21.4 Bipolar Diodes
563
For a one-sided (p
+ n-) diode, the saturation current is (here N B = N D )
j
d
s
∼ =
e D p p n 0
L p
∼ = e
D p
τ p
1/2 n
2
i
N B
.
(21.133)
The saturation depends via D p /τ p weakly on the temperature. The term n
2
i depends on T , proportional
to T
3 exp(−E g /kT ), which is dominated by the exponential function.
If the minority carrier lifetime is given by the radiative recombination (10.19), the hole diffusion
length is
L p =
D p
B n n 0
.
(21.134)
For GaAs (Tables 8.2 and 10.1) with N D = 10
18 cm
−3 , we find τ p = 10 ns and L p ≈ 3 µm. For L n we
find 14 µm, however, the lifetime at room temperature can be significantly shorter due to nonradiative
recombination and subsequently also the diffusion length will be shorter (by about a factor of 10). For
L ∼ 1 µm, the diffusion saturation current is j
d
s ∼ 4 × 10
−20 A/cm
2 .
The radiative recombination rate (band-band recombination, b–b) in the neutral n-region (as relevant
for LEDs, see Sect. 23.3) is B(np − n
2
i ) ≈ Bn n 0 ( p n (x) − p n 0 ). Therefore, the recombination current
j
b−b
d,n in the neutral n-region is (using (21.128))
j
b−b
d,n = e
∞
x n
B n
2
i
exp (βV ) − 1
exp
−
x − x n
L p
dx
= e B L p n
2
i
exp (βV ) − 1
.
(21.135)
For the neutral region on the p-side a similar expression results. The total radiative recombination
current from the neutral regions is
j
b−b
d
= e B (L n + L p ) n
2
i
exp (βV ) − 1
.
(21.136)
For GaAs, the saturation current for the radiative recombination in the neutral region
j
r,b−b
s
= e B (L n + L p ) n
2
i
(21.137)
is (Tables 7.1 and 10.1) for a diffusion length of 1 µm of j
r,b−b
s
∼ 4 × 10
−21 A/cm
2 .
Since the (radiative) minority carrier lifetime is inversely proportional to the majority carrier density,
the relevant diffusion length is that of the side with the lower doping level and is given by
L =
1
n i
D B N B
B
,
(21.138)
where D B is the minority carrier diffusion coefficient on the lowly doped side. The radiative recombination current from the neutral region can be written as
j
b−b
d
= e
B D B N B n i
exp (βV ) − 1
.
(21.139)
The I –V characteristic for pn diodes from two semiconductors with different band gap are shown in
Fig. 21.47a (for Ge and Si). The Si diode has the much smaller saturation current. The saturation current
563
For a one-sided (p
+ n-) diode, the saturation current is (here N B = N D )
j
d
s
∼ =
e D p p n 0
L p
∼ = e
D p
τ p
1/2 n
2
i
N B
.
(21.133)
The saturation depends via D p /τ p weakly on the temperature. The term n
2
i depends on T , proportional
to T
3 exp(−E g /kT ), which is dominated by the exponential function.
If the minority carrier lifetime is given by the radiative recombination (10.19), the hole diffusion
length is
L p =
D p
B n n 0
.
(21.134)
For GaAs (Tables 8.2 and 10.1) with N D = 10
18 cm
−3 , we find τ p = 10 ns and L p ≈ 3 µm. For L n we
find 14 µm, however, the lifetime at room temperature can be significantly shorter due to nonradiative
recombination and subsequently also the diffusion length will be shorter (by about a factor of 10). For
L ∼ 1 µm, the diffusion saturation current is j
d
s ∼ 4 × 10
−20 A/cm
2 .
The radiative recombination rate (band-band recombination, b–b) in the neutral n-region (as relevant
for LEDs, see Sect. 23.3) is B(np − n
2
i ) ≈ Bn n 0 ( p n (x) − p n 0 ). Therefore, the recombination current
j
b−b
d,n in the neutral n-region is (using (21.128))
j
b−b
d,n = e
∞
x n
B n
2
i
exp (βV ) − 1
exp
−
x − x n
L p
dx
= e B L p n
2
i
exp (βV ) − 1
.
(21.135)
For the neutral region on the p-side a similar expression results. The total radiative recombination
current from the neutral regions is
j
b−b
d
= e B (L n + L p ) n
2
i
exp (βV ) − 1
.
(21.136)
For GaAs, the saturation current for the radiative recombination in the neutral region
j
r,b−b
s
= e B (L n + L p ) n
2
i
(21.137)
is (Tables 7.1 and 10.1) for a diffusion length of 1 µm of j
r,b−b
s
∼ 4 × 10
−21 A/cm
2 .
Since the (radiative) minority carrier lifetime is inversely proportional to the majority carrier density,
the relevant diffusion length is that of the side with the lower doping level and is given by
L =
1
n i
D B N B
B
,
(21.138)
where D B is the minority carrier diffusion coefficient on the lowly doped side. The radiative recombination current from the neutral region can be written as
j
b−b
d
= e
B D B N B n i
exp (βV ) − 1
.
(21.139)
The I –V characteristic for pn diodes from two semiconductors with different band gap are shown in
Fig. 21.47a (for Ge and Si). The Si diode has the much smaller saturation current. The saturation current