320
Compact Models for Integrated Circuit Design
and
n
N
n
v
i
a
i
B
kT
2
=
−
exp
φ
(9.8)
where:
f B is the bulk potential
Typically, FinFETs are undoped or lightly doped channel devices; therefore,
we consider only the inversion carrier electron concentration n(x, y) at any
point (x, y) given by Equation 9.6 and uniformly doped p-type body doping
concentration, N a (x, y) ≡ N b . Let us assume that V ch (y) is the channel potential
at any point y and GCA as described in Chapter 4 is valid [57]. Then for a
double-gate FET (DG-FET) shown in Figure 9.5, we can express Poisson’s
Equation 9.5 as
d x y
dx
q
K
n
x y
V y
v
N
si
i
B
c h
kT
b
2
2
0
φ
ε
φ
φ
( , )
exp
( , )
=
− − ( )
+
(9.9)
where:
V ch (y) is given by V ch (0) = V s at the source and V ch (L) = V d at the drain
From Equation 9.9, the electrostatic potential f(x, y) at any point (x, y) in the
channel can be written as
φ
φ
φ
( , )
( , )
( , )
x y
x y
x y
≅
+
1
2
(9.10)
In Equation 9.10, f 1 (x, y) is the contribution to f(x, y) due to the inversion carriers without the effect of the ionized body dopants, and f 2 (x, y) is the contribution to f(x, y) due to body dopants, N b . Therefore, we have
d
x y
x
qn
K
x y
V y
v
i
si
B
c h
kT
2
1
2
0
φ
ε
φ
φ
( , )
exp
( , )
∂
=
− − ( )
(9.11)
d
x y
x
qN
K
b
si
2
2
2
0
φ
ε
( , )
∂
=
(9.12)
If t fin is less than the width of the depletion region, then for a certain gate bias
V g , the silicon fin is fully depleted and consequently the inversion carriers
are spread throughout the entire body. Thus, Q i >> Q b , and therefore, we can
safely neglect the term containing N b in Equation 9.9 and the channel potential is obtained by solving Equation 9.11.
We know that for a symmetric double-gate structure, the vertical component of the electric field E x is zero at the center, that is, at x = 0, df 1 /dx = 0
Compact Models for Integrated Circuit Design
and
n
N
n
v
i
a
i
B
kT
2
=
−
exp
φ
(9.8)
where:
f B is the bulk potential
Typically, FinFETs are undoped or lightly doped channel devices; therefore,
we consider only the inversion carrier electron concentration n(x, y) at any
point (x, y) given by Equation 9.6 and uniformly doped p-type body doping
concentration, N a (x, y) ≡ N b . Let us assume that V ch (y) is the channel potential
at any point y and GCA as described in Chapter 4 is valid [57]. Then for a
double-gate FET (DG-FET) shown in Figure 9.5, we can express Poisson’s
Equation 9.5 as
d x y
dx
q
K
n
x y
V y
v
N
si
i
B
c h
kT
b
2
2
0
φ
ε
φ
φ
( , )
exp
( , )
=
− − ( )
+
(9.9)
where:
V ch (y) is given by V ch (0) = V s at the source and V ch (L) = V d at the drain
From Equation 9.9, the electrostatic potential f(x, y) at any point (x, y) in the
channel can be written as
φ
φ
φ
( , )
( , )
( , )
x y
x y
x y
≅
+
1
2
(9.10)
In Equation 9.10, f 1 (x, y) is the contribution to f(x, y) due to the inversion carriers without the effect of the ionized body dopants, and f 2 (x, y) is the contribution to f(x, y) due to body dopants, N b . Therefore, we have
d
x y
x
qn
K
x y
V y
v
i
si
B
c h
kT
2
1
2
0
φ
ε
φ
φ
( , )
exp
( , )
∂
=
− − ( )
(9.11)
d
x y
x
qN
K
b
si
2
2
2
0
φ
ε
( , )
∂
=
(9.12)
If t fin is less than the width of the depletion region, then for a certain gate bias
V g , the silicon fin is fully depleted and consequently the inversion carriers
are spread throughout the entire body. Thus, Q i >> Q b , and therefore, we can
safely neglect the term containing N b in Equation 9.9 and the channel potential is obtained by solving Equation 9.11.
We know that for a symmetric double-gate structure, the vertical component of the electric field E x is zero at the center, that is, at x = 0, df 1 /dx = 0