A 1D nonlinear model for piezoelectric semiconductor fibers incorporating thermal and thermoelectric effects

Abstract

This paper studies multi-physical fields in a piezoelectric semiconductor (PS) fiber with consideration of heat conduction, pyroelectric, and thermoelectric effects. Based on the three-dimensional (3D) framework of piezoelectricity, drift–diffusion theory, Seebeck effect, and Peltier effect, we develop a highly nonlinear one-dimensional (1D) model that incorporates axial deformation, axial variations of the electric field, the redistribution of carriers, and temperature deviation. Combining the 1D nonlinear governing equations and the corresponding boundary conditions, the influence of the thermoelectric coefficients and axial forces on electrostatic potential, carrier redistribution, and temperature variation are solved numerically. Due to the nonlinearity of the model, these solutions are observed without symmetry or asymmetry. Our research shows that the temperature will increase near the action point of the axial force. Therefore, the temperature deviation in the fiber can be controlled by applying axial force at different points. Finally, we examine how the axial forces applied in the fiber affect the current–voltage relation. The presented study provides a potential application for mechanical switches, sensors, or thermal control for PSs.

1 Introduction

During the past few decades, interest in PSs has grown tremendously, owing to the development of the electronic industry. Since PS materials have both piezoelectric and semiconductor properties, the mechanical loads can generate piezoelectric potentials through piezoelectric effects, which, in turn, significantly affect the distribution of carriers within the materials. Therefore, semiconductor devices based on functional piezoelectric materials can play a significant role in electroacoustics, energy harvesting, and biomedicine [1,2,3,4]. With low-cost, high-energy conversion, efficiency, and environmental friendliness benefits, PS devices such as self-powered sensors, smart wearable electronics, and portable electronics are becoming increasingly prevalent in people’s daily lives [5,6,7,8,9].

Researchers in mechanics have been making significant contributions to the theoretical development of PSs. By combining the piezoelectric theory of dielectrics with the linearized semiconductor drift–diffusion theory, Yang theoretically solved the distribution of carriers in typical PS structures under mechanical loads and summarized the linear theoretical framework of PSs [10]. Therefore, by merging this 3D theoretical framework with the tensile, bending, shear, and torsional deformation theories of the 1D structures, the potential and carrier distribution of PS nanowires under mechanical loads are explored [11,12,13,14,15]. The propagation of elastic waves in PS rods and plates has been investigated in several studies under diverse situations [16,17,18,19,20,21,22]. By using the displacement discontinuity method, researchers have studied the failure analysis of various types of crack propagation and other failure analysis problems under transient loading of axial force, lateral force, and other loads [23,24,25,26,27,28]. Additionally, in order to expand the selection of materials, researchers have increasingly focused on composite fiber, which is made by fusing non-piezoelectric semiconductor materials with piezoelectric dielectrics. The behavior of PS composite fibers under bending moment, torque, shear stress, and tensile stress was investigated by the researchers [29,30,31,32].

When the current flows through the semiconductor, the thermal effect of the current cannot be ignored. Therefore, the influence of temperature has been explored by integrating thermal effects into the macroscopic theory of PSs, further enriching our understanding of the electromechanical responses [33,34,35]. By combining 1D structure theory and the semi-coupling theory between temperature deviation and piezoelectric field, researchers have conducted several investigations on the impact of temperature variations on the electrical behavior of PS fibers and composite fibers and proposed the thermal manipulation of moving charges in semiconductors [35,36,37,38,39]. Yang et al. studied the physical response of a composite PS cylindrical shell under a uniform temperature load [40]. Jin et al. analyzed the multi-field response of the n-type PS film under the temperature gradient [41]. Based on the two-dimensional equation of the PS film, Qu et al. revealed the current bypass phenomenon of p-type doped ZnO film under thermal load [42]. In terms of a fully coupled thermo-piezo-elastic model, Zhang et al. established a nonlinear plane model based on piezoelectric electronic theory and Fourier diffusion equation and clarified the mechanism of in-plane mechanical loads on the current and heat flow of PS films [43]. By following the Coleman–Noll procedure [44,45,46], a self-consistent continuum theory for elastic thermoelectric semiconductors was established by Qu et al. [47], marking an advancement in the field and providing a comprehensive theoretical framework for future research.

Currently, many studies of PS fibers do not consider heat conduction and the complex thermoelectric effects [36,37,38, 48]. Therefore, it is impossible to accurately describe the influence of temperature change on mechanical deformation and current density distribution. The unidirectional coupling between the temperature field and other fields that leads to the secondary and multiple responses of the structure also cannot be observed. In this paper, we concentrate on studying the extension behavior of PS fibers with consideration of the Seebeck and Peltier effects. This nonlinear thermal-electromechanical fully coupled model is used to demonstrate the interactions between axial deformation, axial electric field, mobile charge distributions, and temperature deviation when axial forces with different numbers and different strengths are applied to the fiber. In Sect. 2, a 1D model for the extension of the PS fiber is presented. In Sect. 3, the nonlinear governing equations of the 1D model are solved numerically to show how axial forces are coupled to the axial motion of charge carriers and temperature deviation through axial deformation in the fiber. The I-V characteristic curves of the fiber with different axial forces are also examined in this section, which means that the PS fiber can work like a switch to manipulate the current. Finally, we draw some conclusions in Sect. 4.

