1 Introduction

Piezoelectric materials are widely used to make electromechanical transducers for converting electric energy to mechanical energy or vice versa, and acoustic wave devices for frequency operation and sensing. In most cases, piezoelectric crystals and ceramics are treated as non-conducting dielectrics, but in reality there is no sharp division separating conductors from dielectrics. Real materials always show some conductance (Tiersten and Sham, 1998). For example, in acoustic wave devices made from quartz crystals, the small ohmic conductance and the related dissipative effects need to be considered when calculating the Q value (quality factor) of the devices (Lee et al., 2004; Yong et al., 2010; Wang et al., 2011), because other dissipative effects in quartz, such as material damping and radiation damping, are very small. Another origin of conduction in piezoelectric crystals is that some of them are in fact semiconductors with charge carriers of electrons and/or holes (Auld, 1973), e.g., the widely used ZnO and AlN films and fibers. In these materials, in addition to carrier drift under an electric field, carrier diffusion also contributes to the electric current. Piezoelectric semiconductors have been used to make devices for acoustic wave amplification (White, 1962; Yang and Zhou, 2004; Ghosh, 2006; Willatzen and Christensen, 2014) and acoustic charge transport (Schülein et al., 2013; Büyükköse et al., 2014) based on the acoustoelectric effect, i.e., the motion of carriers under the electric field produced by an acoustic wave through piezoelectric coupling. Recently, the electric field produced by mechanical fields in a piezoelectric semiconductor has been used to manipulate the operation of semiconductor devices, which forms the foundation of piezotronics (Wang et al., 2006; Wang, 2007; 2010). Piezoelectric semiconductors, such as ZnO, are also used for mechanical energy harvesting and conversion to electric energy (Hiralal et al., 2012; Kumar and Kim, 2012; Graton et al., 2013; Yin et al., 2013).

The basic behaviors of piezoelectric semiconductors can be described by the conventional theory which consists of the equations of linear piezoelectricity and the equations for the conservation of charges of electrons and holes (Hutson and White, 1962). This theory has been used to study some of the applications mentioned above: the inclusion problem for composites (Yang et al., 2006), the fracture of piezoelectric semiconductors (Hu et al., 2007; Sladek et al., 2014a; 2014b), the electromechanical energy conversion in these materials (Li et al., 2015), the vibrations of plates (Wauer and Suherman, 1997), and to develop low-dimensional theories of piezoelectric semiconductor plates and shells (Yang and Zhou, 2005; Yang et al., 2005). Researchers have also developed more general and fully nonlinear theories (de Lorenzi and Tiersten, 1975; McCarthy and Tiersten, 1978; Maugin and Daher, 1986).

This paper is concerned with the carrier distribution and electromechanical fields in the thin piezoelectric semiconductor rod, which is often used in piezoelectric semiconductor devices. The rod is free from externally applied mechanical and electrical loads. Because of the presence of carriers and the electromechanical coupling in the material, electromechanical fields develop and the carriers assume a certain spatial distribution. This problem is fundamental to the applications of piezoelectric semiconductor rods or wires. However, there exists some nonlinearity associated with the drift current, which is proportional to the product of the unknown carrier density and the unknown electric field (Pierret, 1988), and hence the problem presents some mathematical challenges. We performed a theoretical analysis using a 1D model and obtained simple and useful results for carrier distribution and electromechanical fields.

2 Equations for piezoelectric semiconductors

For a piezoelectric semiconductor, the 3D phenomenological theory consists of the linear momentum equation of motion, the charge equation of electrostatics, and the conservation of charge for electrons and holes (continuity equations) (Hutson and White, 1962; Pierret, 1988):

$$\begin{array}{*{20}c} {{T_{ji,j}} = \rho {{\ddot u}_i},\quad \quad \quad \quad \quad \quad \quad }\\ {{D_{i,i}} = q(p - n + N_D^ + - N_A^ -)\;,\quad }\\ {J_{i,i}^n = q\dot n, \quad \quad \quad \quad \quad \quad \quad \quad }\\ {J_{i,i}^p = - q\dot p,\quad \quad \quad \quad \quad \quad \quad \quad } \end{array}$$
(1)

