Virtual Work Principle for Piezoelectric Semiconductors and Its Application on Extension and Bending of ZnO Nanowires

Abstract

This paper presents the principle of virtual work (PVW) for piezoelectric semiconductors (PSs), which extends the piezoelectric dielectrics to involve the semiconducting effect. As an application of the PVW, a one-dimensional (1D) approximation theory for the extension and bending of PS nanowires is established by directly applying the PVW and Bernoulli–Euler beam theory with the aid of the second-order approximation of electrostatic potential. To illustrate the new model, the mechanical displacement, electrostatic potential, and concentration of electrons for extension and bending deformation of n-type ZnO nanowires are analytically determined. Additionally, numerical results show that, for n-type Zinc Oxide nanowires, the distribution of electrostatic potential is anti-symmetric along the thickness direction for extension deformation. In contrast, the bending deformation causes a symmetric distribution of electrostatic potential characterized by the zeroth-order and the second-order electrostatic potential. Furthermore, these two different deformations result in the redistribution of electrons. The electrostatic potential can be tuned by adjusting the amplitude of the applied mechanical load. Moreover, we find that the increase in doping level will reduce the magnitude of electrostatic potential due to the screening effect. The presented PVW provides a general approach to establishing structural theories and an effective way of implementing numerical methods.

1. Introduction

Piezoelectric semiconductors (PSs) are materials with both semiconducting and piezoelectric properties [1], whose crystal structure usually belongs to group III-V zinc-blende (e.g., GaAs) and group II-VI wurtzite (e.g., ZnO). In PSs, stress/strain produces electric polarization through piezoelectric coupling, thereby resulting in the redistribution of free carriers. As early as the 1960s, researchers made their efforts on the application of PSs on acoustoelectric devices. Meanwhile, bulk acoustic waves, delay lines, traveling wave amplifiers, and oscillators were developed [2]. The research of piezoelectric semiconductor materials and devices encountered rapid development in the early twenty-first century. Various one-dimensional (1D) and two-dimensional (2D) structures of PSs, including fibers, spirals, belts, and films [3,4,5], have been synthesized using modern material processing techniques. These PS structures have a broad application in energy harvesting, sensing, transduction, and electronics. Studies on PS structures have formed new and rapidly developing research fields called piezotronics and piezo-phototronics [6]. Based on the piezotronics effect, Wang et al. [7] proposed a nanogenerator by using zinc oxide nanowire arrays and achieved the conversion between mechanical energy and electrical energy.

The development of PSs has generated significant interest among researchers in the field of mechanics, leading to extensive theoretical investigations. For example, Sharma et al. [8] studied the surface waves in a PS composite structure. Tian et al. [9] investigated the characteristics of elastic waves propagating in an anisotropic n-type PSs plate. Cao et al. [10] investigated the generalized Rayleigh surface waves in a PS half space. Gokhale and Rais-Zadeh [11] studied the vibrations of acoustic wave resonators. Sladek et al. [12] investigated a dynamic analysis of an anti-plane crack in PSs. Fan et al. [13] proposed a piezoelectric-conductor iterative method for the analysis of cracks in PSs. Qin et al. [14] investigated the fracture behavior of GaN PS ceramics under combined mechanical and electric loading. In addition, the PS rods [15,16,17,18], beams [19,20], plates [21,22,23], PN junction [24], and Schottky junction [25] were analyzed. Recently, Qu et al. [26] proposed a continuum theory for analyzing the interaction between the thermoelectric and mechanical fields in PSs. In addition, nanowires made from Zinc Oxide typically exhibit slender structures (they typically have diameters of 20–100 nm and lengths of a few micrometers [27]). As a result, they are often approximated as a Bernoulli–Euler beam. Researchers have conducted extensive studies on the application of the Bernoulli–Euler beam theory to PSs. For example, Liang et al. [28] developed a 1D static buckling model for PSs. Zhang et al. [29] proposed a 1D multi-field coupling model incorporating surface effects for PS nanostructures based on surface elasticity theory. Zhang et al. [30] investigated the dynamic buckling and free bending vibration of an axially compressed simply supported PS rod. Luo et al. [31] studied the bending of a composite fiber of piezoelectric dielectrics and nonpiezoelectric semiconductors.

The aforementioned models are constructed by using differential methods. It is noted that using differential methods to derive field equations and boundary conditions is ambiguous and difficult in dealing with higher-order structural theories. The principle of virtual work (PVW) is another method for developing structural theories that allows for the simultaneous derivation of field equations and boundary conditions more simply and conveniently [32,33,34]. The PVW offers a crucial framework employed for deriving solutions in mechanics, electrostatics, and so on, serving as the foundation for the finite element method. Using the PVW, researchers investigate structures like beams and plates. For example, Falsone and La Valle [35] used the PVW to derive the field equations of functionally graded beams subjected to axial and transversal loads. Ascione et al. [36] derived the nonlinear equilibrium equations of geometrically imperfect composite beams based on the PVW.

The objective of this work is to propose the PVW for PSs and analyze the electric behavior of the PS nanowires when axial and transverse loadings are applied. The rest of the paper is arranged as follows: In Section 2, the three-dimensional (3-D) framework for PSs is established, including formulations for PVW and constitutive relations. In Section 3, the field equations for PS nanowires and corresponding boundary conditions are derived with the aid of the PVW and the Bernoulli–Euler beam theory which consider the second-order approximation of electrostatic potential. In addition, the 1-D constitutive relations are obtained by taking into account the stress relaxation procedure. Finally, the governing equations of PS nanowires are derived. In Section 4, as the application, the extension and bending for ZnO nanowires are explored. The concluding remarks are given in Section 5.

2. Three-Dimensional Framework for PSs

2.1. The PVW for PSs

The field equations and corresponding boundary conditions for PSs can be derived from the following PVW:

$$\delta {𝕎}_{(d)}+{\delta 𝕎}_{(c)}+{\delta 𝕎}_{(i)}=0,$$

(1)

where δ𝕎_(d), δ𝕎_(c), and δ𝕎_(i) represent the virtual work of actions at a distance, contact forces, and internal forces [37], as

$$\begin{array}{l}{\delta 𝕎}_{(d)}=-{\displaystyle {\int }_{\mathsf{\Omega }}[(f_i-\rho {\ddot{u}}_i)\delta u_i-q(p-n+N_D^{+}-N_A^{-})\delta \phi }\\ \hspace{1em}\hspace{1em}+\lambda q(\dot{p}\delta p-\dot{n}\delta n)]\mathrm{d}v,\end{array}$$

(2)

$${\delta 𝕎}_{(c)}=-{\displaystyle {\int }_{\partial \mathsf{\Omega }}[(t_i\delta u_i)+(\omega \delta \phi )+\lambda (J^p\delta p+J^n\delta n)]\mathrm{d}A},$$

(3)

$${\delta 𝕎}_{(i)}={\displaystyle {\int }_{\mathsf{\Omega }}[T_{ij}\delta S_{ij}-D_i\delta E_i+\lambda (J_i^p\delta P_i+J_i^n\delta N_i)]\mathrm{d}v},$$

(4)

where Ω is the region occupied by PSs; dv is the volume element; f_i is the mechanical body force; ρ is the mass density; u_i is mechanical displacement; q is the elementary charge; p and n are the concentrations of holes and electrons; $N_D^{+}$ and $N_A^{-}$ are the concentrations of ionized donors and acceptors; φ is electrostatic potential; λ is a constant with a unit J·m⁶·s·C⁻¹, which satisfies all terms associated with concentrations of free-carriers having units J; t_i, ω, J^p, and Jⁿ are surface traction, surface charge, and surface electric currents for holes and electrons, respectively; T_ij, D_i, $J_i^p$ and $J_i^n$ are Cauchy stress, electric displacement, electric current densities for holes and electrons, respectively; and S_ij, E_i, P_i, and N_i are the strain, electric field, gradient of concentration of holes and electrons.

In addition, S_ij, E_i, P_i, and N_i are related to u_i, φ, p, and n through

$$S_{ij}=(u_{i,j}+u_{j,i})/2,\hspace{1em}{E}_i=-{\phi }_{,i},\hspace{1em}{P}_i=p_{,i},\hspace{1em}{N}_i=n_{,i}.$$

(5)

It is noted that Equation (1) is a weak form for differential equations of semiconducting systems with elastic considerations. Due to the dissipation nature (the first-order time derivative of concentrations), no functionals can be obtained, and δ𝕎 represents a symbol of virtual work rather than variation of 𝕎.

