Numerical Analysis on the Electromechanical Behaviors and Piezotronic Effects Controlled by the Applied of Initial Stress

Based on the linear three-dimensional elastic theory, shear horizontal (SH) wave propagation along the normal direction to the piezoelectric semiconductor plate (PSC) was obtained in the presence of initial horizontal and vertical stresses. The purpose behind obtaining the numerical solutions was to evaluate the effect of stress on dimensionless frequency and phase velocity. To verify the exactness of the process, the dispersion curves of phase velocity in a piezoelectric semiconductor plate without loaded initial stresses were calculated beforehand by the proposed approach to compare with those reported in the literature. Next, a series of cases of initial stress were further executed to explore the effects of horizontal initial stress and vertical initial stress on the SH0 and SH1 wave modes. The results revealed that the positive initial tensile stress led to the increase in the phase velocity, while the preliminary negative compressive stress decreased the phase velocity. Finally, the effect of the initial stress on the electromechanical fields was explained in detail. The obtained results offer a predictable and theoretical fundament for the development of piezoelectric semiconductor structures under the influence of initial stress to acoustic wave devices.

Keywords: Piezoelectric Semiconductor; Shear Horizontal Waves; Initial Stress; Dispersion Curves; Phase Velocity; Electromechanical Fields

Piezoelectric (PS) materials can be recognized as dielectrics or semiconductors. In PS semiconductors (PS conductors), since the induced electric field produces electric current, these smart materials possess piezoelectric effect and semiconductor properties [1,2]. Due to their prevalent physical properties, PSCs have been broadly used in surface acoustic wave devices, electronic sensors, and energy harvesting [3]. At the same time, a great deal of research has been realized on the physical properties of PSCs, such as acoustoelectric, photoelectric, photothermal and piezoelectric effects [4-6]. Moreover, novel piezoelectric semiconductor nanostructures, such as ZnO fibers, tubes, belts, spirals and films have been synthesized [7-10]. In this regard zinc oxide (ZnO) is in fact one of the most important multi-functional semiconductors, which has drawn much attention thanks to its many unique properties as low toxicity, high fluorescence activity and relatively strong piezoelectric effect [11]. This makes it an interesting electromechanical coupling material for various piezoelectric applications [12]. Numerous kinds of one-dimensional (1D) ZnO nanostructures, such as nanowires, nanorods, nanotubes, nano-belts, etc. have attracted much interest as building blocks for various fascinating nanodevices applications such as electronics, optics, and nano-electromechanical systems (NEMS) [13]. In recent years, diverse applications based on piezoelectric properties of ZnO nanostructures grown on a variety of conventional substrates, namely piezoelectric field effect transistors, the piezoelectric diode, piezoelectric sensors and the piezoelectric nano-generator, [14-16] have been demonstrated.

For device applications, the basic behaviors of piezoelectric semiconductors can be described by a phenomenological theory consisting of the equations of linear piezoelectricity [17] and the conservation of charge for electrons and holes [18]. The theory has been used in a series of useful applications of PSC, including thickness vibration of plates [19], propagation of plate and surface waves [10,20,21], electromechanical fields in an inclusion [22], fields near cracks [23,24], static extension of a fiber [25], bending of a fiber [26], and static fields near stress-free PN junctions [27].

