k
0
ð Þ
2 ¼
p
0
0 μ
0 p
D
0 p þ
n
0
0 μ
0 n
D
0 n
q
ε 0 T
33
:
ð7:63Þ
Similarly, for 0 < x < L, we have
Δp À Δn ¼ A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ,
ð7:64Þ
φ ¼ À
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 7 x þ C 8 ,
ð7:65Þ
u ¼
e
00
c 00
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 9 x þ C 10 , ð7:66Þ
Δp ¼
p
00
0 μ
00 p
D
00 p
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þC 11 x þ C 12 ,
ð7:67Þ
where A 2 , B 2 , and C 7 -C 12 are undetermined constants and
k
00
ð Þ
2 ¼
p
00
0 μ
00 p
D
00 p þ
n
00
0 μ
00 n
D
00 n
q
ε 00 T
33
:
ð7:68Þ
Equations (7.59), (7.60), (7.61), and (7.62) and Eqs. (7.64), (7.65), (7.66), and (7.67)
are substituted into Eqs. (7.53), (7.54), (7.55), (7.56), (7.57), and (7.58), resulting in
a system of linear equations for the undetermined constants. These equations are
solved on a computer. The axial electric polarization P¼P 3 is calculated from
Eq. (7.41). Then the related volume effective polarization charge density ρ
P , the
sum of the doping and mobile charge densities ρ
e , and the total charge density ρ
t can
be calculated from the following expressions:
ρ
P
¼ ÀP k,k ¼ ÀP 3,3 ,
ρ
e
¼ q Δp À Δn
ð
Þ ,
ρ
t
¼ ρ
e
þ ρ
P
:
ð7:69Þ
For a numerical example, consider a rod with 2L ¼ 30 μm. The doping profile is
given by p 0
0
¼ 10
21 m
À3 , n 0
0
¼ 7 Â 10
20 m
À3
, n 0
00
¼ 10
21 m
À3 , and
p 0
00
¼ 7 Â 10
20 m
À3 . It is chosen to be antisymmetric about the origin for simplicity.
Consider three cases of temperature change with θ ¼ 0 K, 0.3 K, and 0.5 K, among
which the case of θ ¼ 0 K is used as a reference.
Figure 7.9 shows the temperature effects on the basic built-in fields near the
junction for a heterogeneous junction with opposite c-axes in the two halves. In the
case of a homogeneous junction, the corresponding figures show no visible temperature effects and therefore are not presented. Only a portion of the rod within
|x| < 1 μm is shown, which is sufficient for examining the fields because they are
7.3 Temperature Effects on PN Junctions
191
0
ð Þ
2 ¼
p
0
0 μ
0 p
D
0 p þ
n
0
0 μ
0 n
D
0 n
q
ε 0 T
33
:
ð7:63Þ
Similarly, for 0 < x < L, we have
Δp À Δn ¼ A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ,
ð7:64Þ
φ ¼ À
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 7 x þ C 8 ,
ð7:65Þ
u ¼
e
00
c 00
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þ C 9 x þ C 10 , ð7:66Þ
Δp ¼
p
00
0 μ
00 p
D
00 p
q
k
00
ð Þ
2 ε 00 T
33
A 2 sinh k
00 x À L
ð
ÞþB 2 sinh k
00 x þ L
ð
Þ
½
þC 11 x þ C 12 ,
ð7:67Þ
where A 2 , B 2 , and C 7 -C 12 are undetermined constants and
k
00
ð Þ
2 ¼
p
00
0 μ
00 p
D
00 p þ
n
00
0 μ
00 n
D
00 n
q
ε 00 T
33
:
ð7:68Þ
Equations (7.59), (7.60), (7.61), and (7.62) and Eqs. (7.64), (7.65), (7.66), and (7.67)
are substituted into Eqs. (7.53), (7.54), (7.55), (7.56), (7.57), and (7.58), resulting in
a system of linear equations for the undetermined constants. These equations are
solved on a computer. The axial electric polarization P¼P 3 is calculated from
Eq. (7.41). Then the related volume effective polarization charge density ρ
P , the
sum of the doping and mobile charge densities ρ
e , and the total charge density ρ
t can
be calculated from the following expressions:
ρ
P
¼ ÀP k,k ¼ ÀP 3,3 ,
ρ
e
¼ q Δp À Δn
ð
Þ ,
ρ
t
¼ ρ
e
þ ρ
P
:
ð7:69Þ
For a numerical example, consider a rod with 2L ¼ 30 μm. The doping profile is
given by p 0
0
¼ 10
21 m
À3 , n 0
0
¼ 7 Â 10
20 m
À3
, n 0
00
¼ 10
21 m
À3 , and
p 0
00
¼ 7 Â 10
20 m
À3 . It is chosen to be antisymmetric about the origin for simplicity.
Consider three cases of temperature change with θ ¼ 0 K, 0.3 K, and 0.5 K, among
which the case of θ ¼ 0 K is used as a reference.
Figure 7.9 shows the temperature effects on the basic built-in fields near the
junction for a heterogeneous junction with opposite c-axes in the two halves. In the
case of a homogeneous junction, the corresponding figures show no visible temperature effects and therefore are not presented. Only a portion of the rod within
|x| < 1 μm is shown, which is sufficient for examining the fields because they are
7.3 Temperature Effects on PN Junctions
191