136
Compact Models for Integrated Circuit Design
∇ = −
2
0
φ
ρ
ε
K si
(4.2)
Generally, a MOSFET is a 3D (three-dimensional) problem; however, for all
practical purposes (except for very small W and L), we can treat the system
as a 2D problem in the x and y directions only. We can further convert the 2D
problem to 1D (one-dimensional) by a set of simplifying assumptions. First
of all, we assume that the variation of the electrical field E y in the y direction
along the channel is much less than the corresponding variation of the electrical field E x in the x direction down to the substrate. Then we have
∂
∂
<<
∂
∂
∴
∂
∂
<<
∂
∂
E
y
E
x
y
x
y
x ;
2
2
2
2
φ
φ
(4.3)
Equation 4.3 is referred to as the gradual channel approximation (GCA) [2].
Therefore, like an MOS capacitor, we solve for f in the x direction along the
depth of the channel only to obtain the total charge, Q s , in the semiconductor. For MOSFETs in inversion (f B < f s < 2f B ), the total charge Q s = Q s (y) due
to the channel potential V ch (y) can be derived from the MOS capacitor theory
(Equation 3.52). In Equation 3.52, we observe that for f s > 0, exp(−f s /v kT ) is
negligibly small, the term “−1” is negligibly small since exp(f s /v kT ) >> −1 in
strong inversion, and the term (−f s /v kT ) is negligibly small in weak inversion.
Therefore, from Equation 3.52, Q s (y) for MOSFETs at inversion can be shown as
Q y
K qN
y v
n
N
e
s
s i
b
s
k T
i
b
y V y v
s
c h
k T
( )
( )
( )
( ) /
= −
+
(
)
−
(
)
2
0
2
2
ε
φ
φ
1 1 2
0
2
1 2
2
/
( )
( ) /
/
( )
= −
+
(
)
−
−
(
)
K qN
y v e
si
b
s
kT
y
V y v
s
B
ch
kT
ε
φ
φ
φ
= = −
(
)
2
0
K qN f
y
V y
si
b
s
B
ch
ε
φ
φ
( ), , ( )
(4.4)
V bs
V s
V gs
V ds
p-Substrate, N b
y = 0
y = L
X j
T ox
n+
n+
x
y
z
FIGURE 4.4
Schematic 2D cross section of an n-channel MOSFET showing the biasing conditions and the
coordinate system; x, y, and z represent the distances along the depth, length, and width of the
device, respectively.
Compact Models for Integrated Circuit Design
∇ = −
2
0
φ
ρ
ε
K si
(4.2)
Generally, a MOSFET is a 3D (three-dimensional) problem; however, for all
practical purposes (except for very small W and L), we can treat the system
as a 2D problem in the x and y directions only. We can further convert the 2D
problem to 1D (one-dimensional) by a set of simplifying assumptions. First
of all, we assume that the variation of the electrical field E y in the y direction
along the channel is much less than the corresponding variation of the electrical field E x in the x direction down to the substrate. Then we have
∂
∂
<<
∂
∂
∴
∂
∂
<<
∂
∂
E
y
E
x
y
x
y
x ;
2
2
2
2
φ
φ
(4.3)
Equation 4.3 is referred to as the gradual channel approximation (GCA) [2].
Therefore, like an MOS capacitor, we solve for f in the x direction along the
depth of the channel only to obtain the total charge, Q s , in the semiconductor. For MOSFETs in inversion (f B < f s < 2f B ), the total charge Q s = Q s (y) due
to the channel potential V ch (y) can be derived from the MOS capacitor theory
(Equation 3.52). In Equation 3.52, we observe that for f s > 0, exp(−f s /v kT ) is
negligibly small, the term “−1” is negligibly small since exp(f s /v kT ) >> −1 in
strong inversion, and the term (−f s /v kT ) is negligibly small in weak inversion.
Therefore, from Equation 3.52, Q s (y) for MOSFETs at inversion can be shown as
Q y
K qN
y v
n
N
e
s
s i
b
s
k T
i
b
y V y v
s
c h
k T
( )
( )
( )
( ) /
= −
+
(
)
−
(
)
2
0
2
2
ε
φ
φ
1 1 2
0
2
1 2
2
/
( )
( ) /
/
( )
= −
+
(
)
−
−
(
)
K qN
y v e
si
b
s
kT
y
V y v
s
B
ch
kT
ε
φ
φ
φ
= = −
(
)
2
0
K qN f
y
V y
si
b
s
B
ch
ε
φ
φ
( ), , ( )
(4.4)
V bs
V s
V gs
V ds
p-Substrate, N b
y = 0
y = L
X j
T ox
n+
n+
x
y
z
FIGURE 4.4
Schematic 2D cross section of an n-channel MOSFET showing the biasing conditions and the
coordinate system; x, y, and z represent the distances along the depth, length, and width of the
device, respectively.