## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures  
The basic 2D equations of a rectangular PS thin-film structure with length \(2a\), width \(2b\), and thickness \(2h\), as well as electrons and holes, as shown in Figure 1, are derived using the Mindlin plate theory. Here, a rectangular coordinate system, denoted by \(Ox_{1}x_{2}x_{3}\), is set in a thin-film structure with the \(Ox_{1}x_{3}\) plane being the middle plane. For a finite thin-film structure shown in Figure 1, the thickness is much smaller than sizes in the other two directions, and referring to the Mindlin plate theorem, the mechanical displacement \(u_{i}\), electric potential \(\phi\), weak induced perturbations \(\Delta n\) and \(\Delta p\) for electron and hole concentrations, and temperature change field \(\theta\) can be expressed as the power series of thickness coordinate \(x_{2}\) as follows [50]:  
\[ \begin{split}& u_{i}(x_{1},x_{2},x_{3},t)=\sum\limits_{n=0}^{ \infty}\ x_{2}^{n}u_{i}^{(n)}(x_{1},x_{3},t),\phi(x_{1},x_{2},x_{3},t)=\sum \limits_{n=0}^{\infty}\ x_{2}^{n}\phi^{(n)}(x_{1},x_{3},t),\\ &\Delta n(x_{1},x_{2},x_{3},t)=\sum\limits_{n=0}^{\infty}\ x_{2}^{n} \Delta n^{(n)}(x_{1},x_{3},t),\Delta p(x_{1},x_{2},x_{3},t)=\sum\limits_{n=0} ^{\infty}\ x_{2}^{n}\Delta p^{(n)}(x_{1},x_{3},t),\\ &\theta(x_{1},x_{2},x_{3},t)=\sum\limits_{n=0}^{\infty}x_{2}^{n} \theta^{(n)}(x_{1},x_{3},t).\end{split}  \tag{4}\]  
in which \(i=1,2,3\), and \(t\) is the time variable. In addition to nth-order displacement \(u_{i}^{(n)}\), electrical potential \(\phi^{(n)}\), weak induced perturbations (i.e., \(\Delta n^{(n)}\) and \(\Delta p^{(n)}\)) about electron and hole concentrations, and temperature change function \(\theta^{(n)}\), here, it should be emphasized that \(u_{i}^{(0)}(i=1,3)\) and \(u_{2}^{(0)}\) are extensional and flexural displacements, respectively. And \(u_{i}^{(1)}(i=1,3)\) and \(u_{2}^{(1)}\) stand for thickness-shear and thickness-stretch displacements, respectively. Referring to Equation (4) and its expressions, the other physical fields, i.e., strain, electric field, weak induced perturbations gradient for electron and hole concentrations, and heat flux, can also be expanded into a similar power series form for thickness coordinate \(x_{2}\), and an example of this is given below:  
\[S_{ij}=\sum\limits_{n=0}^{\infty}S_{ij}^{(n)}x_{2}^{n},\ E_{i}=\sum\limits_{n=0 }^{\infty}E_{i}^{(n)}x_{2}^{n},\ N_{i}=\sum\limits_{n=0}^{\infty}N_{i}^{(n)}x _{2}^{n},\ P_{i}=\sum\limits_{n=0}^{\infty}P_{i}^{(n)}x_{2}^{n},\ h_{i}=\sum \limits_{n=0}^{\infty}h_{i}^{(n)}x_{2}^{n}. \tag{5}\]  
in which  
Figure 1: A rectangular PS thin-film structure with the dimensions of \(2a\times 2b\times 2h\) and mobile electrons and holes. \[ \begin{array}{l}S_{ij}^{(n)}=\frac{1}{2}\Big{[}u_{ij}^{(n)}+u_{jj}^{(n)}+(n+1) \Big{(}\delta_{2j}u_{i}^{(n+1)}+\delta_{2i}u_{j}^{(n+1)}\Big{)}\Big{]},E_{i}^{(n )}=-\Big{[}\phi_{j}^{(n)}+(n+1)\delta_{2i}\phi^{(n+1)}\Big{]},\\ N_{i}^{(n)}=\Big{[}\Delta n_{j}^{(n)}+(n+1)\delta_{2i}\Delta n^{(n+1)}\Big{]},P_ {i}^{(n)}=\Big{[}\Delta p_{j}^{(n)}+(n+1)\delta_{2i}\Delta p^{(n+1)}\Big{]},\\ h_{i}^{(n)}=-\kappa_{ij}\Big{[}\theta_{j}^{(n)}+(n+1)\delta_{2j}\theta^{(n+1) }\Big{]}.\end{array}  \tag{6}\]  
where \(S_{ij}^{(n)}\), \(E_{i}^{(n)}\), \(N_{i}^{(n)}\), \(P_{i}^{(n)}\), and \(h_{i}^{(n)}\) are the nth-order physical field quantities mentioned above. \(\delta_{ij}\) is Kronecker delta. Substituting Equation (4) into the following variational equation,  
\[\int_{V}\left\{ \begin{array}{l}\Big{(}T_{ij,j}-\rho\tilde{u_{i}}\Big{)} \delta u_{i}+\big{[}D_{i,i}-q(\Delta p-\Delta n)\big{]}\delta\phi+\\ \Big{(}I_{i,i}^{n}-q\Delta\tilde{n}\big{)}\delta\Delta n+\Big{(}I_{i,i}^{p}+q \Delta\tilde{p}\Big{)}\delta\Delta p+\big{(}h_{i,i}+\Theta_{0}\tilde{\eta} \big{)}\delta\theta\end{array} \right\}\mathrm{d}V=0. \tag{7}\]  
and setting \(\mathrm{d}V=\mathrm{d}x_{2}\mathrm{d}A\), the obtained dynamic equations with respect to the nth-order physical field components are as follows:  
\[