2.2. Constitutive Relations for PSs

The constitutive relations (for Cauchy stress T_ij, electric displacement D_i, electric current densities for holes and electrons, $J_i^p$ and $J_i^n$) of PSs read [21]:

$$T_{ij}=c_{ijkl}{S}_{kl}-e_{kij}{E}_k,$$

(6)

$$D_i={\epsilon }_{ij}{E}_j+e_{ijk}{S}_{jk},$$

(7)

$$J_i^p=qp{\mu }_{ij}^p{E}_j-q{D}_{ij}^p{P}_j,$$

(8)

$$J_i^n=qn{\mu }_{ij}^n{E}_j+q{D}_{ij}^n{N}_j,$$

(9)

where c_ijkl, e_ijk, and ε_ij are, respectively, the elastic stiffness constants, piezoelectric constants, and dielectric constants; ${\mu }_{ij}^p$ and ${\mu }_{ij}^n$ are, respectively, the carrier mobilities for holes and electrons; and $D_{ij}^p$ and $D_{ij}^n$ are, respectively, the carrier diffusion constants for holes and electrons.

Concentration perturbations for holes and electrons can be expressed as:

$$\Delta p=p-p_0,\hspace{1em}\Delta n=n-n_0,\hspace{1em}{p}_0=N_A^{-},\hspace{1em}{n}_0=N_D^{+},$$

(10)

where p₀ and n₀ denote doping level. For the case of small Δp and Δn, the constitutive relations (8) and (9) can be linearized into

$$\begin{array}{l}{J}_i^p\cong q{p}_0{\mu }_{ij}^p{E}_j-q{D}_{ij}^p{P}_j,\\ J_i^n\cong q{n}_0{\mu }_{ij}^n{E}_j+q{D}_{ij}^n{N}_j.\end{array}$$

(11)

3. One-Dimensional Model for PS Nanowires

3.1. Field Equations for PS Nanowires

Consider a PS nanowire as shown in Figure 1. The length of PS nanowires is L, which is along the x₁ direction. The width and height of PS nanowires are, respectively, 2b and 2h. The surface traction t_i, surface charge ω, and surface electric currents for holes and electrons J^p and Jⁿ are not considered here. Note that although this paper primarily discusses nanowires with rectangular cross-sections, it is also applicable to circular, elliptical, and other cross-sectional shapes.

With the aid of the Bernoulli–Euler beam theory, the displacements of PS nanowires [38] can be given by

$$u_1(\mathit{x},t)\cong u_1^{(0)}(x_1,t)-x_3{u}_{3,1}^{(0)}(x_1,t),\hspace{1em}{u}_2(\mathit{x},t)=0,\hspace{1em}{u}_3(\mathit{x},t)\cong u_3^{(0)}(x_1,t),$$

(12)

where $u_1^{(0)}$ describes the in-plane extensional displacement and $u_3^{(0)}$ represents the bending displacement.

Moreover, the second-order approximation of electrostatic potential, the concentration of holes and electrons can be given by [39]

$$\begin{array}{l}\phi (\mathit{x},t)\cong {\phi }^{(0)}(x_1,t)+x_3{\phi }^{(1)}(x_1,t)+(x_3^2-h^2){\phi }^{(2)}(x_1,t),\\ p(\mathit{x},t)\cong p^{(0)}(x_1,t)+x_3{p}^{(1)}(x_1,t)+(x_3^2-h^2)p^{(2)}(x_1,t),\\ n(\mathit{x},t)\cong n^{(0)}(x_1,t)+x_3{n}^{(1)}(x_1,t)+(x_3^2-h^2)n^{(2)}(x_1,t),\end{array}$$

(13)

where ${\phi }^{(0)}$, ${\phi }^{(1)}$, and ${\phi }^{(2)}$ are, respectively, the zeroth-, first-, and second-order electrostatic potentials. Similarly, $p^{(0)}$ ($n^{(0)}$), $p^{(1)}$ ($n^{(1)}$), and $p^{(2)}$ ($n^{(2)}$) are the zeroth-, first-, and second-order holes (electrons) concentration.

Inserting Equations (12) and (13) into Equation (5), we have the nonzero components of strain, electrostatic potential, and gradient of concentration of holes and electrons:

$$S_{11}=u_{1,1}^{(0)}-x_3{u}_{3,11}^{(0)},$$

(14)

$$\begin{array}{l}{E}_1=-[{\phi }_{,1}^{(0)}+x_3{\phi }_{,1}^{(1)}+(x_3^2-h^2){\phi }_{,1}^{(2)}],\\ E_3=-({\phi }^{(1)}+2x_3{\phi }^{(2)}),\end{array}$$

(15)

$$\begin{array}{l}{P}_1=p_{,1}^{(0)}+x_3{p}_{,1}^{(1)}+(x_3^2-h^2)p_{,1}^{(2)},P_3=p^{(1)}+2x_3{p}^{(2)},\\ N_1=n_{,1}^{(0)}+x_3{n}_{,1}^{(1)}+(x_3^2-h^2)n_{,1}^{(2)},N_3=n^{(1)}+2x_3{n}^{(2)}.\end{array}$$

(16)

Substituting Equations (12) and (13) into Equation (2), the virtual work of actions at a distance yields

$$\begin{array}{l}{\delta 𝕎}_{(d)}={\displaystyle {\int }_0^L[-(f_1^{(0)}-{\rho }^{(0)}{\ddot{u}}_1^{(0)})\delta u_1^{(0)}}-(f_3^{(0)}+f_{1,1}^{(1)}-{\rho }^{(0)}{\ddot{u}}_3^{(0)}\\ +{\rho }^{(2)}{\ddot{u}}_{3,11}^{(0)})\delta u_3^{(0)}+(q^{(0)}{p}^{(0)}+q^{(2)}{p}^{(2)}-q^{(0)}{n}^{(0)}-q^{(2)}{n}^{(2)}\\ +q^{(0)}{N}_D^{+}-q^{(0)}{N}_A^{-})\delta {\phi }^{(0)}+({\tilde{q}}^{(2)}{p}^{(1)}-{\tilde{q}}^{(2)}{n}^{(1)})\delta {\phi }^{(1)}\\ +(q^{(2)}{p}^{(0)}+q^{(4)}{p}^{(2)}-q^{(2)}{n}^{(0)}-q^{(4)}{n}^{(2)}+q^{(2)}{N}_D^{+}\\ -q^{(2)}{N}_A^{-})\delta {\phi }^{(2)}-\lambda (q^{(0)}{\dot{p}}^{(0)}+q^{(2)}{\dot{p}}^{(2)})\delta p^{(0)}\\ -\lambda {\tilde{q}}^{(2)}{\dot{p}}^{(1)}\delta p^{(1)}-\lambda (q^{(2)}{\dot{p}}^{(0)}+q^{(4)}{\dot{p}}^{(2)})\delta p^{(2)}\\ +\lambda (q^{(0)}{\dot{n}}^{(0)}+q^{(2)}{\dot{n}}^{(2)})\delta n^{(0)}+\lambda {\tilde{q}}^{(2)}{\dot{n}}^{(1)}\delta n^{(1)}+\lambda (q^{(2)}{\dot{n}}^{(0)}\\ +q^{(4)}{\dot{n}}^{(2)})\delta n^{(2)}]\mathrm{d}{x}_1+{[(f_1^{(1)}+{\rho }^{(2)}{\ddot{u}}_{3,1}^{(0)})\delta u_3^{(0)}]}_0^L,\end{array}$$

(17)

where

$$\begin{array}{l}\{q^{(0)},{\tilde{q}}^{(2)}\}=2b{\displaystyle {\int }_{-h}^h\{q,q{x}_3^2\}\mathrm{d}{x}_3}=\{I^{(0)},I^{(2)}\}q,\\ \{{\rho }^{(0)},{\rho }^{(2)}\}=2b{\displaystyle {\int }_{-h}^h\{\rho ,\rho x_3^2\}\mathrm{d}{x}_3}=\{I^{(0)},I^{(2)}\}\rho ,\\ q^{(2)}=2b{\displaystyle {\int }_{-h}^{h}q(x_3^2-h^2)\mathrm{d}{x}_3}=-2I^{(2)}q,\\ q^{(4)}=2b{\displaystyle {\int }_{-h}^{h}q{(x_3^2-h^2)}^2\mathrm{d}{x}_3}=I^{(4)}q,\\ I^{(0)}=4bh,I^{(2)}=\frac{4b{h}^3}{3},I^{(4)}=\frac{32b{h}^5}{15},\end{array}$$

