## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Theoretical Solutions
Correspondingly, the blue region \(A_{b}\) with \(\theta^{(0)}(x_{1},x_{3})=-\theta_{0}^{(0)}\) or yellow region \(A_{y}\) with \(\theta^{(0)}(x_{1},x_{3})=+\theta_{0}^{(0)}\) stands for a lower or higher local and uniform temperature loading area compared to reference temperature area denoted by green \(A_{g}\) and other regions with \(\theta^{(0)}(x_{1},x_{3})=0\) K. Then, in the total PS thin film, a non-uniform temperature distribution, denoted by different colors of loading areas, can be exactly described and designed via Equation (19). This is just the design idea of temperature loading in this paper. It can be verified that Equation (17) automatically satisfies boundary conditions, i.e., Equation (15). Substituting Equation (17) into Equation (16) without other excitation fields(i.e., \(F_{1}^{(0)}=F_{3}^{(0)}=D^{(0)}=J^{n(0)}=J^{p(0)}=0\)) yields a system of linear algebraic equations, as shown below:  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}A_{mn}\cos (\xi_{m}x_{1})\sin(\eta_{n}x_{3})=0,\sum\limits_{m=1}^{\infty}\sum\limits_{n=1} ^{\infty}B_{mn}\sin(\xi_{m}x_{1})\cos(\eta_{n}x_{3})=0,\\ \sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}C_{mn}\sin(\xi_{m}x_{1}) \cos(\eta_{n}x_{3})=0,\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}D_{ mn}\sin(\xi_{m}x_{1})\cos(\eta_{n}x_{3})=0,\\ \sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}E_{mn}\sin(\xi_{m}x_{1}) \cos(\eta_{n}x_{3})=0.\end{array}  \tag{20}\]  
where  
\[\begin{array}{l}A_{mn}=\left[-\left(\overline{c}_{11}\xi_{m}^{2}+c_{55} \overline{\eta}_{n}^{2}\right)U_{mn}-(\overline{c}_{13}+c_{55})\bar{\xi}_{m} \eta_{n}W_{mn}-(\overline{c}_{31}+e_{15})\bar{\xi}_{m}\eta_{n}\Psi_{mn}- \overline{\lambda}_{11}\bar{\xi}_{m}F_{mn}\right],\\ B_{mn}=\left[-(c_{55}+\overline{c}_{13})\xi_{m}\eta_{n}U_{mn}-\left(c_{55} \xi_{m}^{2}+\overline{c}_{33}\eta_{n}^{2}\right)W_{mn}-\left(e_{15}\xi_{m}^{2 }+\overline{c}_{33}\eta_{n}^{2}\right)\Psi_{mn}-\overline{\lambda}_{33}\eta_{ n}F_{mn}\right],\\ C_{mn}=\left[ \begin{array}{l}-(e_{15}+\overline{c}_{31})\bar{\xi}_{m}\eta_{ n}U_{mn}-(e_{15}\xi_{m}^{2}+\overline{c}_{33}\eta_{n}^{2})W_{mn}+\left( \epsilon_{11}\xi_{m}^{2}+\overline{c}_{33}\eta_{n}^{2}\right)\[PERSON]_{mn}\\ +\overline{p}_{3}\eta_{n}F_{mn}+qN_{mn}-qP_{mn}\end{array} \right],\\ D_{mn}=\left[ \begin{array}{l}\eta_{0}\left(\mu_{11}^{\prime\prime}\xi_{m} ^{2}+\mu_{33}^{\prime\prime}\eta_{n}^{2}\right)\[PERSON]_{mn}-\left(D_{11}^{\prime \prime}\xi_{m}^{2}+D_{33}^{\prime\prime}\eta_{n}^{2}\right)N_{mn}\right],\\ E_{mn}=\left[\begin{array}{l}p_{0}\left(\mu_{11}^{\prime}\xi_{m}^{2}+\mu_{33 }^{\prime\prime}\eta_{n}^{2}\right)\Psi_{mn}+\left(D_{11}^{\prime}\xi_{m}^{2}+ D_{33}^{\prime\prime}\eta_{n}^{2}\right)P_{mn}\end{array}\right].\end{array}  \tag{21}\]  
In the following analysis, for a PS thin film that is excited by the applied thermal field, the undetermined constants \(U_{mn}\), \(W_{mn}\), \(\Psi_{mn}\), \(N_{mn}\), and \(P_{mn}\) can be obtained from Equation (20) with the aid of Equations (18) and (19). Accordingly, the various electromechanical fields can be calculated via Equation (17). To further reveal the effect of local temperature loading on current density near local loading areas, we introduce electron and hole current density vectors \(\mathbf{J}^{n(0)}\) and \(\mathbf{J}^{p(0)}\):  
\[\mathbf{J}^{n(0)}=J_{1}^{n(0)}\mathbf{e}_{1}+J_{3}^{n(0)}\mathbf{e}_{3},\ \mathbf{J}^{p(0)}=J_{1}^{p(0)}\mathbf{e}_{1}+J_{3}^{p(0)}\mathbf{e}_{3}. \tag{22}\]  
where \(\mathbf{e}_{1}\) and \(\mathbf{e}_{3}\) are unit vectors along positive \(x_{1}\) and \(x_{3}\) directions, respectively. Based on the theoretical solution established above, the basic mechanical analysis of a rectangular PS thin-film device with local temperature loading areas is carried out.