ε ¼ ε þ
e
2
c
:
ð2:27Þ
In polar coordinates,
∇
2
¼
∂
2
∂r 2 þ
1
r
∂
∂r
þ
1
r 2
∂
2
∂θ
2
:
ð2:28Þ
At the interface between the p- and n-doped regions, we have the following
continuity conditions:
u a
À
ð Þ ¼ u a
þ
ð Þ, T 3r a
À
ð Þ ¼ T 3r a
þ
ð Þ,
φ a
À
ð Þ ¼ φ a
þ
ð Þ, D r a
À
ð Þ ¼ D r a
þ
ð Þ,
ð2:29Þ
n a
À
ð Þ ¼ n a
þ
ð Þ, p a
À
ð Þ ¼ p a
þ
ð Þ,
J
n
r a
À
ð Þ ¼ J
n
r a
þ
ð Þ, J
p
r a
À
ð Þ ¼ J
p
r a
þ
ð Þ:
ð2:30Þ
The above equations and boundary/continuity conditions are invariant under a rigidbody displacement 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 and impose
u a
ð Þ ¼ 0, φ a
ð Þ ¼ 0:
ð2:31Þ
We also have the following additional conditions representing the global conservation of holes and electrons:
Z a
0
Δpdr þ
Z 1
a
Δpdr ¼ 0,
Z a
0
Δndr þ
Z 1
a
Δndr ¼ 0:
ð2:32Þ
In addition, the fields should be finite at r ¼ 0. There are restrictions on the fields at
infinity too.
We use a prime for the material parameters in the p-doped central region and a
double prime for the material parameters in the n-doped outer region. For the central
region, the solution finite at the origin is given by
Δp À Δn ¼ C 1 I 0 k
0 r
ð Þ,
ð2:33Þ
2.3 Circular PN Junction
21