Analysis of nonlinear multi-field coupling responses of piezoelectric semiconductor rods via machine learning

ABSTRACT Piezoelectric semiconductors (PSs) have widespread applications in semiconductor devices due to the coexistence of piezoelectricity and semiconducting properties. It is very important to conduct a theoretical analysis of PS structures. However, the present of nonlinearity in the partial differential equations (PDEs) that describe those multi-field coupling mechanical behaviors of PSs poses a significant mathematical challenge when studying these PS structures. In this paper, we present a novel approach based on machine learning for solving multi-field coupling problems in PS structures. A physics-informed neural networks (PINNs) is constructed for predicting the multi-field coupling behaviors of PS rods with extensional deformation. By utilizing the proposed PINNs, we evaluate the multi-field coupling responses of a ZnO rod under static and dynamic axial forces. Numerical results demonstrate that the proposed PINNs exhibit high accuracy in solving both static and dynamic problems associated with PS structures. It provides an effective approach to predicting the nonlinear multi-field coupling phenomena in PS structures. GRAPHICAL ABSTRACT


Introduction
Due to dual physical properties of piezoelectricity and semiconductor, piezoelectric semiconductor (PS) materials have emerged as optimal building blocks for the next generation of multifunctional electronic devices.The interaction between deformation, polarization, and charge carriers, known as the deformation-polarization-carrier (DPC) coupling effect, makes them far superior to traditional semiconductors and piezoelectric (PE) dielectrics.Over the past two decades, researchers have successfully fabricated various types of PS structures, exhibiting excellent mechanical and electronical properties, such as ZnO nanostructures [1][2][3] and two-dimensional MoS 2 ultrathin films [4].These structures, featuring the DPC coupling effect, are highly suitable for the ongoing trend of miniaturization in future semiconductor devices [5][6][7].As a result, they have been used in smart devices such as energy harvesters [8], strain-gated devices [9], field effect transistors [10,11], and medical sensors [12].
Indeed, the performances of semiconductor devices based on traditional semiconductors without piezoelectricity rely primarily on applied bias voltage.However, the performances of PS devices can be effectively controlled not only by an applied bias voltage but also by mechanical forces and elastic waves through the DPC coupling effect.This unique characteristic opens up new possibilities for manipulating and enhancing the performance of PS structures.Several researchers have indeed conducted experiments to demonstrate the potential of manipulating and enhancing of the performances of PS structures using mechanical forces.For instance, the strain-induced piezoelectric polarization field has been used to enhance the light emission of ZnO microwire-based diodes [13].Additionally, the coherent light emission wavelengths of single ZnO microwires have been dynamically controlled using mechanical strain [14].Furthermore, the potential barrier in ZnO bicrystals has been modified through the application of mechanical strain [15].These experimental findings highlight the versatility and potential of using mechanical forces to optimize the performance of PS structures.
From a continuum perspective, the analysis of PS structures presents a nonlinear problem.This poses a significant challenge in obtaining analytical solutions.To simplify the analysis, the linearization assumption has been employed in the aforementioned works for the sake of simplicity.However, it is important to note that the linearized method is only applicable to PS structures subjected to relatively small loads.In practical engineering applications, conducting nonlinear analyses on PS structures becomes crucial and necessary.While researchers have carried out a few simple nonlinear analyses on PS structures [41,42], the current progress in this area remains limited.There is a need for further advancements to tackle the complexity of the nonlinear behavior exhibited by PS structures.This calls for the development of advanced analytical and computational techniques that can accurately capture and predict the performance and behavior of PS structures.Such advancements will enable more precise design optimizations and predictions in practical engineering applications.
The growing machine learning (ML) technology, a branch of computer science, offers a very powerful approach to predicting the future unknown data based on existing data using various algorithms.ML has been widely used to provide a prediction of physical quantities in diverse fields, including mathematics [43][44][45], materials science [46], mechanics [47,48], and biomedicine [49].One notable development in this area is the introduction of physics-informed neural networks (PINNs), which was first proposed by Raissi and his collaborators [50].PINNs offers an effective and efficient approach to solving nonlinear partial differential equations (PDEs).The core idea behind PINNs is training a neural network using automatic differentiation to calculate and minimize the residual, typically with the inclusion of initial and boundary conditions.This approach overcomes two main limitations of commonly used deep, convolutional, and recurrent neural networks: the lack of robustness and the absence of convergence guarantees.PINNs provides a new and powerful method for predicting and understanding the nonlinear coupling of multi-physical fields governed by nonlinear PDEs in various fields, such as fluid and solid mechanics.By incorporating physical principles into the training process, PINNs can capture the underlying physics and improve the accuracy and reliability of predictions.This opens up new possibilities for solving complex problems and gaining insights into the behavior of nonlinear systems in different scientific and engineering domains.
In this paper, we aim to extend PINNs to the prediction of multi-physical field coupling responses of PS structures with the DPC coupling effect under different loads.In Section II, a PINNs-based model for investigating multi-field coupling behaviors of PS rods is presented based on one-dimensional (1-D) rod equations.In Section III, we employe the proposed PINNs-based model to study static and dynamic extensions of PS rods.Finally, some conclusions are drawn in Section IV.

