566
21 Diodes
with j
mr
s = eσ v th N t wn i /2 and the approximation being valid for eV /kT 1. Thus the nonradiative
band-impurity recombination is often said to cause an ideality factor of n = 2.
For a better approximation of χ , the dependence of the potential ϕ(x) can be approximated as linear
(constant-field approximation), i.e. using the local field E mr at the position where n = p [1766]. For a
symmetric diode with n n 0 = p p 0 , this position is at x = 0; for a one-sided junction on the lower-doped
side. E mr is given for p p 0 ≤ n n 0 by
E mr = −
√
2
w
(V bi − V )
1 +
1
β (V bi − V )
ln
p p 0
n n 0
1 +
p p 0
n n 0
.
(21.150)
For a symmetric diode (21.151a) holds, for a one-sided diode the approximation in (21.151b) holds
E mr = −
2
w
(V bi − V ) ∝
V bi − V
(21.151a)
E mr ∼ = −
√
2
w
(V bi − V ) ∝
V bi − V .
(21.151b)
We note that for zero bias (V = 0), (21.108) is recovered from (21.151a). Using the above approximation ζ is given by
ζ(x) =
1
2
1
1 + exp (βV /2) cosh(β E mr x)
.
(21.152)
Since ζ decreases sufficiently fast within the depletion layer, the integration over the depletion layer
can be extended to ±∞ and we obtain
χ =
2
β E mr
1
exp(βV ) − 1
arctan
exp(βV /2) − 1
exp(βV /2) + 1
.
(21.153)
We note that for V = 0, the integral takes the value χ = (β E mr )
−1 . The recombination current is now
given by [1766]
j r,b−i =
2 σ v th N t n i kT
E mr
arctan
exp(βV /2) − 1
exp(βV /2) + 1
exp (βV ) − 1 .
(21.154)
For large voltage the arctan term becomes π/4. For eV /kT 1 the nonradiative recombination
current can be written as
j r,b−i = j
r,b−i
s
exp
βV
n
,
(21.155)
with j
r,b−i
s
= eσ v th N t n i kT π/(2E mr ). The voltage-dependent ideality factor n (semilogarithmic slope
n = β j r (V )/j
r (V )) is close but not identical to 2 and is shown in Fig. 21.49 for various values of V bi .
The built-in voltage influences the logarithmic slope via the factor 1/E mr in (21.154).
In the case of radiative band-band (b–b) recombination, the recombination rate is given by (10.14).
Together with (21.125) and integrated over the depletion layer, the recombination current in the depletion layer is given by (cf. (21.136))
j r,b−b = e B w n
2
i
exp (βV ) − 1
,
(21.156)
and exhibits an ideality factor of n = 1. Comparing (21.136) and (21.156), the dominating radiative-
21 Diodes
with j
mr
s = eσ v th N t wn i /2 and the approximation being valid for eV /kT 1. Thus the nonradiative
band-impurity recombination is often said to cause an ideality factor of n = 2.
For a better approximation of χ , the dependence of the potential ϕ(x) can be approximated as linear
(constant-field approximation), i.e. using the local field E mr at the position where n = p [1766]. For a
symmetric diode with n n 0 = p p 0 , this position is at x = 0; for a one-sided junction on the lower-doped
side. E mr is given for p p 0 ≤ n n 0 by
E mr = −
√
2
w
(V bi − V )
1 +
1
β (V bi − V )
ln
p p 0
n n 0
1 +
p p 0
n n 0
.
(21.150)
For a symmetric diode (21.151a) holds, for a one-sided diode the approximation in (21.151b) holds
E mr = −
2
w
(V bi − V ) ∝
V bi − V
(21.151a)
E mr ∼ = −
√
2
w
(V bi − V ) ∝
V bi − V .
(21.151b)
We note that for zero bias (V = 0), (21.108) is recovered from (21.151a). Using the above approximation ζ is given by
ζ(x) =
1
2
1
1 + exp (βV /2) cosh(β E mr x)
.
(21.152)
Since ζ decreases sufficiently fast within the depletion layer, the integration over the depletion layer
can be extended to ±∞ and we obtain
χ =
2
β E mr
1
exp(βV ) − 1
arctan
exp(βV /2) − 1
exp(βV /2) + 1
.
(21.153)
We note that for V = 0, the integral takes the value χ = (β E mr )
−1 . The recombination current is now
given by [1766]
j r,b−i =
2 σ v th N t n i kT
E mr
arctan
exp(βV /2) − 1
exp(βV /2) + 1
exp (βV ) − 1 .
(21.154)
For large voltage the arctan term becomes π/4. For eV /kT 1 the nonradiative recombination
current can be written as
j r,b−i = j
r,b−i
s
exp
βV
n
,
(21.155)
with j
r,b−i
s
= eσ v th N t n i kT π/(2E mr ). The voltage-dependent ideality factor n (semilogarithmic slope
n = β j r (V )/j
r (V )) is close but not identical to 2 and is shown in Fig. 21.49 for various values of V bi .
The built-in voltage influences the logarithmic slope via the factor 1/E mr in (21.154).
In the case of radiative band-band (b–b) recombination, the recombination rate is given by (10.14).
Together with (21.125) and integrated over the depletion layer, the recombination current in the depletion layer is given by (cf. (21.136))
j r,b−b = e B w n
2
i
exp (βV ) − 1
,
(21.156)
and exhibits an ideality factor of n = 1. Comparing (21.136) and (21.156), the dominating radiative-