n ¼ n 0 þ
n 0 μ
n
D
n
b eF
b εb c þ b e
2
k cosh kL
sinh kz,
ð6:64Þ
P ¼
b eF
b c cosh kL
À
1
A
1
ð Þ
þ A
2
ð Þ
þ
ε 0
εA
2
ð Þ
cosh kz þ
b eF
b c A
1
ð Þ
þ A
2
ð Þ
À
Á,
ð6:65Þ
ρ
P
¼
b eFk
b c cosh kL
1
A
1
ð Þ
þ A
2
ð Þ
þ
ε 0
εA
2
ð Þ
sinh kz,
ð6:66Þ
u ¼ À
b e
2 F
b c b εb c þ b e
2
k cosh kL
sinh kz þ
F
b c
z,
ð6:67Þ
S ¼ À
b e
2 F
b c b εb c þ b e
2
cosh kL
cosh kz þ
F
b c
,
ð6:68Þ
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
b e
2
b c
:
ð6:69Þ
We are mainly interested in Δp and Δn produced by F. Consider Δn as an example.
From Eq. (6.64) we obtain
Δn
n 0
¼ γ
F
A
1
ð Þ
þ A
2
ð Þ
sinh kz
cosh kL
,
ð6:70Þ
where
γ ¼
q
k B T
b e
b εe ck
, e c ¼
b c þ b e
2 =b ε
A
1
ð Þ
þ A
2
ð Þ
:
ð6:71Þ
F/(A
(1) + A
(2) ) is the average extensional stress. e c is an averaged, piezoelectrically
stiffened elastic constant. At the end of a long rod where kL is large, sinhkL ffi cosh kL.
Therefore γ describes the change of carrier concentration near the end of the rod per
unit stress normalized by n 0 and can be used as a measure of the strength of the effect
of F on Δn.
As a numerical example, consider a rod of PZT-5A and silicon doped into an ntype semiconductor with n 0 ¼ 10
22 /m
3 . The geometric parameters are that L ¼ 6 μm,
h ¼ c ¼ 0.05 μm, and b ¼ 0.2 μm. Figure 6.4 shows the effect of interest, i.e.,
mechanically induced redistribution of mobile charges in a composite rod of
152
6 Composite Structures
n 0 μ
n
D
n
b eF
b εb c þ b e
2
k cosh kL
sinh kz,
ð6:64Þ
P ¼
b eF
b c cosh kL
À
1
A
1
ð Þ
þ A
2
ð Þ
þ
ε 0
εA
2
ð Þ
cosh kz þ
b eF
b c A
1
ð Þ
þ A
2
ð Þ
À
Á,
ð6:65Þ
ρ
P
¼
b eFk
b c cosh kL
1
A
1
ð Þ
þ A
2
ð Þ
þ
ε 0
εA
2
ð Þ
sinh kz,
ð6:66Þ
u ¼ À
b e
2 F
b c b εb c þ b e
2
k cosh kL
sinh kz þ
F
b c
z,
ð6:67Þ
S ¼ À
b e
2 F
b c b εb c þ b e
2
cosh kL
cosh kz þ
F
b c
,
ð6:68Þ
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
b e
2
b c
:
ð6:69Þ
We are mainly interested in Δp and Δn produced by F. Consider Δn as an example.
From Eq. (6.64) we obtain
Δn
n 0
¼ γ
F
A
1
ð Þ
þ A
2
ð Þ
sinh kz
cosh kL
,
ð6:70Þ
where
γ ¼
q
k B T
b e
b εe ck
, e c ¼
b c þ b e
2 =b ε
A
1
ð Þ
þ A
2
ð Þ
:
ð6:71Þ
F/(A
(1) + A
(2) ) is the average extensional stress. e c is an averaged, piezoelectrically
stiffened elastic constant. At the end of a long rod where kL is large, sinhkL ffi cosh kL.
Therefore γ describes the change of carrier concentration near the end of the rod per
unit stress normalized by n 0 and can be used as a measure of the strength of the effect
of F on Δn.
As a numerical example, consider a rod of PZT-5A and silicon doped into an ntype semiconductor with n 0 ¼ 10
22 /m
3 . The geometric parameters are that L ¼ 6 μm,
h ¼ c ¼ 0.05 μm, and b ¼ 0.2 μm. Figure 6.4 shows the effect of interest, i.e.,
mechanically induced redistribution of mobile charges in a composite rod of
152
6 Composite Structures