S 33 ¼ À
e 33 G 0 θ
c 33
cosh kx 3
cosh kL
þ
λ 33 θ
c 33
,
E 3 ¼ ÀG 0 θ
cosh kx 3
cosh kL
,
ð7:24Þ
D 3 ¼ G 1 1 À
cosh kz
cosh kL
θ,
P 3 ¼ D 3 À ε 0 E 3 ¼ G 1 θ
ε 0
b ε 33
À 1
cosh kx 3
cosh kL
þ 1
!
,
ð7:25Þ
p ¼ p 0 1 À
q
k B Θ 0
G 0 θ
k
sinh kx 3
cosh kL
,
n ¼ n 0 1 þ
q
k B Θ 0
G 0 θ
k
sinh kx 3
cosh kL
,
ð7:26Þ
ρ
P
¼ ÀP k,k ¼ ÀP 3,3 ¼ G 1 θ 1 À
ε 0
b ε 33
k
sinh kx 3
cosh kL
:
ð7:27Þ
where
G 1 ¼
e 33 λ 33
c 33
þ p 3 :
ð7:28Þ
As a numerical example, consider a ZnO piezoelectric semiconductor rod. It is
doped into an n-type semiconductor with p ffi 0. The length of the rod is
2L ¼ 1.2 μm. The cross-sectional area is 0.04 μm
2 . Figure 7.2 shows the basic effect
of interest, i.e., the temperature-induced carrier concentration perturbation. Under a
temperature change, the mobile charges in the rod redistribute themselves under the
electric field produced by the pyroelectric effect and the combination of
θ=0.05K
θ=0.10K
θ=0.20K
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-1.0
-0.5
0.0
0.5
1.0
Δn
(m
-3
)
×10
20
x 3 (µm)
θ
Fig. 7.2 Distribution of
electron concentration
perturbation along the rod
for different temperature
change. n 0 ¼ 10
21
/m
3
7.1 Extension of Homogeneous Rods
181