3.5 Periodic Doping
In this section we consider a rod with a periodically varying doping profile described
by [2]:
N
À
A x 3
ð Þ ¼ a 1 þ b 1 sin λx 3 ,
N
þ
D x 3
ð Þ ¼ a 2 þ b 2 sin λx 3 :
ð3:62Þ
Hence
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ ¼ a þ b sin λx 3 ,
a ¼ a 2 À a 1 , b ¼ b 2 À b 1 :
ð3:63Þ
Then, for the particular solution of Eq. (3.19), we have
φ
p
¼
q
ε 33
a
k
2
þ
b
k
2
þ λ
2
sin λx 3
!
:
ð3:64Þ
In this case, it can be shown [2] that p
0 and n
0 , i.e., the values of p and n at zero
potential, are determined by the following equations:
p
0
À n
0
þ a ¼ 0,
p
0
¼
n
0 a 1
2n 0 À a 2
:
ð3:65Þ
In the special case when a 2 ¼ a 1 , Eq. (3.63) implies that a ¼ 0. Then, from
Eq. (3.65), we obtain p
0
¼ n
0
¼ a 1 . For a numerical example, we consider the
case when a 1 ¼ 10
21 m
À3 , a 2 ¼ a 1 , b 1 ¼ 0.2a 1 , b 2 ¼ À b 1 , and λ ¼ 0.2k. Then 2π/
λ ffi 3 μm which can be viewed as some wavelength of the doping. We choose
2L ¼ 30 μm which is about ten times the doping wavelength 2π/λ. The doping and
mobile charge density ρ
e whose definition is in Eq. (3.42), the electric field E 3 (x 3 ),
and the axial strain S 3 (x 3 ) in this case are shown in Fig. 3.6. They are essentially
periodic except at x 3 ¼ Æ L where there are some edge effects. ρ
e has some
concentration near the ends.
44
3 Extension of Rods
In this section we consider a rod with a periodically varying doping profile described
by [2]:
N
À
A x 3
ð Þ ¼ a 1 þ b 1 sin λx 3 ,
N
þ
D x 3
ð Þ ¼ a 2 þ b 2 sin λx 3 :
ð3:62Þ
Hence
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ ¼ a þ b sin λx 3 ,
a ¼ a 2 À a 1 , b ¼ b 2 À b 1 :
ð3:63Þ
Then, for the particular solution of Eq. (3.19), we have
φ
p
¼
q
ε 33
a
k
2
þ
b
k
2
þ λ
2
sin λx 3
!
:
ð3:64Þ
In this case, it can be shown [2] that p
0 and n
0 , i.e., the values of p and n at zero
potential, are determined by the following equations:
p
0
À n
0
þ a ¼ 0,
p
0
¼
n
0 a 1
2n 0 À a 2
:
ð3:65Þ
In the special case when a 2 ¼ a 1 , Eq. (3.63) implies that a ¼ 0. Then, from
Eq. (3.65), we obtain p
0
¼ n
0
¼ a 1 . For a numerical example, we consider the
case when a 1 ¼ 10
21 m
À3 , a 2 ¼ a 1 , b 1 ¼ 0.2a 1 , b 2 ¼ À b 1 , and λ ¼ 0.2k. Then 2π/
λ ffi 3 μm which can be viewed as some wavelength of the doping. We choose
2L ¼ 30 μm which is about ten times the doping wavelength 2π/λ. The doping and
mobile charge density ρ
e whose definition is in Eq. (3.42), the electric field E 3 (x 3 ),
and the axial strain S 3 (x 3 ) in this case are shown in Fig. 3.6. They are essentially
periodic except at x 3 ¼ Æ L where there are some edge effects. ρ
e has some
concentration near the ends.
44
3 Extension of Rods