(18)

and the resultants of external body force over a cross-section are defined by:

$$\{f_1^{(0)},f_3^{(0)},f_1^{(1)}\}=2b{\displaystyle {\int }_{-h}^h\{f_1,f_3,f_1{x}_3\}\mathrm{d}{x}_3}.$$

(19)

Substituting Equations (14)–(16) into Equation (4), the virtual work of internal forces yields

$$\begin{array}{l}{\delta 𝕎}_{(i)}={\displaystyle {\int }_0^L\{-T_{11,1}^{(0)}\delta u_1^{(0)}-T_{11,11}^{(1)}\delta u_3^{(0)}-D_{1,1}^{(0)}\delta {\phi }^{(0)}+(D_3^{(0)}-D_{1,1}^{(1)})\delta {\phi }^{(1)}}\\ +(2D_3^{(1)}-D_{1,1}^{(2)})\delta {\phi }^{(2)}+\lambda [-J_{1,1}^{p(0)}\delta p^{(0)}+(J_3^{p(0)}-J_{1,1}^{p(1)})\delta p^{(1)}\\ +(2J_3^{p(1)}-J_{1,1}^{p(2)})\delta p^{(2)}]+\lambda [-J_{1,1}^{n(0)}\delta n^{(0)}+(J_3^{n(0)}-J_{1,1}^{n(1)})\delta n^{(1)}\\ +(2J_3^{n(1)}-J_{1,1}^{n(2)})\delta n^{(2)}]\}\mathrm{d}{x}_1+{[T_{11}^{(0)}\delta u_1^{(0)}]}_0^L-{[T_{11}^{(1)}\delta u_{3,1}^{(0)}]}_0^L\\ +{[T_{11,1}^{(1)}\delta u_3^{(0)}]}_0^L+{[D_1^{(0)}\delta {\phi }^{(0)}]}_0^L+{[D_1^{(1)}\delta {\phi }^{(1)}]}_0^L+{[D_1^{(2)}\delta {\phi }^{(2)}]}_0^L\\ +{[\lambda J_1^{p(0)}\delta p^{(0)}]}_0^L+{[\lambda J_1^{p(1)}\delta p^{(1)}]}_0^L+{[\lambda J_1^{p(2)}\delta p^{(2)}]}_0^L\\ +{[\lambda J_1^{n(0)}\delta n^{(0)}]}_0^L+{[\lambda J_1^{n(1)}\delta n^{(1)}]}_0^L+{[\lambda J_1^{n(2)}\delta n^{(2)}]}_0^L,\end{array}$$

(20)

where the resultants of stress, electric displacement, and electric current densities for holes and electrons are defined as:

$$\{T_{11}^{(0)},T_{11}^{(1)}\}=2b{\displaystyle {\int }_{-h}^h\{T_{11},T_{11}{x}_3\}\mathrm{d}{x}_3},$$

(21)

$$\begin{array}{l}\{D_1^{(0)},D_3^{(0)},D_1^{(1)},D_3^{(1)}\}=2b{\displaystyle {\int }_{-h}^h\{D_1,D_3,D_1{x}_3,D_3{x}_3\}\mathrm{d}{x}_3},\\ D_1^{(2)}=2b{\displaystyle {\int }_{-h}^h[D_1(x_3^2-h^2)]\mathrm{d}{x}_3},\end{array}$$

(22)

$$\begin{array}{l}\{J_1^{p(0)},J_3^{p(0)},J_1^{p(1)},J_3^{p(1)}\}=2b{\displaystyle {\int }_{-h}^h\{J_1^p,J_3^p,J_1^p{x}_3,J_3^p{x}_3\}\mathrm{d}{x}_3},\\ J_1^{p(2)}=2b{\displaystyle {\int }_{-h}^h[J_1^p(x_3^2-h^2)]\mathrm{d}{x}_3},\\ \{J_1^{n(0)},J_3^{n(0)},J_1^{n(1)},J_3^{n(1)}\}=2b{\displaystyle {\int }_{-h}^h\{J_1^n,J_3^n,J_1^n{x}_3,J_3^n{x}_3\}\mathrm{d}{x}_3},\\ J_1^{n(2)}=2b{\displaystyle {\int }_{-h}^h[J_1^n(x_3^2-h^2)]\mathrm{d}{x}_3}.\end{array}$$

(23)

Substituting Equations (17) and (20) into Equation (1) then leads to, with the aid of the fundamental lemma of the calculus of variation [40,41],

$$\begin{array}{l}{T}_{11,1}^{(0)}+f_1^{(0)}={\rho }^{(0)}{\ddot{u}}_1^{(0)},\\ T_{11,11}^{(1)}+f_{1,1}^{(1)}+f_3^{(0)}={\rho }^{(0)}{\ddot{u}}_3^{(0)}-{\rho }^{(2)}{\ddot{u}}_{3,11}^{(0)},\\ D_{1,1}^{(0)}=q^{(0)}(p^{(0)}-n^{(0)}+N_D^{+}-N_A^{-})+q^{(2)}(p^{(2)}-n^{(2)}),\\ D_{1,1}^{(1)}-D_3^{(0)}={\tilde{q}}^{(2)}(p^{(1)}-n^{(1)}),\\ D_{1,1}^{(2)}-2D_3^{(1)}=q^{(2)}(p^{(0)}-n^{(0)}+N_D^{+}-N_A^{-})+q^{(4)}(p^{(2)}-n^{(2)}),\\ J_{1,1}^{p(0)}=-q^{(0)}{\dot{p}}^{(0)}-q^{(2)}{\dot{p}}^{(2)},\\ J_{1,1}^{n(0)}=q^{(0)}{\dot{n}}^{(0)}+q^{(2)}{\dot{n}}^{(2)},\\ J_{1,1}^{p(1)}-J_3^{p(0)}=-{\tilde{q}}^{(2)}{\dot{p}}^{(1)},\\ J_{1,1}^{n(1)}-J_3^{n(0)}={\tilde{q}}^{(2)}{\dot{n}}^{(1)},\\ J_{1,1}^{p(2)}-2J_3^{p(1)}=-q^{(2)}{\dot{p}}^{(0)}-q^{(4)}{\dot{p}}^{(2)},\\ J_{1,1}^{n(2)}-2J_3^{n(1)}=q^{(2)}{\dot{n}}^{(0)}+q^{(4)}{\dot{n}}^{(2)},\end{array}$$

(24)

as the field equations, and

$$\begin{array}{l}{T}_{11}^{(0)}=0\hspace{1em}\mathrm{or}\hspace{1em}{u}_1^{(0)}={\overline{u}}_1^{(0)},\hspace{1em}{T}_{11}^{(1)}=0\hspace{1em}\mathrm{or}\hspace{1em}{u}_{3,1}^{(0)}={\overline{u}}_{3,1}^{(0)},\\ T_{11,1}^{(1)}+f_1^{(1)}=0\hspace{1em}\mathrm{or}\hspace{1em}{u}_3^{(0)}={\overline{u}}_3^{(0)},\hspace{1em}{D}_1^{(0)}=0\hspace{1em}\mathrm{or}\hspace{1em}{\phi }^{(0)}={\overline{\phi }}^{(0)},\\ D_1^{(1)}=0\hspace{1em}\mathrm{or}\hspace{1em}{\phi }^{(1)}={\overline{\phi }}^{(1)},\hspace{1em}{D}_1^{(2)}=0\hspace{1em}\mathrm{or}\hspace{1em}{\phi }^{(2)}={\overline{\phi }}^{(2)},\\ J_1^{p(0)}=0\hspace{1em}\mathrm{or}\hspace{1em}{p}^{(0)}={\overline{p}}^{(0)},\hspace{1em}{J}_1^{p(1)}=0\hspace{1em}\mathrm{or}\hspace{1em}{p}^{(1)}={\overline{p}}^{(1)},\\ J_1^{p(2)}=0\hspace{1em}\mathrm{or}\hspace{1em}{p}^{(2)}={\overline{p}}^{(2)},\hspace{1em}{J}_1^{n(0)}=0\hspace{1em}\mathrm{or}\hspace{1em}{n}^{(0)}={\overline{n}}^{(0)},\\ J_1^{n(1)}=0\hspace{1em}\mathrm{or}\hspace{1em}{n}^{(1)}={\overline{n}}^{(1)},\hspace{1em}{J}_1^{n(2)}=0\hspace{1em}\mathrm{or}\hspace{1em}{n}^{(2)}={\overline{n}}^{(2)},\end{array}$$

(25)

as the boundary conditions for x₁ = 0 or L.

