Two-Dimensional Linear Elasticity Equations of Thermo-Piezoelectric Semiconductor Thin-Film Devices and Their Application in Static Characteristic Analysis

Abstract

Based on the three-dimensional (3D) linear elasticity theory of piezoelectric semiconductor (PS) structures, inspired by the variational principle and the Mindlin plate theory, a two-dimensional (2D) higher-order theory and equations for thin-film devices are established for a rectangular coordinate system, in which Newton’s law (i.e., stress equation of motion), Gauss’s law (i.e., charge equation of electrostatics), Continuity equations (i.e., conservation of charge for holes and electrons), drift–diffusion theory for currents in semiconductors, and unavoidable thermo-deformation-polarization-carrier coupling response in external stimulus field environment are all considered. As a typical application of these equations, the static characteristic analysis of electromechanical fields for the extensional deformation of a PS thin-film device with thermal field excitations is carried out by utilizing established zeroth-order equations and the double trigonometric series solution method. It is revealed that the extensional deformations, electric potential, electron and hole concentration perturbations, and their current densities can be controlled actively via artificially tuning thermal fields of external stimuli. Especially, a higher temperature rise can induce a deeper potential well and a higher potential barrier, which can play a vital role in driving effectively motions and redistributions of electrons and holes. Overall, the derived 2D equations as well as the quantitative results provide us some useful guidelines for investigating the thermal regulation behavior of PS thin-film devices.

1. Introduction

With the improvement in material processing technology and development of 3D printing, various new PS materials and their one-dimensional (1D) and 2D structures, including fibers, spirals, helices, belts, tubes, and films, have been developed and synthesized using the so-called third-generation semiconductor materials represented by ZnO, GaN, and MoS₂ because of their excellent features, i.e., a wider band gap, higher thermal conductivity, and a higher electron saturation rate [1,2]. Different from conventional semiconductors, in these PS materials and structures, mechanical fields can produce electric polarization through piezoelectric coupling, which acts on mobile holes and electrons and affects their motion, distribution, and transport behavior [3,4]. This unique interaction among mechanical fields, polarization, and mobile charges is the main characteristic of PS, which can be used to develop novel electronic devices. Furthermore, the investigation of PS devices is growing rapidly in the form of single structures or arrays. Based on their unique structures and coupling of semiconductor and piezoelectricity properties, an increasing number of PS multifunctional devices (such as, nanogenerators, piezotronic acoustic tweezers, field-effect transistors, logic nanodevices, piezotronic and chemical sensors, etc.) have been successfully developed and applied in phototronics and electronics [1,2].

For various PS devices, the basic behavior can be described through a coupled-field phenomenological theory [5], consisting of the theory of linear piezoelectricity, the conservation of charge for electrons and holes, and the drift–diffusion theory of semiconductors. Here, it should be stressed that the macroscopic theory of PS is inherently nonlinear because of drift current terms, i.e., products of unknown electric field and carrier concentrations [6]. Especially, from material physics to mechanics, the theoretical analyses of PS structures and devices are highly interdisciplinary and multi-physical, which have formed new research areas called piezotronics and piezophototronics [1,2], simultaneously presenting sizeable mechanical and mathematical challenges.

Due to the coexistence of semiconduction and piezoelectricity of PS materials and structures, the multi-field coupling interactions related to physical and mechanical mechanisms of novel PS structures and devices between the mechanical deformation, electric polarization, and carriers require a comprehensive understanding from the perspective of device research and development and applied science. Here, some representative research studies and results are listed and reported one by one from the existing literature. For example, in PS materials with advanced structures and unique features, to reveal the novel indentation characteristics induced via the interaction between semiconduction and piezoelectric coupling, Gao and his coauthors [7] investigated the indentation responses of a PS half-space subjected to an insulating and rigid spherical indenter, and the theoretical and numerical results indicated that the existence of semiconducting property (i.e., steady carrier concentration) can decrease the indentation resistance of PS materials and further affect the indentation force and electric potential. By considering the principle of virtual work, Cao et al. [8,9] established PS circular cylindrical shell models based on the Kirchhoff–Love assumption and the first-order shear deformation theory, analyzed the static bending and forced vibration problems, and then obtained the fundamental coupling relations between mechanical deformation, piezoelectric polarization, and mobile charges. In Ref. [10], by simultaneously considering the imperfect interface, piezoelectricity, and flexoelectricity in a PS bilayer composite beam system subjected to uniform temperature variation, Ren et al. obtained the closed-form solutions for the electromechanical fields based on the Newmark model, in which the effects of interfacial parameter, flexoelectricity, and initial carrier concentration have been taken into account. To capture the electric nonlinear property between the electric field and carriers that is ignored in most of the related studies, a novel size-dependent nonlinear model of PS nanofiber with asymmetric crystal structures related to flexoelectricity and semiconducting properties was developed by Yang et al. [11]. In addition to the above existing efforts, a groundbreaking theoretical framework about the thermoelectric effects of PS was also taken into consideration in the existing literature [12]. From the view of mechanics, to comprehensively understand the underlying mechanism of PS macroscopic structures, a series of other mechanics problems has also been assessed by an increasing number of researchers from the mechanics community, such as transient analysis of surface-heated PS plate [13], dynamic wave particle drag effect investigation related to free carrier (i.e., electron and hole) motion and flexoelectric waves [14], wave propagation [15,16,17,18,19,20,21,22,23], static torsion of PS and flexoelectric semiconductor rod [24,25], surface effect [26,27,28], warping deformations [29], nonlinear analysis [30,31], analytical solutions related to stress-induced coupled extensional and thickness-stretch deformations and electric potential barriers in PS plate [32], vibration of PS plate [33], buckling modeling and critical load analysis of flexoelectric silicon semiconductor beams [34], semi-analytical solutions related to power series expansion method for composite PS fibers and PN junctions [35], 2D theoretical modeling of transport behavior and redistributions of mobile charges in thermo-PS plates [36], double trigonometric series solution method for electromechanical behaviors of PS and flexoelectric semiconductor plate [37,38], 1D theoretical model for interaction between static and dynamic bending deformations and electromechanical fields in composite flexoelectric semiconductor beams [39], electromechanical field distribution and evolution in non-uniform PS fibers [40,41], thermally induced electric potential distribution and free carrier redistribution [42], cracks [43,44], modulation of local electromechanical coupling field distributions in PS structures [45,46,47], and other modulation methodologies of PS devices [48,49]. Clearly, the theoretical development of PS structures with rich results in both fundamental theory and applied science has been rapid, and the comprehensive studies of mechanical behavior of PS have become already an active area in the field of theoretical and applied mechanics, which can provide a theoretical basis for structural design, performance optimization, and device fabrication.

