2.2 Abrupt PN Junction
Consider the interface between two piezoelectric half spaces of ZnO in Fig. 2.1
[1]. The two half spaces are doped into a p region where p > n, and an n region where
n > p. Then diffusion occurs at the interface to form a PN junction. Since the doping
is uniform separately in the p and n regions, respectively, it has a jump discontinuity
at the interface and the corresponding PN junction is called an abrupt junction [2].
Mathematically, we need to treat the two half spaces separately and then apply
boundary conditions at infinity as well as continuity conditions at the interface. The
problem is one-dimensional without dependence on x 1 and x 2 . The governing
equations are taken from Eqs. (1.26), (1.24), (1.23), and (1.21) 1 :
Δp À Δn
ð
Þ ,33 ¼ k
2
Δp À Δn
ð
Þ ,
ð2:2Þ
φ ,33 ¼ À
q
k
2
ε 33
Δp À Δn
ð
Þ ,33 ,
ð2:3Þ
Δp
ð Þ ,33 ¼ À
p 0 μ
p
33
D
p
33
φ ,33 ,
Δn
ð Þ ,33 ¼
n 0 μ
n
33
D
n
33
φ ,33 ,
ð2:4Þ
u 3,33 ¼ À
e 33
c 33
φ ,33 ,
ð2:5Þ
where
k
2
¼
p 0 μ
p
33
D
p
33
þ
n 0 μ
n
33
D
n
33
q
ε 33
, ε 33 ¼ ε 33 þ
e
2
33
c 33
:
ð2:6Þ
At infinity we have the following boundary conditions:
T 33 Æ1
ð
Þ ¼0, D 3 Æ1
ð
Þ¼ 0,
J
p
3 Æ1
ð
Þ¼ 0, J
n
3 Æ1
ð
Þ¼ 0:
ð2:7Þ
n-doped
x3
p-doped
Interface
Fig. 2.1 An interface
between two piezoelectric
semiconductor half spaces
14
2 Exact Solutions