1 Introduction

Owing to their capacity of facilitating the conversion between mechanical and electric energies, piezoelectric materials have been manufactured into various kinds of surface acoustic wave (SAW) and bulk acoustic wave (BAW) devices^[1-6], e.g., actuators and sensors for wave generation and reception^[2-3], transformers for raising or lowering voltages^[4], energy harvesters for energy conversion and handling^[5], and gyroscopes for detecting the vibration of moving objects^[6]. During the past decades, many efforts have been concentrated on the inner physical and mechanical properties of these piezoelectric devices, including temperature stability^[7], initial bias^[8], material coefficient inhomogeneity^[9], dissipation^[10], viscous effect^[11], and large deformation and nonlinearity^[5]. Simplifying piezoelectric materials into dielectrics (insulators) is a well-known theoretical methodology^[12], in which small electrical conduction is usually ignored. Generally, no real material can be considered as perfect insulators^[13], since all existing materials show certain levels of electrical conduction. Therefore, the zero charge equation of electrostatics and boundary conditions has been improved^[14], which can effectively simulate the effect of real current. It has been revealed that low ohmic conduction and related dissipative effects should also be considered in quartz crystals, which are usually treated as good insulators, especially when the Q values (quality factor) of these devices are calculated^[14]. To some extent, the effect of semi-conduction on the performance of piezoelectric devices is unknown, which is just the origin of the present contribution. Hence, in the present work, we consider the piezoelectric semiconductor beam as a research target, and discuss its basic mechanical and physical behaviors to reveal the semiconduction effect on the behaviors of piezoelectric materials.

As we know, piezoelectric materials belong to anisotropic media, and extension may induce mechanical deformations, e.g., flexure, shear, and torsion. The deformation in piezoelectric materials is very complex, and the static and dynamic properties are difficult to be analyzed. Even for an infinite plate or a finite beam, few exact solutions can be obtained theoretically at present. In other words, classical three-dimensional (3D) equations are difficult to be used in the direct analysis of single vibration modes^[15], and improvement or approximation should be provided^[16-17]. Hence, under the framework of continuum mechanics, we will propose a double power series expansion technique in this paper to derive the approximate one-dimensional (1D) equations of a piezoelectric semiconductor beam. Some necessary stress relaxation relations are introduced, and the extension, flexure, and shear constitutive relations are revised correspondingly. For piezoelectric semiconductor materials, the carrier diffusion not only contributes to the current, but also contributes to the carrier drift under an electric field associated with the ohmic conduction^[18]. The current boundary condition is rewritten. Based on the derived equations, the extensional motion of a ZnO beam is considered as the numerical simulation, and the effect of the initial carrier density on the extensional behavior is discussed in detail. The theoretical equations and numerical outcome are general, which can clearly interpret the inner physical mechanism of the semiconductors in the piezoelectrics and provide theoretical guidance for further experimental design of piezoelectric semiconductor devices.

2 Basic 3D equations

Similar to piezoelectrics, the basic coupled electro-mechanical behavior of piezoelectric semiconductors can be analyzed theoretically in the framework of continuum mechanics^[19-20]. We consider a homogeneous one-carrier piezoelectric semiconductor under a uniform direct current (DC) electric field E_j with the carrier charge q and the initial steady state carrier density n. In the linear theory of piezoelectricity, the corresponding displacement component u_i, electrical potential function ϕ, and perturbation of the carrier density n should, respectively, satisfy Newton's law, Gauss's law of electrostatics, and the conservation of charge^[18-20].

