1 Introduction

For a semiconductor with piezo-effect, polarized electric fields will be aroused when it is subjected to the deformation(s) due to electromechanical coupling, and charge carriers will be driven to motion/redistribution. The carrier motion/redistribution will in turn produce a screening effect on the polarized electric fields^[1-3]. Such an effect, from the electromechanical fields on carriers, can be used to develop tuning functions of microelectronic semiconductor devices. In recent years, various piezoelectric semiconductor structures have been synthesized, e.g., ZnO nanofibers, tubes, belts, spirals, and films^[4-6], and used to make generators for converting mechanical energy into electrical energy^[7-8], piezoelectric diodes^[9], piezoelectric field effect transistors^[10], piezoelectric chemical transistors^[11] and so on. Piezoelectric semiconductors are also used in static and dynamic characteristic analyses^[12] as well as nano-structures such as quantum wells, dots, and wires^[13-15] due to their electromechanical couplings^[16-17]. Studies on piezoelectric semiconductor materials and devices are growing rapidly. Because of the anisotropy in piezoelectric materials, the electromechanical couplings in them, and the drift currents of electrons and holes which are inherently nonlinear as the products of the unknown carrier concentrations and the unknown electric field, the theoretical analyses for piezoelectric semiconductor devices present considerable mathematical challenges. According to a recent review^[13], the electromechanical phenomena in piezoelectric semiconductors have still been poorly studied from a fundamental and applied science point of view, even though significant strides have been made in the last decade.

Parts of the recent studies on piezoelectric semiconductors have formed a new research area called piezotronics and piezo-phototronics^[18-19]. In piezotronic devices, the motion and redistribution of charge carriers are modulated by mechanical loads through the electric field associated with the polarization or effective polarization charges produced by the mechanical loads via piezoelectric couplings. The effects of end forces on the mobile charges in the bending and extension of piezoelectric semiconductor nanofibers were studied in Refs. [20]-[24] and Refs. [25]-[29]. The stress effects on PN junctions and metal-semiconductor junctions were studied in Refs. [18], [19], [30], and [31], where the effective polarization charges were approximated by block charge distributions in the spirit of the well-known depletion layer approximation.

In this paper, we will study the stress induced charge distributions and the resulting potential barriers in piezoelectric semiconductor fibers in a fundamental way, beginning with the effects of local extension produced by a pair of equal and opposite forces in a fiber. Section 2 specializes the general equations into a one-dimensional (1D) model for the extension of the fiber. Section 3 presents the theoretical and numerical results for a few basic cases of stress induced potential barriers and charge distributions. Finally, some conclusions are given in Section 4.

2 Method for establishing governing equations for a tensile fiber

We consider a piezoelectric semiconductor fiber, whose length is much greater than the dimension of its cross section with an arbitrary shape (see Fig. 1). It is made of a piezoelectric semiconductor crystal of class 6mm. The c-axis of the fiber is along the direction of x₃, and the lateral surface of the fiber is free.

For the 1D problem in this paper, tensile forces are applied on a nanofiber symmetrically. When we take its piezoelectric property into consideration, we write the equilibrium equation and the well-known piezoelectric equation as follows^[25]:

(1)

where T₃ is the applied axial stress, f is the body force of the fiber, and the rest of the coefficients are^[25]

(2)

in which s₃₃^E, d₃₃, and ε₃₃^T are the compliance constant, the piezoelectric strain constant, and the dielectric constant, respectively. Besides, for the strain and electric fields in the fiber, there exist

(3)

where u and $φ$ denote the displacement and the electric potential produced by the tensile stress, respectively.

Moreover, carriers including holes and electrons will redistribute due to the existing electric field when we apply stresses on the fiber. To describe the behavior of the carrier redistribution of a tensile fiber, we introduce the basic equations in the aspect of semiconducting properties. First, we write the Gauss law to establish the relationship between the electric field and the electric charges as follows:

(4)

where D and q represent the electric displacement and the elementary charge, respectively, and p and n are the carrier concentrations of holes and electrons, respectively. N_D⁺ is the donor doping concentration, while N_A^- is the accepter doping concentration. The redistribution of the carriers will generate electric fields, which will produce drift current as well as concentration gradient induced diffusion current, which can be written as the current densities of holes and electrons as follows^[25]:

(5)

where μ₃₃^p and μ₃₃ⁿ represent the mobility of holes and electrons, respectively, and D₃₃^p and D₃₃ⁿ represent the diffusion coefficients of holes and electrons, respectively. p₀ and n₀ denote the concentrations of holes and electrons in the original state without applied stresses on the fiber, respectively. Here, p₀ =N_A^-, and n₀ =N_D⁺. Δp and Δn denote the fluctuating concentrations of holes and electrons due to the applied tensile stress, respectively.

In this paper, we discuss a static problem, where the stresses are independent of time. The continuity equation of semiconductors describing the process of carriers associated with time degrades into

(6)

Equations (1)-(6) describe the behaviors of the carrier redistribution by coupling the mechanical field and the electrical field, and such equations together produce the equation for the fluctuating concentration of holes and electrons as follows:

(7)

where

in which k_B is the Boltzmann constant, and T is the Kelvin temperature.

3 Results and discussion 3.1 A fiber under a pair of local concentrated forces

To explore more meaningful conclusions for the 1D problem, we consider a ZnO fiber with n-type dopants, where (p, Δp)=0. We assume that the fiber is unbounded and there is no body force. Besides, we set the position zero in the middle of the fiber to guarantee the symmetry of the applied pair of local forces F (see Fig. 2).

