Carrier distribution and electromechanical fields in a free piezoelectric semiconductor rod
J Zhejiang Univ-Sci A (Appl Phys & Eng)
free piezoelectric semiconductor rod*
Chun-li ZHANG 0 2 3 5
Xiao-yuan WANG 2 5
Wei-qiu CHEN 0 2 3 5
Jia-shi YANG 1 2 5
0 Soft Matter Research Center, Zhejiang University , Hangzhou 310027, China) (
1 Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln , Lincoln, NE 68588-0526 , USA) (
2 Department of Engineering Mechanics, Zhejiang University , Hangzhou 310027, China) (
3 Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province , Hangzhou 310027 , China)
4 Project supported by the National Natural Science Foundation of China (Nos. 11202182, 11272281, and 11321202) ORCID: Chun-li ZHANG
5 References Auld, B.A., 1973. Acoustic Fields and Waves in Solids. John Wiley and Sons , New York, p.357-382. Büyükköse, S., Hernández-Mínguez, A., Vratzov, B., et al. , USA
We made a theoretical study of the carrier distribution and electromechanical fields in a free piezoelectric semiconductor rod of crystals of class 6 mm. Simple analytical expressions for the carrier distribution, electric potential, electric field, electric displacement, mechanical displacement, stress, and strain were obtained from a 1D nonlinear model reduced from the 3D equations for piezoelectric semiconductors. The distribution and fields were found to be either symmetric or antisymmetric about the center of the rod. They are qualitatively the same for electrons and holes. Numerical calculations show that the carrier distribution and the fields are relatively strong near the ends of the rod than in its central part. They are sensitive to the value of the carrier density near the ends of the rod.
Piezoelectricity; Semiconductor; Rod; Carrier distribution http; //dx; doi; org/10; 1631/jzus; A1500213 CLC number; O33; TB1
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
(Lee et al., 2004; Yong et al., 2010; Wang et
, 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
Yang and Zhou, 2004; Ghosh, 2006; Willatzen and
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
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,
. 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
Zhou, 2005; Yang et al., 2005)
. Researchers have
also developed more general and fully nonlinear
(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
, 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
2 Equations for piezoelectric semiconductors
For a piezoelectric semiconductor, the 3D
phenomenological theory consists of the linear
where Sij is the component of strain, Ei the
component of electric field, siEjkl the elastic compliance
constant, dkij the piezoelectric constants, iTj the
dielectric constant, μij the carrier mobility, and Dij the
carrier diffusion constants. The superscripts “E” and
tum equation of motion, the charge equation of
electrostatics, and the conservation of charge for
electrons and holes (continuity equations)
White, 1962; Pierret, 1988)
Tji, j ui ,
Jip,i q p,
Di,i q( p n ND N A ) ,
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,
ND and N A the densities of impurities of donors
and accepters, and Jip and Jin 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 ND 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:
Sij siEjklTkl dkij Ek ,
Di diklTkl iTk Ek ,
Jin qnij E j qDij n, j ,
“T” in siEjkl and iTj as well as the superscript “n” in
Jin 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
E j , j .
(ui, j u j,i ),
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.
The deformation of the thin rod is mainly an
axial extension or contraction which can be described
by a 1D model
. 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
D3,3 q(n ND ) ,
With the use of Eq. (6), Eqs. (7) and (5)3
which are the 1D constitutive relations ready to be
used in Eq. (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 charge N D in Eq. (4) and focus on the
mobile electrons described by n. In the compact matrix
, the relevant constitutive
relations from Eq. (2) for the axial fields are
The relevant strain-displacement relation and
the electric field-potential relation are the following
ones from Eq. (3):
Eq. (5)1,2 can be solved for T3 and rewritten as
where the effective 1D elastic, piezoelectric, and
dielectric constants are:
4 Carrier distribution and fields
The substitution of Eq. (9)1,2 in Eq. (4)1,2 yields
where C2 is another integration constant. Let
,3 E3 E,
then Eq. (14) takes the following form
2 33E2 D33E,3 C1x3 C2 D33.
For the static and free rod we are considering,
J3=0 and hence C1=0. Eq. (16) reduces to
E,3 aE2 b,
where we have denoted
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
b tan ab (x3 C3 )
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
ln cos( ab x3 ) C4 ,
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:
From Eqs. (11)1 and (20), we can obtain
S3 u3,3 e33 b tan( ab x3 ) C5 ,
u3 e33 1 ln cos( ab x3 ) C5 x3 C6 ,
where C5 and C6 are integration constants. C6
represents a rigid-body displacement and can be set to
zero. Then, from Eq. (7)1,
For a free rod without an axial force, C5=0.
From Eq. (7)2,
b tan( ab x3 ) .
In summary, the electromechanical fields and
the carrier distribution are:
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
5 Numerical results and discussion
Denoting X ab L, we write Eq. (28) as the
following equation for X:
cos X gX ,
u3 ce3333 a1 ln cos( ab x3 ) ,
S3 e33 b tan( ab x3 ) ,
ln cos( ab x3 ) ,
b tan( ab x3 ) ,
b tan( ab x3 ) ,
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
q cos2 ( ab L)
n0 makes the boundary value problem
nonhomogeneous 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
J3 qp33E3 qD33 p,3.
As a numerical example, consider a rod with
L=10 mm. At room temperature kT/q=0.026 V
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. 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. 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. 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.
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.
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