φ ¼ À
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2 ,
ð2:34Þ
u ¼ À
e
0
c 0 À
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2
"
#
þ C 3 ,
ð2:35Þ
Δn ¼
n
0
0 μ
0 n
D
0 n
À
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2
"
#
þ C 4 ,
ð2:36Þ
where I 0 is the zero-order modified Bessel function of the first kind. C 1 through C 4
are undetermined constants. Similarly, for r > a, we have
Δp À Δn ¼ C 5 K 0 k
00 r
ð Þ,
ð2:37Þ
φ ¼ À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7 ,
ð2:38Þ
u ¼ À
e
00
c 00 À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7
"
#
þ C 8 ln r þ C 9 ,
ð2:39Þ
Δn ¼
n
00
0 μ
00 n
D
00 n
À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7
"
#
þ C 10 ln r þ C 11 , ð2:40Þ
where K 0 is the zero-order modified Bessel function of the second kind. It vanishes at
infinity. The following properties of I 0 and K 0 are needed in the numerical
calculation:
dI 0 ξ
ð Þ
dξ
¼ I 1 ξ
ð Þ,
dK 0 ξ
ð Þ
dξ
¼ ÀK 1 ξ
ð Þ:
ð2:41Þ
C 1 through C 11 are determined by Eqs. (2.29), (2.30), (2.31), and (2.32). Some of
them can be determined in a simple manner. At the end eight constants are determined by solving eight linear algebraic equations on a computer using MATLAB.
As a numerical example, consider the doping profile shown in Fig. 2.8 with
p
0
0 ¼ 10 Â 10
20 m
À3 , p
00
0 ¼ 7 Â 10
20 m
À3
¼ n
0
0 ,
n
0
0 ¼ 7 Â 10
20 m
À3 , n
00
0 ¼ 10 Â 10
20 m
À3
¼ p
0
0 :
ð2:42Þ
Under Eq. (2.42), we have k
0 ¼ k
00 ¼ k. We introduce
k
2
0 ¼
p 0 μ
p
33
D
p
33
þ
n 0 μ
n
33
D
n
33
q
ε 33
¼ p 0 þ n 0
ð
Þ
q
k B T
q
ε 33
:
ð2:43Þ
22
2 Exact Solutions
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2 ,
ð2:34Þ
u ¼ À
e
0
c 0 À
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2
"
#
þ C 3 ,
ð2:35Þ
Δn ¼
n
0
0 μ
0 n
D
0 n
À
q
k
0
ð Þ
2 ε
0
C 1 I 0 k
0 r
ð Þ þ C 2
"
#
þ C 4 ,
ð2:36Þ
where I 0 is the zero-order modified Bessel function of the first kind. C 1 through C 4
are undetermined constants. Similarly, for r > a, we have
Δp À Δn ¼ C 5 K 0 k
00 r
ð Þ,
ð2:37Þ
φ ¼ À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7 ,
ð2:38Þ
u ¼ À
e
00
c 00 À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7
"
#
þ C 8 ln r þ C 9 ,
ð2:39Þ
Δn ¼
n
00
0 μ
00 n
D
00 n
À
q
k
00
R
À Á 2 ε
00
C 5 K 0 k
00 r
ð ÞþC 6 ln r þ C 7
"
#
þ C 10 ln r þ C 11 , ð2:40Þ
where K 0 is the zero-order modified Bessel function of the second kind. It vanishes at
infinity. The following properties of I 0 and K 0 are needed in the numerical
calculation:
dI 0 ξ
ð Þ
dξ
¼ I 1 ξ
ð Þ,
dK 0 ξ
ð Þ
dξ
¼ ÀK 1 ξ
ð Þ:
ð2:41Þ
C 1 through C 11 are determined by Eqs. (2.29), (2.30), (2.31), and (2.32). Some of
them can be determined in a simple manner. At the end eight constants are determined by solving eight linear algebraic equations on a computer using MATLAB.
As a numerical example, consider the doping profile shown in Fig. 2.8 with
p
0
0 ¼ 10 Â 10
20 m
À3 , p
00
0 ¼ 7 Â 10
20 m
À3
¼ n
0
0 ,
n
0
0 ¼ 7 Â 10
20 m
À3 , n
00
0 ¼ 10 Â 10
20 m
À3
¼ p
0
0 :
ð2:42Þ
Under Eq. (2.42), we have k
0 ¼ k
00 ¼ k. We introduce
k
2
0 ¼
p 0 μ
p
33
D
p
33
þ
n 0 μ
n
33
D
n
33
q
ε 33
¼ p 0 þ n 0
ð
Þ
q
k B T
q
ε 33
:
ð2:43Þ
22
2 Exact Solutions