A PS material may also demonstrate an initial stress effect, i.e., an initial stress in the multi-layered structures can lead to delamination, microcracking, debonding and degradation of the layer. It can also lead to the dramatic change of dispersion relation corresponding to the wave propagation in the above-mentioned structures. Meanwhile, the effects of initial stresses on bulk wave propagation in layered piezomagnetic/piezoelectric plates, with a special focus on the phase velocities and frequency spectrum have been investigated in [28]. Furthermore, the effects of initial stress on the propagation behavior and dispersion relation of shear horizontal (SH) waves in piezoelectric plates have been studied numerically [29-32]. Concerning the study of the effects of initial stress on the reflection and transmission of waves at the interface between two piezoelectric half spaces, it has been carried out in [33]. The influence of compressive and tensile initial stresses on the behaviors of Lamb and SH waves in a prestressed multilayered PZT-4/PZT-5A composites system has also been considered [34]. Moreover, the effects of initial stress on the electromechanical coupling factor of SH wave in layered piezoelectric structures have been investigated as well [35]. The performance of Sc43% AlN57%/ZnO/diamond piezoelectric hetero-structures using matrix and polynomial methods under uniaxial stress have been developed [36]. In the same context, Shams [37] has demonstrated the effects of initial stress on the phase velocities of surface Love wave in hetero-structures, while Sahu et al. [38] has explored a similar problem in the piezo-composite structure composed of functionally graded piezoelectric material (FGPM).

Considering all these facts, it is also necessary that the effects of initial stresses be analyzed exhaustively as has been realized in a piezoelectric semiconductor (PSC) fiber under local extensional or compressive stress [39] and under axial force [6-16]. Unlike most of the existing literature in which only the effects of normal initial stress have been examined, in the present research work, the authors have explored the effects of two types of initial stresses, namely normal and shear initial stresses on the propagation behavior of shear waves (SH-waves) in a piezoelectric semiconductor plate.

Part of the recent studies on piezoelectric semiconductors has formed a new area of research called piezotronics and piezophototronics [40,41]. In piezotronic devices, the motion of charge carriers is manipulated by mechanical charges through the accompanying electric field via piezoelectric coupling. For piezotronic applications, the effects of end forces on moving loads in the bending and extension of piezoelectric semiconductor fibers have been investigated [42,43]. The study of stress effects on PN junctions and metal-semiconductor junctions has also been caried out [44]. It should be mentioned that in the relatively recent literature, some researchers have used the above theory of piezoelectric semiconductors to study the propagation of elastic waves in piezoelectric semiconductors [10,14,30-32].

To the authors’ best knowledge, however, no work on wave propagation in piezoelectric semiconductor plate under an initial stress has been reported, which motivates the present study. This paper focusses on the propagation characteristic of SH waves in the PSC plate under an initial horizontal and vertical stress.

This paper is organized as follows. In sections 2 and 3, the problem to be solved via the phenomenological theory of piezoelectric semiconductor materials is described. It consists of the equation of motion, the charge equation of electrostatics, and the conservation of charge for electrons and holes. In section 4, the dispersion equation in PSC plate under an initial stress is derived based on numerical resolution by a system to the eigenvalues and eigenvectors and the interface and boundary conditions. In section 5, the effect of initial horizontal and vertical stress on the dispersion curves of SH wave in a PSC plate are presented in detail. The effect of initial stress on the electromechanical fields of PSC plate are investigated in Section 5.4. Finally, conclusions are drawn in Section 6.

In this section, a brief summary of the general 3D equations for piezoelectric semiconductors with initial stresses are presented. From the basic behavior of PSC is described by a phenomenological, coupled-field theory consisting of the equations of linear piezoelectricity [10] and the conservation of charge for holes and electrons. The Cartesian tensor notation is employed in this paper. Indices i,j,k and l assume 1, 2, and 3. A comma followed by an index indicates partial differentiation with respect to the coordinate associated with the index. A superimposed dot “.” represents a time derivative. The theory consists of [10]:

where T_ij,u_i and D_i are the components of stresses, mechanical and electric displacements, respectively. p is the mass density, q is the electronic charge with 1.6×10_-19 Coulomb, and the concentrations of holes and electrons,N_D⁺and N_A^- are the concentrations of impurities of donors and acceptors, respectively. Finally,J^p_i and Jⁿ_i are the hole and electron current densities, respectively.