(1)

where T_ij, D_i, and J_i denote the components of the stress, the electric field, and the electric current, respectively. ρ is the mass density, and f_i is the body force per unit mass. A superimposed dot represents differentiation with respect to the time t. The summation convention for repeated tensor indices is used. Equations (2) and (3) are only suitable for positive holes, which are just as an example in the present contribution. From the mathematical view, if electrons are considered, the two equations are still applicable after n (or q) is replaced by -n (or -q). The generalized constitutive relations of piezoelectric semiconductors are^[18-20]

(2)

where s_ijkl and e_kij represent the elastic compliances, ε_ik is the piezoelectric constant, µ _ij is the dielectric permittivity, and d_ij represents the diffusion constant. For electrons, Eq.(2) should be improved as follows:

Here, the strain tensor S_ij, electric field E_i, and carrier density gradient N_i are, respectively, defined by

(3)

In order to solve the above differential equations, necessary boundary conditions should be adopted. If the unit outward normal vector of the boundary is denoted by ζ, the mechanical displacement component u_i or the traction vector T_ijζ_j, the electric potential function ϕ or the normal component of the electric displacement vector D_iζ_i, and the carrier density n or the normal current J_iζ_i may be prescribed to obtain the closed-form solution to the problem^[21]. The boundary condition containing the normal current J_iζ_i should be improved into for the piezoelectric semiconductor^[12].

3 Double power series expansion technique

The considered piezoelectric semiconductor beam with a rectangular cross section is shown in Fig. 1, where the length is much larger than the width and thickness, i.e., c» a, b. Under this condition, the reduced 1D equations for the piezoelectric semiconductor beam can be derived with the mechanical displacement component u_i, electrical potential function ϕ, and perturbation of the carrier density n written by the following double power series expansions^[22-25]:

(4)

It should be stressed here that, the full mechanical properties of the piezoelectric semiconductor beam have been included by the expansions in Eq.(4) and the general derivation. Some particular motion for extension can be deduced from this expression. For example, the usual zeroth-order theory can be obtained easily when the series is truncated with only one term, i.e., m=p=0. The first-order theory for the coupled extensional and flexural motion corresponds to the cases where m and p are smaller than 2. Similarly, the related higher-order theory with larger m and p, which could depict various deformations of the cross section, can also be captured^[23-25]. Based on the double power series, the corresponding strain tensor S_ij, electric field E_i, and carrier density gradient N_i in Eq.(3) can be written as follows:

(5)

where

(6)

Correspondingly, the generalized constitutive relations of the piezoelectric semiconductors can be expressed as follows:

(7)

where the stress resultant T_ij^{(m, p)}, electric displacement resultant D_i^{(m, p)}, and electric current resultant J_i^{(m, p)} are, respectively, defined by

(8)

In the above equations, A=4ab is the area of the cross section, and

(9)

Multiplying Eq.(1) by x₁^mx₂^p and integrating, we can obtain the final 1D governing equations as follows^{[16-18, 22-25]}:

(10)

where

Meanwhile, the corresponding surface traction resultant T_i^{(m, p)}, electric displacement resultant D^{(m, p)}, and electric current resultant J^{(m, p)} of various orders are

(11)

4 1D theory for the piezoelectric semiconductor beam

Corresponding to different truncations of the double power series, different cases degenerated from the 1D theory will be introduced and explained in detail in the following content.

4.1 Zeroth-order theory for extension

The zeroth-order theory for extension can be deduced easily when the series is truncated with only one term, i.e., m=p=0. It is valid when the concerned wavelength is much larger than the dimensions of the rectangular cross section^[26]. The main characteristics of the beam will be controlled by u₃^{(0, 0)}, ϕ^{(0, 0)}, and n^{(0, 0)}. The major strain component is

(12)

Considering the Poisson effect, other zeroth-order strain components cannot be set to zero. Nevertheless, they will be eliminated instead. This process is taken as the stress relaxation procedure^{[16, 18, 24-26]}. For convenience, the usual compact matrix notation^[24] with u, v=1, 2, …, 6 will be adopted, so that Eq.(7) can be written as follows:

(13)

(14)

(15)

where A₍₀₀₀₀₎ =A=4ab. Meanwhile, considering that the extension of the beam is mainly represented by the major stress T₃^{(0, 0)} and the beam features a slender shape with c» a, b, we can set^{[16, 18, 24-25]}

(16)

which is just the stress relaxation condition. Then, the expression of T₃^{(0, 0)} can be degenerated from Eq.(13), i.e.,