2 A one-dimension model for the PS fiber

For PSs considering the Seebeck effect and Peltier effect, the 3D mechanical equilibrium equations, quasi-static Gaussian equations, charge continuity equations, and heat conduction equations in PS materials are represented as follows [47, 49, 50]

$$\begin{gathered} T_{ji,j} + f_{i} = 0, \hfill \\ D_{i,i} = q(p - n + N_{D}^{ + } - N_{A}^{ - } ), \hfill \\ J_{i,i}^{p} = 0,\quad J_{i,i}^{n} = 0, \hfill \\ q_{i,i} = (J_{i}^{p} + J_{i}^{n} )E_{i} , \hfill \\ \end{gathered}$$

(1)

where f_i are the body forces; q = 1.6 × 10^–19 C is the elementary charge; p and n are the concentrations of holes and electrons; and $N_{D}^{ + }$ and $N_{A}^{ - }$ are the concentrations of ionized acceptors and donors from doping. The 3D constitutive relations for Cauchy stress (T_ij), electric displacement (D_i), the hole and electron current densities ($J_{i}^{p}$ and $J_{i}^{n}$), and the heat flux density (q_i) can be expressed as [47, 51,52,53]

$$\begin{gathered} T_{ij} = c_{ijkl} S_{kl} - e_{kij} E_{k} - \lambda_{ij} \theta , \hfill \\ D_{i} = e_{ijk} S_{jk} + \varepsilon_{ij} E_{j} + p_{i} \theta , \hfill \\ J_{i}^{p} = qp\mu_{ij}^{p} E_{j} - qD_{ij}^{p} p_{,j} , \hfill \\ J_{i}^{n} = qn\mu_{ij}^{n} E_{j} + qD_{ij}^{n} n_{,j} , \hfill \\ q_{i} = - k_{ij} \theta_{,j} + P_{ij}^{p} J_{j}^{p} + P_{ij}^{n} J_{j}^{n} , \hfill \\ \end{gathered}$$

(2)

where c_ijkl are the elastic constants; e_ijk are the piezoelectric constants; $\theta$ is the small temperature deviation from the reference temperature $\Theta_{0}$; $\lambda_{ij}$ are the thermoelastic constants; $\varepsilon_{ij}$ are the dielectric constants; p_i are the pyroelectric constants; $\mu_{ij}^{p}$ and $\mu_{ij}^{n}$ are the mobilities for holes and electrons; $D_{ij}^{p}$ and $D_{ij}^{n}$ are the diffusion constants for holes and electrons; k_ij are the thermal conductivity constants; and $P_{ij}^{p}$ and $P_{ij}^{n}$ are the Peltier constants for holes and electrons. S_kl represent the strain components, whereas E_j are the electric fields. They can be characterized by the subsequent relations:

$$S_{ij} = \frac{1}{2}(u_{i,j} + u_{j.i} ),\quad E_{i} = - \varphi_{,i} - \alpha_{ij} \theta_{,j} ,$$

(3)

where $\alpha_{ij}$ represent the Seebeck constants; u_i(x₁, x₂, x₃) are the i-th components of the mechanical displacement vector u of a point (x₁, x₂, x₃); and φ(x₁, x₂, x₃) stand for and electrostatic potential.

In this paper, the thermo-electromechanical coupling problem of a PS fiber, which incorporates in-plane displacement, electric field, temperature, and n-type charge carriers, will be considered. As shown in Fig. 1, the PS fiber occupies A with $A = \left\{ {(x_{1} ,x_{2} )|\left| {x_{1} } \right| \le \frac{b}{2},\left| {x_{2} } \right| \le \frac{h}{2}} \right\}$, where the thermo-piezoelectric semiconductor of crystals is the 6mm point group (e.g., ZnO). According to references [34, 42], if the polarization axis c corresponds with the x₃-direction, axial deformation has a considerable effect on the axial potential, temperature change, and carrier disturbance. If the polarization axis c coincides with the vertical axis (x₁ or x₂-axis), axial deformation will be associated with the thickness potential, temperature change, and carrier disturbance. In this paper, we consider the influence of the axial loads on the axial current in the thermo-piezoelectric semiconductor rod. Therefore, the polar axis c is parallel to the x₃-direction.

Fig. 1

The schematic diagram of the PS fiber

We only consider a PS fiber with thermo-electromechanical loads along the x₃-direction. Therefore, the displacement u_i, the electrostatic potential $\varphi$ and the temperature deviation $\theta$ in the PS fiber can be approximately described by [35, 48, 54, 55]

$$u_{1} = u_{2} \simeq 0,\quad u_{3} \simeq u_{3}^{(0)} (x_{3} ),\quad \varphi \simeq \varphi_{{}}^{(0)} (x_{3} ),\quad \theta \simeq \theta_{{}}^{(0)} (x_{3} ),$$

(4)

where $u_{3}^{(0)}$ represent the axial displacement of the PS fiber, which is assumed to be only dependent on x₃; $\varphi^{(0)}$ and $\theta^{(0)}$ represent the axial potential change and the axial temperature deviation, respectively, which are closely incorporated with the axial deformation $u_{3}^{(0)}$. For n-type PS fibers, we omit the variation of minority carriers, which assumes that p = 0. The concentration of electrons in the PS fiber is denoted as $n = n_{0} + \Delta n$, where n₀ is the concentration of ionized acceptors, and the carrier concentration perturbation $\Delta n$ is approximated by

$$\Delta n \simeq n^{(0)} (x_{3} ).$$

(5)

Substituting Eq. (4) into Eq. (3) leads to the strain and electric field components in the PS fiber, which reads

$$S_{33} = u_{3,3}^{(0)} ,\quad E_{3} = - \varphi_{,3}^{(0)} - \alpha \theta_{,3}^{(0)} .$$

(6)

Since $u_{3}^{(0)}$, $\varphi^{(0)}$, $\theta^{(0)}$, $n^{(0)}$ only depend on x₃, we can observe S₃₃ and E₃ only rely on x₃. Substituting these variables into Eq. (2), we can find that $T_{ij}$, $D_{i}$, $J_{i}^{n}$, $q_{i}$ are the univariate functions of x₃.