where Tji is the component of stress, ρ the mass density, ui the component of mechanical displacement, Di the component of electric displacement, q=1.6 ×10−19 C the magnitude of the electronic charge, p and n the number densities of holes and electrons, \(N_D^ + \) and \(N_A^ - \) the densities of impurities of donors and accepters, and \(J_i^p\) and \(J_i^n\) the hole and electron current densities, respectively. The reference state is the state before doping when the charges are not present. When the charges are introduced, they assume certain distributions and electromechanical fields develop. In Eq. (1), we have neglected carrier recombination and generation. Cartesian tensor notation has been used. A comma followed by an index indicates a partial derivative with respect to the coordinate associated with the index. A superimposed dot represents a time derivative. We consider doped n-type semiconductors for which n and \(N_D^ + \) are much greater than p and \(N_A^ - \). Therefore, we neglect p and \(N_A^ - \) in this paper. In this case Eq. (1)1–3 become uncoupled to the p in Eq. (1)4. Constitutive relations for Eq. (1)1–3 describing material behaviors can be written in the following form:

$$\begin{array}{*{20}c} {{S_{ij}} = s_{ijkl}^E{T_{kl}} + {d_{kij}}{E_k},\quad }\\ {{D_i} = {d_{ikl}}{T_{kl}} + \varepsilon _{ik}^T{E_k},\quad \;}\\ {J_i^n = qn{\mu _{ij}} {E_j} + q{D_{ij}}{n_{,j}},} \end{array}$$
(2)

where Sij is the component of strain, Ei the component of electric field, \(s_{ijkl}^E\) the elastic compliance constant, dkij the piezoelectric constants, \(\varepsilon _{ij}^T\) the dielectric constant, μij the carrier mobility, and Dij the carrier diffusion constants. The superscripts “E” and “T” in \(s_{ijkl}^E\) and \(\varepsilon _{ij}^T\) as well as the superscript “n” in \(J_i^n\) will be dropped in the rest of the paper. The strain Sij and the electric field Ei are related to the mechanical displacement ui and the electric potential ϕ through

$$\begin{array}{*{20}c} {{S_{ij}} = {1 \over 2}({u_{i,j}} + {u_{j,i}}),}\\ {{E_j} = - {\phi _{,j}}{.}\quad \quad \quad \quad } \end{array}$$
(3)

3 1D model for a rod

Specifically, we consider a cylindrical rod of length 2L as shown in Fig. 1. It is within ∣x3∣<L. The cross section of the rod is arbitrary. Its surface is traction free. The rod is made from crystals of class 6 mm which includes the widely used crystals of ZnO and AlN. The c-axis of the crystal is along the axis of the rod, i.e., the x3 axis. The rod is assumed to be long and thin. The electric field in the surrounding free space is neglected. The rod has electrons described by its density n varying along x3 only. The electric field produced by the electrons causes deformation in the rod through piezoelectric coupling.

Fig. 1
figure 1

A piezoelectric semiconductor rod of crystals of class 6 mm

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The deformation of the thin rod is mainly an axial extension or contraction which can be described by a 1D model (Yang, 2005). For the 1D model, corresponding to Eq. (1)1–3, the field equations for the static case we want to consider when all the time derivatives vanish are

$$\begin{array}{*{20}c} {{T_{33,3}} = 0,\quad \quad \quad \quad }\\ {{D_{3,3}} = q(- n + N_D^ +)\;,}\\ {{J_{3,3}} = 0{.}\quad \quad \quad \quad \;\;} \end{array}$$
(4)

For a thin rod, as will be seen later, the electrons tend to concentrate near the two ends of the rod. If the impurity is essentially uniform along the rod, then, near the two ends of the rod, which is the region of interest, n is much larger than \(N_D^ + \). Therefore, in the following we will neglect the non-mobile impurity chargein \(N_D^ + \) Eq. (4) and focus on the mobile electrons described by n. In the compact matrix notation (Yang, 2005), the relevant constitutive relations from Eq. (2) for the axial fields are

$$\begin{array}{*{20}c} {{S_3} = {s_{33}}{T_3} + {d_{33}}{E_3},\quad \quad }\\ {{D_3} = {d_{33}}{T_3} + {\varepsilon _{33}}{E_3},\quad \quad }\\ {{J_3} = qn{\mu _{33}}{E_3} + q{D_{33}}{n_{,3}}{.}} \end{array}$$
(5)