(17)

where

Furthermore, the corresponding zeroth-order constitutive relations governing D_i^{(0, 0)} and J_i^{(0, 0)} can be obtained via Eqs.(13)-(17) as follows:

(18)

(19)

where

On the basis of stress relaxation, the final governing equations for the zeroth-order theory of the piezoelectric semiconductor beam (see Fig. 1) can be summarized as follows:

(20)

(21)

(22)

To obtain the closed-form solutions, u₃^{(0, 0)} or T₃^{(0, 0)}, ϕ^{(0, 0)} or D₃^{(0, 0)}, and n^{(0, 0)} or (+J₃^{(0, 0)}) need to be prescribed at the end of x₃ =± c.

4.2 First-order theory for the coupled extensional and flexural motions

Actually, if a finite piezoelectric semiconductor beam suffers from the external homogeneous pressure, extensional motion will occur, accompanied with inevitable flexural and torsional deformations. This is due to the anisotropy of the material. The piezoelectric semiconductor beam shown in Fig. 1 is transversely isotropic with the polarization along the z-axis, and does not exhibit coupling to torsional modes^[24-25]. Therefore, the aforementioned equations can be reduced to a first-order theory for coupled extensional and flexural motion without torsion. In this case, the equations containing extension (u₃^{(0, 0)}), flexure (u₁^{(0, 0)} and u₂^{(0, 0)}), and shear deformations (u₃^{(1, 0)} and u₃^{(0, 1)}) with electrical potential functions (ϕ^{(0, 0)}, ϕ^{(1, 0)}, and ϕ ^{(0, 1)}) and the perturbation of the carrier density (n^{(0, 0)}, n^{(1, 0)}, and n^{(0, 1)}) can be retained from Eq.(10) as follows:

(23)

(24)

(25)

(26)

(27)

The corresponding strain components and electric fields of various orders are, respectively, stated as follows:

(28)

(29)

(30)

(31)

Similarly, the relevant zeroth-and first-order gradients of the carrier density are stated, respectively, as follows:

(32)

(33)

In the zeroth-order constitutive relations, the shear force resultants T₄^{(0, 0)} and T₅^{(0, 0)}, which are caused by flexure, cannot be set to zero, while the other stress resultant components in Eq.(16) are null^{[16, 18, 24-25]}, i.e.,

(34)

Equation (34) is the stress relaxation condition for the zeroth-order components, in which the extension with the Poisson effect has been considered. Introducing α, β =3, 4, 5 and µ =1, 2, 6, it can be expressed as T_µ^{(0, 0)} =0. Then, Eq.(13) can be rewritten as follows:

(35)

Similarly, Eq.(35) can be expressed in an inverted form as follows:

(36)

where

Substituting Eq.(36) into Eqs.(14) and (15) yields

(37)

(38)

where

In the following, the first-order constitutive relations will be introduced. The flexural deformations emerging in the x₁ -and x₂ -directions should be considered separately. For example, the major first-order resultant corresponding to the flexure in the x₁ -direction is the bending moment T₃^{(1, 0)}. Therefore, it is suitable that the following first-order stress resultants are set to zero^{[16, 18, 24-25]}:

(39)

If

the following relations can be obtained from Eq.(7):

(40)

where

Hence, the first-order constitutive relations in the x₁ -direction are

(41)

Similarly, the major first-order resultant corresponding to the flexure in the x₂ -direction is the bending moment T₃^{(0, 1)}. Therefore, the stress relaxation condition is

(42)

Correspondingly, the first-order constitutive relations in the x₂-direction can be obtained as follows:

(43)