(1a) is the stress equation of motion or the linear momentum equation (Newton's law). (1b) is the charge equation of electrostatics (Gauss's law). (1c) and (1d) are the conservation of charge for holes and electrons, respectively, which are also called continuity equations. Constitutive relations accompanying (1a), (1b), (1c) and (1d) can be written in the following form [6,10]:

In which S_Kl and E_k represent the strain and electric field, respectively.c_ijkl, e_kij and ɛ_ik and are elastic, piezoelectric and dielectric constants, respectively. Moreover,μ_ijⁿand μ_ijⁿ,D_ijⁿand D_ij^p are the carrier mobility and the carrier diffusion constants, respectively, and the indices and correspond to the two cases of holes and electrons, respectively. (2a) and (2b) are the usual constitutive relations for piezoelectric materials. (2c) and (2d) are for hole and electron currents including both drift currents under an electric field and diffusion currents due to concentration gradients, respectively. The strain and the electric field are related to the mechanical displacement μ_i and the electric potential ∅ through.

By considering the disturbance of the carriers, the concentrations of holes and electrons can be written as

The crystals of 6mm symmetry include widely used materials like Zinc Oxide (ZnO) [10]. For this crystal, the material tensors in Eqs. (2a, b, c and d) can be represented by the following matrices under the compact matrix notation

where c₆₆=(c₁₁-c₁₂)/2,and the superscript T is for matrix transpose. μ_ijand D_ij have the same structure as .We consider motions with ∂/∂x₃=0.Then Eq.(4) splits into a plane strain problem for u₁ and u₂,and an anti-plane problem for

For the anti-plane problem, the strain and electric field components are

where Δ=i₁∂₁+i₂∂₂ is the two-dimensional gradient operator. The nontrivial components of T_ij,D_i,J_iⁱand J_iⁿ are

The nontrivial ones of the equations of motion and charge Eqs. (1a and b) take the following form:

And for the Eqs. (1c and d) can be written

As shown in Figure 1, a piezoelectric semiconductor plate is taken into account. It involves a PSC plate with the thickness h=1μm of with the total thickness of 2μm. We take the origin of cartesian coordinate system ox₁x₂x₃ at any point on the mid-plane of the semiconductor layer and x₂-axis along the thickness of the layer as shown in Figure 1 The piezoelectric semiconductor plate is polarized along the x₃-axis direction, perpendicular to the x₁x₂plane. It is assumed that there exists constant initial horizontal and vertical stress (Tensile and compression) in the piezoelectric semiconductor plate.

The harmonic solutions to (14-15) are considered when the waves propagating in the x₁ direction, and they have the following forms

where A, B, C, and D are constants. w is real and positive. The real part of is positive. Substituting (16) into (14) and (15) results in four linear homogeneous algebraic equations for A, B, C and D. Then, combining Eqs. (14) and (15) to (16), the following lineareigenvalue system is obtained:

where a=[u₃ ϕ Δp Δn]^T is related to the generalized displacement vector, b=[T₂₃D₂,J^PJⁿ]^T to the generalized traction vector, where superscript T denotes the vector transpose. The matrices [K_ij] are defined as

The explicit expressions of elements in the matrix[k_ij] are given in Appendix A. Eq. (18) is an eigenvalue/eigenvector system for the given wave number_kand frequency_w . Solving Eq. (18) gives eight complex eigenvalues and their corresponding eigenvectors (a, b) in PSC plate. The eight eigenvalues are divided into two groups as λ⁺and λ^-(also for the corresponding eigenvectors) such that λ⁺and λ^- have positive and negative imaginary parts, respectively. All eigenvalues are classified into two categories: If Re(λ_i)>0or Re(λ_i)<0 and lm(λ_i)>0 or Re(λ_i)<0 , denoting the real and imaginary parts of eigenvalues, respectively, where i∈[1...8].In order to study the wave propagation in the PSC plate, we need the proper boundary conditions on the plate surfaces as considered below.

