164
Compact Models for Integrated Circuit Design
subthreshold or weak inversion current and occurs when V gs < V th or f B < f s < 2f B .
Unlike the inversion region where drift current dominates, the subthreshold
region conduction is dominated by diffusion current as shown in Figure 4.9.
The subthreshold region current is important since this is a major contributor
to device leakage current that affects the dynamic circuit performance and
determines CMOS standby power. In this region of operation, the assumption
I ds = 0 in Equations 4.87 and 4.100 (by assumption 6) is not valid.
In the subthreshold region of operation, Q i << Q b , and therefore, the surface potential f s (or band bending) is nearly constant from the source to drain
end of the device. This means that we can replace f s (y) in the subthreshold
region by some constant value, f ss . Then, the bulk charge Q b in Equation 4.38
can be expressed as
Q
C
y
C
b
o x
s
ox
ss
= −
≅ −
γ φ
γ φ
( )
(4.101)
Again, since Q i << Q b , we have Q s ≈ Q b , so that Equation 4.6 becomes
V
V
Q
C
gb
fb
ss
b
ox
=
+ −
φ
(4.102)
Substituting for Q b from Equation 4.101 to Equation 4.93, we get
V
V
V V
gb
fb
ss
ss
ss
ss
gb
fb
=
+ +
( ) +
−
−
(
) =
φ
γ φ
φ
γ φ
or
2
0
(4.103)
Solving the quadratic Equation 4.103 we can show
φ
γ
γ
ss
gb
fb
V V
= − +
+
−
2
4
2
2
(4.104)
Equation 4.104 shows that f ss is almost linearly dependent on V gb = V gs for
V sb = 0. It should be emphasized that f ss is a constant in the subthreshold
region for long channel devices only. As the channel length becomes shorter,
f ss no longer remains constant over the entire channel length.
Since f ss is a constant, E y = −df ss /dy = 0. Therefore, the only current that can
flow is the diffusion current as can be seen from Equation 4.16 and is given by
J x y qD
dn
dy
q v
dn
dy
n
n
s kT
( , ) =
= (
)
µ
(4.105)
where from Einstein relation D n = μ s v kT . Integrating J n (x,y) from x = 0 at the
Si/SiO 2 interface to x = X inv at the end of the inversion layer width, we get for
an nMOSFET device of channel length, L, and width, W
Compact Models for Integrated Circuit Design
subthreshold or weak inversion current and occurs when V gs < V th or f B < f s < 2f B .
Unlike the inversion region where drift current dominates, the subthreshold
region conduction is dominated by diffusion current as shown in Figure 4.9.
The subthreshold region current is important since this is a major contributor
to device leakage current that affects the dynamic circuit performance and
determines CMOS standby power. In this region of operation, the assumption
I ds = 0 in Equations 4.87 and 4.100 (by assumption 6) is not valid.
In the subthreshold region of operation, Q i << Q b , and therefore, the surface potential f s (or band bending) is nearly constant from the source to drain
end of the device. This means that we can replace f s (y) in the subthreshold
region by some constant value, f ss . Then, the bulk charge Q b in Equation 4.38
can be expressed as
Q
C
y
C
b
o x
s
ox
ss
= −
≅ −
γ φ
γ φ
( )
(4.101)
Again, since Q i << Q b , we have Q s ≈ Q b , so that Equation 4.6 becomes
V
V
Q
C
gb
fb
ss
b
ox
=
+ −
φ
(4.102)
Substituting for Q b from Equation 4.101 to Equation 4.93, we get
V
V
V V
gb
fb
ss
ss
ss
ss
gb
fb
=
+ +
( ) +
−
−
(
) =
φ
γ φ
φ
γ φ
or
2
0
(4.103)
Solving the quadratic Equation 4.103 we can show
φ
γ
γ
ss
gb
fb
V V
= − +
+
−
2
4
2
2
(4.104)
Equation 4.104 shows that f ss is almost linearly dependent on V gb = V gs for
V sb = 0. It should be emphasized that f ss is a constant in the subthreshold
region for long channel devices only. As the channel length becomes shorter,
f ss no longer remains constant over the entire channel length.
Since f ss is a constant, E y = −df ss /dy = 0. Therefore, the only current that can
flow is the diffusion current as can be seen from Equation 4.16 and is given by
J x y qD
dn
dy
q v
dn
dy
n
n
s kT
( , ) =
= (
)
µ
(4.105)
where from Einstein relation D n = μ s v kT . Integrating J n (x,y) from x = 0 at the
Si/SiO 2 interface to x = X inv at the end of the inversion layer width, we get for
an nMOSFET device of channel length, L, and width, W