Following the similar steps in [10, 13, 56, 57], the 1D basic equations of the PS fiber under extension can be obtained by integrating the products of (1) with $x_{1}^{0} x_{2}^{0}$ over the cross section A:

$$\begin{gathered} T_{33,3}^{(0)} + {\mathcal{F}}_{3}^{(0)} = 0, \hfill \\ D_{3,3}^{(0)} + {\mathcal{D}}^{(0)} = - q^{(0)} n^{(0)} , \hfill \\ J_{3,3}^{n(0)} + {\mathcal{J}}^{n(0)} = 0, \hfill \\ q_{3,3}^{(0)} + {\mathcal{Q}}^{(0)} = E_{3} J_{3}^{n(0)} , \hfill \\ \end{gathered}$$

(7)

where the thermo-electromechanical resultants over a cross section are defined by

$$\{ T_{ij}^{(0)} ,D_{i}^{(0)} ,J_{i}^{n(0)} ,q_{i}^{(0)} \} = \int_{A}^{{}} {x_{1}^{0} x_{2}^{0} \{ T_{ij} ,D_{i} ,J_{i}^{n} ,q_{i} \} {\text{d}}x_{1} {\text{d}}x_{2} } ,$$

(8)

and ${\mathcal{F}}_{3}^{(0)}$, ${\mathcal{D}}^{(0)}$, ${\mathcal{J}}^{n(0)}$, ${\mathcal{Q}}^{(0)}$ represent the contributions from the internal body force loads, the surface electromechanical loads and the surface heat fluxes of PS fibers, which can be expressed by

$$\begin{gathered} {\mathcal{F}}_{3}^{(0)} = \int_{A}^{{}} {f_{3} {\text{d}}A} + \oint_{\partial A} {T_{3i} n_{i} {\text{d}}l} ,\quad {\mathcal{D}}^{(0)} = \oint_{\partial A} {D_{i} n_{i} {\text{d}}l} , \hfill \\ {\mathcal{J}}^{n(0)} = \oint_{\partial A} {J_{i}^{n} n_{i} {\text{d}}l} ,\quad {\mathcal{Q}}^{(0)} = \oint_{\partial A} {q_{i} n_{i} {\text{d}}l} , \hfill \\ \end{gathered}$$

(9)

where n_i represents the i-th component of the normal vector on the lateral surface of the rod.

Substituting Eqs. (4), (5), and (6) into the constitutive relations (2), following the stress release procedure $T_{11} \simeq 0$ and $T_{22} \simeq 0$, and then integrating along the cross section A, we obtain the 1D constitutive relations as (for more information, see Appendix A)

$$\begin{gathered} T_{33}^{(0)} = \tilde{c}_{33}^{(0)} u_{3,3}^{(0)} + \tilde{e}_{33}^{(0)} (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ) - \tilde{\lambda }_{33}^{(0)} \theta^{(0)} , \hfill \\ D_{3}^{(0)} = \tilde{e}_{33}^{(0)} u_{3,3}^{(0)} - \tilde{\varepsilon }_{33}^{(0)} (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ) + \tilde{p}_{3}^{(0)} \theta^{(0)} , \hfill \\ J_{3}^{n(0)} = q^{(0)} D_{33}^{n} n_{,3}^{(0)} - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ), \hfill \\ q_{3}^{(0)} {\kern 1pt} = - k_{33}^{(0)} \theta_{,3}^{(0)} + \alpha_{33}^{{}} (\Theta_{0}^{{}} + \theta^{(0)} )\{ q^{(0)} D_{33}^{n} n_{,3}^{(0)} \; - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} )\} . \hfill \\ \end{gathered}$$

(10)

Due to the nonlinearity of the current drift–diffusion equation and the thermoelectric effect, the obtained 1D constitutive relations are highly nonlinear. Taking the 1D constitutive relations (10) into the 1D basic Eq. (7) of the PS fiber under extension, therefore, the nonlinear governing equations regarding the axial displacement $u_{3}^{(0)}$, the electrostatic potential $\varphi^{(0)}$, the temperature deviation $\theta^{(0)},$ and the carrier concentration perturbation $n^{(0)}$ can be written as