Consider the plate is unelectroded. In the top and bottom free space of the plate, the electrical potential should satisfy [10,47]

The free space electric displacement is given by

According to Eq. (19), the harmonic solutions for electrical potential have the following forms

where G and H are the amplitudes. It is noted that the real part of must be positive so that the second relation in (19) can besatisfied. Substitute (21) into the equation (20), and then it can be obtained that

At the top and bottom of the composite plate where x₂=±h, there are six boundary and four continuity conditions:

For nontrivial solutions, the determinant of the coefficient matrix of the equations has to vanish, which leads to a polynomial equation of degree eight for λ . The eight roots of this equation as λ^(m)(k,w) and the corresponding nontrivial solution of A,B,C and D as A^(m),B^(m),C^(m) and D^(m) should be represented. It is to be noted that only the ratios among A^(m),B^(m),C^(m) and D^(m)and can be determined. Then, the general solution of (14) and (15) can be written as

where F_(m) are undetermined constants.

Substituting Eqs.(24),(21)and(22)into Eq.(23), we obtain ten linear algebraic equations for F^(m),G and H . At x₂=±h ,the resulting equations will be:

where [M_ij] is a matrix 10 ×10, whose elements are given in Appendix B. The dispersion relation of the problem can be found by equating the determinant of matrix [M_ij] to zero, which gives an equation that determines the dispersion relations of versus , for which numerical solutions are presented in the section that follows.

The constitutive equations of the piezoelectric semiconductor plate Eq. (1a) under an initial stress can be written an developed as [29–33]

In this section, we present the discussion of the results obtained through computer simulations from the numerical developments in the previous sections for a piezoelectric semiconductor plate (ZnO), whose material parameters and constants are defined as [10]:

K_Bis the Botlzmann constant and T is the absolute temperature. At room temperature K_BT/q=0.026 V . We set the h=1um with the total thickness is 2um as shown in Figure 1.

The numerical computations have been performed with the help of MATLAB software employing the procedure outlined in section 4 for the different modes of SH waves. Here the quantity kh denotes the non-dimensional wave number of acoustic waves traveling in the piezoelectric semiconductor plate.

To check the validity and the efficiency of our approach, a comparison between the obtained results for SH wave and those found in the literature [10]. While Figure 2(a) is the dispersion curves of our results, Figure 2(b) represents those obtained by [10]. It can be seen that our results are in accordance with those of the published data, proving that our approach and program are valid.

As previously mentioned in Eq. (1a), the presence of the initial stress in the piezoelectric semiconductor plate is often inevitable due to chemical shrinkage, material growth, enhancing fracture toughness, mismatching thermal expansion, machining at different temperature values, creep deformation, etc. This initial mechanical stress for the PSC plate ZnO might lead to the dramatic change in the characteristics of the guided waves. Based on the above numerical approach, we calculate the effect of initial stress on the characteristics of SH waves in PSC plate using MATLAB program codes.

Figure 3 shows the dispersion curves of SH waves with and without initial stress, where the left-hand side is for imaginary wavenumber case and the right-hand side is for real wavenumber case. For the curves in Figure 3, there is only one group of dispersion curves, i.e., the one going to the zero frequency as wave number changes from real to pure imaginary. This leads to an energy trapping phenomenon because the real wave number represents a cyclical wave while the imaginary wave number represents a decayed wave.

Here, only one initial horizontal stress component either positive or negative, denoted by , exists in the piezoelectric semiconductor plate along the propagating direction. These two applied cases are considered and shown in Figure 3, where the initial stress is tensile of + 0.1 GPa and + 0.5 GPa, and the initial stress is compressive of - 0.1 GPa and - 0.5 GPa are used in the horizontal direction, respectively, In other words, the direction of wave propagation is parallel to the initial stress direction. Here, while the “+” symbol shows a tensile stress, the “-” symbol means compressive one. It is clear at this level that the effects of the initial stress on the dispersion curves are different for the tensile stress case and the compression stress case. The dispersive curves emanating from tensile stress shift to the right side of the dispersive curves with zero stress, while the dispersive curves coming from the compression stresses shift to the left side of the dispersive curves without initial stress in real wave number range, which is reversed in the imaginary wave number range. Meanwhile, it can be observed that despite a slight drift for higher order wavenumber, the dispersion curves of three modes have almost no visible changes for the lowest wavenumber in general.

