Δp À Δn ¼ C 8 exp Àk
00 x 3
ð
Þ,
ð2:16Þ
φ ¼ À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10 ,
ð2:17Þ
u 3 ¼ À
e
00
33
c 00
33
À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10
"
#
þ C 11 x 3 þ C 12 , ð2:18Þ
Δn ¼
n
00
0 μ
00 n
33
D
00 n
33
À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10
"
#
þ C 13 x 3 þ C 14 : ð2:19Þ
C 1 through C 14 are determined by Eqs. (2.7), (2.8), (2.9), (2.10), and (2.11). 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. 2.2 described
by step functions. Specifically, we limit ourselves to the following doping profile
with some symmetry or antisymmetry:
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 :
ð2:20Þ
The jumps of the step functions are not large so that the assumption of the linear
theory is not violated. When Eq. (2.20) is true, k
0
¼ k
00 ¼ k. It can be seen from the
above expressions that k is an important combination of parameters in piezoelectric
semiconductors. For the parameters in Eq. (2.20), let
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
,
ð2:21Þ
which will be used as a unit for k.
Figure 2.3 shows the concentrations of holes and electrons which were initially
determined by the step functions in Fig. 2.2 but are now continuous because of
diffusion. They are smooth functions after diffusion although they still change
rapidly near the interface. When k varies, effectively, p 0 + n 0 varies when other
parameters are fixed. From the expressions of the fields in the above, we expect that
x
3
Holes
Interface
Electrons
0
p
0
n
0
p
0
n
Fig. 2.2 Doping profile
16
2 Exact Solutions
00 x 3
ð
Þ,
ð2:16Þ
φ ¼ À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10 ,
ð2:17Þ
u 3 ¼ À
e
00
33
c 00
33
À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10
"
#
þ C 11 x 3 þ C 12 , ð2:18Þ
Δn ¼
n
00
0 μ
00 n
33
D
00 n
33
À
q
k
00
ð Þ
2 ε
00
33
C 8 exp Àk
00 x 3
ð
ÞþC 9 x 3 þ C 10
"
#
þ C 13 x 3 þ C 14 : ð2:19Þ
C 1 through C 14 are determined by Eqs. (2.7), (2.8), (2.9), (2.10), and (2.11). 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. 2.2 described
by step functions. Specifically, we limit ourselves to the following doping profile
with some symmetry or antisymmetry:
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 :
ð2:20Þ
The jumps of the step functions are not large so that the assumption of the linear
theory is not violated. When Eq. (2.20) is true, k
0
¼ k
00 ¼ k. It can be seen from the
above expressions that k is an important combination of parameters in piezoelectric
semiconductors. For the parameters in Eq. (2.20), let
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
,
ð2:21Þ
which will be used as a unit for k.
Figure 2.3 shows the concentrations of holes and electrons which were initially
determined by the step functions in Fig. 2.2 but are now continuous because of
diffusion. They are smooth functions after diffusion although they still change
rapidly near the interface. When k varies, effectively, p 0 + n 0 varies when other
parameters are fixed. From the expressions of the fields in the above, we expect that
x
3
Holes
Interface
Electrons
0
p
0
n
0
p
0
n
Fig. 2.2 Doping profile
16
2 Exact Solutions