u 3 ffi u 3 z, t
ð Þ, φ ffi φ z, t
ð Þ:
ð7:74Þ
In the semiconductor layer only, we also assume that
Δp ffi Δp z, t
ð Þ, Δn ffi Δn z, t
ð Þ:
ð7:75Þ
Then the axial strain and electric field can be approximated by
S 3 ¼ S 33 ¼ u 3,3 , E 3 ¼ Àφ ,3 ,
ð7:76Þ
which are common to all layers. To simplify the notation of the axial fields, we write
u ¼ u 3 , S ¼ S 3 , T ¼ T 33 ¼ T 3 ,
E ¼ E 3 , P ¼ P 3 , D ¼ D 3 ,
J
p
¼ J
p
3 , J
n
¼ J
n
3 :
ð7:77Þ
We also denote the relevant material constants by
s ¼ s
E
3333 ¼ s
E
33 , d ¼ d 333 ¼ d 33 , ε ¼ ε
T
33 ,
μ
p
¼ μ
p
33 , μ
n
¼ μ
n
33 , D
p
¼ D
p
33 , D
n
¼ D
n
33 ,
α ¼ α 33 :
ð7:78Þ
The fields and material constants for the piezoelectric layers will carry a superscript
1 in parentheses and those of the semiconductor layer a superscript 2 in parentheses.
(2) Si
(1) PZT
z
2L
c
c
(1) PZT
x
y
b
2c
h
h
Fig. 7.13 Side view and
cross section of a composite
rod of piezoelectric
dielectrics and
nonpiezoelectric
semiconductors
7.4 Extension of Composite Rods
197