$$\begin{gathered} \tilde{c}_{33}^{(0)} u_{3,33}^{(0)} + \tilde{e}_{33}^{(0)} (\varphi_{,33}^{(0)} + \alpha_{33}^{{}} \theta_{,33}^{(0)} ) - \tilde{\lambda }_{33}^{(0)} \theta_{,3}^{(0)} + {\mathcal{F}}_{3}^{(0)} = 0, \hfill \\ \tilde{e}_{33}^{(0)} u_{3,33}^{(0)} - \tilde{\varepsilon }_{33}^{(0)} (\varphi_{,33}^{(0)} + \alpha_{33}^{{}} \theta_{,33}^{(0)} ) + \tilde{p}_{3}^{(0)} \theta_{,3}^{(0)} + {\mathcal{D}}^{(0)} = - q^{(0)} n^{(0)} , \hfill \\ q^{(0)} D_{33}^{n} n_{,33}^{(0)} - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,33}^{(0)} + \alpha_{33}^{{}} \theta_{,33}^{(0)} ) - q^{(0)} \mu_{33}^{n} n_{,3}^{(0)} (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ) + {\mathcal{J}}^{n(0)} = 0, \hfill \\ - k_{33}^{(0)} \theta_{,33}^{(0)} + \alpha_{33}^{{}} \{ (\Theta_{0}^{{}} + \theta^{(0)} )[q^{(0)} D_{33}^{n} n_{,3}^{(0)} \; - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} )]\}_{,3} + {\mathcal{Q}}^{(0)} \hfill \\ \qquad \quad = - (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} )[q^{(0)} D_{33}^{n} n_{,3}^{(0)} - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} )]. \hfill \\ \end{gathered}$$

(11)

Hence, coupled with the associated boundary conditions at the interface

$$\begin{gathered} T_{33}^{(0)} = N,\quad {\text{or}}\quad u_{3}^{(0)} = \overline{u}_{3} , \hfill \\ D_{3}^{(0)} = \overline{D},\quad {\text{or}}\quad \varphi_{{}}^{(0)} = \overline{\varphi }, \hfill \\ J_{3}^{n(0)} = \overline{J},\quad {\text{or}}\quad n^{(0)} = \overline{n}, \hfill \\ q_{3}^{(0)} = \overline{q},\quad {\text{or}}\quad \theta^{(0)} = \overline{\theta }. \hfill \\ \end{gathered}$$

(12)

Then, the partial differential Eq. (11) with the variables $u_{3}^{(0)}$, $\varphi^{(0)}$, $\theta^{(0)}$, $n^{(0)}$ are uniquely determined. If the electron current is linearized and the Seebeck effect and Peltier effect are ignored in this model, the governing Eq.  (11) _1,2,3 degenerate into the linear equations

$$\begin{gathered} \tilde{c}_{33}^{(0)} u_{3,33}^{(0)} + \tilde{e}_{33}^{(0)} \varphi_{,33}^{(0)} - \tilde{\lambda }_{33}^{(0)} \theta_{,3}^{(0)} + {\mathcal{F}}_{3}^{(0)} = 0, \hfill \\ \tilde{e}_{33}^{(0)} u_{3,33}^{(0)} - \tilde{\varepsilon }_{33}^{(0)} \varphi_{,33}^{(0)} + \tilde{p}_{3}^{(0)} \theta_{,3}^{(0)} + {\mathcal{D}}^{(0)} = - q^{(0)} n^{(0)} , \hfill \\ q^{(0)} D_{33}^{n} n_{,33}^{(0)} - q^{(0)} \mu_{33}^{n} n_{0} \varphi_{,33}^{(0)} + {\mathcal{J}}^{n(0)} = 0, \hfill \\ \end{gathered}$$

(13)

which is consistent with the governing equations of the semi-coupling theory between temperature deviation and PS fibers [36, 37].

3 Numerical results and discussion

In this section, we will solve the nonlinear governing equations presented in Eq. (11) with corresponding boundary conditions. In the model established in Sect. 2, one can observe that the electric currents give rise to the Joule heating effect; the Peltier effect induces heat fluxes to flow through the structure and accumulate at the boundaries; at the same time, temperature changes will, in turn, affect the distribution of stress and electric fields, thereby altering the current. Therefore, there is a strong coupling between heat, force, and electric fields. Due to the high nonlinearity of the model, we adopted COMSOL Multiphysics to obtain numerical results.

In this research, we will focus on n-type doped ZnO PS fibers with the concentration of impurities n₀ = 10²¹ m⁻³ subjected to static extensional deformations. Consider the rod model under axial mechanical loads as shown in Fig. 1. Geometric parameters in this section are L = 10 μm, b = h = 1 μm. The material parameters employed in the numerical examples are given in Table 1 (under the reference temperature $\Theta_{0} = 300\;{\text{K}}$).

Table 1 Material parameters of n-type ZnO PS fibers

3.1 The influence of thermoelectric effects on fiber

In this section, we will investigate the influence of the Peltier effect and Seebeck effect on the thermo-electromechanical fields of the n-type doped ZnO PS fiber, as shown in Fig. 2. From reference [58,59,60,61], we can observe that the boundary conditions will influence the Seebeck and Peltier effects in the rod. In order to highlight the influence of the thermoelectric effect, we assume that the temperature deviation $\theta^{(0)}$ at x₃ = 0 is zero and x₃ = L is adiabatic. Electrically, the voltage $\varphi^{(0)}$ is applied at x₃ = L with $\overline{\varphi } = 0.2\;{\text{V}}$. The boundary conditions are as follows:

$$\begin{gathered} u_{3}^{(0)} (0) = u_{3}^{(0)} (L) = 0,\quad \varphi_{{}}^{(0)} (0) = 0,\quad \varphi_{{}}^{(0)} (L) = \overline{\varphi }, \hfill \\ n_{{}}^{(0)} (0) = n_{{}}^{(0)} (L) = 0,\quad \theta_{{}}^{(0)} (0) = 0,\quad q_{{}}^{(0)} (L) = 0. \hfill \\ \end{gathered}$$

(14)

Fig. 2

The simply supported PS fiber under a driven voltage $\overline{\varphi }$ with constant temperature at the left end and adiabatic at the right end

