188
Compact Models for Integrated Circuit Design
In order to model V bs dependence of NWE, a model parameter K 3B can be used.
Thus, Equation 5.30 can be expressed to include the effect of body bias as
∆V
K K V
T
W
W
thW
Bbs
ox
eff
s
=
+
(
) ′ +
3
3
0
φ
(5.31)
Thus, three fitting parameters K 3 , K 3B , and W 0 are required to model NWE.
Here ′
W eff is the effective channel width with the additional fitting parameter
W 0 for accurate fitting of the measured data. Equation 5.31 models NWE in
MOSFETs; however it does not model SCE of the narrow devices. In order
to model SCE in narrow channel devices, we use Equations 5.24 and 5.25 to
obtain the shift in V th for narrow- and short channel devices as
∆V
W
W L W l
V
thWL
eff eff tw
bs
s
=
(
) −
−
(
)
0 5
0
1
1
.
cosh
.
DVT
DVT
φ
(5.32)
where:
l
K T W K
WV
tw
si ox d
o x
b s
=
+
(
)
1
2
DVT
.
is the characteristic length for short
and narrow devices
Equation 5.32 models V th shift in the short- and narrow channel devices
whereas Equation 5.31 models that in narrow- and long channel devices. The
final expression for V th including nonuniform substrate concentration and
small geometry effects is given by
V V n
V
V
V
th
th
th
th
=
−
+
+
+
(
)
(
)
(
,
)
on uniform substarate
NWE
N WE SCE
∆
∆
∆ t th
th
th
V
V
(
)
(
)
(
)
SCE
D IBL
D ITS
+
+
∆
∆
(5.33)
Thus, combining Equations 5.18, 5.19, 5.25, 5.28, 5.31, and 5.32, we can show
the expression for V th as used in BSIM4 model for circuit CAD.
V V
K
V
L
L
K
L
L
th
TH
ox
s
b seff
s
PEB
eff
ox
PE
=
+
−
−
(
) +
+
+
0
1
1
0
1
1
φ
φ .
e eff
s
o x bseff
bseff
ox
eff
s
K V
K K B V
T
W
W
−
−
+
+
(
) ′ +
−
1
3
3
1
2
0
φ
φ
.
2 2
0
1
1
0
1
DVT
DVT
DVT
DVT
W
W L W l
L l
eff
eff tw
eff t
cosh
cosh
.
(
)
−
+
(
)
−
−
(
)
−
+
(
)
1
1
2
0
V
L
bseff
s
bseff
eff
φ
EAT EATB V
DSUB
.
cosh
l l
V nv
L
L
e
t
ds
kT
eff
eff
Vds
0
1
1
0 1
(
)
(
) −
−
+
+
(
)
−
.
. ln
.
DVTP
DVTP
(5.34)
Compact Models for Integrated Circuit Design
In order to model V bs dependence of NWE, a model parameter K 3B can be used.
Thus, Equation 5.30 can be expressed to include the effect of body bias as
∆V
K K V
T
W
W
thW
Bbs
ox
eff
s
=
+
(
) ′ +
3
3
0
φ
(5.31)
Thus, three fitting parameters K 3 , K 3B , and W 0 are required to model NWE.
Here ′
W eff is the effective channel width with the additional fitting parameter
W 0 for accurate fitting of the measured data. Equation 5.31 models NWE in
MOSFETs; however it does not model SCE of the narrow devices. In order
to model SCE in narrow channel devices, we use Equations 5.24 and 5.25 to
obtain the shift in V th for narrow- and short channel devices as
∆V
W
W L W l
V
thWL
eff eff tw
bs
s
=
(
) −
−
(
)
0 5
0
1
1
.
cosh
.
DVT
DVT
φ
(5.32)
where:
l
K T W K
WV
tw
si ox d
o x
b s
=
+
(
)
1
2
DVT
.
is the characteristic length for short
and narrow devices
Equation 5.32 models V th shift in the short- and narrow channel devices
whereas Equation 5.31 models that in narrow- and long channel devices. The
final expression for V th including nonuniform substrate concentration and
small geometry effects is given by
V V n
V
V
V
th
th
th
th
=
−
+
+
+
(
)
(
)
(
,
)
on uniform substarate
NWE
N WE SCE
∆
∆
∆ t th
th
th
V
V
(
)
(
)
(
)
SCE
D IBL
D ITS
+
+
∆
∆
(5.33)
Thus, combining Equations 5.18, 5.19, 5.25, 5.28, 5.31, and 5.32, we can show
the expression for V th as used in BSIM4 model for circuit CAD.
V V
K
V
L
L
K
L
L
th
TH
ox
s
b seff
s
PEB
eff
ox
PE
=
+
−
−
(
) +
+
+
0
1
1
0
1
1
φ
φ .
e eff
s
o x bseff
bseff
ox
eff
s
K V
K K B V
T
W
W
−
−
+
+
(
) ′ +
−
1
3
3
1
2
0
φ
φ
.
2 2
0
1
1
0
1
DVT
DVT
DVT
DVT
W
W L W l
L l
eff
eff tw
eff t
cosh
cosh
.
(
)
−
+
(
)
−
−
(
)
−
+
(
)
1
1
2
0
V
L
bseff
s
bseff
eff
φ
EAT EATB V
DSUB
.
cosh
l l
V nv
L
L
e
t
ds
kT
eff
eff
Vds
0
1
1
0 1
(
)
(
) −
−
+
+
(
)
−
.
. ln
.
DVTP
DVTP
(5.34)