The relevant strain-displacement relation and the electric field-potential relation are the following ones from Eq. (3):

$$\begin{array}{*{20}c} {{S_3} = {u_{3,3}},}\\ {{E_3} = - {\phi _{,3}}{.}} \end{array}$$
(6)

Eq. (5)1,2 can be solved for T3 and rewritten as

$$\begin{array}{*{20}c} {{T_3} = {{\bar c}_{33}}{S_3} - {{\bar e}_{33}}{E_3},\;}\\ {{D_3} = {{\bar e}_{33}}{S_3} + {{\bar \varepsilon }_{33}}{E_3},} \end{array}$$
(7)

where the effective 1D elastic, piezoelectric, and dielectric constants are:

$$\begin{array}{*{20}c} {{{\bar c}_{33}} = 1/{s_{33}},\quad \quad \quad }\\ {{{\bar e}_{33}} = {d_{33}}/{s_{33}},\quad \quad }\\ {{{\bar \varepsilon }_{33}} = {\varepsilon _{33}} - d_{33}^2/{s_{33}}{.}} \end{array}$$
(8)

With the use of Eq. (6), Eqs. (7) and (5)3 become:

$$\begin{array}{*{20}c} {{T_3} = {{\bar c}_{33}}{u_{3,3}} + {{\bar e}_{33}}{\phi _{,3}},\quad \quad }\\ {{D_3} = {{\bar e}_{33}}{u_{3,3}} - {{\bar \varepsilon }_{33}}{\phi _{,3}},\quad \quad }\\ {{J_3} = - qn{\mu _{33}}{\phi _{,3}} + q{D_{33}}{n_{,3}},} \end{array}$$
(9)

which are the 1D constitutive relations ready to be used in Eq. (4).

4 Carrier distribution and fields

The substitution of Eq. (9)1,2 in Eq. (4)1,2 yields

$$\begin{array}{*{20}c} {{{\bar c}_{33}}{u_{3,33}} + {{\bar e}_{33}}{\phi _{,33}} = 0,\quad }\\ {{{\bar e}_{33}}{u_{3,33}} - {{\bar \varepsilon }_{33}}{\phi _{,33}} = - qn,} \end{array}$$
(10)

which can be rearranged to

$$\begin{array}{*{20}c} {{u_{3,33}} = - {{{{\bar e}_{33}}} \over {{{\bar c}_{33}}}}{\phi _{,33}},\;}\\ {n = {{{\varepsilon _{33}}} \over q}{\phi _{,33}}{.}\quad \quad } \end{array}$$
(11)

Eq. (4)3 can be integrated to give

$${J_3} = - qn{\mu _{33}}{\phi _{,3}} + q{D_{33}}{n_{,3}} = {C_1}{\varepsilon _{33}},$$
(12)

where Eq. (9)3 has been used and C1 is an integration constant. Substituting Eq. (11)2 into Eq. (12), we can obtain

$$ - {\mu _{33}}{\phi _{,33}}{\phi _{,3}} + {D_{33}}{\phi _{,333}} = {C_1}{.}$$
(13)

The integration of Eq. (13) leads to

$$ - {1 \over 2}{\mu _{33}}{({\phi _{,3}})^2} + {D_{33}}{\phi _{,33}} = {C_1}{x_3} + {C_2}{D_{33}},$$
(14)

where C2 is another integration constant. Let

$$ - {\phi _{,3}} = {E_3} = E,$$
(15)

then Eq. (14) takes the following form

$$ - {1 \over 2}{\mu _{33}}{E^2} - {D_{33}}{E_{,3}} = {C_1}{x_3} + {C_2}{D_{33}}{.}$$
(16)

For the static and free rod we are considering, J3=0 and hence C1=0. Eq. (16) reduces to

$${E_{,3}} + a{E^2} = - b,$$
(17)

where we have denoted

$$a = {1 \over 2}{{{\mu _{33}}} \over {{D_{33}}}} = {1 \over 2}{q \over {kT}},\;\;b = {C_2}{.}$$
(18)

In Eq. (18), we have used the Einstein relationship (Navon, 1986) that

$${{{\mu _{33}}} \over {{D_{33}}}} = {q \over {kT}}{.}$$
(19)