Construction of PINNs for PS rods
In this section, we establish PINNs for predicting the multi-field coupling mechanical behaviors of rod-like PS structures undergoing extensional deformation.This working mode is commonly observed in semiconductor devices.Consider an n-type PS rod with length of 2L and radius of R under a pair of opposite axial forces F as shown in Figure 1.It is with uniform donor concentration of N þ D and is polarized along the x 3 direction.The axial middle point of the PS rod is chosen as the original point of its axial coordinate.For the extension of the considered PS rod, the dominant physical quantities include the axial displacement u, the electric potential φ, the concentration of electrons n, the axial stress T 33 , the axial electric displacement D 3 , and the axial current density of electrons J n 3 , which are functions of only x 3 and time t.
For the considered n-type PS rod, the 1-D governing equations [22] are where ρ is the mass density, q is the elementary charge.In Eq. ( 1), � c 33 , � e 33 , and � ε 33 are the effective elastic, piezoelectric, dielectric constants [22], respectively, μ n 33 and D n 33 are the mobility and diffusion constants of electrons.In addition, the Cartesian tensor notation is used in Eq. ( 1).For the case of static extension, the time-related terms in Eq. ( 1) should be deleted.
Eq. ( 1) is a set of nonlinear differential equations.We write uðt; xÞ ¼ ½uðt; xÞ; φðt; xÞ; nðt; xÞ� T , a superscript of T denotes transposition of a vector or matrix, and, use parameter λ i (i = 1 ~ 5) to replace constants � c 33 , � e 33 , � ε 33 , D n 33 and μ n 33 .Then, Eq. ( 1) can be rewritten in the matrix form as In Eq. (2), A i (i = 1 ~ 7) and M are the parameter matrixes composed of known material parameters, they are The nonlinear system of Eq. ( 2) describes the multi-field coupling behaviors of a PS rod with extension deformation.It can be numerically solved by constructing the classical PINNs.We introduce a learning function f as Following the same idea and procedure as in Ref [46], the PINNs for a PS rod with extension deformation constructed and be trained to satisfy both the initial/boundary conditions and the physical laws by imposing constraints derived from the dataset u and the learning function f.The details are shown in Figure 2, where ANNS is the Artificial Neural Networks.
The parameters of neural networks can be learned by minimizing the following two loss functions In Eq. ( 5), u i predict and f i predict are the predicted values at the set of collocation points, u i training denotes the training data at the same points as u i predict .L u and L f correspond to the initial and boundary conditions and the intrinsic physical law of the studied problem.

Numerical examples and discussion
In this section, we numerically evaluate the multi-field coupling responses of an n-type ZnO PS rod under static and dynamic axial forces with the constructed PINNs as shown in Figure 2. The material constants of ZnO are shown in Ref [22].The training dataset is obtained by using COMSOL.In the PINNs, the number of hidden layers is 8, neuron count in each hidden layer is 20, and activation function is tanh function.Assume that the considered ZnO rod' length and cross-sectional area are 6 μm and 1 μm 2 , the initial concentration of electrons is n 0 ¼ 10 22 /m 3 .The analytical solutions of static and dynamical extension of PS rod have been presented in Ref. [20,22] with the linearized method.For the static extension of PS rods, the analytical solutions of axial displacement, electric potential, and concentration of electrons are [22] where 33 n 0 and A is the cross-sectional area of the PS rod, n 0 ¼ N þ D denotes the initial concentration of electrons in the PS rod.For the extensional vibration of the PS rod under the time-dependent axial force of F 0 expðiωtÞ, the analytical solutions of axial displacement, electric potential, and concentration of electrons are [20] where 1 and B m are determined by the boundary conditions.The data processing method of the normalization is adopted in the computation of PINNs.
Firstly, we consider the static extension of the ZnO rod under a pair of small axial force F = 9.43 × 10 −7 N with free at both ends.With the established PINNs, we can obtain the physical fields in the ZnO rod.The corresponding analytical results obtained from Eq. ( 5) are also provided here for comparison with those predicted by the PINNs with randomly selecting the training data from the entire domain.Figure 3 shows the nephogram of the displacement, electric field and carrier concentration in the domain (0, L) and (0, t) of the PS rod. Figure 4 shows a comparison between the linearized model and the PINNs.It can be observed that the results obtained from the two approaches agree well for a small mechanical loading.
Indeed, randomly selecting training data can yield accurate results in certain cases.However, this selection method is not consistent with the general approach of solving partial differential equations (PDEs).In most cases, the boundary and initial conditions for the PDE can be easily obtained or specified.To highlight the difference between randomly selected training data and using the boundary conditions, we provide predicted results using the PINNs with a training dataset derived from the boundary conditions.These results are shown in Figure 5.It can be observed that only the displacement obtained by the boundary-condition-data-trained PINNs has a slight error, but the overall trend remains consistent.On the other hand, the electric potential and electron concentration agree well with the analytical results.Comparatively, the random selection method performs better to some extent than using the boundary and initial conditions for training data.However, from a practical standpoint, it is generally more appropriate to select the boundary and initial conditions as the training data set.This ensures that the PINNs capture the specific behavior dictated by the known physical laws and constraints of the problem, leading to more reliable and accurate predictions.
To assess the robustness of the developed PINNs, an evaluation is conducted using training data that includes different levels of disturbance.The accuracy error of the PINNs' output is used as a reference for this evaluation.The error function is expressed as follows:  The results of this evaluation are presented in Tables 1 and 2, showing the performance of the PINNs under disturbance levels, indicating their robustness or sensitivity to changes in the training data.From Tables 1 and 2, we also find that for the noisy input into training data the output error is always at a low level of less than 10%, and the accuracy will increase when the structure of networks becomes more complicated.
Next, we use the proposed PINNs to evaluate the multi-field coupling responses of the ZnO rod under larger axial forces.The boundary conditions are the same as those in Figure 3.It is noted that the analytical results obtained from the linearization theory are not accurate in these cases.Figure 6 shows the displacement, electric potential, and electron concentration in the ZnO rod with axial stresses (T ¼ F=A) of 1 MPa and 100 MPa and length of 1.2 μm, which are predicted by the proposed PINNs.It can be observed that for the larger applied axial forces, the electrical quantities (electric potential and electron concentration) obtained from the linearized method exhibit a relatively larger deviation from those predicted by the proposed PINNs.In fact, the linearized method fails to provide accurate results for this case.Additionally, the comparison of the maximum   This discrepancy may arise due to the training sets being obtained through a linear method, whereas the results from the finite element method are obtained using a nonlinear approach.The polarization electric field obtained by the finite element model may be larger, leading to a higher electron concentration.These differences highlight the impact of nonlinear effects on the behavior of the PS rod and suggest that further refinements in the training methodology could improve the accuracy of the PINNs in capturing these nonlinear phenomena.Finally, we use the proposed PINNs to investigate the multi-field coupling responses of the ZnO rod under a time-dependent axial force of F=F 0 e iωt .Here, F 0 = 9.43 × 10 −7 N and is small.The displacement, electric potential, and electron concentration predicted by the PINNs are depicted in Figure 7.At the same time, for comparison, the results obtained from the linear analytical solution are also presented in Figure 7.It can be seen that the PINNs can also be used for the dynamic case.