Compared to 1D PS fibers, an increasing number of PS thin-film devices are emerging with the development of 2D materials. Usually, these multi-field coupled materials and devices work in a complex environment, such as coupled mechanical, thermal, electrical, and semiconductive fields. Owing to material anisotropy, these structural devices inevitably exhibit obvious thermo-deformation-polarization-carrier coupling characteristics [27,28], which creates a great challenge of exactly calculating static and dynamic properties and simulating their working performances. From material physics to mechanics, the theoretical analyses of PS structures and devices are highly interdisciplinary and multi-physical, which simultaneously present sizeable mechanical and mathematical challenges. So far, despite the existing efforts, to the best of authors’ knowledge, the understanding of the 2D higher-order theory of PS thin-film device structures considering the thermo-deformation-polarization-carrier coupling effect is still relatively low and scattered. Especially, in a 2D PS thin-film device with thermal field excitations, for active manipulation methodology and mechanism of electromechanical fields (including deformations, electric potential, potential barriers/wells, electron and hole concentrations, drift and diffusion currents, and their evolution versus external stimulus field and various physical and geometric parameters), there seem to be few reported results. This is precisely the deficiency of the current research situation and research motivation of this article. Considering this, our research aims to fill the gap in the 2D higher-order theory of PS thin-film device structure space and provide an effective and reliable theoretical solution. Additionally, the obtained theory is used to explain and analyze above-mentioned mechanical and electrical behaviors as well as mechanical tuning methodology related to extensional deformations of 2D PS thin-film device with thermal field excitations, which is just the main highlight, academic contribution, and innovation point of this work, and significantly different from the existing research studies.

For solving the above bottleneck and exploring the local modulation and evolution of electromechanical fields caused by thermal field excitations, an essential, insightful, and comprehensive study, titled “Two-dimensional linear elasticity equations of thermo-piezoelectric semiconductor thin-film devices and their application in static characteristic analysis”, is carried out. The outline of this paper is organized as follows. In Section 2, the basic 3D linear elasticity equations for PS structures have been obtained in the rectangular coordinate system. After that, by using the Mindlin plate theory, the 2D higher-order theory and equations of PS thin-film structure have been established theoretically in Section 3, which can be used to completely describe the mechanical behaviors of the thermo-PS thin-film devices from the viewpoints of thermoelastic, pyroelectric, and piezoelectric couplings. These 2D equations not only can be used to investigate the macroscopic behaviors of thermo-piezoelectric (if both diffusion and mobility constants of holes and electrons, and uniform electron and hole concentrations in the reference state, are ignored) or elastic (if piezoelectric constants are ignored) thin-film devices, but also can be used to analyze the static and dynamic behaviors of conventional PS devices by neglecting heat conduction effect. Therefore, in the present contribution, the established 2D higher-order theory and equations of PS thin-film structure are more general and suitable for many cases for the design of the smart devices. Subsequently, as a typical application, based on the reduced 2D equations (i.e., zeroth-order equations), the mechanical tuning methodology of electromechanical fields in extensional deformation of PS thin-film device with thermal field excitations is systematically investigated in Section 4. Meanwhile, some quantitative simulations and qualitative discussions about mechanical tuning methodology and evolution related to electromechanical fields are performed. Finally, we summarize some of the main conclusions in Section 5.

2. 3D Linear Elasticity Theory of Thermo-PS Structures

To accurately depict physical and mechanical properties, and further reveal the underlying physics of PS structures, a general theoretical framework, including dynamic governing equations and constitutive equations, is given first in this section [13,36,50]:

$$T_{ji,j}=\rho {\ddot{u}}_i,D_{i,i}=q\left(\Delta p-\Delta n\right),J_{i,i}^n=q\Delta \dot{n},J_{i,i}^p=-q\Delta \dot{p},h_{i,i}=-{\Theta }_0\dot{\eta }.$$

(1)

in which $i,j=1,2,3$. $T_{ij}$ is the component of stress tensor related to mechanical displacement $u_i$ and electric displacement $D_i$. The mass density and elementary charge are denoted by $\rho$ and $q$. $\Delta n$ and $\Delta p$ stand for weak induced perturbations of electron and hole concentrations, respectively. $J_i^n$ and $J_i^p$ denote electron and hole current densities, respectively. It should be emphasized that $h_i$, ${\Theta }_0$, and $\eta$ are heat flux, reference temperature, and entropy density, respectively. Accompanying Equation (1), the corresponding constitutive equations can be written in the following form:

$$\begin{array}{l}{T}_{ij}=c_{ijkl}{S}_{kl}-e_{kij}{E}_k-{\lambda }_{ij}\theta ,D_i=e_{ikl}{S}_{kl}+{\epsilon }_{ik}{E}_k+p_i\theta ,\\ J_i^n=q{n}_0{\mu }_{ij}^n{E}_j+q{D}_{ij}^n{N}_j,J_i^p=q{p}_0{\mu }_{ij}^p{E}_j-q{D}_{ij}^p{P}_j,\\ \eta ={\lambda }_{kl}{S}_{kl}+p_k{E}_k+\alpha \theta .\end{array}$$

(2)

where $i,j,k,l=1,2,3$, and the above multi-field coupling material constants are depicted by elastic $c_{ijkl}$, piezoelectric $e_{kij}$, and dielectric ${\epsilon }_{ik}$ constants, respectively. $n_0$ and $p_0$ denote uniform electron and hole concentrations in the reference state, respectively. $D_{ij}^p$($D_{ij}^n$) and ${\mu }_{ij}^p$(${\mu }_{ij}^n$) are diffusion and mobility constants of holes (electrons), respectively. It should be stressed that ${\lambda }_{ij}$ and $p_i$ are thermal stress and pyroelectric moduli, respectively. Compared to absolute temperature $\Theta$ and reference temperature ${\Theta }_0$, $\alpha$ related to specific heat $C_{\upsilon }^E$ is depicted by $\alpha =\rho C_{\upsilon }^E{\Theta }_0^{-1}$, and $\theta =\Theta -{\Theta }_0$ represents a small temperature change, i.e., $\left|\theta \right|\ll {\Theta }_0$. Additionally, $S_{ij}$, $E_i$, $N_j$, $P_j$, and $h_i$ denote strain, electric field, weak induced perturbation concentration gradients for electrons and holes, and heat flux related to heat conduction coefficient ${\kappa }_{ij}$, respectively, and are defined as follows:

$$S_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right),E_i=-{\varphi }_{,i},N_i=\Delta n_{,i},P_i=\Delta p_{,i},h_i=-{\kappa }_{ij}{\theta }_{,j}.$$