Since a>0, from Eq. (17) it can be seen that b>0 if E,3<0 and dominates aE2, which is always positive. In this case ab>0 and the solution to Eq. (17) can be found as

$$E = - {\phi _{,3}} = - {{b\tan \left[ {\sqrt {ab} \,({x_3} - {C_3})} \right]} \over {\sqrt {ab} }},$$
(20)

where C3 is an integration constant. Because of the symmetry in the problem, we expect that E=0 when x3=0 which can be satisfied by choosing C3=0 in Eq. (20). Then, integrating Eq. (20), we have

$$\phi = - {1 \over a}\ln \left[ {\cos (\sqrt {ab} \,{x_3})} \right] + {C_4},$$
(21)

where C4 is an integration constant and is immaterial. The substitution of Eq. (20) in Eq. (11)2 gives the carrier distribution which is always positive:

$$n = {{{\varepsilon _{33}}} \over q}{b \over {{{\cos }^2}(\sqrt {ab} \,{x_3})}}{.}$$
(22)

From Eqs. (11)1 and (20), we can obtain

$${S_3} = {u_{3,3}} = - {{{{\bar e}_{33}}} \over {{{\bar c}_{33}}}}{{b\tan (\sqrt {ab} \,{x_3})} \over {\sqrt {ab} }} + {C_5},$$
(23)

and

$${u_3} = {{{{\bar e}_{33}}} \over {{{\bar c}_{33}}}}{1 \over a}\ln \left[ {\cos (\sqrt {ab} \,{x_3})} \right] + {C_5}{x_3} + {C_6},$$
(24)

where C5 and C6 are integration constants. C6 represents a rigid-body displacement and can be set to zero. Then, from Eq. (7)1,

$${T_3} = {\bar c_{33}}{C_5}{.}$$
(25)

For a free rod without an axial force, C5=0. From Eq. (7)2,

$${D_3} = - {\varepsilon _{33}}{{b\tan (\sqrt {ab} \,{x_3})} \over {\sqrt {ab} }}{.}$$
(26)

In summary, the electromechanical fields and the carrier distribution are:

$$\begin{array}{*{20}c} {{u_3} = {{{{\bar e}_{33}}} \over {{{\bar c}_{33}}}}{1 \over a}\ln \left[ {\cos (\sqrt {ab} \,{x_3})} \right],}\\ {{S_3} = - {{{{\bar e}_{33}}} \over {{{\bar c}_{33}}}}{{b\tan (\sqrt {ab} \,{x_3})} \over {\sqrt {ab} }},\quad }\\ {{T_3} = 0,\;\quad \quad \quad \quad \quad \quad \quad \quad }\\ {\phi = - {1 \over a}\ln \left[ {\cos (\sqrt {ab} \,{x_3})} \right],\quad }\\ {{E_3} = - {{b\tan (\sqrt {ab} \,{x_3})} \over {\sqrt {ab} }},\quad \quad \quad }\\ {{D_3} = - {\varepsilon _{33}}{{b\tan (\sqrt {ab} \,{x_3})} \over {\sqrt {ab} }},\quad \quad }\\ {n = {{{\varepsilon _{33}}} \over q}{b \over {{{\cos }^2}(\sqrt {ab} \,{x_3})}},\quad \quad \quad }\\ {{J_3} = 0{.}\quad \quad \quad \quad \quad \quad \quad \quad \quad } \end{array}$$
(27)

The traction-free mechanical end conditions are satisfied by T3=0. The non-conducting end conditions are satisfied by J3=0. b is related to C2 through Eq. (18)2 and is the only integration constant left. To determine b, we impose the boundary condition that the carrier density at the ends of the rod where x3=L is known to be n0 and

$${{{\varepsilon _{33}}} \over q}{b \over {{{\cos }^2}(\sqrt {ab} \,L)}} = {n_0}{.}$$
(28)

n0 makes the boundary value problem non-homogeneous and may be viewed as a driving term or a load.

