62
Compact Models for Integrated Circuit Design
From Equation 2.85, the equilibrium carrier concentrations in a pn-junction
are given by the expressions
φ bi
kT
no
po
kT
po
no
v
n
n
v
p
p
=
ln
ln
(2.109)
Therefore, from Equation 2.109, we can write for a pn-junction at equilibrium
n
n
v
p
p
v
n
p
bi
kT
p
n
bi
kT
o
o
o
o
=
=
exp
exp
φ
φ
(2.110)
Now, under the applied bias V d , we replace f bi by (f bi ± V d ); therefore, from
Equation of 2.110, the nonequilibrium carrier concentrations are given by
n n
V
v
p
p
V
v
n
p
bi
d
kT
p
n
bi
d
kT
=
−
=
−
exp
exp
φ
φ
(2.111)
where:
n p is the nonequilibrium minority electron concentration at the edge of the
depletion region in the neutral p-region
p n is the nonequilibrium hole concentration at the edge of the depletion
region in the neutral n-region as shown in Figure 2.21b
p
n
p
n
p po
p po
n no
n no
n po
p no
npo
p no
n p
p n
Depletion region
Depletion region
(a)
(b)
FIGURE 2.21
Carrier concentrations at the edge of depletion region: (a) pn-junction at equilibrium where p po
and n no are the equilibrium majority carrier hole and electron concentrations in the p-type and
n-type regions, respectively, whereas n po and p no are the equilibrium minority carrier electron
and hole concentrations in the p-type and n-type regions, respectively and (b) pn-junction after
minority carrier n p and p n injection in the bulk p-region and n-region, respectively.
Compact Models for Integrated Circuit Design
From Equation 2.85, the equilibrium carrier concentrations in a pn-junction
are given by the expressions
φ bi
kT
no
po
kT
po
no
v
n
n
v
p
p
=
ln
ln
(2.109)
Therefore, from Equation 2.109, we can write for a pn-junction at equilibrium
n
n
v
p
p
v
n
p
bi
kT
p
n
bi
kT
o
o
o
o
=
=
exp
exp
φ
φ
(2.110)
Now, under the applied bias V d , we replace f bi by (f bi ± V d ); therefore, from
Equation of 2.110, the nonequilibrium carrier concentrations are given by
n n
V
v
p
p
V
v
n
p
bi
d
kT
p
n
bi
d
kT
=
−
=
−
exp
exp
φ
φ
(2.111)
where:
n p is the nonequilibrium minority electron concentration at the edge of the
depletion region in the neutral p-region
p n is the nonequilibrium hole concentration at the edge of the depletion
region in the neutral n-region as shown in Figure 2.21b
p
n
p
n
p po
p po
n no
n no
n po
p no
npo
p no
n p
p n
Depletion region
Depletion region
(a)
(b)
FIGURE 2.21
Carrier concentrations at the edge of depletion region: (a) pn-junction at equilibrium where p po
and n no are the equilibrium majority carrier hole and electron concentrations in the p-type and
n-type regions, respectively, whereas n po and p no are the equilibrium minority carrier electron
and hole concentrations in the p-type and n-type regions, respectively and (b) pn-junction after
minority carrier n p and p n injection in the bulk p-region and n-region, respectively.