(3)

Until now, the basic 3D linear elasticity equations for thermo-PS structures have been obtained in a rectangular coordinate system, which can be used to theoretically analyze multi-field coupling effects. Mathematically, the coupled field equations of PS thin-film devices are very complicated such that an exact theoretical solution from 3D equations can hardly be obtained in most cases, even when mode coupling characteristics are not considered. Fortunately, the Mindlin plate theory [50] concerning rotatory inertia and shear deformations is proposed. This may provide some opportunities and hints for us to solve similar problems in terms of computational methods.

3. Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures

The basic 2D equations of a rectangular PS thin-film structure with length $2a$, width $2b$, and thickness $2h$, as well as electrons and holes, as shown in Figure 1, are derived using the Mindlin plate theory. Here, a rectangular coordinate system, denoted by $O{x}_1{x}_2{x}_3$, is set in a thin-film structure with the $O{x}_1{x}_3$ plane being the middle plane.

For a finite thin-film structure shown in Figure 1, the thickness is much smaller than sizes in the other two directions, and referring to the Mindlin plate theorem, the mechanical displacement $u_i$, electric potential $\varphi$, weak induced perturbations $\Delta n$ and $\Delta p$ for electron and hole concentrations, and temperature change field $\theta$ can be expressed as the power series of thickness coordinate $x_2$ as follows [50]:

$$\begin{array}{l}{u}_i\left(x_1,x_2,x_3,t\right)={\displaystyle \sum _{n=0}^{\infty }{x}_2^n}{u}_i^{\left(n\right)}\left(x_1,x_3,t\right),\varphi \left(x_1,x_2,x_3,t\right)={\displaystyle \sum _{n=0}^{\infty }{x}_2^n}{\varphi }^{\left(n\right)}\left(x_1,x_3,t\right),\\ \Delta n\left(x_1,x_2,x_3,t\right)={\displaystyle \sum _{n=0}^{\infty }{x}_2^n}\Delta n^{\left(n\right)}\left(x_1,x_3,t\right),\Delta p\left(x_1,x_2,x_3,t\right)={\displaystyle \sum _{n=0}^{\infty }{x}_2^n}\Delta p^{\left(n\right)}\left(x_1,x_3,t\right),\\ \theta \left(x_1,x_2,x_3,t\right)={\displaystyle \sum _{n=0}^{\infty }{x}_2^n{\theta }^{\left(n\right)}\left(x_1,x_3,t\right)}.\end{array}$$

(4)

in which $i=1,2,3$, and $t$ is the time variable. In addition to nth-order displacement $u_i^{\left(n\right)}$, electrical potential ${\varphi }^{\left(n\right)}$, weak induced perturbations (i.e., $\Delta n^{\left(n\right)}$ and $\Delta p^{\left(n\right)}$) about electron and hole concentrations, and temperature change function ${\theta }^{\left(n\right)}$, here, it should be emphasized that $u_i^{\left(0\right)}\left(i=1,3\right)$ and $u_2^{\left(0\right)}$ are extensional and flexural displacements, respectively. And $u_i^{\left(1\right)}\left(i=1,3\right)$ and $u_2^{\left(1\right)}$ stand for thickness-shear and thickness-stretch displacements, respectively.

Referring to Equation (4) and its expressions, the other physical fields, i.e., strain, electric field, weak induced perturbations gradient for electron and hole concentrations, and heat flux, can also be expanded into a similar power series form for thickness coordinate $x_2$, and an example of this is given below:

$$S_{ij}={\displaystyle \sum _{n=0}^{\infty }{S}_{ij}^{\left(n\right)}{x}_2^n},E_i={\displaystyle \sum _{n=0}^{\infty }{E}_i^{\left(n\right)}{x}_2^n},N_i={\displaystyle \sum _{n=0}^{\infty }{N}_i^{\left(n\right)}{x}_2^n},P_i={\displaystyle \sum _{n=0}^{\infty }{P}_i^{\left(n\right)}{x}_2^n},h_i={\displaystyle \sum _{n=0}^{\infty }{h}_i^{\left(n\right)}{x}_2^n}.$$

(5)

in which

$$\begin{array}{l}{S}_{ij}^{\left(n\right)}={\displaystyle \frac{1}{2}}\left[u_{i,j}^{\left(n\right)}+u_{j,i}^{\left(n\right)}+\left(n+1\right)\left({\delta }_{2j}{u}_i^{\left(n+1\right)}+{\delta }_{2i}{u}_j^{\left(n+1\right)}\right)\right],E_i^{\left(n\right)}=-\left[{\varphi }_{,i}^{\left(n\right)}+\left(n+1\right){\delta }_{2i}{\varphi }^{\left(n+1\right)}\right],\\ N_i^{\left(n\right)}=\left[\Delta n_{,i}^{\left(n\right)}+\left(n+1\right){\delta }_{2i}\Delta n^{\left(n+1\right)}\right],P_i^{\left(n\right)}=\left[\Delta p_{,i}^{\left(n\right)}+\left(n+1\right){\delta }_{2i}\Delta p^{\left(n+1\right)}\right],\\ h_i^{\left(n\right)}=-{\kappa }_{ij}\left[{\theta }_{,j}^{\left(n\right)}+\left(n+1\right){\delta }_{2j}{\theta }^{\left(n+1\right)}\right].\end{array}$$

(6)

where $S_{ij}^{\left(n\right)}$, $E_i^{\left(n\right)}$, $N_i^{\left(n\right)}$, $P_i^{\left(n\right)}$, and $h_i^{\left(n\right)}$ are the nth-order physical field quantities mentioned above. ${\delta }_{ij}$ is Kronecker delta.

