215
Compact Models for Small Geometry MOSFETs
E y E
y
l
E
y l
y l
E
y
l
c
i
c
i
i
c
i
( )
cosh
exp( / ) exp( / )
exp
=
=
−
−
≅
2
1
2
(5.119)
Since l i is very small and y/l i is a very large number, exp(−y/l i ) is negligibly
small; then differentiating Equation 5.119 we get
dE
dy
E
y
l
l
E
l
c
i
i
i
=
=
1
2
1
exp
.
(5.120)
Therefore,
−
= −
=
E
dy
dE
E
l
E
l E
i
i
2
2
(5.121)
Substituting Equation 5.121 in Equation 5.118, we get
I
I A lE
B
E y
d E
sub
d s i
i
i
Ec
Em
= −
−
∫
exp
( )
1
(5.122)
Since the exponential term in Equation 5.122 has a pronounced peak at
E = E m , we evaluate it at E = E m and let it be constant over the region so that
we can remove it from the integral. After this simplification, Equation 5.122
can be solved for I sub as
I
I A l E
B
E y
d E
sub
d s i i m
i
Ec
Em
= −
−
∫
exp
( )
1
(5.123)
After integration and simplification, we can show assuming E c << E m ,
I
I
A
B
l E
B
E
sub
d s
i
i
i m
i
m
≅
−
exp
(5.124)
Substituting for E m from Equation 5.117 and Equation 5.124 can be expressed
in terms of drain voltage as
I
I
A
B
V V
l B
V V
sub
d s
i
i
ds
dsat
i i
ds
dsat
≅
−
(
)
−
−
exp
(5.125)
Equation 5.125 is used for substrate current modeling. Note that Equation 5.125
is independent of device geometry. In order to model channel length dependence of I sub , the ratio A i /B i can be replaced by (α 0 + α 1 /L eff ) to express
I
L
V V
V V
I
sub
eff
ds
dsat
ds
dsat
dsa
≅
+
−
(
)
−
−
α
α
β
0
1
exp
.
(5.126)
Compact Models for Small Geometry MOSFETs
E y E
y
l
E
y l
y l
E
y
l
c
i
c
i
i
c
i
( )
cosh
exp( / ) exp( / )
exp
=
=
−
−
≅
2
1
2
(5.119)
Since l i is very small and y/l i is a very large number, exp(−y/l i ) is negligibly
small; then differentiating Equation 5.119 we get
dE
dy
E
y
l
l
E
l
c
i
i
i
=
=
1
2
1
exp
.
(5.120)
Therefore,
−
= −
=
E
dy
dE
E
l
E
l E
i
i
2
2
(5.121)
Substituting Equation 5.121 in Equation 5.118, we get
I
I A lE
B
E y
d E
sub
d s i
i
i
Ec
Em
= −
−
∫
exp
( )
1
(5.122)
Since the exponential term in Equation 5.122 has a pronounced peak at
E = E m , we evaluate it at E = E m and let it be constant over the region so that
we can remove it from the integral. After this simplification, Equation 5.122
can be solved for I sub as
I
I A l E
B
E y
d E
sub
d s i i m
i
Ec
Em
= −
−
∫
exp
( )
1
(5.123)
After integration and simplification, we can show assuming E c << E m ,
I
I
A
B
l E
B
E
sub
d s
i
i
i m
i
m
≅
−
exp
(5.124)
Substituting for E m from Equation 5.117 and Equation 5.124 can be expressed
in terms of drain voltage as
I
I
A
B
V V
l B
V V
sub
d s
i
i
ds
dsat
i i
ds
dsat
≅
−
(
)
−
−
exp
(5.125)
Equation 5.125 is used for substrate current modeling. Note that Equation 5.125
is independent of device geometry. In order to model channel length dependence of I sub , the ratio A i /B i can be replaced by (α 0 + α 1 /L eff ) to express
I
L
V V
V V
I
sub
eff
ds
dsat
ds
dsat
dsa
≅
+
−
(
)
−
−
α
α
β
0
1
exp
.
(5.126)