φ
p
¼
q
ε 33 k
2
d 1 ,
φ ¼ C 1 exp k x 3 þ w
ð
Þþ
q
k
2
ε 33
p
0
À n
0
À
Á þ φ
p x 3
ð Þ,
p ffi p
0 1 À
q
k B T
φ
, n ffi n
0 1 þ
q
k B T
φ
,
u 3 ¼ À
e 33
c 33
φ þ C 2 ,
D 3 ¼ Àε 33 φ ,3 , T 3 ¼ 0, J
p
3 ¼ 0, J
n
3 ¼ 0,
ð3:36Þ
and, similarly, for x 3 >w,
φ
p
¼
q
ε 33 k
2
d 2 ,
φ ¼ C 3 exp k w À x 3
ð
Þþ
q
k
2
ε 33
p
0
À n
0
À
Á þ φ
p x 3
ð Þ,
p ffi p
0 1 À
q
k B T
φ
, n ffi n
0 1 þ
q
k B T
φ
,
u 3 ¼ À
e 33
c 33
φ þ C 4 ,
D 3 ¼ Àε 33 φ ,3 , T 3 ¼ 0, J
p
3 ¼ 0, J
n
3 ¼ 0:
ð3:37Þ
Inside the transition zone where jx 3 j < w, we have
φ
p
¼
q
ε 33 k
2
d 3 þ d 4 x 3
ð
Þ ,
φ ¼ C 5 sinh k x 3 þ w
ð
ÞþC 6 sinh k x 3 À w
ð
Þ
þ
q
k
2
ε 33
p
0
À n
0
À
Á þ φ
p x 3
ð Þ,
p ffi p
0 1 À
q
k B T
φ
, n ffi n
0 1 þ
q
k B T
φ
,
u 3 ¼ À
e 33
c 33
φ þ C 7 ,
D 3 ¼ Àε 33 φ ,3 , T 3 ¼ 0, J
p
3 ¼ 0, J
n
3 ¼ 0:
ð3:38Þ
C 1 –C 7 , p
0 , and n
0 are determined by Eqs. (3.29), (3.30), (3.31), (3.32), and (3.33)
using a computer.
As an example, consider the following case with some symmetry and/or
antisymmetry:
a 1 ¼ b 2 ¼ 1:0 Â 10
21 m
À3 ,
a 2 ¼ b 1 ¼ 0:8a 1 ,
w ¼ 0:1 μm:
ð3:39Þ
38
3 Extension of Rods