21.4 Bipolar Diodes
567
(a)
(b)
Fig. 21.49 a Integral χ (21.147) multiplied by exp(βV /2) in order to extract the differences on a linear scale. Solid line:
Exact numerical calculation, dash-dotted line: standard approximation with constant maximum rate, dashed line: this
work (approximation with constant field). As material parameters we have used room temperature and n i = 10 10 cm −3
(Si), n n0 = 10 18 cm −3 and p p0 = 10 17 cm −3 . b Logarithmic slope of band-impurity recombination current in the forward
bias regime for various values of the built-in voltage V bi = 0.6, 1.0, and 1.4 eV and in the limit V bi → ∞. Adapted
from [1766]
Fig. 21.50 Theoretical
modeling of charge-carrier
concentration, intrinsic
Fermi level (potential) ψ
and quasi-Fermi levels
(with arbitrary offset) for a
Si p + n diode for various
current densities: a
10 A cm −2 , b 10 3 A cm −2
and c 10 4 A cm −2 .
N A = 10 18 cm −3 ,
N D = 10 16 cm −3 ,
τ n = 0.3 ns, τ p = 0.84 ns.
Adapted from [1767]
10
18
10
10
10
10
10
16
14
12
10
8
Carrier
-3
n
p
p
n
n
p
p
n
n
p
p
n
2
1
0 (a)
(c)
(b)
recombination current is determined by the ratio of w and L n + L p . Since in the forward direction,
the depletion-layer width tends towards zero (for flat-band conditions), the radiative-recombination
current is dominated by the recombination in the neutral region(s).
For high injection current (under forward bias), the injected minority carrier density can become
comparable with the majority carrier density. In this case, diffusion and drift need to be considered. At
large current density, the voltage drop across the junction is small compared to the ohmic voltage drop
across the current path. In the simulation (Fig. 21.50), the high-injection effects start on the n-doped
side because it has been modeled with the lower doping (N D < N A ).
The series resistance R s (typically a few Ohms) also effects the characteristic at low injection. The
voltage drop across the junction is reduced by R s I . Thus, the I –V characteristic taking into account
the effect of the series resistance is
I = I s
exp
e (V − R s I )
n kT
− 1
.
(21.157)
567
(a)
(b)
Fig. 21.49 a Integral χ (21.147) multiplied by exp(βV /2) in order to extract the differences on a linear scale. Solid line:
Exact numerical calculation, dash-dotted line: standard approximation with constant maximum rate, dashed line: this
work (approximation with constant field). As material parameters we have used room temperature and n i = 10 10 cm −3
(Si), n n0 = 10 18 cm −3 and p p0 = 10 17 cm −3 . b Logarithmic slope of band-impurity recombination current in the forward
bias regime for various values of the built-in voltage V bi = 0.6, 1.0, and 1.4 eV and in the limit V bi → ∞. Adapted
from [1766]
Fig. 21.50 Theoretical
modeling of charge-carrier
concentration, intrinsic
Fermi level (potential) ψ
and quasi-Fermi levels
(with arbitrary offset) for a
Si p + n diode for various
current densities: a
10 A cm −2 , b 10 3 A cm −2
and c 10 4 A cm −2 .
N A = 10 18 cm −3 ,
N D = 10 16 cm −3 ,
τ n = 0.3 ns, τ p = 0.84 ns.
Adapted from [1767]
10
18
10
10
10
10
10
16
14
12
10
8
Carrier
-3
n
p
p
n
n
p
p
n
n
p
p
n
2
1
0 (a)
(c)
(b)
recombination current is determined by the ratio of w and L n + L p . Since in the forward direction,
the depletion-layer width tends towards zero (for flat-band conditions), the radiative-recombination
current is dominated by the recombination in the neutral region(s).
For high injection current (under forward bias), the injected minority carrier density can become
comparable with the majority carrier density. In this case, diffusion and drift need to be considered. At
large current density, the voltage drop across the junction is small compared to the ohmic voltage drop
across the current path. In the simulation (Fig. 21.50), the high-injection effects start on the n-doped
side because it has been modeled with the lower doping (N D < N A ).
The series resistance R s (typically a few Ohms) also effects the characteristic at low injection. The
voltage drop across the junction is reduced by R s I . Thus, the I –V characteristic taking into account
the effect of the series resistance is
I = I s
exp
e (V − R s I )
n kT
− 1
.
(21.157)