For the piezoelectric layers, in terms of the elastic compliance, the constitutive
relation for the axial strain and electric displacement are
S ¼ s
1
ð Þ T þ d
1
ð Þ E þ α
1
ð Þ
θ,
D ¼ d
1
ð Þ T þ ε
1
ð Þ E þ p
1
ð Þ
3 θ:
ð7:79Þ
From Eq. (7.79) we obtain
T ¼ c
1
ð Þ S À e
1
ð Þ E À λ
1
ð Þ
θ,
D ¼ e
1
ð Þ S þ ε
1
ð Þ E þ p
1
ð Þ
θ,
ð7:80Þ
where
c
1
ð Þ
¼
1
s 1
ð Þ
, e
1
ð Þ
¼
d
1
ð Þ
s 1
ð Þ
, λ
1
ð Þ
¼
α
1
ð Þ
s 1
ð Þ
,
ε
1
ð Þ
¼ ε
1
ð Þ
À
d
1
ð Þ
À Á 2
s 1
ð Þ
, p
1
ð Þ
¼ p
1
ð Þ
3 À
d
1
ð Þ
α
1
ð Þ
s 1
ð Þ
:
ð7:81Þ
Similarly, for the semiconductor layer, it is assumed to be nonpiezoelectric and
hence does not have pyroelectric coupling. The relevant constitutive relations are
S ¼ s
2
ð Þ T þ α
2
ð Þ
θ,
D ¼ ε
2
ð Þ E:
ð7:82Þ
Equation (7.82) can be rewritten into
T ¼ c
2
ð Þ S À λ
2
ð Þ
θ,
D ¼ ε
2
ð Þ E,
ð7:83Þ
where
c
2
ð Þ
¼
1
s 2
ð Þ
, λ
2
ð Þ
¼
α
2
ð Þ
s 2
ð Þ
, ε
2
ð Þ
¼ ε
2
ð Þ
:
ð7:84Þ
The currents exist in the semiconductor layer only and therefore the superscript 2 in
parentheses is neglected:
J
p
ffi qp 0 μ
p E À qD
p d Δp
ð Þ
dz
,
J
n
ffi qn 0 μ
n E þ qD
n d Δn
ð Þ
dz
:
ð7:85Þ
198
7 Thermal Effects