Figure 3 illustrates the influence of the Seebeck coefficient $\alpha_{33}$ (where there is a relationship between the Peltier coefficient and the Seebeck coefficient $P_{33}^{n} = \alpha_{33} \Theta$) on the thermo-electromechanical fields with an applied voltage $\overline{\varphi } = 0.2\;{\text{V}}$. It can be seen that the thermoelectric effects have a significant impact on the temperature field and carrier distribution in PS fibers, while it has little influence on the electrostatic potential in the fiber. As shown in Fig. 3c, for n-type PSs, the negative Seebeck coefficient can induce larger carrier perturbations, while the positive Seebeck coefficient can suppress the carrier perturbations in the rod. It is shown in Eq. (2)₅ that the direction of the heat flow caused by the Peltier effect is determined by the positive and negative Peltier coefficient. It is consistent with the phenomenon shown in Fig. 3d that the heat converges to the adiabatic end with a negative Peltier coefficient, while the positive Peltier coefficient causes the heat flow in the opposite direction, reducing the temperature of the adiabatic end. Therefore, we can see that the impact of the thermoelectric effects is considerable. In the process of semiconductor device design, ignoring thermoelectric effects may lead to simulation misalignment and even risk.

Fig. 3

The influence of the thermoelectric effects on the thermo-electromechanical fields of a n-type PS fiber with the applied voltage $\overline{\varphi } = 0.2\;{\text{V}}$ a axial deformation, b electric potential, c electron concentration perturbation, d temperature deviation

3.2 Static extension of a fiber

In this subsection, we examine the impact of axial forces on thermo-electromechanical fields. As shown in Fig. 4, we consider a rod fixed at both ends and apply a voltage of $\overline{\varphi }$ at the right end of the rod to form a stable current in the fiber. Since the carrier concentration and temperature remain invariant at the endpoint, the following boundary conditions are applied at the left and right ends of the fiber:

$$\begin{gathered} u_{3}^{(0)} (0) = u_{3}^{(0)} (L) = 0,\quad \varphi_{{}}^{(0)} (0) = 0,\quad \varphi_{{}}^{(0)} (L) = \overline{\varphi }, \hfill \\ n_{{}}^{(0)} (0) = n_{{}}^{(0)} (L) = 0,\quad \theta_{{}}^{(0)} (0) = \theta_{{}}^{(0)} (L) = 0. \hfill \\ \end{gathered}$$

(15)

Fig. 4

The simply supported PS fiber under an axial force $N$ and a driven voltage $\overline{\varphi }$ with a constant temperature at both ends

There is an axial force N acting on the rod at x₃ = x₀, which leads to a sudden change in axial stress $T_{33}^{(0)}$. At the same time, the other physical fields, including displacement, potential, electric displacement, and current, as well as the hole and electron concentrations, are all continuous, which means that

$$\begin{gathered} u_{3}^{(0)} (x_{0}^{ - } ) = u_{3}^{(0)} (x_{0}^{ + } ),\quad \varphi_{{}}^{(0)} (x_{0}^{ - } ) = \varphi_{{}}^{(0)} (x_{0}^{ + } ), \hfill \\ n_{{}}^{(0)} (x_{0}^{ - } ) = n_{{}}^{(0)} (x_{0}^{ + } ),\quad \theta_{{}}^{(0)} (x_{0}^{ - } ) = \theta_{{}}^{(0)} (x_{0}^{ + } ), \hfill \\ T_{33}^{(0)} (x_{0}^{ - } ) = T_{33}^{(0)} (x_{0}^{ + } ) + N,\quad D_{3}^{(0)} (x_{0}^{ - } ) = D_{3}^{(0)} (x_{0}^{ + } ), \hfill \\ J_{3}^{n(0)} (x_{0}^{ - } ) = J_{3}^{n(0)} (x_{0}^{ + } ),\quad q_{3}^{(0)} (x_{0}^{ - } ) = q_{3}^{(0)} (x_{0}^{ + } ). \hfill \\ \end{gathered}$$

(16)

Therefore, by combining the boundary conditions (15) and (16) together, the variables $u_{3}^{(0)}$, $\varphi^{(0)}$, $\theta^{(0)}$, $n^{(0)}$ are uniquely determined.

The numerical solution for the fundamental thermo-electromagnetic fields of an n-type PS fiber and the impact of axial force N applied at x₀ = L/2 on these fields with the driven voltage $\overline{\varphi } = 0.2\;{\text{V}}$ are shown in Fig. 5. The maximum axial deformation $u_{3}^{(0)}$, depicted in Fig. 5a, occurs at x₀ = L/2 due to the application of axial force N at this interface. It can be seen from Fig. 5b that the polarization electric field generated by the piezoelectric effect in the middle of the beam leads to a severe potential fluctuation. Because the carrier migration in the semiconductor material is affected by the polarized electric field, under the action of the piezoelectric effect, the free electrons in the n-type doped semiconductor material spontaneously gather near the barrier and away from the potential well. As shown in Fig. 5c, a positive polarization potential is generated under the action of a positive axial force, which leads to the accumulation of the free electrons in the n-type PS fiber around the potential barrier; also, a negative polarization potential is generated under a negative axial force, which leads to a sharp decrease of electrons in the middle of the beam. The distribution of temperature deviation under different axial forces is shown in Fig. 5d. In general, the temperature in the PS fiber increases slightly when a positive axial force is applied, but it decreases greatly when a negative axial force is applied to the beam.

