116
Compact Models for Integrated Circuit Design
3.5.1.1 Accumulation
In strong accumulation, f s   <<  0; then considering the majority carrier
term only in Equation 3.87 and recognizing that exp
/
−
(
)>>
φ s k T
v
1 and
exp
/
/
−
(
)>> (
)
φ
φ
s
k T
s
kT
v
v , we can show after simplification
C
K
L
e
s
si
d
v
s
k T
=
−(
)
ε
φ
0
2
2
(3.88)
Since f s  << 0, for large f s , C s becomes very large; therefore, from Equation
3.86, we get
1
1
1
1
C C
C
C
ox
s
o x
=
+
≅
(3.89)
Thus, in accumulation region
C C ox
≅
(
)
accumulation
(3.90)
3.5.1.2 Flat Band
At flat band condition, f s  = 0, the inversion charge term (i.e., term containing
n po /p po ) in Equation 3.87 can be neglected to get
C
d Q
d
K
L
e
e
v
s
s
s
si
d
v
v
s
t
s t
s t
=
−
= −
−




+ (
)−
(
)


−(
)
−(
)
(
)
φ
ε
φ
φ
φ
2
1
2
1
0
 

 
1 2
/
(3.91)
Since the direct substitution of f s   =  0 in Equation 3.91 results in indeterminate, we simplify Equation 3.91 by series expansion using
e
x
x
x
x
x
= +(
)+ ( ) + ( ) + −∞ < < ∞
1
1
2
3
2
3
!
!
!
,

, where x
v
s
kT
≡ −φ
, then
C
K
L
v
v
v
s
si
d
s
k T
skT
s
k T
≅ −
− −(
)+ ( ) −(
)



 + ( ) −(
)




2
1 1
1 2
1 6
0
2
ε
φ
φ
φ
3 3
2
3
2 1
1 2
1 6
+
{
}






− (
)+ ( ) −(
)



 + ( ) −(
)





φ
φ
φ
s
k T
skT
s
k T
v
v
v
+ +
{
} + ( )−






= −
−(
)+ ( )(
) −

φ
ε
φ
φ
s
k T
si
d
s
k T
s
kT
v
K
L
v
v
1
2
1 2
1 6
1 2
0
2
/
( ( )(
) +

 

 
( )(
) − ( )(
) +

 

 
= −
φ
φ
φ
s
k T
s
k T
s
kT
v
v
v
3
2
3
1 2
2 1 2
1 6


/
2 2
1 1 2
1 6
2
0
2
K
L
v
v
v
si
d
s
k T
skT
s
k T
s
ε
φ
φ
φ
φ
−(
)



 − ( )(
)− ( )(
) +

 

 

v v
v
K
L
v
v
kT
s
k T
si
d
s
k T
s
k
(
) − ( )(
)+




=
− ( )(
)− ( )
1 1 6
1 1 2
1 6
1 2
0
φ
ε
φ
φ

/
T T
s
k T
v
(
) +

 

 
− ( )(
)+




2
1 2
1 1 6


φ
/