3.2. One-Dimensional Constitutive Relations

Consider a PS nanowire with 6mm class crystals that poles along the x₃ direction [21,26], then the constitutive relations Equations (6), (7), and (11) become

$$\begin{array}{l}{T}_{11}=c_{11}{S}_{11}+c_{12}{S}_{22}+c_{13}{S}_{33}-e_{31}{E}_3,\\ T_{22}=c_{12}{S}_{11}+c_{11}{S}_{22}+c_{13}{S}_{33}-e_{31}{E}_3,\\ T_{33}=c_{13}{S}_{11}+c_{13}{S}_{22}+c_{33}{S}_{33}-e_{33}{E}_3,\end{array}$$

(26)

$$\begin{array}{l}{D}_1={\epsilon }_{11}{E}_1,\\ D_3={\epsilon }_{33}{E}_3+e_{31}{S}_{11}+e_{31}{S}_{22}+e_{33}{S}_{33},\end{array}$$

(27)

$$\begin{array}{l}{J}_1^p\cong q{p}_0{\mu }_{11}^p{E}_1-q{D}_{11}^p{P}_1,\\ J_3^p\cong q{p}_0{\mu }_{33}^p{E}_3-q{D}_{33}^p{P}_3,\end{array}$$

(28)

$$\begin{array}{l}{J}_1^n\cong q{n}_0{\mu }_{11}^n{E}_1+q{D}_{11}^n{N}_1,\\ J_3^n\cong q{n}_0{\mu }_{33}^n{E}_3+q{D}_{33}^n{N}_3.\end{array}$$

(29)

Consider the stress relaxation procedure with T₂₂ = T₃₃ = 0 [42], it derives

$$\begin{array}{l}{S}_{22}=-\frac{c_{12}{c}_{33}-c_{13}^2}{c_{11}{c}_{33}-c_{13}^2}{S}_{11}-\frac{c_{13}{e}_{33}-c_{33}{e}_{31}}{c_{11}{c}_{33}-c_{13}^2}{E}_3,\\ S_{33}=\frac{c_{12}{c}_{13}-c_{11}{c}_{13}}{c_{11}{c}_{33}-c_{13}^2}{S}_{11}+\frac{c_{11}{e}_{33}-c_{13}{e}_{31}}{c_{11}{c}_{33}-c_{13}^2}{E}_3.\end{array}$$

(30)

Inserting Equation (30) into Equations (26)–(29) and with the help of Equations (14)–(16), we have

$$T_{11}={\overline{c}}_{11}{u}_{1,1}^{(0)}-{\overline{c}}_{11}{x}_3{u}_{3,11}^{(0)}+{\overline{e}}_{31}{\phi }^{(1)}+2{\overline{e}}_{31}{x}_3{\phi }^{(2)},$$

(31)

$$\begin{array}{l}{D}_1=-{\epsilon }_{11}{\phi }_{,1}^{(0)}-{\epsilon }_{11}{x}_3{\phi }_{,1}^{(1)}-{\epsilon }_{11}(x_3^2-h^2){\phi }_{,1}^{(2)},\\ D_3=-{\overline{\epsilon }}_{33}{\phi }^{(1)}-2{\overline{\epsilon }}_{33}{x}_3{\phi }^{(2)}+{\overline{e}}_{31}{u}_{1,1}^{(0)}-{\overline{e}}_{31}{x}_3{u}_{3,11}^{(0)},\end{array}$$

(32)

$$\begin{array}{l}{J}_1^p\cong -q{p}_0{\mu }_{11}^p[{\phi }_{,1}^{(0)}+x_3{\phi }_{,1}^{(1)}+(x_3^2-h^2){\phi }_{,1}^{(2)}]\\ -q{D}_{11}^p[p_{,1}^{(0)}+x_3{p}_{,1}^{(1)}+(x_3^2-h^2)p_{,1}^{(2)}],\\ J_3^p\cong -q{p}_0{\mu }_{33}^p({\phi }^{(1)}+2x_3{\phi }^{(2)})-q{D}_{33}^p(p^{(1)}+2x_3{p}^{(2)}),\end{array}$$

(33)

$$\begin{array}{l}{J}_1^n\cong -q{n}_0{\mu }_{11}^n[{\phi }_{,1}^{(0)}+x_3{\phi }_{,1}^{(1)}+(x_3^2-h^2){\phi }_{,1}^{(2)}]\\ +q{D}_{11}^n[n_{,1}^{(0)}+x_3{n}_{,1}^{(1)}+(x_3^2-h^2)n_{,1}^{(2)}],\\ J_3^n\cong -q{n}_0{\mu }_{33}^n({\phi }^{(1)}+2x_3{\phi }^{(2)})+q{D}_{33}^n(n^{(1)}+2x_3{n}^{(2)}),\end{array}$$

(34)

where the effective material constants are defined as:

$$\begin{array}{l}{\overline{c}}_{11}=c_{11}+\frac{2c_{12}{c}_{13}^2-c_{33}{c}_{12}^2-c_{11}{c}_{13}^2}{c_{11}{c}_{33}-c_{13}^2},\\ {\overline{e}}_{31}=e_{31}+\frac{(c_{12}-c_{11})c_{13}{e}_{33}+(c_{13}^2-c_{12}{c}_{33})e_{31}}{c_{11}{c}_{33}-c_{13}^2},\\ {\overline{\epsilon }}_{33}={\epsilon }_{33}+\frac{c_{11}{e}_{33}^2-2c_{13}{e}_{31}{e}_{33}+c_{33}{e}_{31}^2}{c_{11}{c}_{33}-c_{13}^2}.\end{array}$$

(35)

According to Equations (21)–(23) and (31)–(34), the correct 1-D constitutive relations can be given by

$$T_{11}^{(0)}={\overline{c}}_{11}^{(0)}{u}_{1,1}^{(0)}+{\overline{e}}_{31}^{(0)}{\phi }^{(1)},T_{11}^{(1)}=-{\overline{c}}_{11}^{(2)}{u}_{3,11}^{(0)}+2{\overline{e}}_{31}^{(2)}{\phi }^{(2)},$$

(36)

$$\begin{array}{l}{D}_1^{(0)}=-{\epsilon }_{11}^{(0)}{\phi }_{,1}^{(0)}+2{\epsilon }_{11}^{(2)}{\phi }_{,1}^{(2)},D_3^{(0)}=-{\overline{\epsilon }}_{33}^{(0)}{\phi }^{(1)}+{\overline{e}}_{31}^{(0)}{u}_{1,1}^{(0)},\\ D_1^{(1)}=-{\epsilon }_{11}^{(2)}{\phi }_{,1}^{(1)},D_3^{(1)}=-2{\overline{\epsilon }}_{33}^{(2)}{\phi }^{(2)}-{\overline{e}}_{31}^{(2)}{u}_{3,11}^{(0)},\\ D_1^{(2)}=2{\epsilon }_{11}^{(2)}{\phi }_{,1}^{(0)}-{\epsilon }_{11}^{(4)}{\phi }_{,1}^{(2)},\end{array}$$

(37)

$$\begin{array}{l}{J}_1^{p(0)}=-{\mu }_{11}^{p(0)}{\phi }_{,1}^{(0)}-D_{11}^{p(0)}{p}_{,1}^{(0)}+2{\mu }_{11}^{p(2)}{\phi }_{,1}^{(2)}+2D_{11}^{p(2)}{p}_{,1}^{(2)},\\ J_3^{p(0)}=-{\mu }_{33}^{p(0)}{\phi }^{(1)}-D_{33}^{p(0)}{p}^{(1)},J_1^{p(1)}=-{\mu }_{11}^{p(2)}{\phi }_{,1}^{(1)}-D_{11}^{p(2)}{p}_{,1}^{(1)},\\ J_3^{p(1)}=-2{\mu }_{33}^{p(2)}{\phi }^{(2)}-2D_{33}^{p(2)}{p}^{(2)},\\ J_1^{p(2)}=2{\mu }_{11}^{p(2)}{\phi }_{,1}^{(0)}+2D_{11}^{p(2)}{p}_{,1}^{(0)}-{\mu }_{11}^{p(4)}{\phi }_{,1}^{(2)}-D_{11}^{p(4)}{p}_{,1}^{(2)},\end{array}$$

