S ¼ À
e e b λ þ b pb c
θ
b εb c þ e 2
ð
Þ b c
cosh kz
cosh kL
þ
b λθ
b c
,
ð7:98Þ
p ¼ p 0 À
p 0 μ
p
D
p
e b λ þ b pb c
θ
b εb c þ e 2
ð
Þk
sinh kz
cosh kL
,
ð7:99Þ
n ¼ n 0 þ
n 0 μ
n
D
n
e b λ þ b pb c
θ
b εb c þ e 2
ð
Þk
sinh kz
cosh kL
,
ð7:100Þ
D ¼
b λe
b c
þ b p
θ
A
1
ð Þ
þ A
2
ð Þ
1 À
cosh kz
cosh kL
,
ð7:101Þ
P ¼
b λe
b c
þ b p
θ
ε 0
e ε
1
A
2
ð Þ
À
1
A
1
ð Þ
þ A
2
ð Þ
cosh kz
cosh kL
þ
b λe
b c
þ b p
θ
A
1
ð Þ
þ A
2
ð Þ
,
ð7:102Þ
ρ
P
¼
b λe
b c
þ b p
1
A
1
ð Þ
þ A
2
ð Þ
À
ε 0
e ε
1
A
2
ð Þ
kθ
sinh κz
cosh κL
,
ð7:103Þ
where
k
2
¼
p 0 μ
p
D
p þ
n 0 μ
n
D
n
q
e ε
,
e εA
2
ð Þ
¼ b ε þ
e
2
b c
:
ð7:104Þ
As a numerical example, consider a composite rod of PZT-5A and silicon with
L ¼ 0.6 μm, h ¼ c ¼ 0.05 μm, and b ¼ 0.2 μm. p 0 ffi 0. n 0 ¼ 10
23 /m
3 . Figure 7.15
shows the redistribution of electrons. It can be seen that the rod considered is
sensitive to temperature and a larger temperature change induces larger changes of
the carrier distribution as expected. Δn is an order of magnitude smaller than n 0 so
that the linearized theory employed is valid.
Figure 7.16 shows the carrier concentration perturbation for different n 0 when
θ¼0.1 K while all other parameters are kept the same as those for Fig. 7.15. The
fields are sensitive to n 0 .
From Eq. (7.100), we obtain
Δn
n 0
¼ γ
θ
Θ 0
sinh kz
cosh kL
,
ð7:105Þ
where
7.4 Extension of Composite Rods
201
e e b λ þ b pb c
θ
b εb c þ e 2
ð
Þ b c
cosh kz
cosh kL
þ
b λθ
b c
,
ð7:98Þ
p ¼ p 0 À
p 0 μ
p
D
p
e b λ þ b pb c
θ
b εb c þ e 2
ð
Þk
sinh kz
cosh kL
,
ð7:99Þ
n ¼ n 0 þ
n 0 μ
n
D
n
e b λ þ b pb c
θ
b εb c þ e 2
ð
Þk
sinh kz
cosh kL
,
ð7:100Þ
D ¼
b λe
b c
þ b p
θ
A
1
ð Þ
þ A
2
ð Þ
1 À
cosh kz
cosh kL
,
ð7:101Þ
P ¼
b λe
b c
þ b p
θ
ε 0
e ε
1
A
2
ð Þ
À
1
A
1
ð Þ
þ A
2
ð Þ
cosh kz
cosh kL
þ
b λe
b c
þ b p
θ
A
1
ð Þ
þ A
2
ð Þ
,
ð7:102Þ
ρ
P
¼
b λe
b c
þ b p
1
A
1
ð Þ
þ A
2
ð Þ
À
ε 0
e ε
1
A
2
ð Þ
kθ
sinh κz
cosh κL
,
ð7:103Þ
where
k
2
¼
p 0 μ
p
D
p þ
n 0 μ
n
D
n
q
e ε
,
e εA
2
ð Þ
¼ b ε þ
e
2
b c
:
ð7:104Þ
As a numerical example, consider a composite rod of PZT-5A and silicon with
L ¼ 0.6 μm, h ¼ c ¼ 0.05 μm, and b ¼ 0.2 μm. p 0 ffi 0. n 0 ¼ 10
23 /m
3 . Figure 7.15
shows the redistribution of electrons. It can be seen that the rod considered is
sensitive to temperature and a larger temperature change induces larger changes of
the carrier distribution as expected. Δn is an order of magnitude smaller than n 0 so
that the linearized theory employed is valid.
Figure 7.16 shows the carrier concentration perturbation for different n 0 when
θ¼0.1 K while all other parameters are kept the same as those for Fig. 7.15. The
fields are sensitive to n 0 .
From Eq. (7.100), we obtain
Δn
n 0
¼ γ
θ
Θ 0
sinh kz
cosh kL
,
ð7:105Þ
where
7.4 Extension of Composite Rods
201