165
Large Geometry MOSFET Compact Models
I y W q v
dn
dy
W v
d qnX
dy
W v
dQ
dy
ds
s kT
X
s kT
inv
s kT
i
inv
( ) =
(
) =
(
)
=
∫
µ
µ
µ
0
(4.106)
where:
Q i = qnX inv is the inversion charge per unit area at any point y along the
channel in the subthreshold region
Now, consider Q is and Q id are the inversion charge densities at y = 0 and
y = L, respectively. Then integrating Equation 4.106 from (y = 0, Q i (y) = Q is )
to (y = L, Q i (y) = Q id ), we get
I
W
L
v Q Q
ds
s
k T
i d
i s
=
−
(
)
µ
(4.107)
Now, in order to calculate the subthreshold current from Equation 4.107, we
need to find the inversion charge in the weak inversion regime, f B < f s < 2f B .
Again, we solve Poisson’s equation to calculate Q s and find an expression
for Q i in the weak inversion following the procedure in Chapter 3 (Equation
3.68). Then for MOSFETs in the weak inversion region, we can show
Q
q K N
v e
s
s i
b ss
kT
ss
V y v
ss
B
c h
k T
≅ −
+
−
−
2
1
0
2
1 2
ε
φ
φ
φ
φ
( )
/
(4.108)
Let us assume that the exponential term in Equation 4.108 is much smaller
than f ss . Then using series expansion 1
1
2
+ ≅ + ( )
x
x , we get for the total
charge Q s in the substrate at weak inversion as
Q
q K N
v e
Q
s
s i
b ss
kT
ss
V y v
b
ss
B
c h
k T
≅ −
+
=
+ −
−
−
2
1 2
0
2
ε
φ
φ
φ
φ
( )
q qK N v e
si
b
ss
kT
V y v
ss
B
c h
k T
ε
φ
φ
φ
0
2
2
−
−
( )
(4.109)
where Q
qK
b
s i
s s
= − 2
0
ε
φ
N b
as shown in Equation 3.64; since, Q s = Q b + Q i ,
from Equation 4.109, the minority carrier charge density at the weak inversion region, f B < f s < 2f B , of nMOSFETs is given by
Q
qK N v e
i
si
b
ss
kT
V y v
ss
B
c h
k T
= −
−
−
ε
φ
φ
φ
0
2
2
( )
(4.110)
Again, from Equation 3.62, the width of the depletion region X
K
qN
d
si
ss
b
= 2
0
ε φ
;
then the depletion capacitance C d (=K si ε 0 /X d ) is given by
Large Geometry MOSFET Compact Models
I y W q v
dn
dy
W v
d qnX
dy
W v
dQ
dy
ds
s kT
X
s kT
inv
s kT
i
inv
( ) =
(
) =
(
)
=
∫
µ
µ
µ
0
(4.106)
where:
Q i = qnX inv is the inversion charge per unit area at any point y along the
channel in the subthreshold region
Now, consider Q is and Q id are the inversion charge densities at y = 0 and
y = L, respectively. Then integrating Equation 4.106 from (y = 0, Q i (y) = Q is )
to (y = L, Q i (y) = Q id ), we get
I
W
L
v Q Q
ds
s
k T
i d
i s
=
−
(
)
µ
(4.107)
Now, in order to calculate the subthreshold current from Equation 4.107, we
need to find the inversion charge in the weak inversion regime, f B < f s < 2f B .
Again, we solve Poisson’s equation to calculate Q s and find an expression
for Q i in the weak inversion following the procedure in Chapter 3 (Equation
3.68). Then for MOSFETs in the weak inversion region, we can show
Q
q K N
v e
s
s i
b ss
kT
ss
V y v
ss
B
c h
k T
≅ −
+
−
−
2
1
0
2
1 2
ε
φ
φ
φ
φ
( )
/
(4.108)
Let us assume that the exponential term in Equation 4.108 is much smaller
than f ss . Then using series expansion 1
1
2
+ ≅ + ( )
x
x , we get for the total
charge Q s in the substrate at weak inversion as
Q
q K N
v e
Q
s
s i
b ss
kT
ss
V y v
b
ss
B
c h
k T
≅ −
+
=
+ −
−
−
2
1 2
0
2
ε
φ
φ
φ
φ
( )
q qK N v e
si
b
ss
kT
V y v
ss
B
c h
k T
ε
φ
φ
φ
0
2
2
−
−
( )
(4.109)
where Q
qK
b
s i
s s
= − 2
0
ε
φ
N b
as shown in Equation 3.64; since, Q s = Q b + Q i ,
from Equation 4.109, the minority carrier charge density at the weak inversion region, f B < f s < 2f B , of nMOSFETs is given by
Q
qK N v e
i
si
b
ss
kT
V y v
ss
B
c h
k T
= −
−
−
ε
φ
φ
φ
0
2
2
( )
(4.110)
Again, from Equation 3.62, the width of the depletion region X
K
qN
d
si
ss
b
= 2
0
ε φ
;
then the depletion capacitance C d (=K si ε 0 /X d ) is given by