(38)

$$\begin{array}{l}{J}_1^{n(0)}=-{\mu }_{11}^{n(0)}{\phi }_{,1}^{(0)}+D_{11}^{n(0)}{n}_{,1}^{(0)}+2{\mu }_{11}^{n(2)}{\phi }_{,1}^{(2)}-2D_{11}^{n(2)}{n}_{,1}^{(2)},\\ J_3^{n(0)}=-{\mu }_{33}^{n(0)}{\phi }^{(1)}+D_{33}^{n(0)}{n}^{(1)},J_1^{n(1)}=-{\mu }_{11}^{n(2)}{\phi }_{,1}^{(1)}+D_{11}^{n(2)}{n}_{,1}^{(1)},\\ J_3^{n(1)}=-2{\mu }_{33}^{n(2)}{\phi }^{(2)}+2D_{33}^{n(2)}{n}^{(2)},\\ J_1^{n(2)}=2{\mu }_{11}^{n(2)}{\phi }_{,1}^{(0)}-2D_{11}^{n(2)}{n}_{,1}^{(0)}-{\mu }_{11}^{n(4)}{\phi }_{,1}^{(2)}+D_{11}^{n(4)}{n}_{,1}^{(2)},\end{array}$$

(39)

where

$$\begin{array}{l}\{{\overline{c}}_{11}^{(m)},{\epsilon }_{11}^{(m)},{\overline{\epsilon }}_{33}^{(m)},{\overline{e}}_{31}^{(m)},{\mu }_{11}^{p(m)},D_{11}^{p(m)},{\mu }_{33}^{p(m)},D_{33}^{p(m)},{\mu }_{11}^{n(m)},D_{11}^{n(m)},{\mu }_{33}^{n(m)},D_{33}^{n(m)}\}\\ =2b{\displaystyle {\int }_{-h}^h\{{\overline{c}}_{11},{\epsilon }_{11},{\overline{\epsilon }}_{33},{\overline{e}}_{31},q{p}_0{\mu }_{11}^p,q{D}_{11}^p,q{p}_0{\mu }_{33}^p,q{D}_{33}^p,q{n}_0{\mu }_{11}^n,q{D}_{11}^n,q{n}_0{\mu }_{33}^n,q{D}_{33}^n\}{x}_3^m\mathrm{d}{x}_3},\hspace{1em}m=0,2,4.\end{array}$$

(40)

The substitution of Equations (36)–(39) into Equation (24) yields the governing equations of PS nanowires, which can be decoupled into two groups,

$$\begin{array}{l}{\overline{c}}_{11}^{(0)}{u}_{1,11}^{(0)}+{\overline{e}}_{31}^{(0)}{\phi }_{,1}^{(1)}+f_1^{(0)}={\rho }^{(0)}{\ddot{u}}_1^{(0)},\\ -{\epsilon }_{11}^{(2)}{\phi }_{,11}^{(1)}-{\overline{e}}_{31}^{(0)}{u}_{1,1}^{(0)}+{\overline{\epsilon }}_{33}^{(0)}{\phi }^{(1)}={\tilde{q}}^{(2)}(p^{(1)}-n^{(1)}),\\ -{\mu }_{11}^{p(2)}{\phi }_{,11}^{(1)}-D_{11}^{p(2)}{p}_{,11}^{(1)}+{\mu }_{33}^{p(0)}{\phi }^{(1)}+D_{33}^{p(0)}{p}^{(1)}=-{\tilde{q}}^{(2)}{\dot{p}}^{(1)},\\ -{\mu }_{11}^{n(2)}{\phi }_{,11}^{(1)}+D_{11}^{n(2)}{n}_{,11}^{(1)}+{\mu }_{33}^{n(0)}{\phi }^{(1)}-D_{33}^{n(0)}{n}^{(1)}={\tilde{q}}^{(2)}{\dot{n}}^{(1)},\end{array}$$

(41)

and

$$\begin{array}{l}-{\overline{c}}_{11}^{(2)}{u}_{3,1111}^{(0)}+2{\overline{e}}_{31}^{(2)}{\phi }_{,11}^{(2)}+f_{1,1}^{(1)}+f_3^{(0)}={\rho }^{(0)}{\ddot{u}}_3^{(0)}-{\rho }^{(2)}{\ddot{u}}_{3,11}^{(0)},\\ -{\epsilon }_{11}^{(0)}{\phi }_{,11}^{(0)}+2{\epsilon }_{11}^{(2)}{\phi }_{,11}^{(2)}\\ \hspace{1em}\hspace{1em}=q^{(0)}(p^{(0)}-n^{(0)}+N_D^{+}-N_A^{-})+q^{(2)}(p^{(2)}-n^{(2)}),\\ 2{\epsilon }_{11}^{(2)}{\phi }_{,11}^{(0)}-{\epsilon }_{11}^{(4)}{\phi }_{,11}^{(2)}+2{\overline{e}}_{31}^{(2)}{u}_{3,11}^{(0)}+4{\overline{\epsilon }}_{33}^{(2)}{\phi }^{(2)}\\ \hspace{1em}\hspace{1em}=q^{(2)}(p^{(0)}-n^{(0)}+N_D^{+}-N_A^{-})+q^{(4)}(p^{(2)}-n^{(2)}),\\ -{\mu }_{11}^{p(0)}{\phi }_{,11}^{(0)}-D_{11}^{p(0)}{p}_{,11}^{(0)}+2{\mu }_{11}^{p(2)}{\phi }_{,11}^{(2)}+2D_{11}^{p(2)}{p}_{,11}^{(2)}\\ \hspace{1em}\hspace{1em}=-q^{(0)}{\dot{p}}^{(0)}-q^{(2)}{\dot{p}}^{(2)},\\ -{\mu }_{11}^{n(0)}{\phi }_{,11}^{(0)}+D_{11}^{n(0)}{n}_{,11}^{(0)}+2{\mu }_{11}^{n(2)}{\phi }_{,11}^{(2)}-2D_{11}^{n(2)}{n}_{,11}^{(2)}\\ \hspace{1em}\hspace{1em}=q^{(0)}{\dot{n}}^{(0)}+q^{(2)}{\dot{n}}^{(2)},\\ 2{\mu }_{11}^{p(2)}{\phi }_{,11}^{(0)}+2D_{11}^{p(2)}{p}_{,11}^{(0)}-{\mu }_{11}^{p(4)}{\phi }_{,11}^{(2)}-D_{11}^{p(4)}{p}_{,11}^{(2)}+4{\mu }_{33}^{p(2)}{\phi }^{(2)}\\ \hspace{1em}\hspace{1em}+4D_{33}^{p(2)}{p}^{(2)}=-q^{(2)}{\dot{p}}^{(0)}-q^{(4)}{\dot{p}}^{(2)},\\ 2{\mu }_{11}^{n(2)}{\phi }_{,11}^{(0)}-2D_{11}^{n(2)}{n}_{,11}^{(0)}-{\mu }_{11}^{n(4)}{\phi }_{,11}^{(2)}+D_{11}^{n(4)}{n}_{,11}^{(2)}+4{\mu }_{33}^{n(2)}{\phi }^{(2)}\\ \hspace{1em}\hspace{1em}-4D_{33}^{n(2)}{n}^{(2)}=q^{(2)}{\dot{n}}^{(0)}+q^{(4)}{\dot{n}}^{(2)}.\end{array}$$

(42)

Equation (41) describes the extensional motion of PS nanowires. Specifically, extension deformation $u_1^{(0)}$ is coupled with ${\phi }^{(1)}$, $p^{(1)}$, and $n^{(1)}$. Equation (42) shows the bending motion of PS nanowires, in which the bending deformation $u_3^{(0)}$ is coupled with ${\phi }^{(0)}$, ${\phi }^{(2)}$, $p^{(0)}$, $p^{(2)}$, $n^{(0)}$, and $n^{(2)}$.

