3.4 Smoothly Graded PN Junction
Consider a rod of length 2L with a general doping profile that varies smoothly
[3]. The rod is in equilibrium without body force and currents, i.e., f 3 ¼ 0, J
p
3 ¼ 0,
and J
n
3 ¼ 0. In this case, from Eqs. (3.16), (3.14), and (3.20), the governing equations
are
φ ,33 À k
2
φ ¼ À
q
ε 33
p
0
À n
0
þ N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
Â
Ã
,
ð3:43Þ
p ffi p
0 1 À
q
k B T
φ
, n ffi n
0 1 þ
q
k B T
φ
,
ð3:44Þ
u 3 ¼
1
c 33
C 1 x 3 À e 33 φ
ð
ÞþC 4 ,
ð3:45Þ
where
k
2
¼
q
ε 33
p
0
þ n
0
À
Á q
k B T
:
ð3:46Þ
We consider a free rod with the following boundary 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:
ð3:47Þ
To make the displacement and the electric potential unique, we impose
u 3 ÀL
ð Þ ¼ 0, φ ÀL
ð Þ ¼ 0:
ð3:48Þ
Since we have set φ ¼ 0 at x 3 ¼ ÀL, we have
p ÀL
ð Þ ¼ p
0 , n ÀL
ð Þ ¼ n
0
:
ð3:49Þ
We are interested in a PN junction near x 3 ¼ 0. Since a PN junction produces a local
charge distribution and a local electric field, when L is sufficiently large, p
0 and n
0 as
defined by Eq. (3.49) can be determined from doping approximately, i.e.,
p
0
ffi N
À
A ÀL
ð Þ, n
0
ffi N
þ
D ÀL
ð Þ:
ð3:50Þ
The general solution to Eq. (3.43) can be written as
φ ¼C 2 sinh k x 3 þ L
ð
ÞþC 3 sinh k x 3 À L
ð
Þ
þ
q
k
2
ε 33
p
0
À n
0
À
Á þ φ
p x 3
ð Þ,
ð3:51Þ
40
3 Extension of Rods