u 3 0
ð Þ ¼ 0, φ 0
ð Þ ¼ 0:
ð3:74Þ
The solution can be obtained in a systematic and straightforward manner.
The electromechanical fields in the rod are found to be
φ ¼
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
E 3 ¼ À
e 33 T
ε T
33 c 33 cosh kL
sinh kx 3 ,
D 3 ¼ À
e 33 T
c 33 cosh kL
cosh kx 3 þ e 33
T
c 33
,
ð3:75Þ
Δn ¼ n 0
μ
n
33
D
n
33
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
Δp ¼ Àp 0
μ
p
33
D
p
33
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
ð3:76Þ
u 3 ¼ À
e
2
33 T
ε T
33 c
2
33 k cosh kL
sinh kx 3 þ
T
c 33
x 3 ,
S 3 ¼ À
e
2
33 T
ε T
33 c
2
33 cosh kL
cosh kx 3 þ
T
c 33
,
T 3 ¼ T,
ð3:77Þ
where
k
2
¼
q
ε T
33
μ
p
33
D
p
33
p 0 þ
μ
n
33
D
n
33
n 0
¼
q
2 p 0 þ n 0
ð
Þ
ε T
33 k B T
:
ð3:78Þ
The fields are either symmetric or antisymmetric about the center of the rod. The
development of the mechanical displacement, stress, strain, electric potential, electric field, and electric displacement is a consequence of elasticity and piezoelectricity. When the rod is a piezoelectric dielectric, these fields are simple linear functions
or constants. Now they are affected by the mobile charges and are described by
hyperbolic functions. It can be verified that when p 0 ! 0, n 0 ! 0, and hence k ! 0,
the above fields reduce to the fields for a piezoelectric dielectric rod under an axial
force.
For numerical results consider an n-type ( p ffi 0) ZnO rod with 2L ¼ 1.2 μm,
n 0 ¼ N
þ
D ¼ 10
21 m
À3 , and f ¼ 8.5 nN or lower which produces and axial stress
T ¼ 3.272 Â 10
5 N/m
2 or lower. For these data, Fig. 3.8 shows the most basic
behavior of the rod as a piezoelectric semiconductor, i.e., under the action of an axial
force, the electrons in the rod redistribute themselves and assume a certain distribution. Numerical calculation also shows that when n 0 increases, k increases, the
hyperbolic functions have larger values, and they change more rapidly near the ends.
3.6 Linear Extension by End Forces
47
ð Þ ¼ 0, φ 0
ð Þ ¼ 0:
ð3:74Þ
The solution can be obtained in a systematic and straightforward manner.
The electromechanical fields in the rod are found to be
φ ¼
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
E 3 ¼ À
e 33 T
ε T
33 c 33 cosh kL
sinh kx 3 ,
D 3 ¼ À
e 33 T
c 33 cosh kL
cosh kx 3 þ e 33
T
c 33
,
ð3:75Þ
Δn ¼ n 0
μ
n
33
D
n
33
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
Δp ¼ Àp 0
μ
p
33
D
p
33
e 33 T
ε T
33 c 33 k cosh kL
sinh kx 3 ,
ð3:76Þ
u 3 ¼ À
e
2
33 T
ε T
33 c
2
33 k cosh kL
sinh kx 3 þ
T
c 33
x 3 ,
S 3 ¼ À
e
2
33 T
ε T
33 c
2
33 cosh kL
cosh kx 3 þ
T
c 33
,
T 3 ¼ T,
ð3:77Þ
where
k
2
¼
q
ε T
33
μ
p
33
D
p
33
p 0 þ
μ
n
33
D
n
33
n 0
¼
q
2 p 0 þ n 0
ð
Þ
ε T
33 k B T
:
ð3:78Þ
The fields are either symmetric or antisymmetric about the center of the rod. The
development of the mechanical displacement, stress, strain, electric potential, electric field, and electric displacement is a consequence of elasticity and piezoelectricity. When the rod is a piezoelectric dielectric, these fields are simple linear functions
or constants. Now they are affected by the mobile charges and are described by
hyperbolic functions. It can be verified that when p 0 ! 0, n 0 ! 0, and hence k ! 0,
the above fields reduce to the fields for a piezoelectric dielectric rod under an axial
force.
For numerical results consider an n-type ( p ffi 0) ZnO rod with 2L ¼ 1.2 μm,
n 0 ¼ N
þ
D ¼ 10
21 m
À3 , and f ¼ 8.5 nN or lower which produces and axial stress
T ¼ 3.272 Â 10
5 N/m
2 or lower. For these data, Fig. 3.8 shows the most basic
behavior of the rod as a piezoelectric semiconductor, i.e., under the action of an axial
force, the electrons in the rod redistribute themselves and assume a certain distribution. Numerical calculation also shows that when n 0 increases, k increases, the
hyperbolic functions have larger values, and they change more rapidly near the ends.
3.6 Linear Extension by End Forces
47