108
Compact Models for Integrated Circuit Design
Using the relation n N
v
i
a
B
kT
2
2
2
/
exp
=
−
(
)
φ
from Equation 3.41 in Equation
3.67, we get for the total induced charge density in the semiconductor as
Q
q K N v
v
e
s
s i
a kT
s
kT
v
s
B
kT
= −
+
−
(
)
2
0
2
1 2
ε
φ
φ
φ
(3.68)
Note that the induced charge represented by Equation 3.68 is the sum of the
inversion charge Q i and the depletion charge Q b , that is,
Q Q Q
s
i
b
= +
(3.69)
Using the expressions for Q
q K N
b
s i
a s
= − 2
0
ε
φ from Equation 3.64 and Q s
from Equation 3.68 in Equation 3.69, we get the inversion charge per unit
area as
Q
q K N
v e
i
s i
a
s
k T
v
s
s
B
kT
= −
+
(
) −
−
(
)
2
0
2
1 2
ε
φ
φ
φ
φ
/
(3.70)
Equation 3.70 shows the relation between the inversion charge density Q i
and surface potential f s for an MOS capacitor system. Figure 3.14 shows the
dependence of Q i , Q b , and Q s on f s . It is observed from Figure 3.14 that Q b
does not vary significantly. On the other hand, Q i and Q s clearly show two
distinct regions of operation depending on the value of f s . These regions
become more apparent on log(Q i ) versus f s plot as shown in Figure 3.15.
These regions are (1) weak inversion for lower values of f s and (2) strong
inversion at higher values of f s . Classically, the condition separating the
0.3
Charge density (C/cm
2
)
0.0E+00
5.0E−08
1.0E−07
1.5E−07
2.0E−07
N a = 1 × 10
16 cm −3
Q s
Q i
Q b
V fb = 0
0.4
0.5
0.6
Surface potential (V)
0.7
0.8
0.9
FIGURE 3.14
Variation of Q b , Q s , and Q i as a function of f s obtained by Equations 3.64, 3.68, and 3.70, respectively, for an MOS capacitor system on a p-type substrate with N a = 1 × 10 16 cm –3 .
Compact Models for Integrated Circuit Design
Using the relation n N
v
i
a
B
kT
2
2
2
/
exp
=
−
(
)
φ
from Equation 3.41 in Equation
3.67, we get for the total induced charge density in the semiconductor as
Q
q K N v
v
e
s
s i
a kT
s
kT
v
s
B
kT
= −
+
−
(
)
2
0
2
1 2
ε
φ
φ
φ
(3.68)
Note that the induced charge represented by Equation 3.68 is the sum of the
inversion charge Q i and the depletion charge Q b , that is,
Q Q Q
s
i
b
= +
(3.69)
Using the expressions for Q
q K N
b
s i
a s
= − 2
0
ε
φ from Equation 3.64 and Q s
from Equation 3.68 in Equation 3.69, we get the inversion charge per unit
area as
Q
q K N
v e
i
s i
a
s
k T
v
s
s
B
kT
= −
+
(
) −
−
(
)
2
0
2
1 2
ε
φ
φ
φ
φ
/
(3.70)
Equation 3.70 shows the relation between the inversion charge density Q i
and surface potential f s for an MOS capacitor system. Figure 3.14 shows the
dependence of Q i , Q b , and Q s on f s . It is observed from Figure 3.14 that Q b
does not vary significantly. On the other hand, Q i and Q s clearly show two
distinct regions of operation depending on the value of f s . These regions
become more apparent on log(Q i ) versus f s plot as shown in Figure 3.15.
These regions are (1) weak inversion for lower values of f s and (2) strong
inversion at higher values of f s . Classically, the condition separating the
0.3
Charge density (C/cm
2
)
0.0E+00
5.0E−08
1.0E−07
1.5E−07
2.0E−07
N a = 1 × 10
16 cm −3
Q s
Q i
Q b
V fb = 0
0.4
0.5
0.6
Surface potential (V)
0.7
0.8
0.9
FIGURE 3.14
Variation of Q b , Q s , and Q i as a function of f s obtained by Equations 3.64, 3.68, and 3.70, respectively, for an MOS capacitor system on a p-type substrate with N a = 1 × 10 16 cm –3 .