where φ
p satisfies
φ
p
,33 À k
2
φ
p
¼ À
q
ε 33
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
Â
Ã
:
ð3:52Þ
Consider the following doping described by a power series in general:
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ ¼
X 1
n¼0
a n x
n
3 :
ð3:53Þ
Let
φ
p
¼
X 1
n¼0
c n x
n
3 :
ð3:54Þ
Substituting Eqs. (3.53) and (3.54) into Eq. (3.52), we obtain
n þ 2
ð
Þ n þ 1
ð
Þc nþ2 À k
2 c n ¼ À
q
ε 33
a n ,
n ¼ 0, 1, 2, Á Á Á:
ð3:55Þ
The solution to Eq. (3.55) is not unique. We choose a specific solution of Eq. (3.55)
with c 0 ¼ 0 and c 1 ¼ 0. Then the rest of c n can be calculated from Eq. (3.55) as a
recurrence relation.
As an example of a PN Junction, we consider the specific doping described by the
error function below:
N
À
A x 3
ð Þ ¼ α 1 þ β 1 erf λx 3
ð Þ,
N
þ
D x 3
ð Þ ¼ α 2 þ β 2 erf λx 3
ð Þ:
ð3:56Þ
Then
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ ¼ a þ berf λx 3
ð Þ,
a ¼ α 2 À α 1 , b ¼ β 2 À β 1 :
ð3:57Þ
Since
erf λx 3
ð Þ ¼
2
ffiffiffi
π
p
X 1
n¼0
À1
ð Þ
n
λ
2nþ1
n! 2n þ 1
ð
Þ
x
2nþ1
3
,
ð3:58Þ
we have, from Eqs. (3.53) and (3.57) 1 ,
3.4 Smoothly Graded PN Junction
41