3.3 Linearly Graded PN Junction
The PN junctions between two piezoelectric semiconductor half spaces and between
a circular cylinder and its surrounding material treated in Sects. 2.2 and 2.3 are
abrupt junctions with piecewise constant doping profiles. In reality, the doping
profiles are gradually changing or graded. In this section we study linearly graded
doping and PN junctions. Consider a rod with the doping profile in Fig. 3.2 [2]. The
left half is mainly p-doped and the right half mainly n-doped. Between the two
halves, there is a finite transition zone of width 2w in which the doping varies
linearly. The PN junctions in Sects. 2.2 and 2.3 correspond to the special case of
w ¼ 0.
Quantitatively, the doping profile in Fig. 3.2 is described by
N
À
A ¼
a 1 ,
À L < x 3 < Àw,
a 1 þ
a 2 À a 1
2w
x 3 þ w
ð
Þ, j x 3 j< w,
a 2 ,
w < x 3 < L,
8
> <
> :
ð3:25Þ
and
N
þ
D ¼
b 1 ,
À L < x 3 < Àw,
b 1 þ
b 2 À b 1
2w
x 3 þ w
ð
Þ, j x 3 j< w,
b 2 ,
w < x 3 < L:
8
> <
> :
ð3:26Þ
Then,
N
þ
D À N
À
A ¼
d 1 , À L < x 3 < Àw,
d 3 þ d 4 x 3 , j x 3 j< w,
d 2 , w < x 3 < L,
8
> <
> :
ð3:27Þ
x
3
A
N
−
a1
D
N
+
0
w
-w
L
-L
a2
b1
b2
Fig. 3.2 Linearly graded
doping
36
3 Extension of Rods