Conclusions
In conclusion, we have developed a physics-informed neural network (PINNs) approach for accurately predicting the multi-field coupling responses of rod-like structures made of PS materials under axial extensional deformation.By leveraging the 1-D governing equations of PS structures, we constructed the PINNs to investigate the extension behavior of ZnO rods under static and time-dependent axial forces.The numerical results demonstrate that the proposed PINNs exhibit high accuracy in predicting the multi-field coupling behaviors of PS rods and can be applied to both static and dynamic problems.We have also examined the robustness of the networks by introducing different levels of noise into the input training dataset, ensuring that the PINNs can handle uncertainties and variations in the data.Furthermore, the proposed PINNs-based method can be extended to solve two-and three-dimensional problems of PSs in the future, enabling a more comprehensive analysis of the multi-field coupling phenomena in these structures.Overall, the PINNs approach provides a promising tool for studying and predicting the complex behavior of piezoelectric semiconductor structures, contributing to the design and optimization of PS devices in various applications.

Figure 1 .
Figure 1.Schematic diagram of a PS rod under external load.

Figure 2 .
Figure 2. The schematic diagram of the piezoelectric problems solving via PINNs.(a) configuration of PINNs.(b) flow chart of the program.

Figure 3 .
Figure 3.The displacement, electric potential, and electron concentration in the PS rod with static extension obtained from (a) the linearized model and (b) the PINNs training with 2000 points.

Figure 4 .
Figure 4.The comparison of the displacement (a), electric potential (b), and electron concentration (c) in the PS rod with static extension between the linearized model and the PINNs.

Figure 5 .
Figure 5.The displacement (a), electric potential (b), and electron concentration (c) predicted by the PINNs with the training date from the boundary conditions.
displacement, electric potential and electron concentration of the PS rod obtained by the PINNs and the COMSOL model under T = 1 MPa is presented in Table3.The results indicate that the PINNs yield good agreement with the COMSOL model.The displacement values predicted by the PINNs are slightly smaller than those obtained from the FEM model, while the electric potential and electron concentration show the opposite trend.

Figure 6 .
Figure 6.The distribution of electromechanical fields in a PS rod under different levels of external loads.The red line indicates the PINNs output result, and the black lines are calculated by linearization theory.

Figure 7 .
Figure 7.The displacement, electric potential, and electron concentration in the ZnO rod under the time-dependent axial force predicted by the linearized method (a) and the proposed PINNs training with 2000 points (b).

Table 2 .
The mean square deviation of the predicted value for different architecture of hidden layers.

Table 1 .
The mean square deviation of the predicted value for different levels of noise and the size of the training set N.

Table 3 .
The maximum displacement, electric potential and electron concentration of a PS rod obtained by the PINNs and COMSOL under T = 1 MPa.