Substituting Equation (4) into the following variational equation,

$${\displaystyle {\int }_V\left\{\begin{array}{l}\left(T_{ji,j}-\rho {\ddot{u}}_i\right)\delta u_i+\left[D_{i,i}-q\left(\Delta p-\Delta n\right)\right]\delta \varphi +\\ \left(J_{i,i}^n-q\Delta \dot{n}\right)\delta \Delta n+\left(J_{i,i}^p+q\Delta \dot{p}\right)\delta \Delta p+\left(h_{i,i}+{\Theta }_0\dot{\eta }\right)\delta \theta \end{array}\right\}\mathrm{d}V=0.}$$

(7)

and setting $\mathrm{d}V=\mathrm{d}{x}_2\mathrm{d}A$, the obtained dynamic equations with respect to the nth-order physical field components are as follows:

$$\begin{array}{l}{T}_{ij,j}^{\left(n\right)}-n{T}_{i2}^{\left(n-1\right)}+F_i^{\left(n\right)}={\displaystyle \sum _{m=0}^{\infty }\rho B_{mn}{\ddot{u}}_i^{\left(m\right)}},D_{i,i}^{\left(n\right)}-n{D}_2^{\left(n-1\right)}+D^{\left(n\right)}={\displaystyle \sum _{m=0}^{\infty }q{B}_{mn}\left(\Delta p^{\left(m\right)}-\Delta n^{\left(m\right)}\right),}\\ J_{i,i}^{n\left(n\right)}-n{J}_2^{n\left(n-1\right)}+{J^n}^{\left(n\right)}={\displaystyle \sum _{m=0}^{\infty }q{B}_{mn}\Delta {\dot{n}}^{\left(m\right)}},J_{i,i}^{p\left(n\right)}-n{J}_2^{p\left(n-1\right)}+{J^p}^{\left(n\right)}=-{\displaystyle \sum _{m=0}^{\infty }q{B}_{mn}\Delta {\dot{p}}^{\left(m\right)},}\\ H_{i,i}^{\left(n\right)}-n{H}_2^{\left(n-1\right)}+H^{\left(n\right)}=-{\Theta }_0{\dot{\eta }}^{\left(n\right)}.\end{array}$$

(8)

here, $T_{ij}^{\left(n\right)}$, $D_i^{\left(n\right)}$, $J_i^{n\left(n\right)}$, $J_i^{p\left(n\right)}$, $H_i^{\left(n\right)}$, and ${\eta }^{\left(n\right)}$ stand for nth-order stress, electric displacement, electron and hole current densities, heat flux, and entropy density, respectively, and are defined by the following:

$$\left\{T_{ij}^{\left(n\right)},D_i^{\left(n\right)},J_i^{n\left(n\right)},J_i^{p\left(n\right)},H_i^{\left(n\right)},{\eta }^{\left(n\right)}\right\}={\displaystyle {\int }_{-h}^h\left\{T_{ij},D_i,J_i^n,J_i^p,h_i,\eta \right\}{x}_2^n\mathrm{d}{x}_2}.$$

(9)

Additionally, various loads on the surface, i.e., $F_i^{\left(n\right)}$, $D^{\left(n\right)}$, ${J^n}^{\left(n\right)}$, ${J^p}^{\left(n\right)}$, and $H^{\left(n\right)}$, as well as the existing integration constant $B_{mn}$ in Equation (8), respectively, are substituted in the following equation:

$$\left\{F_i^{\left(n\right)},D^{\left(n\right)},{J^n}^{\left(n\right)},{J^p}^{\left(n\right)},H^{\left(n\right)}\right\}={\left[\left\{T_{2i},D_2,J_2^n,J_2^p,h_2\right\}{x}_2^n\right]}_{-h}^h.$$

(10)

$$B_{mn}={\displaystyle {\int }_{-h}^h{x}_2^{m+n}}\mathrm{d}{x}_2={\left[{\displaystyle \frac{x_2^{m+n+1}}{m+n+1}}\right]}_{-h}^h.$$

(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., Newton’s law), charge equation of electrostatics (i.e., Gauss’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.

4. 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-used extensional deformation working modes of PS devices, the zeroth-order displacement modes $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$ 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.

4.1. Design for Loading Regions of Static and Local Temperature Loads

As shown in Figure 2, according to the applied external stimulus fields related to design for loading regions of static and local temperature loads in the rectangular PS thin-film with dimensions $a\times b\times 2h$, the applied loads are divided into four types, which are defined as the first, second, third, and fourth patterns of temperature loads, respectively, for convenience. Specifically, in Figure 2a (or Figure 2b), the applied load type with ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=-{\theta }_0^{\left(0\right)}$ (or ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=+{\theta }_0^{\left(0\right)}$) in the blue $A_b$ (or yellow $A_y$) regions of PS thin-film with dimensions $a\times b\times 2h$ is defined as the first (or second) patterns of temperature loads. Similarly, in Figure 2c (or Figure 2d), the third (or fourth) patterns of temperature loads are described by ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=-{\theta }_0^{\left(0\right)}$ in the blue $A_b$ and ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=+{\theta }_0^{\left(0\right)}$ in the yellow $A_y$ regions. Here, it should be emphasized that ${\theta }_0^{\left(0\right)}\ge 0$ is a very small temperature load intensity parameter. For every rectangular element with same dimensions $2c\times 2d\times 2h$, it can be seen from Figure 2 that $2c$ and $2d$ are known from the first rectangular element center coordinate $\left(x_{01},x_{03}\right)$ via $2c=\left(a-2x_{01}\right)/4$ and $2d=\left[b-2\left(b-x_{03}\right)\right]/4$.

4.2. Theoretical Solutions

The relevant zeroth-order divergence equations, which are sufficient to predict extensional deformation-dependent electromechanical fields, can be obtained from the theory established in the previous section

$$\begin{array}{l}{T}_{11,1}^{\left(0\right)}+T_{13,3}^{\left(0\right)}+F_1^{\left(0\right)}=2h\rho {\ddot{u}}_1^{\left(0\right)},T_{31,1}^{\left(0\right)}+T_{33,3}^{\left(0\right)}+F_3^{\left(0\right)}=2h\rho {\ddot{u}}_3^{\left(0\right)},\\ D_{1,1}^{\left(0\right)}+D_{3,3}^{\left(0\right)}+D^{\left(0\right)}=2hq\left(\Delta p^{\left(0\right)}-\Delta n^{\left(0\right)}\right),J_{1,1}^{n\left(0\right)}+J_{3,3}^{n\left(0\right)}+{J^n}^{\left(0\right)}=2hq\Delta {\dot{n}}^{\left(0\right)},\\ J_{1,1}^{p\left(0\right)}+J_{3,3}^{p\left(0\right)}+{J^p}^{\left(0\right)}=-2hq\Delta {\dot{p}}^{\left(0\right)}.\end{array}$$

