## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Applied Local Temperature Fields  
In Figure 5, the total distributions of applied external stimulus fields \(\theta^{(0)}(x_{1},x_{3})\) for the first, second, third, and fourth patterns of temperature loads are displayed one by one, which are described by a piecewise constant function (i.e., Equation (19)) with some finite discontinuities at the edges of thermal loading areas, and thus, their trigonometric series representations show small Gibbs oscillations near discontinuities. Compared to the temperature field \(\theta^{(0)}(x_{1},x_{3})\), the other physical fields without Gibbs oscillations are continuous functions. Theoretically, from the viewpoints of thermoelastic, pyroelectric, and piezoelectric couplings, we can conclude from Equation (13) that a locally applied temperature load can produce extensional resultant forces \(T_{11}^{(0)}\) and \(T_{33}^{(0)}\) everywhere. Meanwhile, electric potential \(\phi^{(0)}\) is closely related to \(T_{11}^{(0)}\) and \(T_{33}^{(0)}\), and pyroelectric effect directly contributes to \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\), and thermoelastic coupling does so through piezoelectric coupling. Therefore, according to the coupled field Equation (16), the electromechanical fields, including 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 current densities \(\textbf{J}^{n(0)}\) and \(\textbf{J}^{p(0)}\), caused by local temperature field \(\theta^{(0)}\), can spread through the entire thin film, where extensional resultant forces \(T_{11}^{(0)}\) and \(T_{33}^{(0)}\) associated with higher \(\theta^{(0)}\) near thermal loading region center are larger. Figure 5: Cont. Induced Local Mechanical Displacement Fields  
To prove a static and local temperature load \(\theta^{(0)}\) can be used to produce and manipulate a local disturbance of electromechanical fields in a PS thin-film device, and considering the extensional deformations \(u_{3}^{(0)}\) as an example, in Figure 6, when local loads change from first to second, third, and fourth patterns of temperature loads, the extensional displacements \(u_{3}^{(0)}\) have more variations along the \(x_{1}\) and \(x_{3}\) directions. These are typical and familiar behaviors of higher-order modes. Clearly, in Equation (16), due to effective multi-field couplings via thermal stress modulus \(\lambda_{3}\), \(\lambda_{2}\), and \(\lambda_{1}\), pyroelectric modulus \(p_{3}\), and piezoelectric constants \(e_{33}\) and \(e_{31}\), it can be concluded that elastic deformations \(u_{1}^{(0)}\) and \(u_{3}^{(0)}\) are accompanied by the distributions of temperature field \(\theta^{(0)}\), electric potential \(\phi^{(0)}\), electron and hole concentration perturbations \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\), and current densities. Inspired by the above analytical results, various in-plane distribution patterns and local deformations of extensional modes \(u_{1}^{(0)}\) and \(u_{3}^{(0)}\) can be adjusted effectively by applying matched temperature load patterns. Figure 5: Total distributions of applied external stimulus fields \(\theta^{(0)}(x_{1},x_{3})\) for first (**a**), second (**b**), third (**c**), and fourth (**d**) patterns of temperature loads. Figure 6: _Cont_. Induced Local Electric Potential Barriers/Wells  
Next, from the practical applications of PS devices, via the external stimuli depicted in Figure 5, we concentrate on the thermal manipulation and control of electric potential barrier/well behavior. As shown in Figure 7, the potential barriers/wells are produced via temperature-induced polarization along the in-plane c axis or positive \(x_{3}\) axis direction. When the applied temperature load is local, the various induced potential barriers/wells are also localized.