## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Applied Local Temperature Fields
In particular, the third and fourth patterns of temperature loads create more localized potential barriers/wells, which can play a vital role in effectively driving motions and redistributions of electrons and holes, enabling the thermal regulation of PS behavior. Figure 6: Total distributions of extensional deformations \(u_{3}^{(0)}(x_{1},x_{3})\) under first (**a**), second (**b**), third (**c**), and fourth (**d**) patterns of temperature loads. Figure 7: Total distributions of electric potential barriers/wells \(\phi^{(0)}(x_{1},x_{3})\) under first (**a**), second (**b**), third (**c**), and fourth (**d**) patterns of temperature loads. Motions and Redistributions of Electrons and Holes  
As shown in Figures 8 and 9, the electric potential wells attract holes and expel electrons, and electric potential barriers attract electrons and expel holes, and the in-plane distributions and motions of electrons and holes described by \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\) are very sensitive to the electric potential barriers/wells. Taking Figures 8a and 9a as an example, the electrons are either attracted or expelled by electric potential barriers or electric potential wells. On the contrary, the holes are either expelled or attracted via electric potential barriers or electric potential wells. Meanwhile, \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\) have opposite signs within the plane of the thin film because of the electric potential \(\phi^{(0)}\) drives holes and electrons in opposite directions in the thin film. From the experimental analysis viewpoint, \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\) describe charges produced by temperature change. Therefore, they can be utilized as a method of measurement and measuring standard for coupling strength related to the interaction between motion, distribution, and transport behavior of mobile charges and applied thermal field excitations. Figure 8: Total distributions of electron concentration perturbations \(\Delta n^{(0)}(x_{1},x_{3})\) under first **(a)**, second **(b)**, third **(c)**, and fourth **(d)** patterns of temperature loads. Induced Local Drift and Diffusion Current Densities  
In PS thin-film device applications, the PS film structure is usually used as a circuit element. When the PS thin film device works under the action of local heat load, the local potential barriers/wells form and further affect the distribution of current density. Consequently, it is also necessary to study the influence mechanism of temperature load on current. According to the drift-diffusion theory of current density, as shown in Equation (13), the total current densities consist of two terms, i.e., the drift current term produced by the electric field driving the carrier motion and the diffusion current term produced by the carrier concentration gradient. Theoretically, by utilizing Equation (22), the current density characteristics of PS thin-film devices with extensional deformation can be described completely and accurately under the thermal field excitations. Numerically, it can be inferred from Equation (22) that the total current density related to the sum of drift and diffusion currents is equal to zero, and the drift and diffusion currents themselves are not zero. For this reason, taking holes as an example, our main interest is the drift and diffusion current density distributions under first, second, third, and fourth patterns of temperature loads. From the linear analysis viewpoint, the induced electromechanical fields depend largely on the applied thermal fields owing to the multi-field coupling effect. The reversal of applied loads causes a change in the sign of electromechanical fields, i.e., the load changes sign in either one or both directions, i.e., \(x_{1}\) or \(x_{3}\) directions, and this leads to sign changes in the extensional deformations, electric potential, electron and hole concentration perturbations, and drift and diffusion current densities accordingly.