143
Large Geometry MOSFET Compact Models
At the source end of the channel, V ch (0) = 0, and at the drain-end of
the channel, V ch (L) = (V sb + V ds ). Thus, compared to the case of an MOS
capacitor, the quasi-Fermi potential is lowered by an amount V ch (y) at
the surface region of a MOSFET device. As a result, the surface electron
concentration (n s ) is lowered by a factor exp(−V ch (y)/v kT ). Then following
the derivation of minority carrier density expression (Equation 3.42)
for an MOS capacitor, we can write the minority carrier surface electron concentration at any point y in a MOSFET device as
n y N
y
V y
v
b
B
c h
kT
( )
exp
( )
( )
=
−
−
φ
φ
2
(4.24)
where the parameters have their usual meanings as defined in
Section 3.4.1. The minority carrier concentration changes due to
the applied bias; however, the majority carrier hole concentration
does not change with bias, and therefore, following MOS capacitor
Equation 3.39, we can write for the majority carrier concentration in
MOSFETs as p = N b exp[(−f(y)/v kT ].
Then using Equation 4.23, Equation 4.22 can be written as:
I y
W
dV y
dy
qn x y
x y dx
ds
ch
s
( )
( )
( , ) ( , )
= −
∝
∫
µ
0
(4.25)
Assumption 5: For the simplicity of long channel I ds calculation, we
assume μ s = constant at some average gate and drain electric field;
however, μ s depends on both E x and E y , as we will discuss in Chapter 5.
With this assumption, we can write Equation 4.25 as:
I y
W
dV y
dy
qn x y dx
ds
s
ch
( )
( )
( , )
= −
∝
∫
µ
0
(4.26)
Now, we define Q i as the mobile minority carrier charge density, that is
Q y
q n x y dx
i ( )
( , )
= −
∝
∫
0
(4.27)
Using Equation 4.27 in Equation 4.26, we get the general expression
for I ds (y) as
I y dy W Q y dV y
ds
s i
ch
( )
( )
( )
= µ
(4.28)
Again, assuming GCA is valid along the entire length of the channel, we get after integrating Equation 4.28 along the channel length
from y = 0 to y = L
Large Geometry MOSFET Compact Models
At the source end of the channel, V ch (0) = 0, and at the drain-end of
the channel, V ch (L) = (V sb + V ds ). Thus, compared to the case of an MOS
capacitor, the quasi-Fermi potential is lowered by an amount V ch (y) at
the surface region of a MOSFET device. As a result, the surface electron
concentration (n s ) is lowered by a factor exp(−V ch (y)/v kT ). Then following
the derivation of minority carrier density expression (Equation 3.42)
for an MOS capacitor, we can write the minority carrier surface electron concentration at any point y in a MOSFET device as
n y N
y
V y
v
b
B
c h
kT
( )
exp
( )
( )
=
−
−
φ
φ
2
(4.24)
where the parameters have their usual meanings as defined in
Section 3.4.1. The minority carrier concentration changes due to
the applied bias; however, the majority carrier hole concentration
does not change with bias, and therefore, following MOS capacitor
Equation 3.39, we can write for the majority carrier concentration in
MOSFETs as p = N b exp[(−f(y)/v kT ].
Then using Equation 4.23, Equation 4.22 can be written as:
I y
W
dV y
dy
qn x y
x y dx
ds
ch
s
( )
( )
( , ) ( , )
= −
∝
∫
µ
0
(4.25)
Assumption 5: For the simplicity of long channel I ds calculation, we
assume μ s = constant at some average gate and drain electric field;
however, μ s depends on both E x and E y , as we will discuss in Chapter 5.
With this assumption, we can write Equation 4.25 as:
I y
W
dV y
dy
qn x y dx
ds
s
ch
( )
( )
( , )
= −
∝
∫
µ
0
(4.26)
Now, we define Q i as the mobile minority carrier charge density, that is
Q y
q n x y dx
i ( )
( , )
= −
∝
∫
0
(4.27)
Using Equation 4.27 in Equation 4.26, we get the general expression
for I ds (y) as
I y dy W Q y dV y
ds
s i
ch
( )
( )
( )
= µ
(4.28)
Again, assuming GCA is valid along the entire length of the channel, we get after integrating Equation 4.28 along the channel length
from y = 0 to y = L