We need to find the solutions for each of the three regions, i.e., x < -a, |x| < a, and x>a, and apply boundary conditions and continuity (or jump) conditions. The general solution to Eq. (7) is

(8)

where C₁, C₂, …, C₆ are undetermined constants. Equation (8) is valid for each of the three regions in Fig. 2 with corresponding undetermined constants for each region. There are eighteen undetermined constants all together. The fiber is stress-free and electrically open at infinity. The boundary and continuity conditions are as follows:

(9)

(10)

(11)

(12)

Equations (9)-(12) show eighteen conditions. However, some of them are not independent. For example, Jⁿ(-∞)=0 shows that Jⁿ is continuous at ±a, and implies that Jⁿ(∞)=0. Hence, to uniquely determine the displacement and potential fields, we need to choose -∞ as a reference and set

(13)

Substituting Eq. (8) into Eqs. (11) and (12) yields a system of linear equations for the undetermined constants. They are solved on a computer with MATLAB. Then, various electromechanical fields can be calculated. Particularly, the relevant components of the polarization vector and the effective polarization charge density ρ^p are calculated according to

(14)

Figure 3 shows various electromechanical fields produced by two different values of F when n₀ =10²¹ m^-3 and a=600 nm. The elastic, piezoelectric, and dielectric constants of ZnO are from Ref. [32]. At room temperature, k_BT/q is about 0.026 V. There are a few effects of elasticity and piezoelectricity in the figure. However, they are modified by the semiconduction, i.e., the development of the mechanical displacement u in Fig. 3(a), the electric potential $φ$ in Fig. 3(b), the electric field E in Fig. 3(c), the electric displacement D in Fig. 3(d), the polarization P in Fig. 3(e), and the effective polarization charge density ρ^p in Fig. 3(f). If the fiber is a piezoelectric dielectric medium, these fields would be much simpler, and could be described by the piecewise constant and the piecewise linear functions. Now, they are more or less affected by the mobile charges indicated by the exponential terms in Eq. (8). The change of the carrier concentration in Fig. 3(g) under mechanical loads and the screening of polarization charges by mobile charges in Fig. 3(h) rely on semiconduction, and are unique effects of piezoelectric semiconductors. We note that the electric field in Fig. 3(c) and the polarization in Fig. 3(e) in fact have discontinuities at x=±a, but are automatically connected by MATLAB using vertically lines. In Figs. 3(f) and 3(h), the values of ρ^p and ρ^p-qΔn have opposite signs. This seems to suggest that the mobile charges are more neutralized than the polarization charges, which is in fact untrue. This is because, in addition to ρ^p, there also exist effective interface polarization charges at x=±a, corresponding to the discontinuities of P caused by the discontinuity of the axial stress. In the figure, as expected, a larger F produces a stronger field. For the purpose of mechanically modulating electrical behaviors of the fiber, the most basic effects of F are to produce a potential well followed by a potential barrier in Fig. 3(b) and a screened electric dipole in Fig. 3(h). With the presence of the potential well and barrier in Fig. 3(b), a mobile charge cannot travel through them unless it has sufficient initial velocity or kinetic energy. Therefore, a local stress distribution forbids the passing of low-energy mobile charges.

Figure 4 shows the effects of n₀. As n₀ increases, there are more mobile charges in the fiber with a stronger screening effect implied by the larger Δn (see Fig. 4(g)), which tends to lower the electric potential in Fig. 4(b) and the corresponding electric field in Fig. 4(c). At the same time, a lower electric field reduces the piezoelectric stiffening effect. The fiber is more effectively compliant with a larger extensional strain in the central part implied by Fig. 4(a), a larger polarization in Fig. 4(e), and hence a larger electric displacement in Fig. 4(d), and has more polarization charges (see Fig. 4(f)).

3.2 A fiber under two local pairs of concentrated forces

The fields and charge distributions obtained in the previous section by one pair of local stresses can be used to build blocks for constructing more complicated fields and charge distributions. As an example, we consider a fiber under two local pairs of forces (see Fig. 5).

The solution to this problem is similar to that in Fig. 2, with more algebra because of more regions, more undetermined constants, and more boundary and continuity conditions in the solutions. It is not presented here.

Figure 6 shows the results directly for two different values of n₀. In this case, the discontinuities of E in Fig. 6(c) and P in Fig. 6(e) at x=±a are more visible. Our main interest is in the big potential well with two small potential barriers in Fig. 6(b), which is close to a conventional potential well.

3.3 Periodic extension and compression under distributed forces

In this section, we consider the effects of the distributed force f in Eq. (7). As an example, we consider the case of a sinusoidal distribution of f=Bsin(ξx), where B and ξ are constants. It is straightforward to find the corresponding fields as follows:

(15)

We plot the potential and charges in Fig. 7 when B=8.553~6×10¹¹ N·m^-3 and ξ=3.490~7×10⁶ m^-1. The potential and charge distributions are also sinusoidal. A series of periodically alternating potential barriers and wells as well as charged regions are produced. In this case, for the effective polarization charges, there is ρ^p only without effective interface polarization charges at the discontinuity surfaces because all fields are continuous everywhere. Figures 7(b) and 7(d) show that ρ^p is significantly but not completely screened by electrons.

4 Conclusions

The electromechanical fields produced by the local stresses in a piezoelectric semiconductor fiber are obtained, which can be used to construct fields from more complicated stress fields and study the modulation of electrical behaviors of the piezoelectric semiconductor fibers in mechanical fields. Specifically, under the action of a pair of equal and opposite forces, a screened electric dipole develops, which is responsible for a potential barrier next to a potential well, which forbids the passing of low-energy charge carriers. Similarly, neighboring extensional and compressive stresses can produce a deep potential well with two low potential barriers next to it and three charge regions locally. Mathematically, distributed forces play a role similar to doping. Specifically, a sinusoidal distribution of force produces periodic distributions of alternating potential barriers and wells as well as alternating charges.