where
k
2
¼ p 0 þ n 0
ð
Þ
q
k B Θ 0
q
b ε 33
,
b ε 33 ¼ ε 33 À
e 33 e 33
c 33
¼ ε 33 :
ð7:18Þ
In Eqs. (7.16) and (7.17), A 1 through A 8 are undetermined constants. It can be shown
that only five of the eight boundary conditions in Eq. (7.15) are independent. We
need to impose the global charge neutrality conditions:
Z L
ÀL
Δpdx 3 ¼ 0,
Z L
ÀL
Δndx 3 ¼ 0,
ð7:19Þ
among which only one is independent. To determine the displacement and potential
fields uniquely, we set
u 3 0
ð Þ ¼ 0, φ 0
ð Þ ¼ 0:
ð7:20Þ
Thus there are eight independent conditions for determining the eight unknown
constants. The final solution is
u 3 ¼ À
e 33 G 0 θ
c 33 k
sinh kx 3
cosh kL
þ
λ 33 θ
c 33
x 3 ,
φ ¼
G 0 θ
k
sinh kx 3
cosh kL
,
ð7:21Þ
Δp ¼ Àp 0
q
k B Θ 0
G 0 θ
k
sinh kx 3
cosh kL
,
Δn ¼ n 0
q
k B Θ 0
G 0 θ
k
sinh kx 3
cosh kL
,
ð7:22Þ
where
G 0 ¼
e 33 λ 33 þ p 3 c 33
ε 33 c 33 þ e 33 e 33
,
ð7:23Þ
which shows that the pyroelectric coupling contributes to the carrier distribution
directly and that the thermoelastic coupling does so through piezoelectric coupling.
Then the following fields can be calculated:
180
7 Thermal Effects