4. Examples

4.1. Extension of ZnO Nanowires

As an application of Equation (41), consider the extension of the ZnO nanowires subjected to an axial load. The semiconductor is of n-type with p₀ = 0 and Δp = 0. The uniform doping is adopted here. All terms related to time and holes will be omitted in Equation (27) later.

The boundary conditions of ZnO nanowires are Ohmic contacts at x₁ = 0 and L, namely,

$$T_{11}^{(0)}=0,\hspace{1em}{\phi }^{(1)}=0,\hspace{1em}{n}^{(1)}=0.$$

(43)

The axial load with amplitude T₀ along the x₁ direction can be written as:

$$f_1^{(0)}=T_0\cos (\frac{\pi x_1}{L}).$$

(44)

Let the Navier solutions that satisfy boundary conditions Equation (43) be:

$$\begin{array}{l}\{{\phi }^{(1)},n^{(1)}\}={\displaystyle \sum _{m=1}^{\infty }\{{\mathsf{\Phi }}_m^{(1)},N_m^{(1)}\}\sin ({\xi }_m{x}_1)},\\ \{u_1^{(0)},f_1^{(0)}\}={\displaystyle \sum _{m=1}^{\infty }\{U_m^{(0)},T_m^{(0)}\}\cos ({\xi }_m{x}_1)},\\ {\xi }_m=\frac{m\pi }{L},m=1,2,3\dots \end{array}$$

(45)

where ${\mathsf{\Phi }}_m^{(1)}$, $N_m^{(1)}$, and $U_m^{(0)}$ are the undetermined Fourier coefficients. The Fourier coefficient $T_m^{(0)}$ is given by [43]:

$$T_1^{(0)}=T_0,\hspace{1em}{T}_m^{(0)}=0\hspace{1em}(m\ne 1).$$

(46)

Substituting Equation (45) into Equation (41), and noticing that there exist nontrivial solutions only when m = 1, we have:

$$\left[\begin{array}{ccc}{𝕂}_{11}& {𝕂}_{12}& 0\\ {𝕂}_{12}& {𝕂}_{22}& {𝕂}_{23}\\ 0& {𝕂}_{32}& {𝕂}_{33}\end{array}\right]\left\{\begin{array}{c}{U}_1^{(0)}\\ {\mathsf{\Phi }}_1^{(1)}\\ N_1^{(1)}\end{array}\right\}=\left\{\begin{array}{c}-T_1^{(0)}\\ 0\\ 0\end{array}\right\},$$

(47)

where

$$\begin{array}{l}{𝕂}_{11}=-{\overline{c}}_{11}^{(0)}{(\frac{\pi }{L})}^2,{𝕂}_{12}={\overline{e}}_{31}^{(0)}\frac{\pi }{L},\\ {𝕂}_{22}={\epsilon }_{11}^{(2)}{(\frac{\pi }{L})}^2+{\overline{\epsilon }}_{33}^{(0)},{𝕂}_{23}={\tilde{q}}^{(2)},\\ {𝕂}_{32}={\mu }_{11}^{n(2)}{(\frac{\pi }{L})}^2+{\mu }_{33}^{n(0)},{𝕂}_{33}=-[D_{11}^{n(2)}{(\frac{\pi }{L})}^2+D_{33}^{n(0)}].\end{array}$$

(48)

Solving the linear algebraic equation system in Equation (47) will yield the Fourier coefficients:

$$\begin{array}{l}{U}_1^{(0)}=\frac{L^2{T}_0}{{\pi }^2[{\overline{c}}_{11}^{(0)}+\frac{{({\overline{e}}_{31}^{(0)})}^2}{{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]},\\ {\mathsf{\Phi }}_1^{(1)}=-\frac{L{\overline{e}}_{31}^{(0)}{T}_0}{\pi \{{({\overline{e}}_{31}^{(0)})}^2+{\overline{c}}_{11}^{(0)}[{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}},\\ N_1^{(1)}=-\frac{q{n}_{0}L{\overline{e}}_{31}^{(0)}{T}_0}{k_{B}T\pi \{{({\overline{e}}_{31}^{(0)})}^2+{\overline{c}}_{11}^{(0)}[{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}},\end{array}$$

(49)

where the Debye–Hückel length of charge systems λ_D is defined as

$${\lambda }_D=\sqrt{\frac{{\epsilon }_{11}{k}_{B}T}{q^2{n}_0}}.$$

(50)

Additionally, the Einstein relation is utilized in deriving Equation (49), i.e.,

$$\frac{{\mu }_{11}}{D_{11}}=\frac{q}{k_{B}T},$$

(51)

where k_B is the Boltzmann constant and T is absolute temperature.

According to Equations (12), (13), (45), and (49), the displacement field u₁, electrostatic potential φ, and the concentration of electrons n reads:

$$u_1=\frac{L^2{T}_0}{{\pi }^2[{\overline{c}}_{11}^{(0)}+\frac{{({\overline{e}}_{31}^{(0)})}^2}{{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]}\cos (\frac{\pi }{L}{x}_1),$$

(52)

$$\phi =-\frac{L{\overline{e}}_{31}^{(0)}{T}_0}{\pi \{{({\overline{e}}_{31}^{(0)})}^2+{\overline{c}}_{11}^{(0)}[{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}}{x}_3\sin (\frac{\pi }{L}{x}_1),$$

(53)

$$n=n_0-\frac{q{n}_{0}L{\overline{e}}_{31}^{(0)}{T}_0}{k_{B}T\pi \{{({\overline{e}}_{31}^{(0)})}^2+{\overline{c}}_{11}^{(0)}[{\overline{\epsilon }}_{33}^{(0)}+{\epsilon }_{11}^{(2)}(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}}{x}_3\sin (\frac{\pi }{L}{x}_1).$$

(54)

From Equation (52), it is observed that the displacement u₁ increases with the increase of the doping level (i.e., ${\lambda }_D$ decrease with the increase of $n_0$, lead to the increase of $u_1$). It is seen that the extension displacement is always greater than that of piezoelectric dielectric, as a result of semiconducting effect.

From Equation (53), the electrostatic potential decreases with the increase of doping level n₀, i.e., the increase doping level n₀ causes the decrease of ${\lambda }_D$ and leads to the decrease of the electrostatic potential. This indicates that electrons have a screening effect on the electrostatic potential induced by the piezoelectric effect. Furthermore, for a high doping level with $n_0\to \infty$, ${\lambda }_D\to 0$, we have $\phi =0$. This indicates that the electrostatic potential induced by the piezoelectric effect is totally screened, and the PS turns into a conductor.

Particularly, when the semiconducting effect is not considered, we have $n_0=0$ and ${\lambda }_D\to \infty$. The current PS nanowires reduce to that for piezoelectric dielectric, and the corresponding solutions are

$$u_1=\frac{L^2{T}_0}{{\pi }^2[{\overline{c}}_{11}^{(0)}+\frac{L^2{({\overline{e}}_{31}^{(0)})}^2}{L^2{\overline{\epsilon }}_{33}^{(0)}+{\pi }^2{\epsilon }_{11}^{(2)}}]}\cos (\frac{\pi }{L}{x}_1),$$

(55)

$$\phi =-\frac{L^3{\overline{e}}_{31}^{(0)}{T}_0}{\pi [L^2{({\overline{e}}_{31}^{(0)})}^2+{\overline{c}}_{11}^{(0)}(L^2{\overline{\epsilon }}_{33}^{(0)}+{\pi }^2{\epsilon }_{11}^{(2)})]}{x}_3\sin (\frac{\pi }{L}{x}_1).$$

(56)

To further investigate the solution in the current model, a numerical example is taken into account. The material properties of ZnO nanowires are shown as follows [1]:

$$\begin{array}{l}{c}_{11}=c_{22}=209.7 \mathrm{GPa},c_{33}=210.9 \mathrm{GPa},c_{12}=121.1 \mathrm{GPa},\\ c_{13}=c_{23}=105.1 \mathrm{GPa},c_{44}=42.47 \mathrm{GPa},e_{31}=-0.573{ \mathrm{C}/\mathrm{m}}^2,\\ e_{33}=1.32{ \mathrm{C}/\mathrm{m}}^2,{\epsilon }_{11}=8.1\times {10}^{-11}\mathrm{C}/(\mathrm{V}\cdot \mathrm{m}),\\ {\epsilon }_{33}=1.12\times {10}^{-10}\mathrm{C}/(\mathrm{V}\cdot \mathrm{m}).\end{array}$$

(57)

The electron mobilities are from Sze and Ng [44]: ${\mu }_{11}^n={\mu }_{33}^n=200\times {10}^{-4}{ \mathrm{m}}^2/(\mathrm{V}\cdot \mathrm{s})$. The diffusion constants are $D_{11}^n=D_{33}^n=4.6575\times {10}^{-4}{ \mathrm{m}}^2/\mathrm{s}$. Moreover, the elementary charge and doping level are q = 1.602 × 10⁻¹⁹ C and n₀ = 1 × 10²¹ m⁻³. The Boltzmann constant is 1.381 × 10^–23 J/K and the absolute temperature takes 300 K. The sizes of ZnO nanowires are L = 600 nm, b = 50 nm, and h = 30 nm, respectively. In addition, the amplitude T₀ of the axial load is 0.1 N/m.

