Δp ¼
p
00
0 μ
00 p
D
00 p
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 11 x
þ C 12 ,
ð3:210Þ
where A 2 , B 2 , and C 7 –C 12 are undetermined constants, and
k
00
ð Þ
2 ¼
p
00
0 μ
00 p
D
00 p þ
n
00
0 μ
00 n
D
00 n
q
ε 00 T
33
:
ð3:211Þ
Equations (3.202), (3.203), (3.204), (3.205) and Eqs. (3.207), (3.208), (3.209),
(3.210) are subsituted into Eqs. (3.196), (3.197), (3.198), (3.199), (3.200), and
(3.201). This results in a system of linear equations for the undetermined constants.
The equations are solved on a computer.
As an example, consider a long ZnO fiber with 2L ¼ 30 μm, p 0
0
¼ 10
21 m
À3 ,
n 0
0
¼ 7Â10
20 m
À3 , n 0
00
¼ 10
21 m
À3 , p 0
00
¼ 7Â10
20 m
À3 , and a ¼ À1 μm. We
organize the numerical results below according to increasing magnitude of F. For
small values of F, the linear analytical solution and the nonlinear COMSOL numerical solution produce the same result. For medium F, the two solutions begin to show
differences. For large F, the linear analytical solution is invalid.
In the case of a small F ¼ 0–0.2 MPa, since the linear analytical solution and the
nonlinear COMSOL solution agree well, only the analytical solution is presented
below. The polarization fields are shown in Fig. 3.27. Away from the junction and on
its both sides, the P in (a) for the homogeneous junction has the same value but, that
in (b) for the heterogeneous has opposite signs.
The different behaviors of P in Fig. 3.27 have implications on the effective
polarization charge density ρ
P they produce according to Eq. (3.195) 2 , which are
shown in Fig. 3.28. For the homogeneous junction in (a), F has no effect, and there
are no net polarization charges near the junction. For the heterogeneous junction in
(b), F produces negative net polarization charges.
In addition to the distributed effective polarization charge density descibed by ρ
P ,
there also exists a surface effective polarization charge density σ
P on the interface of
the PN junction due to the discontinuity of P there. σ
P can be calculated from
σ
P
¼ P
À
À P
þ ,
P
À
¼ P 0
À
ð Þ, P
þ
¼ P 0
þ
ð Þ:
ð3:212Þ
We list σ
P at the junction interface in Table 3.1. For the homogeneous junction, we
always have σ
P
ffi 0. However, for the heterogeneous junction, the magnitude of σ
P
increases with F. σ
P is negative and contributes further to the negative net polarization charges near the junction produced by ρ
P in Fig. 3.28b.
For a small F ¼ 0–0.2 MPa, Fig. 3.29 shows the effects of F on the built-in
potential and electric field near the heterogeneous junction. For the homogeneous
junction F has no visible effects. Only the central part of the rod within |x|<1 μm is
3.14 Extension of a PN Junction
79
p
00
0 μ
00 p
D
00 p
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 11 x
þ C 12 ,
ð3:210Þ
where A 2 , B 2 , and C 7 –C 12 are undetermined constants, and
k
00
ð Þ
2 ¼
p
00
0 μ
00 p
D
00 p þ
n
00
0 μ
00 n
D
00 n
q
ε 00 T
33
:
ð3:211Þ
Equations (3.202), (3.203), (3.204), (3.205) and Eqs. (3.207), (3.208), (3.209),
(3.210) are subsituted into Eqs. (3.196), (3.197), (3.198), (3.199), (3.200), and
(3.201). This results in a system of linear equations for the undetermined constants.
The equations are solved on a computer.
As an example, consider a long ZnO fiber with 2L ¼ 30 μm, p 0
0
¼ 10
21 m
À3 ,
n 0
0
¼ 7Â10
20 m
À3 , n 0
00
¼ 10
21 m
À3 , p 0
00
¼ 7Â10
20 m
À3 , and a ¼ À1 μm. We
organize the numerical results below according to increasing magnitude of F. For
small values of F, the linear analytical solution and the nonlinear COMSOL numerical solution produce the same result. For medium F, the two solutions begin to show
differences. For large F, the linear analytical solution is invalid.
In the case of a small F ¼ 0–0.2 MPa, since the linear analytical solution and the
nonlinear COMSOL solution agree well, only the analytical solution is presented
below. The polarization fields are shown in Fig. 3.27. Away from the junction and on
its both sides, the P in (a) for the homogeneous junction has the same value but, that
in (b) for the heterogeneous has opposite signs.
The different behaviors of P in Fig. 3.27 have implications on the effective
polarization charge density ρ
P they produce according to Eq. (3.195) 2 , which are
shown in Fig. 3.28. For the homogeneous junction in (a), F has no effect, and there
are no net polarization charges near the junction. For the heterogeneous junction in
(b), F produces negative net polarization charges.
In addition to the distributed effective polarization charge density descibed by ρ
P ,
there also exists a surface effective polarization charge density σ
P on the interface of
the PN junction due to the discontinuity of P there. σ
P can be calculated from
σ
P
¼ P
À
À P
þ ,
P
À
¼ P 0
À
ð Þ, P
þ
¼ P 0
þ
ð Þ:
ð3:212Þ
We list σ
P at the junction interface in Table 3.1. For the homogeneous junction, we
always have σ
P
ffi 0. However, for the heterogeneous junction, the magnitude of σ
P
increases with F. σ
P is negative and contributes further to the negative net polarization charges near the junction produced by ρ
P in Fig. 3.28b.
For a small F ¼ 0–0.2 MPa, Fig. 3.29 shows the effects of F on the built-in
potential and electric field near the heterogeneous junction. For the homogeneous
junction F has no visible effects. Only the central part of the rod within |x|<1 μm is
3.14 Extension of a PN Junction
79