110
Compact Models for Integrated Circuit Design
Strong inversion is defined when the band bending at the surface is such
that f s  >> 2f B so that the inversion charge Q i is large compared to the depletion region charge Q b , that is,
Q
Q
i
b
>>
(
)
strong inversion
(3.73)
In this case, the exponential term in Equation 3.70 is large compared to f s and
f s  >> 2f B . Thus, at strong inversion we get
Q
q K N v e
i
s i
a kT
v
s
k T
≅ −
(
)
(
)
2
0
2
ε
φ
strong inversion
(3.74)
Thus, the inversion charge is an exponential function of the surface potential. Therefore, a small increment of the surface potential induces a large change in
the inversion layer charge.
Let us now find out the relation between the gate voltage V g and surface
potential f s . From Equation 3.23 we get
V V
Q
C
g
f b
s
s
ox
=
+ −
φ
(3.75)
Now, substituting for Q s from Equation 3.68 in Equation 3.23, we get
V V
qK N v
C
v
e
g
f b
s
si
a kT
ox
s
kT
v
s
B
kT
=
+ +
+






−
(
)
φ
ε
φ
φ
φ
2
0
2
1 2
(3.76)
0.3
1.E−14
1.E−13
1.E−12
1.E−11
1.E−10
1.E−09
Inversin charge (C/cm
2
)
1.E−08
1.E−07
1.E−06
1.E−05
1.E−04
0.4
0.5
Weak
inversion
Moderate
inversion
Strong
inversion
0.6
0.7
Surface potential (V)
0.8
0.9
1.0
1.1
2ϕ B
2ϕ B + 6v kT
FIGURE 3.16
Variation of Q i as function of f s for an MOS capacitor with p-type silicon substrate showing the weak, moderate, and strong inversion regions; Q i is obtained by Equation 3.70; here
N b  = 1 × 10 16  cm –3 and V fb  = 0.