{G}^{\prime} \big{)}_{x}^{2}(\mathbf{k}+\mathbf{G}^{\prime\prime})_{x}^{2}+\big{(}\mathbf{ k}+\mathbf{G}^{\prime\prime})_{  
(1) Definition of the objective function and the fitness: In order to achieve broadband and low-frequency vibration energy harvesting, we hope to maximize the width of the first-order bandgap.