114
Compact Models for Integrated Circuit Design
where:
Δz is the shift in the centroid of inversion charge
Since the peak of the inversion charge is away from the surface due to QM
effect, a higher V g overdrive is required to produce the same level of inversion
charge density predicted by classical theory. In other words, QM effect can
be considered to reduce the net inversion charge density. Thus, the inversion
layer quantization can be modeled as bandgap widening due to an increase in
the effective bandgap energy, E g , by an amount ΔE g [13]. Then from Equation 2.14
relating E g and the intrinsic carrier concentration, we can show that the intrinsic carrier concentration n i
QM
due to QM effect is given by
n
n
E
kT
i
i
g
QM
CL
=
−
exp
∆
2
(3.83)
where:
∆E
E
E
g
g
g
=
−
(
)
QM
CL is the increase in the apparent value of E g due to QM
effect; here, E g
QM
is the energy gap due to QM effect and E g
CL
and n i
CL
are the energy gap and intrinsic carrier concentration, respectively,
without the QM effect denoting the classical expression
Equation 3.83 shows that the inversion layer quantization decreases the
intrinsic concentration compared to the classical value. We know from
Equation 3.72, Q i is proportional to n i through the term exp(–2f B /v KT ). Thus,
Q i decreases due to QM effect. This decrease in Q i due to QM effect has severe
consequences on MOS transistor device performance as we will discuss in
Chapter 9.
3.5 Capacitance of MOS Structure
In the previous section, we developed the mathematical foundation of MOS
capacitor system relating the charge and potential under different gate biasing conditions. In this section, we will discuss the basic characteristics of
MOS capacitor system under an applied bias. We know that the capacitance
of any system is the ratio of the variation in charge due to the corresponding variation in the small signal voltage. Thus, the total capacitance (C) of an
MOS structure in equilibrium is given by
C
d Q
dV
s
g
=
−
( )
(3.84)
From Equation 3.17, the applied bias of an MOS system is V V V
g
f b
o x
s
=
+
+ φ ;
since V fb is a constant, we can write
Compact Models for Integrated Circuit Design
where:
Δz is the shift in the centroid of inversion charge
Since the peak of the inversion charge is away from the surface due to QM
effect, a higher V g overdrive is required to produce the same level of inversion
charge density predicted by classical theory. In other words, QM effect can
be considered to reduce the net inversion charge density. Thus, the inversion
layer quantization can be modeled as bandgap widening due to an increase in
the effective bandgap energy, E g , by an amount ΔE g [13]. Then from Equation 2.14
relating E g and the intrinsic carrier concentration, we can show that the intrinsic carrier concentration n i
QM
due to QM effect is given by
n
n
E
kT
i
i
g
QM
CL
=
−
exp
∆
2
(3.83)
where:
∆E
E
E
g
g
g
=
−
(
)
QM
CL is the increase in the apparent value of E g due to QM
effect; here, E g
QM
is the energy gap due to QM effect and E g
CL
and n i
CL
are the energy gap and intrinsic carrier concentration, respectively,
without the QM effect denoting the classical expression
Equation 3.83 shows that the inversion layer quantization decreases the
intrinsic concentration compared to the classical value. We know from
Equation 3.72, Q i is proportional to n i through the term exp(–2f B /v KT ). Thus,
Q i decreases due to QM effect. This decrease in Q i due to QM effect has severe
consequences on MOS transistor device performance as we will discuss in
Chapter 9.
3.5 Capacitance of MOS Structure
In the previous section, we developed the mathematical foundation of MOS
capacitor system relating the charge and potential under different gate biasing conditions. In this section, we will discuss the basic characteristics of
MOS capacitor system under an applied bias. We know that the capacitance
of any system is the ratio of the variation in charge due to the corresponding variation in the small signal voltage. Thus, the total capacitance (C) of an
MOS structure in equilibrium is given by
C
d Q
dV
s
g
=
−
( )
(3.84)
From Equation 3.17, the applied bias of an MOS system is V V V
g
f b
o x
s
=
+
+ φ ;
since V fb is a constant, we can write