where
d 1 ¼ b 1 À a 1 , d 2 ¼ b 2 À a 2 ,
d 3 ¼
b 1 À a 1
2
þ
b 2 À a 2
2
, d 4 ¼
b 2 À b 1
2w
À
a 2 À a 1
2w
:
ð3:28Þ
We consider a free rod with the following boundary and continuity conditions:
T 33 ÆL
ð Þ ¼ 0, D 3 ÆL
ð Þ ¼ 0,
J
p
3 ÆL
ð Þ ¼ 0, J
n
3 ÆL
ð Þ ¼ 0,
ð3:29Þ
u 3 Æw
À
ð
Þ¼u 3 Æw
þ
ð
Þ, φ Æw
À
ð
Þ¼φ Æw
þ
ð
Þ,
p Æw
À
ð
Þ¼p Æw
þ
ð
Þ, n Æw
À
ð
Þ¼n Æw
þ
ð
Þ,
ð3:30Þ
T 33 Æw
À
ð
Þ¼T 33 Æw
þ
ð
Þ, D 3 Æw
À
ð
Þ¼D 3 Æw
þ
ð
Þ,
J
p
3 Æw
À
ð
Þ¼J
p
3 Æw
þ
ð
Þ, J
n
3 Æw
À
ð
Þ¼J
n
3 Æw
þ
ð
Þ:
ð3:31Þ
To make the displacement and potential unique, we impose
u 3 0
ð Þ ¼ 0,
φ 0
ð Þ ¼ 0:
ð3:32Þ
In addition, we also have the following global charge neutrality condition:
Z L
ÀL
p À N
À
A
À
Á
dx 3 ¼ 0:
ð3:33Þ
Since Eq. (3.5) 2 and the relevant boundary as well as continuity conditions imply
that
Z L
ÀL
p À n þ N
þ
D À N
À
A
À
Á
dx 3 ¼ 0,
ð3:34Þ
Eqs. (3.33) and (3.34) together further imply that
Z L
ÀL
Àn þ N
þ
D
À
Á
dx 3 ¼ 0:
ð3:35Þ
Therefore Eq. (3.33) is the only independent charge neutrality condition.
Consider the special and relatively simple case when L ¼ 1. In this case, we
have, for x 3 < À w,
3.3 Linearly Graded PN Junction
37