## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Applied Local Temperature Fields
#### 4.3.4 Potential Barrier/Well Evolutions  
To achieve a deeper understanding of potential barrier/well evolutions versus different external loads, Figure 12 further analyzes qualitatively and quantitatively the mode shapes of electric potential \(\phi^{(0)}\) when \(x_{1}=a/2\). Accompanying the variations in temperature load patterns, the potential barrier/well number, location, and height/depth also change. To be specific, when the thermal load \(\theta^{(0)}(x_{1},x_{3})\) gradually changes from the first to the second pattern of temperature loads, it is found from curves 1 and 2 of Figure 12 that a pair of potential barriers/wells are reversed. This is as expected from a linear theory. Furthermore, when the applied external load field is set to be second and third patterns of temperature loads, respectively, from curves 2 and 3 of Figure 12, it can be seen that the locations of potential barrier/well shift obviously to the left/right of the \(x_{3}\) axis. Moreover, the potential barriers/wells become higher/deeper. By using this relation between patterns of temperature loads and electric potential \(\phi^{(0)}\), the excited potential barriers/wells and their locations can be easily predicted in the experimental design, which is of great significance for the rational optimization and design of PS thin-film devices. Under the fourth pattern of temperature loads, three pairs of potential barriers/wells exist in the \(x_{3}\) axis, and from curve 4, an optimal location corresponding to the highest potential barrier and the deepest potential well can be determined, respectively. Overall, by applying the different patterns of external stimuli, the potential barriers/wells can be manipulated artificially and dramatically enhanced near the loading areas. This is the foundation for the thermal manipulation of electric potential barriers/wells \(\phi^{(0)}\).  
Figure 11: Total distributions of diffusion current density vector plots \(\mathbf{J}^{p(0)}(x_{1},x_{3})\) with mobility constants \(\mu_{11}^{p}=\mu_{33}^{p}=0\) for holes under first (**a**), second (**b**), third (**c**), and fourth (**d**) patterns of temperature loads.  
Next, taking fourth pattern of temperature loads as an example, the dependence and sensibility of potential barriers/wells on various physical and geometric parameters (i.e., temperature load intensity parameter \(\theta_{0}^{(0)}\), initial-state electron and hole concentrations \(n_{0}=p_{0}\), and first rectangular element center coordinate \(x_{03}\)) are examined in Figure 13, which can provide theoretical guidance for potential barrier/well design and evolution. For example, in Figure 13a, as the load intensity \(\theta_{0}^{(0)}\) increases, the deformation of the PS thin film with a stronger temperature-induced polarization becomes larger, and hence, the potential barriers/wells become higher/deeper. In Figure 13b, as initial-state electron and hole concentrations \(n_{0}=p_{0}\) increase, more mobile charges screen temperature-induced effective polarization and cause a lower electric potential. These qualitative analyses and conclusions are physically correct. In Figure 13c, as the first rectangular element center coordinate \(x_{03}\) increases, the thermal loading areas become wider along the \(x_{3}\) direction, and the potential barriers/wells become farther apart from each other. This phenomenon is reasonable because the potential barriers/wells, caused by temperature-induced effective polarization charges, concentrate near the edges of loading areas.  
Figure 12: The potential barrier/well \(\phi^{(0)}(x_{1},x_{3})\) evolutions versus the first, second, third, and fourth patterns of temperature loads when \(x_{1}=a/2\).  
Figure 13: _Cont._