qΔp ¼ À
qp
0
0 μ
p
33
D
p
33
φ þ C
0
1 ,
ÀqΔn ¼ À
qn
0
0 μ
n
33
D
n
33
φ þ C
0
2 :
ð6:147Þ
Similarly, for x 3 > 0, we have
φ ¼ k
00 C
00
3 exp k
00 x 3
ð
ÞÀk
00 C
00
4 exp Àk
00 x 3
ð
Þþ
1
k
00
ð Þ
2
e ε
00
C
00
1 þ C
00
2
À
Á
A
2
ð Þ , ð6:148Þ
u 2 ¼
2Ge
1
ð Þ
33
D
C
00
3 exp k
00 x 3
ð
ÞþC
00
4 exp Àk
00 x 3
ð
Þ
Â
à þ C
00
5 x 3 þ C
00
6 ,
ð6:149Þ
qΔp ¼ À
qp
00
0 μ
p
33
D
p
33
φ þ C
00
1 ,
ÀqΔn ¼ À
qn
00
0 μ
n
33
D
n
33
φ þ C
00
2 :
ð6:150Þ
The boundary conditions at the two ends of the beam and the continuity conditions at
the junction interface are
M ÆL
ð Þ ¼ 0, Q ÆL
ð Þ ¼ 0,
b
D 3 ÆL
ð Þ ¼ 0,
J
p
3 ÆL
ð Þ ¼ 0, J
n
3 ÆL
ð Þ ¼ 0,
ð6:151Þ
u 0 0
À
ð Þ ¼ u 2 0
þ
ð Þ, u 2,3 0
À
ð Þ ¼ u 2,3 0
þ
ð Þ,
M 0
À
ð Þ ¼ M 0
þ
ð Þ, Q 0
À
ð Þ ¼ Q 0
þ
ð Þ,
φ 0
À
ð Þ ¼ φ 0
þ
ð Þ, b
D 3 0
À
ð Þ ¼ b
D 3 0
þ
ð Þ
p 0
À
ð Þ ¼ p 0
þ
ð Þ, n 0
À
ð Þ ¼ n 0
þ
ð Þ,
J
p
3 0
À
ð Þ ¼ J
p
3 0
þ
ð Þ, J
n
3 0
À
ð Þ ¼ J
n
3 0
þ
ð Þ:
ð6:152Þ
To make the displacements and electric potential unique, we impose
u 2 0
ð Þ ¼ 0, u 2,3 0
ð Þ ¼ 0, φ 0
ð Þ ¼ 0:
ð6:153Þ
We also have the following charge neutrality conditions:
Z L
ÀL
Δpdx 3 ¼ 0,
Z L
ÀL
Δndx 3 ¼ 0:
ð6:154Þ
6.5 Bending of a PN Junction
171