In the case when holes dominate, Eqs. (4)2 and (5)3 should be replaced by (Pierret, 1988)

$${D_{3,3}} = qp,$$
(29)

and

$${J_3} = qp{\mu _{33}}{E_3} - q{D_{33}}{p_{,3}}{.}$$
(30)

In this case, formally Eq. (17) is still valid but a becomes negative. When b<0, if E,3>0 and dominates aE2 which is always negative, we still have ab>0 and the solution to Eq. (17) is still given by Eq. (20). Then the carrier distribution and the electromechanical fields are still given by Eq. (27). The only difference is that for holes a and b are both negative.

5 Numerical results and discussion

Denoting we \(X = \sqrt {ab} \;L\), write Eq. (28) as the following equation for X:

$$\cos X = \pm gX,$$
(31)

where

$$g = {1 \over L}\sqrt {{{{\varepsilon _{33}}} \over {qa{n_0}}}} {.}$$
(32)

As a numerical example, consider a rod with L=10 mm. At room temperature kT/q=0.026 V (Pierret, 1988) which determines a through Eq. (18). The material constants for ZnO can be found in Auld (1973). For different values of n0 and hence g, we plot Y=cosX and Y=±gX in Fig. 2. The intersections of the two families of curves determine the roots of Eq. (31). When n0=1012 m−3, there is only one root. When n0=1014 m−3, there are several roots. When n0 increases further, the slopes of the straight lines described by Y=±gX become smaller which implies more roots of Eq. (31). This is not surprising in view of the nonlinear nature of the problem which is associated with strong fields and high carrier density.

Fig. 2
figure 2

Roots of Eq. (31): n0=1012 m−3, Y=±0.603X; n0= 1014 m−3, Y=±0.0603X

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Fig. 3 shows the carrier distribution n(x3) along the rod for different values of n0. For these values of n0, there are multiple roots from Eq. (31). The first root is chosen which presents the physical picture expected. n(x3) is symmetric about the rod center as expected. n(x3) is nearly constant in the central part of the rod, and increases rapidly near the ends. When n0 increases, the carrier density near the rod ends is more sensitive to n0 than the carrier density in the central part. Later, more insight into the carrier distribution will be given when examining the electric field in the rod.

Fig. 3
figure 3

Carrier distribution along the rod

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Fig. 4 shows the axial distributions of the electric field E3, the electric potential φ, and the electric displacement D3. The electric field in Fig. 4a is physically symmetric about x3=0 as it should be, but mathematically it is described by an odd function of x3. As an odd function it vanishes at the origin. It increases its magnitude rapidly near the ends of the rod, which is consistent with the fact shown in Fig. 3 that near the ends the carrier distribution also has a large gradient. The electric field and the gradient of n(x3) have to balance each other to make J3=0. Hence, E3 and n3 have to be large or small together in magnitude. A larger n0 is associated with a higher electric field as expected. The electric potential in Fig. 4b is the spatial integration of E3 along the rod and as a consequence it is an even function of x3. It may have an arbitrary constant which is fixed by choosing φ(0)=0. A larger n0 is also associated with a higher electric potential as expected. The behavior of D3 in Fig. 4c is similar to that of E3 in Fig. 4a.

Fig. 4
figure 4

Axial distributions of the electric fieldE 3(a), elec-tric potentialφ(b), and electric displacementD 3(c)

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Fig. 5 shows the mechanical fields which are present because the material is piezoelectric. For free extension the axial stress is zero. Hence, only the axial strain and the axial displacement are plotted in Fig. 5a and Fig. 5b, respectively. The strain is an odd function of x3, indicating that half of the rod is in axial extension and the other half in contraction. This is related to the fact that the electric field in Fig. 4a is also an odd function of x3. For the displacement in Fig. 5b, the rigid-body axial displacement is removed by choosing the axial displacement at the rod center to be zero. From Fig. 5 it can be seen that the mechanical fields are also relatively large near the ends of the rod and are sensitive to n0 near the ends.

Fig. 5
figure 5

Axial strain (a) and axial displacement (b)

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6 Conclusions

Because of the presence of carriers and piezoelectric coupling, a thin piezoelectric semiconductor rod undergoes axial extension/contraction. The equations for determining the carrier density and electromechanical fields are nonlinear because of the drift current term. The carrier distribution and electromechanical fields are either symmetric or antisymmetric about the center of the rod. They are relatively strong near the ends of the rod than at its central part. They are also sensitive to the number of carriers. The effects of electrons and holes are qualitatively the same.