246
Compact Models for Integrated Circuit Design
From Equations 6.67 and 6.68, we observe that at V ds = 0 and V gs = V th ,
Q D = Q S = 0.5 WLC ox (n − 1)v kT . It is also observed from Equations 6.67 and
6.68 that Q S and Q D depend weakly on V ds . This is due to fact that for V ds
greater than a few v kT , the terms involving V ds become negligible and therefore, Q S = 2Q D .
Since in weak inversion, the bulk charge Q B is virtually independent of the
S/D voltage V ds , we can use Equation 6.24 for Q B , which at the boundary of
the strong inversion can be rewritten as
Q
WLC
V
B
o x
B
sb
= −
+
γ φ
2
(6.69)
Equation 6.69 is the same as the first term of the first expression in Equation
6.59. If the channel charge is assumed zero (Q I = 0) in the subthreshold
region, the gate charge becomes equal to the bulk charge. Thus, Q G = −Q B .
6.3.1.3 Accumulation
In the accumulation region of a MOSFET device operation, V gb < V fb ; thus a thin
layer of majority carriers are formed at the interface, forming a parallel plate
capacitor with the gate. In this case, the bulk charge Q B is simply written as
Q
WLC V V V
B
o x
g s
s b
f b
= −
+
−
(
)
(6.70)
Since there is no current flow, the gate charge is given by
Q
Q WLC V V V
G
B
ox
gs
sb
fb
= − =
+
−
(
)
(6.71)
6.3.2 Long Channel Capacitance Model
We can now derive the expressions for capacitances associated with a
MOSFET using the equations derived for various charges in different regions
of device operation and the definition of C ij in Equation 6.35. The mathematics to derive 12 capacitances is basic, however involved and long. It is left as
an exercise for the readers.
The expressions for C GD and C DG in the linear region are obtained by differentiating Q G (Equation 6.57) with respect to V d (or V ds ) and Q D (Equation 6.54) with
respect to V g or V gs , respectively, and using A and B defined in Equation 6.55,
that is,
C
Q
V
WLC
V V
V
A
V
GD
G
d
ox
gs
th
ds
ds
= −
∂
∂
=
− +
−
(
) −
+
1
2
1
1
1 2
1
3
( )α
α
= −
∂
∂
=
+
−
(
) −
−
(
)
C
Q
V
WLC
A
V V
V
B
DG
D
g
ox
gs
th
ds
1
2
1
1 2
1 4
( )α
(6.72)
Compact Models for Integrated Circuit Design
From Equations 6.67 and 6.68, we observe that at V ds = 0 and V gs = V th ,
Q D = Q S = 0.5 WLC ox (n − 1)v kT . It is also observed from Equations 6.67 and
6.68 that Q S and Q D depend weakly on V ds . This is due to fact that for V ds
greater than a few v kT , the terms involving V ds become negligible and therefore, Q S = 2Q D .
Since in weak inversion, the bulk charge Q B is virtually independent of the
S/D voltage V ds , we can use Equation 6.24 for Q B , which at the boundary of
the strong inversion can be rewritten as
Q
WLC
V
B
o x
B
sb
= −
+
γ φ
2
(6.69)
Equation 6.69 is the same as the first term of the first expression in Equation
6.59. If the channel charge is assumed zero (Q I = 0) in the subthreshold
region, the gate charge becomes equal to the bulk charge. Thus, Q G = −Q B .
6.3.1.3 Accumulation
In the accumulation region of a MOSFET device operation, V gb < V fb ; thus a thin
layer of majority carriers are formed at the interface, forming a parallel plate
capacitor with the gate. In this case, the bulk charge Q B is simply written as
Q
WLC V V V
B
o x
g s
s b
f b
= −
+
−
(
)
(6.70)
Since there is no current flow, the gate charge is given by
Q
Q WLC V V V
G
B
ox
gs
sb
fb
= − =
+
−
(
)
(6.71)
6.3.2 Long Channel Capacitance Model
We can now derive the expressions for capacitances associated with a
MOSFET using the equations derived for various charges in different regions
of device operation and the definition of C ij in Equation 6.35. The mathematics to derive 12 capacitances is basic, however involved and long. It is left as
an exercise for the readers.
The expressions for C GD and C DG in the linear region are obtained by differentiating Q G (Equation 6.57) with respect to V d (or V ds ) and Q D (Equation 6.54) with
respect to V g or V gs , respectively, and using A and B defined in Equation 6.55,
that is,
C
Q
V
WLC
V V
V
A
V
GD
G
d
ox
gs
th
ds
ds
= −
∂
∂
=
− +
−
(
) −
+
1
2
1
1
1 2
1
3
( )α
α
= −
∂
∂
=
+
−
(
) −
−
(
)
C
Q
V
WLC
A
V V
V
B
DG
D
g
ox
gs
th
ds
1
2
1
1 2
1 4
( )α
(6.72)