## 5 Conclusions  
The 2D higher-order theory and equations for completely describing the mechanical behaviors of PS thin-film devices are established in this paper, in which [PERSON]'s law, [PERSON]'s law, Continuity equations, drift-diffusion theory, and unavoidable thermo-deformation-polarization-carrier coupling response are all included. The derived equations are general and widely applicable to macroscopic mechanical analysis, which can be degenerated to some classical cases, including piezoelectric and elastic thin-film structures. Based on this, taking PS thin-film device with static extensional deformations as an example, 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. After convergence validation and validity and accuracy analysis for the current theoretical model, the systematic investigations are carried out sequentially. Finally, some qualitative conclusions are drawn as follows:  
1. The electromechanical fields (i.e., extensional deformations \(u_{1}^{(0)}\) and \(u_{3}^{(0)}\), electric potential \(\phi^{(0)}\), electron and hole concentration perturbations \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\), and their current densities) are temperature-dependent due to the fact that thermoelastic,  
Figure 13: The potential barrier/well \(\phi^{(0)}(x_{1},x_{3})\) evolutions versus various (**a**) load intensity parameters \(\theta_{0}^{(0)}\); (**b**) initial-state electron and hole concentrations \(n_{0}=p_{0}\); (**c**) the first rectangular element center coordinates \(x_{03}\) under the fourth pattern of temperature loads and \(x_{1}=a/2\).  
pyroelectric, and piezoelectric couplings exist in PS thin-film structures, whose evolution can be actively controlled via artificially tuning external stimulus thermal fields, meaning that the sensitivity of a PS device can be improved.
2. In a PS thin-film device, the potential barriers/wells have great impacts on in-plane distributions and motions of electrons and holes, as well as their drift and diffusion currents. To be specific, potential barriers attract diffusion currents and expels drift currents, but the potential well attracts drift currents and expels diffusion currents.
3. Under the fourth pattern of temperature loads, as expected, a higher temperature rise can induce a higher potential barrier and a deeper potential well. Moreover, when there are more electrons and holes with a higher \(n_{0}=p_{0}\), the mobile charges have a stronger screening effect on electric potential. Hence, the potential barriers/wells become lower/shallower, which are partially screened by mobile charges.  
Overall, the present work can be regarded as a useful theoretical tool for accurately depicting the physical and mechanical properties of PS thin-film materials and devices, which can deepen the interpretation of underlying mechanisms. The quantitative outcomes calculated in this paper offer us basic guidelines for the structural design and engineering applications of multi-functional PS thin-film devices.  
Conceptualization, H.H. and W.W.; methodology, M.Z. and X.L.; software, L.Z.; validation, X.L.; formal analysis, M.Z. and M.L.; writing--original draft, H.H. and W.W.; writing--review and editing, M.Z. and M.L.; supervision, W.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (11972176).  
All data generated or analyzed during this study are included in this article. The raw data are available from the corresponding author upon reasonable request.  
The authors thank all the participants in this study for their time and willingness to share their experiences and feelings.  
The authors declare no conflicts of interest.