More detailed analyses of the investigation of the effect of the initial horizontal stress of the three SH modes are shown in Figure 4. Hence, we used Ω=wh as the dimensionless frequency to plot the relative dimensionless frequency defined as ΔΩ =(Ω-Ω^*)/Ω^* , it is considered in Figures 4 and 6, in which is the frequency of PSC plate without initial stress, belongs to PSC plate with initial stress.

Figure 4 presents a comparison of the product of frequency and thickness changes as a function of kh for the SH₀,SH₁ and SH₂ waves modes at fluctuating initial horizontal tensile and compression stresses. This plot indicates broad changes in relative frequency for higher kh value that reduces in the region of larger kh value at SH₁ and SH₂ modes. Furthermore, the initial stress is given in a certain percentage of the average value of relative frequency and expands up to 0.11% and 0.53%, corresponding to a level of 0.1 and 0.5 GPa, respectively. Hence, it can be seen that the change of relative frequency is linear with the applied initial horizontal stress at fundamental mode SH0. The ±0.5 GPa in this case (SH₀) is roughly ±0.58 % of the greatest tensile and compression strength of the plate.

The present section is continuation of the study of the effect of vertical initial stress. Figure 5 is a plot of dispersion curves of SH waves spreading along the normal direction to the piezoelectric semiconductor plate subjected to the initial vertical stress. The initial stress of ± 0.1 GPa and ± 0.5 GPa are equally employed in the vertical direction, where the “+” symbol implies a vertical tensile stress, and the “-” symbol is a conversely compressive stress. As expected, and as shown in Figure 3, these dispersion curves of phase velocity appear to be almost indistinguishable to the ones obtained using the initial horizontal stress. The phase velocity changes due to the applied initial vertical stress are still not visible at this scale for the SH modes.

Figure 6 further gives the relative frequency changes versus the product of wavenumber and thickness (kh) for the SH wave modes at varying initial vertical stress values. Correspondingly with Figure 4, the dramatic changes in relative frequency are seen roughly below the region of kh < 3.5 and the variation approaches to zero in the area of high value. It is clearly noted that the relative frequency is quite large when kh is small, and the variation turns to be much smaller as kh becomes larger. Then the overall magnitude in Figure 6 is similar to those found in Figure 4. Another substantial observation is that SH0 mode is not affected by the loading of initial vertical stress compared to the circumstance of the applied initial horizontal stress in Figure 4a.

In this part, modal analysis is performed for SH waves propagating in a piezoelectric semiconductor plate. The through-thickness distributions of some physical quantities are shown in Figure 7 and 8 for the initial horizontal stress and in Figure 9 and 10 for the initial vertical stress on the SH0 and SH1 modes behaviors. The real wavenumber is taken to be Re(k)=1 μm^-1 . The values of sampling points are shown in Tables 1 and 2 in detail for the different values of the initial horizontal stress σ⁰₁₁=± 0.5 GPa and (Tensile) and initial vertical stress σ⁰₁₁= ± 0.1 GPa and (Compression). These values are extracted from the results of Figure 3 and 5 for the frequency spectra.

We present 3D figures of the displacement, potential, concentrations of holes and concentrations of electrons, namely Figure 5a,b,c and d, respectively, when the values on the range of x₁ are all calculated in one wavelength equal to 2π/Re(k). Thus, the range is from 0 to 6.28 μm with Re(k)=1μm^-1 . The wave modes in the cycle of 2π/Re(k) is enough due to the item exp(ikx₁) in Eq. (24). For the range of x₂,the displacement and concentrations of holes and electrons are calculated along the thickness of the plate ranging from -h to h, i.e. -1μm to 1μm . However, the potential is calculated from -2μm to 2μm including the bottom free space from -2μm to -1μm , the plate from -1μm to 1μm , and the top free space from 1μm to 2μm .

