## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
\begin{array}{l}T_{ij,j}^{(n)}-nT_{i2}^{(n-1)}+F_{i}^{(n)}=\sum\limits_{m=0} ^{\infty}\rho B_{mn}u_{i}^{(m)},D_{i,i}^{(n)}-nD_{2}^{(n-1)}+D^{(n)}=\sum \limits_{m=0}^{\infty}qB_{mn}\Big{(}\Delta p^{(m)}-\Delta n^{(m)}\Big{)},\\ I_{i,i}^{n(n)}-nI_{2}^{n(n-1)}+J^{n(n)}=\sum\limits_{m=0}^{\infty}qB_{mn} \Delta\dot{n}^{(m)},I_{i,i}^{p(n)}-nI_{2}^{p(n-1)}+J^{p(n)}=-\sum\limits_{m=0} ^{\infty}qB_{mn}\Delta\dot{p}^{(m)},\\ H_{i,i}^{(n)}-nH_{2}^{(n-1)}+H^{(n)}=-\Theta_{0}\tilde{\eta}^{(n)}.\end{array}  \tag{8}\]  
here, \(T_{ij}^{(n)}\), \(D_{i}^{(n)}\), \(I_{i}^{n(n)}\), \(I_{i}^{p(n)}\), \(H_{i}^{(n)}\), and \(\eta^{(n)}\) stand for nth-order stress, electric displacement, electron and hole current densities, heat flux, and entropy density, respectively, and are defined by the following:  
\[\Big{\{}T_{ij}^{(n)},D_{i}^{(n)},J_{i}^{n(n)},J_{i}^{p(n)},H_{i}^{(n)},\eta^{( n)}\Big{\}}=\int_{-h}^{h}\Big{\{}T_{ij,j},D_{i},J_{i}^{n},J_{i}^{p},h_{i},\eta \Big{\}}x_{2}^{n}\mathrm{d}x_{2}. \tag{9}\]  
Additionally, various loads on the surface, i.e., \(F_{i}^{(n)}\), \(D^{(n)}\), \(J^{n(n)}\), \(J^{p(n)}\), and \(H^{(n)}\), as well as the existing integration constant \(B_{mn}\) in Equation (8), respectively, are substituted in the following equation:  
\[\Big{\{}F_{i}^{(n)},D^{(n)},J^{n(n)},J^{p(n)},H^{(n)}\Big{\}}=\Big{[}\Big{\{}T _{2i},D_{2},J_{2}^{n},J_{2}^{p},h_{2}\Big{\}}x_{2}^{n}\Big{]}_{-h}^{h}. \tag{10}\]  
\[B_{mn}=\int_{-h}^{h}x_{2}^{m+n}\mathrm{d}x_{2}=\Bigg{[}\frac{x_{2}^{m+n+1}}{m +n+1}\Bigg{]}_{-h}^{h}. \tag{11}\]  
Up to now, the 2D higher-order theory and equations (i.e., Equations (6) and (8)-(10)) of rectangular PS thin-film device structures have been established theoretically, in which the stress equation of motion (i.e., [PERSON]'s law), charge equation of electrostatics (i.e., [PERSON]'s law), conservation of charge for holes and electrons (i.e., Continuity equations), drift-diffusion theory for currents in semiconductors, and the unavoidable thermo-deformation-polarization-carrier coupling response effect in external stimulus field environment are all considered in the established 2D higher-order theory and equations. Application: Static Characteristic Analysis of Electromechanical Fields in Extensional Deformation of PS Thin-Film Device with Thermal Field Excitations  
In the present study, by employing the Hamilton's principle, the basic 3D linear elasticity equations of thermo-PS structures have been obtained. After that, with the aid of the Mindlin plate theory, a series of 2D equations is theoretically established, which can be used to analyze the static and dynamic behaviors of thermo-PS thin-film devices. Therefore, as a typical application of the above equations, taking into account widely-usedextensional deformation working modes of PS devices, the zeroth-order displacement modes \(u_{1}^{(0)}\) and \(u_{3}^{(0)}\) are considered by combining the established 2D nth-order equations (i.e., Equations (6) and (8)-(10)). Correspondingly, in the 2D rectangular PS thin-film structure of crystals of class (6 mm) with thermal field excitations, whose c axis is along the positive \(x_{3}\) direction, the theoretical solution for depicting static and local temperature load-induced electromechanical fields is derived with the aid of established zeroth-order equations and the double trigonometric series solution method. Sequentially, the systematic investigations, i.e., the active manipulation methodology and mechanism about the motion, distribution, and transport behavior of mobile charges (i.e., electrons and holes), as well as their evolution versus external stimulus field and various physical and geometric parameters, are carried out in detail.