Fig. 5

Effects of the axial force applied at x₀ = L/2 on the thermo-electromechanical fields of a n-type PS fiber with the applied voltage $\overline{\varphi } = 0.2\;{\text{V}}$ a axial deformation, b electric potential, c electron concentration perturbation, d temperature deviation

The temperature deviation of the PS fiber with the driven voltage $\overline{\varphi } = 0.5\;{\text{V}}$ under the axial force $N = - 10\;{\upmu N}$ applied at x₀ = L/4, x₀ = L/2, and x₀ = 3L/4 is shown in Fig. 6. As shown in Fig. 5b, under the action of an axial force opposite to the polarization direction, a potential well will be generated near the action point of the force, which results in the carrier accumulation and temperature increasing around the potential well. Therefore, we can observe that the main heating position shifts to the left and right corresponding to the action point of the axial force, which means that the temperature field inside heating components can be effectively controlled with the help of this mechanically induced potential well.

Fig. 6

The influence of axial force at different points on the temperature deviation of the PS fiber a applied at x₀ = L/4, b applied at x₀ = L/2, c applied at x₀ = 3L/4

As shown in Fig. 7, we consider the numerical solution for the fundamental thermo-electromagnetic fields of a n-type PS fiber with the driven voltage $\overline{\varphi } = 0.2\;{\text{V}}$ under the action of two axial forces $N_{1} ,\;N_{2}$ applied at x₀ = L/3 and 2L/3, respectively. The axial deformation of the fiber is shown in Fig. 8a. We can observe that the location of the maximum deformation is consistent with the axial force. As can be seen from Fig. 8b and c, a positive polarization potential will be generated in the fiber if the axial force is compatible with the polarization direction, causing the free electrons in the n-type PS fiber to gather near the potential barrier whether at x₀ = L/3 or x₀ = 2L/3. Conversely, a significant drop in free electrons is observed at x₀ = L/3 or x₀ = 2L/3, while the axial force is directed in the opposite direction from the polarization direction. The distribution of temperature deviation under different axial forces is shown in Fig. 8d. We can observe an increase in temperature near the point of axial force. Because the polarization potential hinders the current flow, the temperature is significantly reduced, when the negative axial forces are applied on the fiber.

Fig. 7

The simply supported PS fiber under two axial forces $N_{1} ,\;N_{2}$ and a driven voltage $\overline{\varphi }$ with a constant temperature at both ends

Fig. 8

Effects of the axial forces applied at x₀ = L/3, 2L/3 on the thermo-electromechanical fields of a n-type PS fiber with the applied voltage $\overline{\varphi } = 0.2\;{\text{V}}$ a axial deformation, b electric potential, c electron concentration perturbation, d temperature deviation

3.3 The tunability of axial forces on current and temperature

A PS element can be employed as a circuit element in piezotronics applications. It is essential for the designer of the device to understand the current–voltage relationship of the element and how this relationship is affected by the axial mechanical loads.

The following boundary conditions are applied at both ends of the fiber:

$$\begin{gathered} u_{3}^{(0)} (0) = u_{3}^{(0)} (L) = 0,\quad \varphi_{{}}^{(0)} (0) = 0,\quad \varphi_{{}}^{(0)} (L) = \overline{\varphi }, \hfill \\ n_{{}}^{(0)} (0) = n_{{}}^{(0)} (L) = 0,\quad \theta_{{}}^{(0)} (0) = \theta_{{}}^{(0)} (L) = 0, \hfill \\ \end{gathered}$$

(17)

where the boundary conditions (17) indicate that a driving voltage is applied at the right end and the left end is electrically grounded. Similarly, at the axial force action surface x = x₀, we have

$$\begin{gathered} u_{3}^{(0)} (x_{0}^{ - } ) = u_{3}^{(0)} (x_{0}^{ + } ),\quad \varphi_{{}}^{(0)} (x_{0}^{ - } ) = \varphi_{{}}^{(0)} (x_{0}^{ + } ), \hfill \\ n_{{}}^{(0)} (x_{0}^{ - } ) = n_{{}}^{(0)} (x_{0}^{ + } ),\quad \theta_{{}}^{(0)} (x_{0}^{ - } ) = \theta_{{}}^{(0)} (x_{0}^{ + } ), \hfill \\ T_{33}^{(0)} (x_{0}^{ - } ) = T_{33}^{(0)} (x_{0}^{ + } ) + {N},\quad D_{3}^{(0)} (x_{0}^{ - } ) = D_{3}^{(0)} (x_{0}^{ + } ), \hfill \\ J_{3}^{n(0)} (x_{0}^{ - } ) = J_{3}^{n(0)} (x_{0}^{ + } ),\quad q_{3}^{(0)} (x_{0}^{ - } ) = q_{3}^{(0)} (x_{0}^{ + } ). \hfill \\ \end{gathered}$$

(18)

The regulation of axial force on the current flowing through the PS fiber calculated by Eq. (10)₃ is studied in Fig. 9. In Fig. 9a, we show the electrostatic potential generated in the fiber with the driven voltage $\overline{\varphi } = 0$ under different axial forces N applied at x₀ = L/2, and the relationship between the current flowing through the PS fiber and the driving voltage, known as the current–voltage characteristic, is represented in Fig. 9b. The potential well generated by an axial force opposite to the polarization direction can hinder the motion of the electrons. As a result, no current is produced when the driving voltage is low, while axial current is produced when the applied voltage is over a particular threshold. Therefore, the axial force works as a switch that can turn on or shut down the current.