(12)

Accompanying Equation (12), the zeroth-order constitutive equations in terms of displacements, electrical potential, and weak induced perturbations of hole and electron concentrations are as follows:

$$\begin{array}{l}{T}_{11}^{\left(0\right)}=2h\left({\overline{c}}_{11}{u}_{1,1}^{\left(0\right)}+{\overline{c}}_{13}{u}_{3,3}^{\left(0\right)}+{\overline{e}}_{31}{\varphi }_{,3}^{\left(0\right)}-{\overline{\lambda }}_1{\theta }^{\left(0\right)}\right),T_{13}^{\left(0\right)}=2h\left(c_{55}\left(u_{1,3}^{\left(0\right)}+u_{3,1}^{\left(0\right)}\right)+e_{15}{\varphi }_{,1}^{\left(0\right)}\right),\\ T_{33}^{\left(0\right)}=2h\left({\overline{c}}_{31}{u}_{1,1}^{\left(0\right)}+{\overline{c}}_{33}{u}_{3,3}^{\left(0\right)}+{\overline{e}}_{33}{\varphi }_{,3}^{\left(0\right)}-{\overline{\lambda }}_3{\theta }^{\left(0\right)}\right),D_3^{\left(0\right)}=2h\left({\overline{e}}_{31}{u}_{1,1}^{\left(0\right)}+{\overline{e}}_{33}{u}_{3,3}^{\left(0\right)}-{\overline{\epsilon }}_{33}{\varphi }_{,3}^{\left(0\right)}+{\overline{p}}_3{\theta }^{\left(0\right)}\right),\\ D_1^{\left(0\right)}=2h\left(e_{15}\left(u_{1,3}^{\left(0\right)}+u_{3,1}^{\left(0\right)}\right)-{\epsilon }_{11}{\varphi }_{,1}^{\left(0\right)}\right),J_1^{n\left(0\right)}=2h\left(-q{n}_0{\mu }_{11}^n{\varphi }_{,1}^{\left(0\right)}+q{D}_{11}^n\Delta n_{,1}^{\left(0\right)}\right),\\ J_3^{n\left(0\right)}=2h\left(-q{n}_0{\mu }_{33}^n{\varphi }_{,3}^{\left(0\right)}+q{D}_{33}^n\Delta n_{,3}^{\left(0\right)}\right),J_1^{p\left(0\right)}=2h\left(-q{p}_0{\mu }_{11}^p{\varphi }_{,1}^{\left(0\right)}-q{D}_{11}^p\Delta p_{,1}^{\left(0\right)}\right),\\ J_3^{p\left(0\right)}=2h\left(-q{p}_0{\mu }_{33}^p{\varphi }_{,3}^{\left(0\right)}-q{D}_{33}^p\Delta p_{,3}^{\left(0\right)}\right).\end{array}$$

(13)

in which, the effective material constants related to the so-called stress relaxation of PS thin-film structure are defined as follows:

$$\begin{array}{l}{\overline{c}}_{11}=c_{11}-{\displaystyle \frac{c_{12}{c}_{21}}{c_{22}}},{\overline{c}}_{13}=c_{13}-{\displaystyle \frac{c_{12}{c}_{23}}{c_{22}}},{\overline{c}}_{33}=c_{33}-{\displaystyle \frac{c_{32}{c}_{23}}{c_{22}}},{\overline{e}}_{31}=e_{31}-{\displaystyle \frac{c_{12}{e}_{32}}{c_{22}}},{\overline{e}}_{33}=e_{33}-{\displaystyle \frac{c_{32}{e}_{32}}{c_{22}}},\\ {\overline{\lambda }}_1={\lambda }_1-{\displaystyle \frac{c_{12}{\lambda }_2}{c_{22}}},{\overline{\lambda }}_3={\lambda }_3-{\displaystyle \frac{c_{32}{\lambda }_2}{c_{22}}},{\overline{\epsilon }}_{33}={\epsilon }_{33}+{\displaystyle \frac{e_{32}{e}_{32}}{c_{22}}},{\overline{p}}_3=p_3+{\displaystyle \frac{e_{32}{\lambda }_2}{c_{22}}}.\end{array}$$

(14)

Here, it should be stressed that thermo-PS can exhibit unique interactions between extensional deformations, polarization, and motion of charge carriers along the in-plane directions of a thin film. As shown in Equation (13), thermal fields produce effective polarization charges, i.e., zeroth-order electric displacements $D_1^{\left(0\right)}$ and $D_3^{\left(0\right)}$, the electric fields $E_1^{\left(0\right)}=-{\varphi }_{,1}^{\left(0\right)}$ and $E_3^{\left(0\right)}=-{\varphi }_{,3}^{\left(0\right)}$ associated with electric potential ${\varphi }^{\left(0\right)}$ or polarization then act on mobile charges, i.e., $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, and further affect their motion and transport behavior, as well as the distribution of current density.

For the static extensional deformation problem considered and investigated in this paper, according to Ref. [45], the mechanical and electric boundary conditions (including extensional deformations $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$, extensional resultant forces $T_{11}^{\left(0\right)}$ and $T_{33}^{\left(0\right)}$, electric potential ${\varphi }^{\left(0\right)}$, electron and hole concentration perturbations $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, electric displacement $D_3^{\left(0\right)}$, and electron and hole current density $J_3^{n\left(0\right)}$ and $J_3^{p\left(0\right)}$) at four edges of the PS thin film are taken to be the following:

$$\begin{array}{l}{u}_3^{\left(0\right)}=0,T_{11}^{\left(0\right)}=0,{\varphi }^{\left(0\right)}=0,\Delta n^{\left(0\right)}=0,\Delta p^{\left(0\right)}=0,\mathrm{at}{x}_1=0\mathrm{and}a,\\ u_1^{\left(0\right)}=0,T_{33}^{\left(0\right)}=0,D_3^{\left(0\right)}=0,J_3^{n\left(0\right)}=0,J_3^{p\left(0\right)}=0,\mathrm{at}{x}_3=0\mathrm{and}b.\end{array}$$

(15)

