## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Theoretical Solutions
\begin{split} u^{(0)}_{3}=0,\ T^{(0)}_{11}=0,\ \phi^{(0)}=0,\ \Delta n^{(0)}=0,\ \Delta p^{(0)}=0,\ \text{at}\ x_{1}=0\ \text{and}\ a,\\ u^{(0)}_{1}=0,\ T^{(0)}_{33}=0,\ D^{(0)}_{3}=0,\ J^{n(0)}_{3}=0,\ J^{p(0)}_{3}=0, \ \text{at}\ x_{3}=0\ \text{and}\ b.\end{split}  \tag{15}\]  
Next, under applied thermal field excitations, substituting Equation (13) into the static form of Equation (12) yields the following:\[ \begin{split}&\overline{c}_{11}u_{1,11}^{(0)}+c_{55}u_{1,33}^{(0)}+( \overline{c}_{13}+c_{55})u_{3,13}^{(0)}+(c_{15}+\overline{c}_{31})\phi_{1,3}^{(0 )}-\overline{\lambda}_{1}\theta_{1,1}^{(0)}+\frac{1}{2h}F_{1}^{(0)}=0,\\ &(c_{55}+\overline{c}_{31})u_{1,3}^{(0)}+c_{55}u_{3,11}^{(0)}+ \overline{c}_{33}u_{3,33}^{(0)}+c_{15}\phi_{1,1}^{(0)}+\overline{c}_{33}\phi_{ 3,3}^{(0)}-\overline{\lambda}_{3}\phi_{3}^{(0)}+\frac{1}{2h}F_{3}^{(0)}=0,\\ &(e_{15}+\overline{c}_{31})u_{1,13}^{(0)}+e_{15}u_{3,11}^{(0)}+ \overline{c}_{33}u_{3,33}^{(0)}-\epsilon_{11}\phi_{1,1}^{(0)}-\overline{c}_{3 3}\phi_{3,3}^{(0)}+\overline{p}_{3}\phi_{3}^{(0)}+\frac{1}{2h}D^{(0)}=q\Big{(} \Delta p^{(0)}-\Delta n^{(0)}\Big{)},\\ &-n_{0}u_{1}^{n}\phi_{1,11}^{(0)}-n_{0}u_{3}^{n}\phi_{3,33}^{(0)} +D_{11}^{n}\Delta n_{11}^{(0)}+D_{33}^{n}\Delta n_{33}^{(0)}+\frac{1}{2h}J^{n (0)}=0,\\ &-p_{0}u_{1}^{p}\phi_{1,11}^{(0)}-p_{0}u_{3}^{p}\phi_{3,33}^{(0)} -D_{11}^{p}\Delta p_{1,1}^{(0)}-D_{33}^{p}\Delta p_{33}^{(0)}+\frac{1}{2h}J^{p (0)}=0.\end{split}  \tag{16}\]  
In Equation (16), thermal fields and charge carriers such as electrons and holes can interact in PS through thermally induced polarization or electric field. Without the loss of generality, the double trigonometric series solution of Equation (16) can be expressed as follows:  
\[ \begin{split}&\theta^{(0)}=\sum\limits_{m=1}^{\infty}\sum \limits_{n=1}^{\infty}F_{mn}\sin(\xi_{m}x_{1})\sin(\eta_{n}x_{3}),\\ & u_{1}^{(0)}=\sum\limits_{n=1}^{\infty}U_{mn}\cos(\xi_{m}x_{1}) \sin(\eta_{n}x_{3}),\\ &\Big{\{}u_{3}^{(0)},\phi^{(0)},\Delta n^{(0)},\Delta p^{(0)} \Big{\}}=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\{W_{mn},\Psi_{ mn},N_{mn},P_{mn}\}\sin(\xi_{m}x_{1})\cos(\eta_{n}x_{3}),\\ &\tilde{\chi}_{m}=\frac{m\pi}{a},\ \eta_{n}=\frac{n\pi}{b}.\end{split}  \tag{17}\]  
where \(U_{mn}\), \(W_{mn}\), \(\Psi_{mn}\), \(N_{mn}\), and \(P_{mn}\) are undetermined constants, and the coefficient of trigonometric series expansion \(F_{mn}\) is known, which is given by the applied temperature field \(\theta^{(0)}(x_{1},x_{3})\) as follows:  
\[F_{mn}=\frac{4}{ab}\iint_{A}\theta^{(0)}(x_{1},x_{3})\sin(\xi_{m}x_{1})\sin( \eta_{n}x_{3})\mathrm{d}x_{1}\mathrm{d}x_{3}. \tag{18}\]  
When a thermal field \(\theta^{(0)}(x_{1},x_{3})\) is applied along the \(Ox_{1}x_{3}\) plane of PS thin-film device, the coupled extensional deformations and electromechanical fields described by Equation (16) are generated owing to multi-field coupling effects. To reveal the effects of local temperature loadings on electromechanical fields, in Equation (18), a particular temperature load \(\theta^{(0)}(x_{1},x_{3})\) is taken into account by a piecewise constant function. \[\theta^{(0)}(x_{1},x_{3})=\left\{ \begin{array}{l}-\theta_{0}^{(0)}\,\ (x_{1},x_{3})\in A_{b},\\ +\theta_{0}^{(0)}\,\ (x_{1},x_{3})\in A_{y},\\ 0\ \mathrm{K},(x_{1},x_{3})\in A_{g},\text{and elsewhere}.\end{array} \right. \tag{19}\]  
in which \(\theta_{0}^{(0)}\geq 0\) is a very small temperature load intensity parameter. Classification discussion is as follows: (1) When \(\theta_{0}^{(0)}=0\), the PS thin film is assumed to be in a natural state, and corresponding concentrations are \(n_{0}\) and \(p_{0}\), respectively. (2) When \(\theta_{0}^{(0)}\
eq 0\), the local temperature rises or falls within the PS thin film, which leads to variations in electron and hole concentration perturbations \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\) owing to the interaction between strain, electric field, and concentration gradients of mobile charges. In this state, their concentrations change to \(p=p_{0}+\Delta p^{(0)}\) and \(n=n_{0}+\Delta n^{(0)}\), respectively.