Up to now, the 1D equations have been presented in detail for the first-order theory. Totally speaking, these equations can be classified into three categories, i.e., the dynamic equilibrium equations, generalized geometric equations, and generalized constitutive equations. The dynamic equilibrium equations consist of extension (23), flexure (24) with shear deformations (25), Gauss's law of electrostatics (26), and conservation of charge (27). The generalized geometric equations include the strain-displacement relationships (28) and (29), electric field-potential function relationships (30) and (31), and the relationships (32) and (33) between the perturbation of the carrier density and its gradient. The constitutive equations contain the zeroth-order constitutive relationships (36)-(38) and the first-order constitutive relationships in the x₁-and x₂-directions (41)-(43). With successive substitutions, the governing equations (23)-(27) can be written as eleven equations containing eleven unknowns, i.e., u₃^{(0, 0)}, u₁^{(0, 0)}, u₂^{(0, 0)}, u₃^{(1, 0)}, u₃^{(0, 1)}, ϕ^{(0, 0)}, ϕ^{(1, 0)}, ϕ^{(0, 1)}, n^{(0, 0)}, n^{(1, 0)}, and n^{(0, 1)}. Similarly, in order to obtain the closed-form solutions, u₃^{(0, 0)} or T₃^{(0, 0)}, u₁^{(0, 0)} or T₅^{(0, 0)}, u₂^{(0, 0)} or T₄^{(0, 0)}, u₃^{(1, 0)} or T₃^{(1, 0)}, u₃^{(0, 1)} or T₃^{(0, 1)}, ϕ^{(m, n)} or D₃^{(m, n)}, and n^{(m, n)} or () should be prescribed at x₃ =± c with (m, n)=(0, 0), (1, 0), and (0, 1) for mechanical and electrical boundary conditions.

4.3 Reduction to elementary flexure

As a reduction of the first-order theory, the case of elementary flexure will be given in this subsection. For the elementary flexure without shear deformations, the corresponding rotatory inertia terms ü₃^(1,0) and ü₃^(0,1) in Eq.(25) are set to be zero^[22-25] so that

(44)

After eliminating T₁₃^{(0, 0)} and T₂₃^{(0, 0)}, the dynamic equilibrium equations for elementary flexure can be obtained as follows:

(45)

Correspondingly, the zeroth-order flexural shear strains S₃₁^{(0, 0)} and S₃₂^{(0, 0)} should be set to be zero. Therefore,

(46)

(47)

Similarly, the theory of extension and elementary flexure also contains three categories of equations, i.e., the dynamic equilibrium equations, generalized geometric equations, and generalized constitutive equations. The dynamic equilibrium equations consist of extension (23), flexure (45), Gauss's law of electrostatics (26), and conservation of charge (27). The generalized geometric equations are composed of extensional strain (28), flexural strains (47), electric field-potential function relationships (30) and (31), and the relationships (32) and (33) between the perturbation of the carrier density and its gradient relation. The constitutive equations are, respectively, the extensional constitutive relationship (36) when α =3, the relationship between the shear force and bending moment (44), and other constitutive relationships (37) and (38). After necessary substitutions and derivations, these equations can be simplified as nine equations with nine unknowns, i.e., u₃^{(0, 0)}, u₁^{(0, 0)}, u₂^{(0, 0)}, ϕ^{(0, 0)}, ϕ^{(1, 0)}, ϕ^{(0, 1)}, n^{(0, 0)}, n^{(1, 0)}, and n^{(0, 1)}. Similarly, nine boundary conditions should be prescribed at x₃ =± c.

5 Static analysis of the extensional motion

As an application of the 1D equations, the static extensional motion of the piezoelectric semiconductor beam is considered firstly. Equations (20)-(22) can be reduced to

(48)

Assuming that E₃ =0 and substituting the generalized constitutive equations for the zeroth-order theory, i.e., Eqs.(17)-(19), into Eq.(48), we can get the following dynamic equilibrium equations:

(49)

S₃^{(0, 0)}, E₃^{(0, 0)}, and n^{(0, 0)} can be decoupled easily from Eq.(49). Therefore,

(50)

where

For the positive hole, i.e., q>0, the solution can be obtained easily as follows:

(51)

where C₁ is an undetermined coefficient, and only the symmetric mode in the x₃-direction is considered. Conversely, if the electron with q < 0 is considered, the second and third expressions in Eq.(49) should be rewritten as follows:

(52)

Using the same procedure, the same expression as Eq.(51) can be derived. Taking the positive hole as an example, the electrical potential and displacement components are stated, respectively, as follows:

(53)

(54)

where C₂ and C₃ are integration constants. Correspondingly,

(55)

Only the symmetric extensional modes are considered, and the boundary condition at x₃ =c (see Fig. 1) is sufficient for solving the static problem. If the symmetric deformation is caused by the carrier density perturbation n₀ at the boundary x₃ =c, i.e.,

(56)

which requires

(57)

(ⅰ) For the electrical open case, D₃^{(0, 0)} should satisfy

(58)

Therefore,

(59)

(ⅱ) However, when the edge is electrically shorted, the electric potential equals zero, i.e.,

(60)

Then, the integration constant C₂ is

(61)

(ⅲ) When the two ends at x₃ =± c are free, T₃₃^{(0, 0)} vanishes at both ends, i.e.,

(62)

which requires

(63)

In fact, it is identically zero along the whole beam.

(ⅳ) If the end of the piezoelectric semiconductor beam is fixed at x₃ =± c, u₃^{(0, 0)} is confined as follows:

(64)

which means

(65)

In general, the boundary conditions come in four types: electrically open and free ends (OF), electrically shorted and free ends (SF), electrically open and simply supported ends (OS), and electrically shorted and simply supported ends (SS). The four cases will be discussed in detail in the following section.

For the numerical results, the parameters of the plate of ZnO beam are as follows^{[18, 28-30]}:

and d₃₃ =µ₃₃k'T/q, where k' is the Boltzmann constant, and T=300 K is the absolute temperature^[21]. The length is 10cm with the external n₀ fixed to 1× 10¹⁶m^-3.

Figure 2 shows the static displacement distribution along the x₃ -direction for different boundary conditions. The cases of SF, OS, and SS have the same displacement distribution, which can be proven from Eqs.(54), (61), (63), and (65). The amplitude of OF is smaller than those of the aforementioned three cases. The deformations are mainly focused on the end of the piezoelectric semiconductor beam. At the region of |x₃| < 4cm, the displacement almost remains at zero. Meanwhile, the initial carrier density n has a significant effect on the displacement distribution (see Fig. 3). A large n leads to a small amplitude, which implies dissipation as a result of semiconduction.

6 Dynamic analysis of the extensional motion

Typically, wave devices, which are made of piezoelectric materials or semiconductors, are very small with the size on the order of centimeter, sometimes even on the order of millimeter. Investigating the dynamic properties of piezoelectric semiconductor beams seems to be more valuable. Taking the aforementioned 1D extensional motion as an example, the dynamic behavior of the piezoelectric semiconductor beam can be controlled by the following equations:

(66)

A solution of Eq.(66) can be explored as follows^[30-33]:

(67)

where A, B, and C are undetermined constants. ξ and ω stand for the circular frequency and the wavenumber, respectively. This solution must satisfy the governing equation (66), which leads to three homogeneous linear equations for A, B, and C, i.e.,

(68)

For the nontrivial solutions of A, B, and C, the determinant of the coefficient matrix of Eq.(3) should be equal to zero, from which an algebraic equation about ξ can be deduced as follows:

(69)

where

Equation (69) comprises six roots, i.e.,

(70)

where

(71)

Considering the structural and loading symmetry, only three roots are enough to construct the solution. Assume

(72)

Then, Eq.(67) can be written as follows:

(73)

where the common term of exp (iωt) has been omitted for brevity,

(74)

and A₁, A₂, and A₃ can be determined with the aid of the boundary conditions. Then, the stress resultant t₃₃^{(0, 0)}, electric displacement resultant D₃^{(0, 0)}, and current resultant J₃^{(0, 0)} can be calculated through

(75)