Next, under applied thermal field excitations, substituting Equation (13) into the static form of Equation (12) yields the following:

$$\begin{array}{l}{\overline{c}}_{11}{u}_{1,11}^{\left(0\right)}+c_{55}{u}_{1,33}^{\left(0\right)}+\left({\overline{c}}_{13}+c_{55}\right)u_{3,13}^{\left(0\right)}+\left(e_{15}+{\overline{e}}_{31}\right){\varphi }_{,13}^{\left(0\right)}-{\overline{\lambda }}_1{\theta }_{,1}^{\left(0\right)}+{\displaystyle \frac{1}{2h}}{F}_1^{\left(0\right)}=0,\\ \left(c_{55}+{\overline{c}}_{31}\right)u_{1,13}^{\left(0\right)}+c_{55}{u}_{3,11}^{\left(0\right)}+{\overline{c}}_{33}{u}_{3,33}^{\left(0\right)}+e_{15}{\varphi }_{,11}^{\left(0\right)}+{\overline{e}}_{33}{\varphi }_{,33}^{\left(0\right)}-{\overline{\lambda }}_3{\theta }_{,3}^{\left(0\right)}+{\displaystyle \frac{1}{2h}}{F}_3^{\left(0\right)}=0,\\ \left(e_{15}+{\overline{e}}_{31}\right)u_{1,13}^{\left(0\right)}+e_{15}{u}_{3,11}^{\left(0\right)}+{\overline{e}}_{33}{u}_{3,33}^{\left(0\right)}-{\epsilon }_{11}{\varphi }_{,11}^{\left(0\right)}-{\overline{\epsilon }}_{33}{\varphi }_{,33}^{\left(0\right)}+{\overline{p}}_3{\theta }_{,3}^{\left(0\right)}+{\displaystyle \frac{1}{2h}}{D}^{\left(0\right)}=q\left(\Delta p^{\left(0\right)}-\Delta n^{\left(0\right)}\right),\\ -n_0{\mu }_{11}^n{\varphi }_{,11}^{\left(0\right)}-n_0{\mu }_{33}^n{\varphi }_{,33}^{\left(0\right)}+D_{11}^n\Delta n_{,11}^{\left(0\right)}+D_{33}^n\Delta n_{,33}^{\left(0\right)}+{\displaystyle \frac{1}{2hq}}{J^n}^{\left(0\right)}=0,\\ -p_0{\mu }_{11}^p{\varphi }_{,11}^{\left(0\right)}-p_0{\mu }_{33}^p{\varphi }_{,33}^{\left(0\right)}-D_{11}^p\Delta p_{,11}^{\left(0\right)}-D_{33}^p\Delta p_{,33}^{\left(0\right)}+{\displaystyle \frac{1}{2hq}}{J^p}^{\left(0\right)}=0.\end{array}$$

(16)

In Equation (16), thermal fields and charge carriers such as electrons and holes can interact in PS through thermally induced polarization or electric field. Without the loss of generality, the double trigonometric series solution of Equation (16) can be expressed as follows:

$$\begin{array}{l}{\theta }^{\left(0\right)}={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{F}_{mn}\sin \left({\xi }_m{x}_1\right)\sin \left({\eta }_n{x}_3\right)}},\\ u_1^{\left(0\right)}={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{U}_{mn}\cos \left({\xi }_m{x}_1\right)\sin \left({\eta }_n{x}_3\right)}},\\ \left\{u_3^{\left(0\right)},{\varphi }^{\left(0\right)},\Delta n^{\left(0\right)},\Delta p^{\left(0\right)}\right\}={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }\left\{W_{mn},{\Psi }_{mn},N_{mn},P_{mn}\right\}\sin \left({\xi }_m{x}_1\right)\cos \left({\eta }_n{x}_3\right)}},\\ {\xi }_m={\displaystyle \frac{m\pi }{a}},{\eta }_n={\displaystyle \frac{n\pi }{b}}.\end{array}$$

(17)

where $U_{mn}$, $W_{mn}$, ${\Psi }_{mn}$, $N_{mn}$, and $P_{mn}$ are undetermined constants, and the coefficient of trigonometric series expansion $F_{mn}$ is known, which is given by the applied temperature field ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ as follows:

$$F_{mn}={\displaystyle \frac{4}{ab}}{\displaystyle {\iint }_A{\theta }^{\left(0\right)}\left(x_1,x_3\right)\sin \left({\xi }_m{x}_1\right)\sin \left({\eta }_n{x}_3\right)\mathrm{d}{x}_1}\mathrm{d}{x}_3.$$

(18)

When a thermal field ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ is applied along the $O{x}_1{x}_3$ plane of PS thin-film device, the coupled extensional deformations and electromechanical fields described by Equation (16) are generated owing to multi-field coupling effects. To reveal the effects of local temperature loadings on electromechanical fields, in Equation (18), a particular temperature load ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ is taken into account by a piecewise constant function.

$${\theta }^{\left(0\right)}\left(x_1,x_3\right)=\left\{\begin{array}{l}-{\theta }_0^{\left(0\right)},\left(x_1,x_3\right)\in A_b,\\ +{\theta }_0^{\left(0\right)},\left(x_1,x_3\right)\in A_y,\\ 0\mathrm{K},\left(x_1,x_3\right)\in A_g,\mathrm{and}\mathrm{elsewhere}.\end{array}\right.$$

(19)

in which ${\theta }_0^{\left(0\right)}\ge 0$ is a very small temperature load intensity parameter. Classification discussion is as follows: (1) When ${\theta }_0^{\left(0\right)}=0$, the PS thin film is assumed to be in a natural state, and corresponding concentrations are $n_0$ and $p_0$, respectively. (2) When ${\theta }_0^{\left(0\right)}\ne 0$, the local temperature rises or falls within the PS thin film, which leads to variations in electron and hole concentration perturbations $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$ owing to the interaction between strain, electric field, and concentration gradients of mobile charges. In this state, their concentrations change to $p=p_0+\Delta p^{\left(0\right)}$ and $n=n_0+\Delta n^{\left(0\right)}$, respectively. Correspondingly, the blue region $A_b$ with ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=-{\theta }_0^{\left(0\right)}$ or yellow region $A_y$ with ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=+{\theta }_0^{\left(0\right)}$ stands for a lower or higher local and uniform temperature loading area compared to reference temperature area denoted by green $A_g$ and other regions with ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=0\mathrm{K}$. Then, in the total PS thin film, a non-uniform temperature distribution, denoted by different colors of loading areas, can be exactly described and designed via Equation (19). This is just the design idea of temperature loading in this paper.

