151
Large Geometry MOSFET Compact Models
I
W
L
Q y dV
ds
s
i
ch
V
V V
sb
sb
ds
=
+
∫
µ
( )
(4.55)
In order to derive simplified regional compact MOSFET models for circuit
CAD, we will make further simplifying assumptions.
Assumption 6: Let us assume that the diffusion current is negligibly
small so that the current flow along the channel in the device is only
due to the drift of minority carriers by the applied drain voltage, V ds .
This is a fairly good assumption provided the device is in strong
inversion; that is, the gate voltage is greater than the threshold voltage (V gs > V th or f s > 2f B ). If the diffusion current is neglected, then
from Equation 4.16 we can write for an nMOSFET device
J y qn x y E y
qn x y
x y
y
y
n
s
s
s
( )
( , ) ( )
( , ) ( , )
( )
≅
= −
∂
∂
µ
µ
φ
(4.56)
where we can safely use ∂ ∂
(
)= ∂ ∂
(
)
φ
φ
n
s
y
y
/
/ , that is, the gradients of
quasi-Fermi potential and surface potential along the channel are
the same in strong inversion. Therefore, for nMOSFET devices at
strong inversion, we can write
φ
φ
s
s
ch
y
V y
( )
( )
( )
=
+
0
(4.57)
where:
f s (0) is the surface potential at y = 0 (source end)
For the simplicity of calculation, it is more convenient to express the
channel potential in terms of the source potential, V sb and channel
voltage V(y) at any point y in the channel due to the applied drain
voltage V ds so that
V y V V y
ch
sb
( )
( )
=
+
(4.58)
where V(y) now varies from 0 at L = 0 at the source end to V ds at y = L
at the drain end of the channel. Then using Equation 4.58, at strong
inversion (f s = 2f B ), Equation 4.57 can be expressed as
φ
φ
s
B
sb
y
V V y
( )
( )
=
+
+
2
(4.59)
where f s (0) = 2f B at strong inversion, and V(y) varies from 0 at the
source end to V ds at the drain end. Now, substituting Equation 4.59
in 4.56, we get
J y qn x y E y
qn x y
x y
dV
dy
n
s
s
( )
( , ) ( )
( , ) ( , )
=
= −
µ
µ
(4.60)
Large Geometry MOSFET Compact Models
I
W
L
Q y dV
ds
s
i
ch
V
V V
sb
sb
ds
=
+
∫
µ
( )
(4.55)
In order to derive simplified regional compact MOSFET models for circuit
CAD, we will make further simplifying assumptions.
Assumption 6: Let us assume that the diffusion current is negligibly
small so that the current flow along the channel in the device is only
due to the drift of minority carriers by the applied drain voltage, V ds .
This is a fairly good assumption provided the device is in strong
inversion; that is, the gate voltage is greater than the threshold voltage (V gs > V th or f s > 2f B ). If the diffusion current is neglected, then
from Equation 4.16 we can write for an nMOSFET device
J y qn x y E y
qn x y
x y
y
y
n
s
s
s
( )
( , ) ( )
( , ) ( , )
( )
≅
= −
∂
∂
µ
µ
φ
(4.56)
where we can safely use ∂ ∂
(
)= ∂ ∂
(
)
φ
φ
n
s
y
y
/
/ , that is, the gradients of
quasi-Fermi potential and surface potential along the channel are
the same in strong inversion. Therefore, for nMOSFET devices at
strong inversion, we can write
φ
φ
s
s
ch
y
V y
( )
( )
( )
=
+
0
(4.57)
where:
f s (0) is the surface potential at y = 0 (source end)
For the simplicity of calculation, it is more convenient to express the
channel potential in terms of the source potential, V sb and channel
voltage V(y) at any point y in the channel due to the applied drain
voltage V ds so that
V y V V y
ch
sb
( )
( )
=
+
(4.58)
where V(y) now varies from 0 at L = 0 at the source end to V ds at y = L
at the drain end of the channel. Then using Equation 4.58, at strong
inversion (f s = 2f B ), Equation 4.57 can be expressed as
φ
φ
s
B
sb
y
V V y
( )
( )
=
+
+
2
(4.59)
where f s (0) = 2f B at strong inversion, and V(y) varies from 0 at the
source end to V ds at the drain end. Now, substituting Equation 4.59
in 4.56, we get
J y qn x y E y
qn x y
x y
dV
dy
n
s
s
( )
( , ) ( )
( , ) ( , )
=
= −
µ
µ
(4.60)