From Figure 2a, it indicates that the top of ZnO nanowires has a high electrostatic potential while the bottom has a low potential. The total potential is distributed anti-symmetrically along the x₃ direction. Figure 2b indicates that the electrons are redistributed under extension deformation. The electrons move toward the top of the ZnO nanowires where they become negative charges, leaving the bottom half of the nanowire positive charges.

To further illustrate the effects of the amplitude of axial load and doping level, the electrostatic potential varying with x₁ of ZnO nanowires is plotted in Figure 3a,b. From Figure 3a, it can be seen that the electrostatic potential shows an enhancement with the increase in mechanical load amplitude. Moreover, the increased doping level lead to a depressed electrostatic potential (see Figure 3b). Driven by mechanically induced polarization, the electrons move to screen the effective polarization charges and thus lower the magnitude of the first-order electrostatic potential [45]; this is consistent with the earlier discussion. It can be predicted that the first-order electrostatic potential ${\phi }^{(1)}$ reaches its maximum value when the semiconducting effect vanishes.

4.2. Bending of ZnO Nanowires

As an application of Equation (42), consider the bending of ZnO nanowires subjected to a transverse load. The semiconductor is of n-type and uniform doping is adopted here. Introduce $\Delta n^{(0)}=n^{(0)}-n_0$, the governing equations for bending of ZnO nanowires can be given by omitting the items related to time and holes,

$$\begin{array}{l}-{\overline{c}}_{11}^{(2)}{u}_{3,1111}^{(0)}+2{\overline{e}}_{31}^{(2)}{\phi }_{,11}^{(2)}+f_3^{(0)}=0,\\ -{\epsilon }_{11}^{(0)}{\phi }_{,11}^{(0)}+2{\epsilon }_{11}^{(2)}{\phi }_{,11}^{(2)}=-q^{(0)}\Delta n^{(0)}-q^{(2)}{n}^{(2)},\\ 2{\epsilon }_{11}^{(2)}{\phi }_{,11}^{(0)}-{\epsilon }_{11}^{(4)}{\phi }_{,11}^{(2)}+2{\overline{e}}_{31}^{(2)}{u}_{3,11}^{(0)}+4{\overline{\epsilon }}_{33}^{(2)}{\phi }^{(2)}\\ \hspace{1em}\hspace{1em}=-q^{(2)}\Delta n^{(0)}-q^{(4)}{n}^{(2)},\\ -{\mu }_{11}^{n(0)}{\phi }_{,11}^{(0)}+D_{11}^{n(0)}\Delta n_{,11}^{(0)}+2{\mu }_{11}^{n(2)}{\phi }_{,11}^{(2)}-2D_{11}^{n(2)}{n}_{,11}^{(2)}=0,\\ 2{\mu }_{11}^{n(2)}{\phi }_{,11}^{(0)}-2D_{11}^{n(2)}\Delta n_{,11}^{(0)}-{\mu }_{11}^{n(4)}{\phi }_{,11}^{(2)}+D_{11}^{n(4)}{n}_{,11}^{(2)}\\ \hspace{1em}\hspace{1em}+4{\mu }_{33}^{n(2)}{\phi }^{(2)}-4D_{33}^{n(2)}{n}^{(2)}=0.\end{array}$$

(58)

For boundary conditions, Ohmic contacts at x₁ = 0 and L are assumed, namely

$$\begin{array}{l}{T}_{11}^{(1)}=0,\hspace{1em}{u}_3^{(0)}=0,\hspace{1em}{\phi }^{(0)}=0,\hspace{1em}{\phi }^{(2)}=0,\\ \mathsf{\Delta }{n}^{(0)}=0,\hspace{1em}{n}^{(2)}=0.\hspace{1em}\mathrm{at}\hspace{1em}{x}_1=0,L.\end{array}$$

(59)

Equation (59)_5,6 indicate $n=n_0$ at the x₁ = 0 and L, which satisfies the acquirement of Ohmic contacts.

Consider the Navier solutions of the following Fourier series form:

$$\begin{array}{l}\{u_3^{(0)},{\phi }^{(0)},{\phi }^{(2)},\mathsf{\Delta }{n}^{(0)},n^{(2)},f_3^{(0)}\}\\ \hspace{1em}={\displaystyle \sum _{m=1}^{\infty }\{W_m^{(0)},{\mathsf{\Phi }}_m^{(0)},{\mathsf{\Phi }}_m^{(2)},N_m^{(0)},N_m^{(2)},F_m^{(0)}\}\sin ({\xi }_m{x}_1)},\\ {\xi }_m=\frac{m\pi }{L},\hspace{1em}m=1,2,3\dots \end{array}$$

(60)

where $W_m^{(0)}$, ${\mathsf{\Phi }}_m^{(0)}$, ${\mathsf{\Phi }}_m^{(2)}$, $N_m^{(0)}$, and $N_m^{(2)}$ are the undetermined Fourier coefficients; $F_m^{(0)}$ is the known Fourier coefficient. Notably, Equation (60) satisfies the boundary conditions (59).

The transverse load with amplitude F₀ can be written as:

$$f_3^{(0)}=F_0\sin (\frac{\pi x_1}{L}),$$

(61)

thus the Fourier coefficient of $f_3^{(0)}$ is [46]:

$$F_1^{(0)}=F_0,F_m^{(0)}=0(m\ne 1).$$

(62)

Using Equations (60) and (62) into Equation (58), and noticing that there exist nontrivial solutions only when m = 1, we have:

$$\left[\begin{array}{ccccc}{𝕄}_{11}& 0& {𝕄}_{13}& 0& 0\\ 0& {𝕄}_{22}& {𝕄}_{23}& {𝕄}_{24}& {𝕄}_{25}\\ {𝕄}_{13}& {𝕄}_{23}& {𝕄}_{33}& {𝕄}_{25}& {𝕄}_{35}\\ 0& {𝕄}_{42}& {𝕄}_{43}& {𝕄}_{44}& {𝕄}_{45}\\ 0& {𝕄}_{43}& {𝕄}_{53}& {𝕄}_{45}& {𝕄}_{55}\end{array}\right]\left\{\begin{array}{c}{W}_1^{(0)}\\ {\mathsf{\Phi }}_1^{(0)}\\ {\mathsf{\Phi }}_1^{(2)}\\ N_1^{(0)}\\ N_1^{(2)}\end{array}\right\}=\left\{\begin{array}{c}-F_1^{(0)}\\ 0\\ 0\\ 0\\ 0\end{array}\right\},$$