It can be verified that Equation (17) automatically satisfies boundary conditions, i.e., Equation (15). Substituting Equation (17) into Equation (16) without other excitation fields (i.e., $F_1^{\left(0\right)}=F_3^{\left(0\right)}=D^{\left(0\right)}={J^n}^{\left(0\right)}={J^p}^{\left(0\right)}=0$) yields a system of linear algebraic equations, as shown below:

$$\begin{array}{l}{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}\cos \left({\xi }_m{x}_1\right)\sin \left({\eta }_n{x}_3\right)=0,{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{B}_{mn}}}\sin \left({\xi }_m{x}_1\right)\cos \left({\eta }_n{x}_3\right)=0,\\ {\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{C}_{mn}}}\sin \left({\xi }_m{x}_1\right)\cos \left({\eta }_n{x}_3\right)=0,{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{D}_{mn}}}\sin \left({\xi }_m{x}_1\right)\cos \left({\eta }_n{x}_3\right)=0,\\ {\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{E}_{mn}}}\sin \left({\xi }_m{x}_1\right)\cos \left({\eta }_n{x}_3\right)=0.\end{array}$$

(20)

where

$$\begin{array}{l}{A}_{mn}=\left[-\left({\overline{c}}_{11}{\xi }_m^2+c_{55}{\eta }_n^2\right)U_{mn}-\left({\overline{c}}_{13}+c_{55}\right){\xi }_m{\eta }_n{W}_{mn}-\left({\overline{e}}_{31}+e_{15}\right){\xi }_m{\eta }_n{\Psi }_{mn}-{\overline{\lambda }}_{11}{\xi }_m{F}_{mn}\right],\\ B_{mn}=\left[-\left(c_{55}+{\overline{c}}_{13}\right){\xi }_m{\eta }_n{U}_{mn}-\left(c_{55}{\xi }_m^2+{\overline{c}}_{33}{\eta }_n^2\right)W_{mn}-\left(e_{15}{\xi }_m^2+{\overline{e}}_{33}{\eta }_n^2\right){\Psi }_{mn}-{\overline{\lambda }}_{33}{\eta }_n{F}_{mn}\right],\\ C_{mn}=\left[\begin{array}{l}-\left(e_{15}+{\overline{e}}_{31}\right){\xi }_m{\eta }_n{U}_{mn}-\left(e_{15}{\xi }_m^2+{\overline{e}}_{33}{\eta }_n^2\right)W_{mn}+\left({\epsilon }_{11}{\xi }_m^2+{\overline{\epsilon }}_{33}{\eta }_n^2\right){\Psi }_{mn}\\ +{\overline{p}}_3{\eta }_n{F}_{mn}+q{N}_{mn}-q{P}_{mn}\end{array}\right],\\ D_{mn}=\left[n_0\left({\mu }_{11}^n{\xi }_m^2+{\mu }_{33}^n{\eta }_n^2\right){\Psi }_{mn}-\left(D_{11}^n{\xi }_m^2+D_{33}^n{\eta }_n^2\right)N_{mn}\right],\\ E_{mn}=\left[p_0\left({\mu }_{11}^p{\xi }_m^2+{\mu }_{33}^p{\eta }_n^2\right){\Psi }_{mn}+\left(D_{11}^p{\xi }_m^2+D_{33}^p{\eta }_n^2\right)P_{mn}\right].\end{array}$$

(21)

In the following analysis, for a PS thin film that is excited by the applied thermal field, the undetermined constants $U_{mn}$, $W_{mn}$, ${\Psi }_{mn}$, $N_{mn}$, and $P_{mn}$ can be obtained from Equation (20) with the aid of Equations (18) and (19). Accordingly, the various electromechanical fields can be calculated via Equation (17). To further reveal the effect of local temperature loading on current density near local loading areas, we introduce electron and hole current density vectors ${\mathbf{J}}^{n\left(0\right)}$ and ${\mathbf{J}}^{p\left(0\right)}$:

$${\mathbf{J}}^{n\left(0\right)}=J_1^{n\left(0\right)}{\mathbf{e}}_1+J_3^{n\left(0\right)}{\mathbf{e}}_3,{\mathbf{J}}^{p\left(0\right)}=J_1^{p\left(0\right)}{\mathbf{e}}_1+J_3^{p\left(0\right)}{\mathbf{e}}_3.$$

(22)

where ${\mathbf{e}}_1$ and ${\mathbf{e}}_3$ are unit vectors along positive $x_1$ and $x_3$ directions, respectively. Based on the theoretical solution established above, the basic mechanical analysis of a rectangular PS thin-film device with local temperature loading areas is carried out.

4.3. Numerical Results and Discussion

During the following numerical simulations, considering the third-generation semiconductor ZnO as an example, the relevant material constants needed are not displayed one by one, which are given in the existing literature [45]. The initial-state electron and hole concentrations and temperature load intensity parameter are set to be $n_0=p_0={10}^{21}{\mathrm{m}}^{-3}$, and ${\theta }_0^{\left(0\right)}=1\mathrm{K}$, respectively. For the numerical calculations, the geometric parameters have been listed clearly by $a=1000\mathrm{nm}$, $b=1200\mathrm{nm}$, $h=50\mathrm{nm}$, $x_{01}=200\mathrm{nm}$, $x_{03}=800\mathrm{nm}$, $2c=150\mathrm{nm}$, and $2d=100\mathrm{nm}$, which are not displayed for simplification in all figure captions below.

4.3.1. Convergence Analysis of Double Trigonometric Series Solutions

For the current theoretical model, the convergence analysis of double trigonometric series solutions should be examined firstly to ensure computational accuracy. Taking the fourth pattern of temperature loads and $x_1=a/2$ as an example, after repeated calculations, as shown in Figure 3, the 50 terms of trigonometric series truncation parameters for $m$ and $n$ are sufficient to ensure acceptable convergent results about electric potential ${\varphi }^{\left(0\right)}$, which are adopted during the following numerical simulations.

