## 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]:  
\[T_{ji,j}=\rho\ddot{u}_{i},D_{i,i}=q(\Delta p-\Delta n),J^{n}_{i,i}=q\Delta i,J^ {p}_{i,i}=-q\Delta j,h_{i,i}=-\Theta_{0}\dot{\eta}. \tag{1}\]  
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. \(J^{n}_{i}\) and \(J^{p}_{i}\) 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:  
\[T_{ij}=c_{ijkl}S_{kl}-e_{kij}E_{k}-\lambda_{ij}\theta,\ D_{i}=e_{ ik}S_{kl}+e_{ik}E_{k}+p_{i}\theta,\] \[J^{n}_{i}=qn_{0}\mu^{n}_{ij}E_{j}+qD^{n}_{i}N_{j},\ J^{p}_{i}=qp_ {0}\mu^{p}_{ij}E_{j}-qD^{p}_{ij}P_{j}, \tag{2}\] \[\eta=\lambda_{kl}S_{kl}+p_{k}E_{k}+\alpha\theta.\]  
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 \(e_{ik}\) constants, respectively. \(n_{0}\) and \(p_{0}\) denote uniform electron and hole concentrations in the reference state, respectively. \(D^{p}_{ij}(D^{n}_{ij})\) and \(\mu^{p}_{ij}(\mu^{n}_{ij})\) 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 \(C^{E}_{b}\) is depicted by \(\alpha=\rho C^{E}_{v}\Theta_{0}^{-1}\), and \(\theta=\Theta-\Theta_{0}\) represents a small temperature change, i.e., \(|\ \theta\ |\ll\Theta_{0}\). 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:  
\[S_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,i}\big{)},\ E_{i}=-\phi_{j},\ N_{i}= \Delta n_{j},\ P_{i}=\Delta p_{j},\ h_{i}=-\kappa_{ij}\theta_{j}. \tag{3}\]  
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.