Neverthless,to further understand the influence of the initial stress, it is crucial to study the shape of the displacement μ₃,potential Φ and concentrations of holes Δp and electrons Δn (in SH0 and SH1 modes) for Re(k)=1 μm^-1.sFigure 7 and 8 illustrate the shapes of u₃,Φ,Δp,Δn in 3D plot, associated with each distribution x₁=0 and x₁=1.5μm for SH0 and SH1 mode, respectively. It can be seen from Figure 7 and 8 that the holes concentration Δp has the same distribution as the displacement u₃, while the electron concentration Δp has a converse distribution. A similar tendency can also be seen for the potential Φ . The point where Re(k) equals 1μm_-1 has a very small Im(k) compared to Re(k) as shown in Tables 1 and 2. Comparing the point when of modes SH0 and SH1, the typical wave mode shapes for SH waves reveal that SH0 is the lowest mode. Besides, while for the latter,the u₃,Φ,Δp,Δnare all constant across the plate, for the SH1 mode, they all have nodes

Figures 8b and 10breveal that when an initial horizontal stress was applied, positive piezoelectric polarization charges appeared at the top surface while negative charges appeared at the bottom surface (caused by the piezoelectric effect of PSC). This resulted in the decrease of the concentration of holes at the top boundary layer and the increase of the concentration at the bottom surface, as shown in Figures 8c and 10c.

Therefore, it is obviously seen from Figures 7a and 9a that the horizontal initial stress increases from -5 GPa to 0.5 GPa. Furthermore, the variation of the displacement u₃ for SH0 mode increases linearly, but in the case of the applied vertical initial stress, it shows the opposite phenomenon. Thus, the effect of vertical initial stress is also observed to be more significant than that of the horizontal initial stress applied in PSC plate for the SH0 mode represented in Figures 7b and 9b.

In the present research work, the propagation of SH-type wave under an initial horizontal and vertical stress of a piezoelectric semiconductor plate was numerically investigated. The phase velocity of SH0, SH1 and SH2 modes waves were calculated for various initial stressσ⁰_ijvalues. The initial horizontal stress was proven to have a significant influence on SH wave behaviors, especially in high wavenumber. It was found that the existence of an initial vertical stress led to the increase in the phase velocity on the range of low wavenumber. From the calculated phase velocity spectra, it was demonstrated that the tensile initial stress would increase the phase velocity of SH waves propagating in the PSC plate, while the compressive initial stress would reduce the phase velocity.

The components of displacement, potential, hole concentration and concentration electrons of SH wave were obtained using a numerical treatment. In fact, the initial horizontal and vertical stresses acting in the plate PSC were confirmed to have a considerable influence on the component of displacement of mode SH0 compared to mode SH1. However, the initial stresses acting in the PSC plate remarkably affected only the top and bottom surfaces of PSC plate on the electromechanical fields of the SH1 mode. Furthermore, it is also established that the initial horizontal and vertical stresses have an opposite effect on the components of displacement, hole concentration and concentration electrons of SH0 mode. These numerical results for the piezoelectric semiconductor plate are significant for the improvement of the piezoelectric nanogenerators devices and the design of ultrasonic transducers in industry for non-destructive testing especially that encounters initial stresses.

• The propagation of the SAW in the piezoelectric nanogenerators is researched by using numerical matrix method approach.

• The phase velocity of SH0, SH1 and SH2 modes waves are calculated for various initial stress values.

• The obtained results reveal the tensile initial stress will increase the phase velocity (increase the kinetic energy of waves) of SH waves propagating in the PSC plate, while the compressive initial stress will reduce the phase velocity.

• The initial horizontal and vertical stresses acting in the plate PSC have a considerable influence on the component of displacement, potential, hole concentration and concentration electrons of mode SH0 compared to mode SH1.