4.3.2. Validity and Accuracy Analysis for Current Theoretical Model

From the linear analysis viewpoint, if we only consider an p-type doped PS thin film without surface loads, i.e., by letting the initial-state electron concentration $n_0$ and its concentration perturbations $\Delta n^{\left(0\right)}$ as well as various loads (including $F_1^{\left(0\right)}$, $F_3^{\left(0\right)}$, $D^{\left(0\right)}$, ${J^n}^{\left(0\right)}$, and ${J^p}^{\left(0\right)}$) on the top and bottom surfaces (i.e., at $x_2=h$ and $x_2=-h$) to be set to zero in Equation (16), then, the present Equation (16) can be directly reduced to the theoretical Equation (17) given by Qu et al. [45]. Further, when considering the second pattern of temperature loads, as shown in Figure 2b, i.e., the applied temperature load ${\theta }^{\left(0\right)}\left(x_1,x_3\right)=+{\theta }_0^{\left(0\right)}$, which is zero elsewhere, is uniform in a small and yellow rectangular area $A_y$. With the rise in temperature load ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ from $0.1\mathrm{K}$ to $0.2\mathrm{K}$ to $0.3\mathrm{K}$, the variation in electric potential ${\varphi }^{\left(0\right)}$ along the $x_3$ axis direction and $x_1=500\mathrm{nm}$ are shown as a comparative example in Figure 4, which are in good agreement with the theoretical results of Qu et al. We can conclude from Figure 4 that the height/depth of electric potential barrier/well is quite sensitive to different temperature loads in a PS thin-film device.

4.3.3. Local Modulation of Electromechanical Fields via First, Second, Third, and Fourth Patterns of Temperature Loads

From the working performance viewpoint, regulating local electromechanical field characteristics inside a PS thin-film device is crucial. Generally, the crucial aspect of local piezoelectric modulation lies in creating localized external stimulus fields within piezoelectric polarization. Inspired by this, for the considered rectangular PS thin film shown in Figure 2, when the applied thermal load is local, the strain produced by a higher or lower temperature change field ${\theta }^{\left(0\right)}$ is always greater than that of films with ${\theta }^{\left(0\right)}=0\mathrm{K}$ under equivalent stress. As is commonly understood, the non-uniform strain is generated in a homogeneous PS thin-film medium by using local thermal field excitations, which can be used to create localized piezoelectric polarization. This is just the research strategy for the active manipulation of non-uniform strain and local polarization.

Applied Local Temperature Fields

In Figure 5, the total distributions of applied external stimulus fields ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ 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 ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$, 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 $T_{11}^{\left(0\right)}$ and $T_{33}^{\left(0\right)}$ everywhere. Meanwhile, electric potential ${\varphi }^{\left(0\right)}$ is closely related to $T_{11}^{\left(0\right)}$ and $T_{33}^{\left(0\right)}$, and pyroelectric effect directly contributes to $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, and thermoelastic coupling does so through piezoelectric coupling. Therefore, according to the coupled field Equation (16), the electromechanical fields, including extensional deformations $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$, electric potential ${\varphi }^{\left(0\right)}$, electron and hole concentration perturbations $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, and current densities ${\mathbf{J}}^{n\left(0\right)}$ and ${\mathbf{J}}^{p\left(0\right)}$, caused by local temperature field ${\theta }^{\left(0\right)}$, can spread through the entire thin film, where extensional resultant forces $T_{11}^{\left(0\right)}$ and $T_{33}^{\left(0\right)}$ associated with higher ${\theta }^{\left(0\right)}$ near thermal loading region center are larger.

Induced Local Mechanical Displacement Fields

To prove a static and local temperature load ${\theta }^{\left(0\right)}$ can be used to produce and manipulate a local disturbance of electromechanical fields in a PS thin-film device, and considering the extensional deformations $u_3^{\left(0\right)}$ as an example, in Figure 6, when local loads change from first to second, third, and fourth patterns of temperature loads, the extensional displacements $u_3^{\left(0\right)}$ 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 $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$ are accompanied by the distributions of temperature field ${\theta }^{\left(0\right)}$, electric potential ${\varphi }^{\left(0\right)}$, electron and hole concentration perturbations $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, and current densities. Inspired by the above analytical results, various in-plane distribution patterns and local deformations of extensional modes $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$ can be adjusted effectively by applying matched temperature load patterns.

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. 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.

Motions and Redistributions of Electrons and Holes

As shown in Figure 8 and Figure 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 $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$ are very sensitive to the electric potential barriers/wells. Taking Figure 8a and Figure 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, $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$ have opposite signs within the plane of the thin film because of the electric potential ${\varphi }^{\left(0\right)}$ drives holes and electrons in opposite directions in the thin film. From the experimental analysis viewpoint, $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$ 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.

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. Thus, the different current vector plot distributions can be achieved, as shown in Figure 10 and Figure 11, and this is just the foundation of the manipulation of current distributions. Distributions of drift and diffusion current vector plots are similar, but the directions of vector arrows are reversed. To be specific, the drift currents flow away from the potential barriers and flow toward the potential wells, but the diffusion currents flow toward the potential barriers and flow away from the potential wells. Therefore, the potential wells attract drift currents and expel diffusion currents, and the potential barriers attract diffusion currents and expel drift currents. For piezotronic applications, the above research results provide a theoretical basis for the thermal manipulation of electromechanical fields in PS thin-film devices.

Overall, the analytical results from Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 theoretically and numerically demonstrate that various electromechanical field distributions can be created, achieved, and manipulated near the loading areas of a thin film by applying different patterns of loads. For instance, the electromechanical fields with the unique coupling of piezoelectricity and semiconductor properties, produced by the locally applied first, second, third, and fourth patterns of temperature loads, are also localized from deformation and polarization to mobile charges. This important research result not only offers more flexibility for developing new electronic devices with electric potential barriers/wells but also gives a mechanical tuning methodology for thermally manipulating mobile charges in PS thin-film devices. Above phenomena can be reasonably explained by examining Equations (13) and (16).

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 ${\varphi }^{\left(0\right)}$ 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 ${\theta }^{\left(0\right)}\left(x_1,x_3\right)$ 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 ${\varphi }^{\left(0\right)}$, 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 ${\varphi }^{\left(0\right)}$.

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 ${\theta }_0^{\left(0\right)}$, 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 ${\theta }_0^{\left(0\right)}$ 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.

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 Newton’s law, Gauss’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:

(a)

The electromechanical fields (i.e., extensional deformations $u_1^{\left(0\right)}$ and $u_3^{\left(0\right)}$, electric potential ${\varphi }^{\left(0\right)}$, electron and hole concentration perturbations $\Delta n^{\left(0\right)}$ and $\Delta p^{\left(0\right)}$, and their current densities) are temperature-dependent due to the fact that thermoelastic, 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.

(b)

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.

(c)

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.