(63)

where the nonzero components of coefficients matrices are

$$\begin{array}{l}{𝕄}_{11}=-{\overline{c}}_{11}^{(2)}{(\frac{\pi }{L})}^4,{𝕄}_{13}=-2{\overline{e}}_{31}^{(2)}{(\frac{\pi }{L})}^2,{𝕄}_{22}={\epsilon }_{11}^{(0)}{(\frac{\pi }{L})}^2,\\ {𝕄}_{23}=-2{\epsilon }_{11}^{(2)}{(\frac{\pi }{L})}^2,{𝕄}_{24}=q^{(0)},{𝕄}_{25}=q^{(2)},{𝕄}_{33}={\epsilon }_{11}^{(4)}{(\frac{\pi }{L})}^2+4{\overline{\epsilon }}_{33}^{(2)},\\ {𝕄}_{35}=q^{(4)},{𝕄}_{42}={\mu }_{11}^{n(0)}{(\frac{\pi }{L})}^2,{𝕄}_{43}=-2{\mu }_{11}^{n(2)}{(\frac{\pi }{L})}^2,\\ {𝕄}_{44}=-D_{11}^{n(0)}{(\frac{\pi }{L})}^2,{𝕄}_{45}=2D_{11}^{n(2)}{(\frac{\pi }{L})}^2,{𝕄}_{53}={\mu }_{11}^{n(4)}{(\frac{\pi }{L})}^2+4{\mu }_{33}^{n(2)},\\ {𝕄}_{55}=-[D_{11}^{n(4)}{(\frac{\pi }{L})}^2+4D_{33}^{n(2)}].\end{array}$$

(64)

Solving the linear algebraic equation system in Equation (63) will yield the Fourier coefficients as,

$$\begin{array}{l}{W}_1^{(0)}=\frac{L^4{F}_0}{{\pi }^4[{\overline{c}}_{11}^{(2)}+\frac{4{({\overline{e}}_{31}^{(2)})}^2}{4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]},\\ {\mathsf{\Phi }}_1^{(0)}=\frac{4L^2{I}^{(2)}{\overline{e}}_{31}^{(2)}{F}_0}{{\pi }^2{I}^{(0)}\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}},\\ {\mathsf{\Phi }}_1^{(2)}=\frac{2L^2{\overline{e}}_{31}^{(2)}{F}_0}{{\pi }^2\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}},\\ N_1^{(0)}=\frac{4q{n}_0{L}^2{I}^{(2)}{\overline{e}}_{31}^{(2)}{F}_0}{k_{B}T{\pi }^2{I}^{(0)}\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}},\\ N_1^{(2)}=\frac{2q{n}_0{L}^2{\overline{e}}_{31}^{(2)}{F}_0}{k_{B}T{\pi }^2\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}}.\end{array}$$

(65)

From Equations (12), (13), (60), and (65), it then follows that u₁, u₃, φ, and n are

$$u_1=-\frac{L^3{F}_0}{{\pi }^3[{\overline{c}}_{11}^{(2)}+\frac{4{({\overline{e}}_{31}^{(2)})}^2}{4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]}{x}_3\cos (\frac{\pi }{L}{x}_1),$$

(66)

$$u_3=\frac{L^4{F}_0}{{\pi }^4[{\overline{c}}_{11}^{(2)}+\frac{4{({\overline{e}}_{31}^{(2)})}^2}{4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]}\sin (\frac{\pi }{L}{x}_1),$$

(67)

$$\phi =\frac{2L^2{\overline{e}}_{31}^{(2)}{F}_0}{{\pi }^2\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}}[2\frac{I^{(2)}}{I^{(0)}}+(x_3^2-h^2)]\sin (\frac{\pi }{L}{x}_1),$$

(68)

$$n=n_0+\frac{2q{n}_0{L}^2{\overline{e}}_{31}^{(2)}{F}_0}{k_{B}T{\pi }^2\{4{({\overline{e}}_{31}^{(2)})}^2+{\overline{c}}_{11}^{(2)}[4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})]\}}[2\frac{I^{(2)}}{I^{(0)}}+(x_3^2-h^2)]\sin (\frac{\pi }{L}{x}_1).$$

(69)

From Equations (66) and (67), it is observed that the displacement u₁ and u₃ increase with the increase of doping level (i.e., $n_0\uparrow$, ${\lambda }_D\downarrow$, lead to $u_1\uparrow$, $u_3\uparrow$). However, when increasing the doping level, the electrostatic potential decrease (i.e., $n_0\uparrow$, ${\lambda }_D\downarrow$, lead to $\phi \downarrow$). The reason is that the electrons have a screening effect on the electrostatic potential induced by the piezoelectric effect. Furthermore, for a high doping level with $n_0\to \infty$, ${\lambda }_D\to 0$, we have $\phi =0$, and the solution of u₁, u₂ become

$$\begin{array}{l}{u}_1=-\frac{L^3{F}_0}{{\pi }^3{\overline{c}}_{11}^{(2)}}{x}_3\cos (\frac{\pi }{L}{x}_1),\\ u_3=\frac{L^4{F}_0}{{\pi }^4{\overline{c}}_{11}^{(2)}}\sin (\frac{\pi }{L}{x}_1).\end{array}$$

(70)

This indicates that the electrostatic potential induced by the piezoelectric effect is totally screened, and the PS turns into a conductor. Moreover, when the semiconducting effect is not considered, the current model reduces to the case of piezoelectric dielectric.

In addition, the bending angle of the Bernoulli–Euler beam is

$$\theta =u_{3,1}=\frac{L^3{F}_0}{{\pi }^3[{\overline{c}}_{11}^{(2)}+\frac{4{({\overline{e}}_{31}^{(2)})}^2}{4{\overline{\epsilon }}_{33}^{(2)}+({\epsilon }_{11}^{(4)}-4\frac{I^{(2)}}{I^{(0)}}{\epsilon }_{11}^{(2)})(\frac{{\pi }^2}{L^2}+\frac{1}{{\lambda }_D^2})}]}\cos (\frac{\pi }{L}{x}_1).$$

(71)

From Equation (71), we can see that the bending angle is proportional to the amplitude of the transverse load. Furthermore, when increasing the doping level n₀, we have ${\lambda }_D\downarrow$, and this leads to the increase of bending angle. Simultaneously, the bending angle is always greater than that of piezoelectric dielectric, as a result of semiconducting effect. When the semiconducting effect takes $n_0=0$ or $n_0\to \infty$, the solution of bending angle turns into that of piezoelectric dielectric or conductor.

To further investigate the solution in the current model, a numerical example is taken into account. The material properties of ZnO nanowires are the same as in Section 4.1, while the load amplitude F₀ is −0.1 N/m. From Figure 4a, it indicates that electrostatic potential has a symmetric distribution along the x₃ direction because of the influence of zeroth-order and second-order electrostatic potential. The high electrostatic potential is exhibited at the top and bottom of ZnO nanowires, while the low potential is exhibited in the middle part. The electrostatic potential will cause the redistribution of electrons (see Figure 4b). The electrons move toward the top and bottom of the ZnO nanowires, leaving the middle part of the ZnO nanowires positive charges.

Figure 5 and Figure 6 depict the influence of load amplitude F₀ and doping level n₀ on the zeroth- and second-order electrostatic potential varying with x₁, respectively. From Figure 5a and Figure 6a, it can be seen that the magnitude of electrostatic potential ${\phi }^{(0)}$ and ${\phi }^{(2)}$ increases with the increase of F₀. Moreover, due to the screening effect of electrons on the electrostatic potential induced by piezoelectric effect, the magnitude of electrostatic potential will decrease when increasing the doping level (see Figure 5b and Figure 6b). In addition, electrostatic potential ${\phi }^{(0)}$ and ${\phi }^{(2)}$ will reach its maximum value when the semiconducting effect vanishes.

5. Conclusions

In this paper, the PVW for PSs is proposed. Different from classical approaches such as differential methods, this PVW is another method to establish structural theories, which will derive the field equations and boundary conditions simultaneously. Then, the field equations and corresponding boundary conditions are derived by directly applying the PVW and the Bernoulli–Euler beam theory with the aid of the second-order approximation of electrostatic potential. The correct 1-D constitutive relations are given by taking into account the relaxation procedure. Then, the 1-D PS nanowires model, which can be decoupled into extensional groups and bending groups, is obtained immediately.

As applications of the 1-D PS nanowires model, extension and bending deformation for ZnO nanowires are investigated. The mechanical displacement, electrostatic potential, and concentration of electrons for extension and bending deformation are analytically derived. From the numerical results, the distribution of electrostatic potential is anti-symmetric along the thickness direction for extension deformation, while bending deformation causes a symmetric distribution of electrostatic potential. Moreover, extension and bending deformation lead to a redistribution of electrons. The electrostatic potential can be tuned by adjusting the amplitude of the applied mechanical load. Finally, electrons redistribute themselves to screen the effective polarization and thus decrease the magnitude of the electrostatic potential.

The use of PVW can help to establish structural theories and provide an effective way of implementing numerical calculation methods.