Similar to the static analysis, the symmetric vibration mode is excited by the carrier density perturbation n₀ exp (iωt) at the boundary x₃ =c. This boundary condition is sometimes difficult to achieve in practice. Nevertheless, investigating and understanding the effect of semiconductor properties is still important and beneficial, which can interpret the inner physical mechanism of semiconductor in piezoelectrics. Therefore,

(76)

The boundary conditions are as follows:

(ⅰ) For the electrically open case, Eq.(58) requires

(77)

(ⅱ) For the electrically shorted case, Eq.(60) can be reduced to

(78)

(ⅲ) When the two ends at x₃ =± c are free, Eq.(62) is equal to

(79)

(ⅳ) If the piezoelectric semiconductor beam is simply supported at x₃ =± c, the following relation can be obtained by use of Eq.(64):

(80)

The displacement signal calculated at x₃ =0.5c versus the driving frequency for different boundary conditions are shown in Fig. 4, from which we can conclude that the displacements assume their own maxima at resonant frequencies. Hence, the piezoelectric semiconductor beam can be viewed as a resonant device, which has better performance at a particular frequency^[32-33]. As pointed out, the maximum amplitude of the first resonance is not shown exactly in Fig. 4. However, the first peak value is larger than the others. Actually, the corresponding modes for high resonances have nodal points along the length direction, which will lead to some voltage cancelation in a piezoelectric semiconductor beam and furthermore smaller displacement response^[34]. Meanwhile, the boundary conditions have significant effects on the dynamic properties of the beam. As shown in Figs. 4(a) and 4(b), six symmetric resonances can be identified in the region of ω ≤2.4× 10⁶ rad·s^-1 when the ends at x₃ =± c are free from traction. However, if the ends at x₃ =± c are simply supported (see Fig. 4(c)), only three resonances can be observed under the same condition. Moreover, the resonance frequency of OF is larger than that of SF (see Figs. 4(a) and 4(b)).

Actually, the length of the piezoelectric semiconductor beam will have a significant effect on the resonance frequencies. Usually, a long beam features a relatively low resonance frequency, which has been proved by Fig. 5. Fortunately, the resonance amplitude is insensitive to the length parameter (see Fig. 5). To some extent, the insensitivity is beneficial for the design of a piezoelectric semiconductor beam. The length can be chosen freely according to the external frequency of the beam, which will not reduce the displacement response.

As a result of the displacement signal achieving its maximum at the first resonance, the effect of the initial carrier density n on the first resonance for the extensional vibration is mainly explored in the following discussion. Both the resonant frequency and the resonance amplitude are sensitive to the initial carrier density n (see Fig. 6). A large n implies great dissipation as a result of the semiconduction, and leads to a weak resonance with a small amplitude. In order to depict the effect of the semiconduction more visually, Fig. 7 presents the variation trend of resonance frequency and resonance amplitude of the excited displacement signal at x₃ =0.5c versus the initial carrier density n when the ends at x₃ =± c are electrically shorted and traction free. The two curves sharply decrease first and then approach some special values when n increases. The resonance amplitude of excited displacement is not zero when the initial carrier density n≥ 1× 10¹⁵m^-3. For instance, when the initial carrier density is assumed to be n=1× 10¹⁵m^-3, the amplitude of the excited displacement is 4.12× 10^-10m, and when n=5× 10¹⁵m^-3, the amplitude is 8.3× 10^-11m. However, our results show that the corresponding displacement magnitude is 2.064× 10^-8m when n=2× 10¹³m^-3.

7 Conclusions

Totally speaking, the reduced 1D equations of a piezoelectric semiconductor beam with a rectangular cross section are proposed with the aid of a double power series expansion technique. These equations are general and widely applicable, which can be degenerated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. Based on these equations, numerical simulations are carried out sequentially to explore the effects of the semiconduction on the static deformation and dynamic extensional behaviors of a slender ZnO beam. It has been revealed that both the resonance frequency and the displacement response evidently decrease, owing to the initial carrier density existing in piezoelectric semiconductor media. The equations derived in the present contribution and the qualitative results about the semiconduction in the piezoelectrics can be viewed as the benchmark for further theoretical and experimental investigation.