Fig. 9

Effects of an axial force applied in the middle of the rod a The potential well with the driven voltage $\overline{\varphi } = 0$, b current–voltage relation

In Fig. 10, we consider the potential well, current–voltage relation, and temperature deviation generated in the n-type PS fiber under the following three conditions: one axial force $N = - 10\;{\upmu N}$ applied at the point x₀ = L/2; two axial forces $N_{1} = N_{2} = - 10\;{\upmu N}$ applied at points x₀ = L/3, 2L/3; and three axial forces $N_{1} = N_{2} = N_{3} = - 10\;{\upmu N}$ applied at points x₀ = L/4, L/2, 3L/4. Figure 10a demonstrates that the polarization potential is only generated near the point of the axial force, and the magnitude of the polarization potential is only proportional to the strength of the axial force. Correspondingly, from Fig. 10b, we can see that the current generated under the same voltage decreases with the increased number of axial forces applied to the fiber. In addition, it can be seen from Fig. 10c–e that the temperature deviation in the fiber is consistent with the prediction of Eq. (7)₄, which indicates that when the current drops, the Joule heat produced in the fiber also drops, resulting in a drop in the temperature produced in the rod. Therefore, in addition to applying a stronger axial force, increasing the number of axial forces applied to the rod can also effectively manipulate the current–voltage relation and the temperature deviation of the PS fiber.

Fig. 10

Effects of different numbers of axial forces applied in the middle of the rod a The potential well with the driven voltage $\overline{\varphi } = 0$, b current–voltage relation, c one axial force with driven voltage $\overline{\varphi } = 0.5\;{\text{V}}$, d two axial forces with driven voltage $\overline{\varphi } = 0.5\;{\text{V}}$, e three axial forces with driven voltage $\overline{\varphi } = 0.5\;{\text{V}}$

4 Conclusions

In this paper, we delve into the thermo-electromechanical fields of a n-type doped ZnO PS fiber with consideration of axial extension deformation. Combining the classic tension and compression theory of fiber and the 3D macroscopic theory of PSs with consideration of thermal conduction, pyroelectric effect, and thermoelectric effects (Seebeck effect and Peltier effect), a 1D model for the extension of PS fibers is established.

Based on this model, we analyze the influence of the Seebeck effect and Peltier effect on the potential, current, carrier, and temperature field in the PS fiber. On this basis, we study the thermo-electromechanical coupling behavior of the PS fiber under the action of axial force. The numerical experiment shows that there is a disturbance of the electrostatic potential, carrier redistribution, and temperature variation near the action point of the axial force and the main heating position moves with the axial force. In addition, we also observe that more applied axial forces will lead to more potential wells generated by piezoelectric polarization, which will increase the dispersal of carriers. These results provide a theoretical foundation for the force-sensitive sensor and force-based adjustment of the current–voltage relation and temperature deviation of the PS fibers, which opens up new design options for PS architectures for force-based piezotronics applications.

Appendix A

Considering the hexagonal crystal (in the 6mm point group), the 1D constitutive relations can be written as

$$\begin{gathered} T_{11} = c_{11} S_{11} + c_{12} S_{22} + c_{13} S_{33} - e_{31} E_{3} - \lambda_{11} \theta , \hfill \\ T_{22} = c_{12} S_{11} + c_{11} S_{22} + c_{13} S_{33} - e_{31} E_{3} - \lambda_{11} \theta , \hfill \\ T_{33} = c_{13} S_{11} + c_{13} S_{22} + c_{33} S_{33} - e_{33} E_{3} - \lambda_{33} \theta , \hfill \\ D_{3} = e_{31} S_{11} + e_{31} S_{22} + e_{33} S_{33} + \varepsilon_{33} E_{3} + p_{3} \theta , \hfill \\ J_{3}^{n} = qn\mu_{33}^{n} E_{3} + qD_{33}^{n} n_{,3} , \hfill \\ q_{3} = - k_{33} \theta_{,3} + P_{33}^{n} J_{3}^{n} . \hfill \\ \end{gathered}$$

(19)

Through the stress release procedure $T_{22} \simeq 0$, the strain in the thickness direction can be expressed as

$$S_{22} = - \frac{{c_{12} }}{{c_{11} }}S_{11} - \frac{{c_{13} }}{{c_{11} }}S_{33} + \frac{{e_{31} }}{{c_{11} }}E_{3} + \frac{{\lambda_{11} }}{{c_{11} }}\theta .$$

(20)

Substituting Eq. (20) into Eq. (19), we can obtain

$$\begin{gathered} T_{{11}} = \left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)S_{{11}} + \left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)S_{{33}} - \left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)E_{3} - \left( {\lambda _{{11}} - \frac{{\lambda _{{11}} c_{{12}} }}{{c_{{11}} }}} \right)\theta , \hfill \\ T_{{33}} = \left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)S_{{11}} + \left( {c_{{33}} - \frac{{c_{{13}}^{2} }}{{c_{{11}} }}} \right)S_{{33}} - \left( {e_{{33}} - \frac{{e_{{31}} c_{{13}} }}{{c_{{11}} }}} \right)E_{3} - \left( {\lambda _{{33}} - \frac{{\lambda _{{11}} c_{{13}} }}{{c_{{11}} }}} \right)\theta , \hfill \\ D_{3} = \left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)S_{{11}} + \left( {e_{{33}} - \frac{{e_{{31}} c_{{13}} }}{{c_{{11}} }}} \right)S_{{33}} + \left( {\varepsilon _{{33}} + \frac{{e_{{31}}^{2} }}{{c_{{11}} }}} \right)E_{3} + \left( {p_{3} + \frac{{e_{{31}}^{{}} \lambda _{{11}} }}{{c_{{11}} }}} \right)\theta . \hfill \\ \end{gathered}$$

