21.4 Bipolar Diodes
565
r ∼ =
σ n σ p v th N t
σ n exp
E t −E i
kT
+ σ p exp
E i −E t
kT
n i ≡
n i
τ e
,
(21.140)
where τ e is the effective electron lifetime. The generation current density is given by
j g =
e n i w
τ e
.
(21.141)
Since the width of the depletion layer varies with the applied reverse bias V , we expect a dependence
j g ∝
V bi + |V | .
(21.142)
The saturation current is given by the sum of the diffusion and generation parts
j s = e
D p
τ p
n
2
i
N D
+
e n i w
τ e
.
(21.143)
In semiconductors with large n i (narrow band gap, e.g. Ge) the diffusion current will dominate; in Si
(larger band gap) the generation current can dominate.
The maximum of the recombination rate is present for E t ≈ E i (10.57). Then n t = p t = n i in
(10.52). Assuming σ = σ n = σ p , the recombination rate is
r b−i = σ v th N t
n p − n
2
i
n + p + 2 n i
.
(21.144)
Using (21.122) we can write
r b−i = σ v th N t n i
n i
n + p + 2 n i
exp (βV ) − 1
.
(21.145)
The term ζ =
n i
n+ p+2n i
is maximal for n = p, which is given (from (21.125)) by
n mr = p mr = n i exp (βV /2) .
(21.146)
The function ζ(x) cannot be integrated analytically. In order to evaluate the integral of ζ over the
depletion layer
χ =
x n
−x p
ζ dx ,
(21.147)
the maximum rate
ζ mr =
n i
n mr + p mr + 2 n i
=
1
2
1
1 + exp (βV /2)
(21.148)
can be integrated over the depletion layer as an approximation [574], χ ≈ ζ mr w. This approach yields
a recombination current
j mr =
e σ v th N t w n i
2
exp (βV ) − 1
exp (βV /2) + 1
∼ = j
mr
s exp
βV
2
,
(21.149)