At the interface between the p- and n-doped regions, we have the following
continuity conditions:
u 3 0
À
ð Þ ¼ u 3 0
þ
ð Þ, T 33 0
À
ð Þ ¼ T 33 0
þ
ð Þ,
φ 0
À
ð Þ ¼ φ 0
þ
ð Þ, D 3 0
À
ð Þ ¼ D 3 0
þ
ð Þ,
ð2:8Þ
n 0
À
ð Þ ¼ n 0
þ
ð Þ, p 0
À
ð Þ ¼ p 0
þ
ð Þ,
J
n
3 0
À
ð Þ ¼ J
n
3 0
þ
ð Þ, J
p
3 0
À
ð Þ ¼ J
p
3 0
þ
ð Þ:
ð2:9Þ
The above equations and boundary/continuity conditions are invariant under a rigidbody translation in the x 3 direction and a shift of the electric potential through a
constant. To fix the rigid-body displacement and the arbitrary constant in the electric
potential so that the displacement and potential fields are unique, we choose the
interface as a reference for the displacement and potential and impose
u 3 0
ð Þ ¼ 0, φ 0
ð Þ ¼ 0:
ð2:10Þ
We have the following additional conditions representing the global conservation of
holes and electrons:
Z 0
À1
Δpdx 3 þ
Z 1
0
Δpdx 3 ¼ 0,
Z 0
À1
Δndx 3 þ
Z 1
0
Δndx 3 ¼ 0:
ð2:11Þ
We use a prime for the material parameters in the p-doped left half space and a
double prime for those in the n-doped region on the right.
Equations (2.2), (2.3), (2.4), and (2.5) are linear ordinary differential equations
with constant coefficients. Their general solution can be obtained in a straightforward and systematic manner. For x 3 < 0,
Δp À Δn ¼ C 1 exp k
0 x 3
ð
Þ,
ð2:12Þ
φ ¼ À
q
k
0
ð Þ
2 ε
0
33
C 1 exp k
0 x 3
ð
ÞþC 2 x 3 þ C 3 ,
ð2:13Þ
u 3 ¼ À
e
0
33
c 0
33
À
q
k
0
ð Þ
2 ε
0
33
C 1 exp k
0 x 3
ð
Þþ C 2 x 3 þ C 3
"
#
þ C 4 x 3 þ C 5 ,
ð2:14Þ
Δn ¼
n
0
0 μ
0 n
33
D
0 n
33
À
q
k
0
ð Þ
2 ε
0
33
C 1 exp k
0 x 3
ð
ÞþC 2 x 3 þ C 3
"
#
þ C 6 x 3 þ C 7 ,
ð2:15Þ
where C 1 through C 7 are undetermined constants. Similarly, for x 3 > 0,
2.2 Abrupt PN Junction
15