(21)

Furthermore, considering stress relaxation $T_{11} \simeq 0$ for the PS fiber, the proper constitutive relations can be rewritten as:

$$\begin{gathered} T_{33} = \tilde{c}_{33} S_{33} - \tilde{e}_{33} E_{3} - \tilde{\lambda }_{33} \theta , \hfill \\ D_{3} = \tilde{e}_{33} S_{33} + \tilde{\varepsilon }_{33} E_{3} + \tilde{p}_{3} \theta , \hfill \\ J_{3}^{n} = qn\mu_{33}^{n} E_{3} + qD_{33}^{n} n_{,3} , \hfill \\ q_{3} = - k_{33} \theta_{,3} + P_{33}^{n} J_{3}^{n}. \hfill \\ \end{gathered}$$

(22)

where the effective material constants are defined by

$$\begin{gathered} \tilde{c}_{{33}} = \left( {c_{{33}} - \frac{{c_{{13}}^{2} }}{{c_{{11}} }}} \right) - \frac{{\left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)\left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)}}{{\left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)}}, \hfill \\ \tilde{e}_{{33}} = \left( {e_{{33}} - \frac{{e_{{31}} c_{{13}} }}{{c_{{11}} }}} \right) - \frac{{\left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)\left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)}}{{\left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)}}, \hfill \\ \tilde{\lambda }_{{33}} = \left( {\lambda _{{33}} - \frac{{\lambda _{{11}} c_{{13}} }}{{c_{{11}} }}} \right) - \frac{{\left( {\lambda _{{11}} - \frac{{\lambda _{{11}} c_{{12}} }}{{c_{{11}} }}} \right)\left( {c_{{13}} - \frac{{c_{{12}} c_{{13}} }}{{c_{{11}} }}} \right)}}{{\left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)}}, \hfill \\ \tilde{\varepsilon }_{{33}} = \left( {\varepsilon _{{33}} + \frac{{e_{{31}}^{2} }}{{c_{{11}} }}} \right) + \frac{{\left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)\left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)}}{{\left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)}}, \hfill \\ \tilde{p}_{3} = \left( {p_{3} + \frac{{e_{{31}}^{{}} \lambda _{{11}} }}{{c_{{11}} }}} \right) + \frac{{\left( {\lambda _{{11}} - \frac{{\lambda _{{11}} c_{{12}} }}{{c_{{11}} }}} \right)\left( {e_{{31}} - \frac{{e_{{31}} c_{{12}} }}{{c_{{11}} }}} \right)}}{{\left( {c_{{11}} - \frac{{c_{{12}}^{2} }}{{c_{{11}} }}} \right)}}. \hfill \\ \end{gathered}$$

(23)

Substituting Eqs. (4), (5), (6) into Eq. (22) and integrating along the cross section A, the 1D constitutive relations can be expressed as

$$\begin{gathered} T_{33}^{(0)} = \tilde{c}_{33}^{(0)} u_{3,3}^{(0)} + \tilde{e}_{33}^{(0)} (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ) - \tilde{\lambda }_{33}^{(0)} \theta^{(0)} , \hfill \\ D_{3}^{(0)} = \tilde{e}_{33}^{(0)} u_{3,3}^{(0)} - \tilde{\varepsilon }_{33}^{(0)} (\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ) + \tilde{p}_{3}^{(0)} \theta^{(0)} , \hfill \\ J_{3}^{n(0)} = q^{(0)} D_{33}^{n} n_{,3}^{(0)} - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} ), \hfill \\ q_{3}^{(0)} {\kern 1pt} = - k_{33}^{(0)} \theta_{,3}^{(0)} + \alpha_{33}^{{}} (\Theta_{0}^{{}} + \theta^{(0)} )\{ q^{(0)} D_{33}^{n} n_{,3}^{(0)} \; - q^{(0)} \mu_{33}^{n} (n_{0} + n^{(0)} )(\varphi_{,3}^{(0)} + \alpha_{33}^{{}} \theta_{,3}^{(0)} )\}. \hfill \\ \end{gathered}$$

(24)

where by introducing absolute temperature $\Theta$, the relationship between the Seebeck effect and the Peltier effect can be written as

$$P_{ij}^{n} = \alpha_{ij} \Theta = \alpha_{ij} (\Theta_{0}^{{}} + \theta^{(0)} ).$$

(25)

and the equivalent material constants over the cross section A are defined by

$$\{ \tilde{c}_{33}^{(0)} ,\tilde{e}_{33}^{(0)} ,\tilde{\lambda }_{33}^{(0)} ,\tilde{\varepsilon }_{33}^{(0)} ,\tilde{p}_{3}^{(0)} ,q^{(0)} ,k_{33}^{(0)} \} = \int_{A}^{{}} {\{ \tilde{c}_{33}^{{}} ,\tilde{e}_{33}^{{}} ,\tilde{\lambda }_{33}^{{}} ,\tilde{\varepsilon }_{33}^{{}} ,\tilde{p}_{3}^{{}} ,q,k_{33}^{{}} \} } {\text{d}}A.$$

(26)