qΔp ¼ À
qp 0 μ
p
33
D
p
33
φ þ C 1 ,
ÀqΔn ¼ À
qn 0 μ
n
33
D
n
33
φ þ C 2 ,
ð7:130Þ
where C 1 and C 2 are undetermined constants and
A
1
ð Þ
¼ h 1 b, A
2
ð Þ
¼ h 2 b,
α ¼
6c
2
ð Þ e
1
ð Þ h 1 h 2 h 1 þ h 2
ð
Þ
c 1
ð Þ c 1
ð Þ h
4
1 þ c 2
ð Þ c 2
ð Þ h
4
2 þ 2c 1
ð Þ c 2
ð Þ h 1 h 2 2h
2
1 þ 3h 1 h 2 þ 2h
2
2
À
Á ,
β ¼ À
e
1
ð Þ h 1 c
1
ð Þ h
3
1 þ c
2
ð Þ h
2
2 3h 1 þ 4h 2
ð
Þ
À
Á
c 1
ð Þ c 1
ð Þ h
4
1 þ c 2
ð Þ c 2
ð Þ h
4
2 þ 2c 1
ð Þ c 2
ð Þ h 1 h 2 2h
2
1 þ 3h 1 h 2 þ 2h
2
2
À
Á ,
ð7:131Þ
e ε ¼ ε
1
ð Þ A
1
ð Þ
þ ε
2
ð Þ A
2
ð Þ
þ
e
1
ð Þ A
1
ð Þ h 1
2
α À e
1
ð Þ A
1
ð Þ
β,
k
2
¼
q
e ε
p 0 μ
p
33
D
p
33
þ
n 0 μ
n
33
D
n
33
A
2
ð Þ
:
ð7:132Þ
The general solution for φ, v, and w are found to be
φ ¼ kC 3 sinh kx 3 þ kC 4 sinh k x 3 À L
ð
Þþ
1
k
2
e ε
C 1 þ C 2
ð
Þ A
2
ð Þ ,
v ¼ α C 3 cosh kx 3 þ C 4 cosh k x 3 À L
ð
Þ
ð
Þ þ C 5 x
2
3 þ C 6 x 3 þ C 7 ,
w ¼ β kC 3 sinh kx 3 þ kC 4 sinh k x 3 À L
ð
Þ
ð
Þ þ C 8 x 3 þ C 9 ,
ð7:133Þ
where C 3 -C 9 are undetermined constants. Then Δp and Δn are determined from
Eq. (7.130). The undetermined constants are governed by Eqs. (7.125), (7.126),
(7.127), and (7.128).
As an example, consider PZT-5A for the piezoelectric layer and Si for the
semiconductor layer with L ¼ 0.6 μm, h 1 ¼ h 2 ¼ 25 nm, and b ¼ 50 μm.
n 0 ¼ p 0 ¼ 10
23 /m
3 . Θ 0 ¼ 298 K. Some of these parameters may be varied later.
We are particularly interested in the temperature-induced redistributions of holes and
electrons. They are described by the Δp and Δn in Fig. 7.20.
Figure 7.21 shows Δp and Δn for different n 0 when θ ¼ 0.5 K while all other
parameters are kept the same as those for Fig. 7.20. The fields are sensitive to p 0 and
n 0 .
Figure 7.22 shows the change of number of electrons in a differential element
with length dz at the left end of the beam, i.e., Δn(0)h 2 bdz which may be used as a
measure of the strength of the effect under consideration. Δn(0)h 2 bdz is plotted
versus h 1 /h 2 when h 1 + h 2 is fixed. The curves all have a maximum when h 1 and h 2
are not very different, which is reasonable.
208
7 Thermal Effects
qp 0 μ
p
33
D
p
33
φ þ C 1 ,
ÀqΔn ¼ À
qn 0 μ
n
33
D
n
33
φ þ C 2 ,
ð7:130Þ
where C 1 and C 2 are undetermined constants and
A
1
ð Þ
¼ h 1 b, A
2
ð Þ
¼ h 2 b,
α ¼
6c
2
ð Þ e
1
ð Þ h 1 h 2 h 1 þ h 2
ð
Þ
c 1
ð Þ c 1
ð Þ h
4
1 þ c 2
ð Þ c 2
ð Þ h
4
2 þ 2c 1
ð Þ c 2
ð Þ h 1 h 2 2h
2
1 þ 3h 1 h 2 þ 2h
2
2
À
Á ,
β ¼ À
e
1
ð Þ h 1 c
1
ð Þ h
3
1 þ c
2
ð Þ h
2
2 3h 1 þ 4h 2
ð
Þ
À
Á
c 1
ð Þ c 1
ð Þ h
4
1 þ c 2
ð Þ c 2
ð Þ h
4
2 þ 2c 1
ð Þ c 2
ð Þ h 1 h 2 2h
2
1 þ 3h 1 h 2 þ 2h
2
2
À
Á ,
ð7:131Þ
e ε ¼ ε
1
ð Þ A
1
ð Þ
þ ε
2
ð Þ A
2
ð Þ
þ
e
1
ð Þ A
1
ð Þ h 1
2
α À e
1
ð Þ A
1
ð Þ
β,
k
2
¼
q
e ε
p 0 μ
p
33
D
p
33
þ
n 0 μ
n
33
D
n
33
A
2
ð Þ
:
ð7:132Þ
The general solution for φ, v, and w are found to be
φ ¼ kC 3 sinh kx 3 þ kC 4 sinh k x 3 À L
ð
Þþ
1
k
2
e ε
C 1 þ C 2
ð
Þ A
2
ð Þ ,
v ¼ α C 3 cosh kx 3 þ C 4 cosh k x 3 À L
ð
Þ
ð
Þ þ C 5 x
2
3 þ C 6 x 3 þ C 7 ,
w ¼ β kC 3 sinh kx 3 þ kC 4 sinh k x 3 À L
ð
Þ
ð
Þ þ C 8 x 3 þ C 9 ,
ð7:133Þ
where C 3 -C 9 are undetermined constants. Then Δp and Δn are determined from
Eq. (7.130). The undetermined constants are governed by Eqs. (7.125), (7.126),
(7.127), and (7.128).
As an example, consider PZT-5A for the piezoelectric layer and Si for the
semiconductor layer with L ¼ 0.6 μm, h 1 ¼ h 2 ¼ 25 nm, and b ¼ 50 μm.
n 0 ¼ p 0 ¼ 10
23 /m
3 . Θ 0 ¼ 298 K. Some of these parameters may be varied later.
We are particularly interested in the temperature-induced redistributions of holes and
electrons. They are described by the Δp and Δn in Fig. 7.20.
Figure 7.21 shows Δp and Δn for different n 0 when θ ¼ 0.5 K while all other
parameters are kept the same as those for Fig. 7.20. The fields are sensitive to p 0 and
n 0 .
Figure 7.22 shows the change of number of electrons in a differential element
with length dz at the left end of the beam, i.e., Δn(0)h 2 bdz which may be used as a
measure of the strength of the effect under consideration. Δn(0)h 2 bdz is plotted
versus h 1 /h 2 when h 1 + h 2 is fixed. The curves all have a maximum when h 1 and h 2
are not very different, which is reasonable.
208
7 Thermal Effects