86
Compact Models for Integrated Circuit Design
q
q
E q
p
sp
s
g
Bp
Φ =
+
+
(
)
χ
φ
2
-type semiconductor
(3.2)
where:
E g is the bandgap energy
f Bp is the bulk or Fermi potential for a p-type semiconductor
Similarly, the work function for an n-type semiconductor is given by
q
q
E q
n
sn
s
g
Bn
Φ =
+ −
(
)
χ
φ
2
-type semiconductor
(3.3)
where:
f Bn is the bulk or Fermi potential for an n-type semiconductor
If the doping concentration for both the n-type and p-type semiconductors is
the same, then φ
φ
φ
Bp
Bn
B
=
≡ and is given by Equation 2.70 as
φ B
k T
b
i
v
N
n
=
ln
(3.4)
where:
v
kT q
kT =
(
)
/ is the thermal voltage at the ambient temperature T
n i is the intrinsic carrier concentration
In v kT , the parameters k and q represent the Boltzmann constant and electronic
charge, respectively. In order to calculate the value of the semiconductor
work function, Φ s , the magnitude of f B is calculated from Equation 3.4 as
shown in the following example.
Let us consider a p-type silicon with N b = N a = 1 × 10 15 cm –3 at room temperature 300 K so that v kT ≅ 0.0259 V. Then using n i = 1.45 × 10 10 cm –3 , we can show
from Equation 3.4 that the value of f B ≅ 0.29 V. Now, considering q s
χ = 4 05
. eV
and E g = 1.12 eV for silicon, we get from Equation 3.2, qΦ sp ≡ qΦ s ≅ 4.90 eV. For
aluminum, qΦ m = 4.1 eV; therefore, for a p-type silicon, Φ m < Φ s , that is, the
energy required to free an electron from the p-type silicon is higher than the
energy required to free an electron from aluminum.
In order to calculate Φ s for polysilicon gate, it is assumed that the polysilicon is degenerately doped so that the Fermi energy lies at the band edges,
that is, E f is at E c for an n-type polysilicon and E f is at E v for a p-type polysilicon. For nanoscale CMOS (complementary metal-oxide-semiconductor)
technology, work function engineering is used to achieve the target value
of metal gate work function [4]. The work functions of commonly used gate
material for IC technology are shown in Table 3.1 [5,6].
Now, let us consider the energy bands of three materials shown in
Figure 3.2a–c are brought in contact to form an MOS capacitor system. It can be
shown that when different materials are in contact with each other, the work
Compact Models for Integrated Circuit Design
q
q
E q
p
sp
s
g
Bp
Φ =
+
+
(
)
χ
φ
2
-type semiconductor
(3.2)
where:
E g is the bandgap energy
f Bp is the bulk or Fermi potential for a p-type semiconductor
Similarly, the work function for an n-type semiconductor is given by
q
q
E q
n
sn
s
g
Bn
Φ =
+ −
(
)
χ
φ
2
-type semiconductor
(3.3)
where:
f Bn is the bulk or Fermi potential for an n-type semiconductor
If the doping concentration for both the n-type and p-type semiconductors is
the same, then φ
φ
φ
Bp
Bn
B
=
≡ and is given by Equation 2.70 as
φ B
k T
b
i
v
N
n
=
ln
(3.4)
where:
v
kT q
kT =
(
)
/ is the thermal voltage at the ambient temperature T
n i is the intrinsic carrier concentration
In v kT , the parameters k and q represent the Boltzmann constant and electronic
charge, respectively. In order to calculate the value of the semiconductor
work function, Φ s , the magnitude of f B is calculated from Equation 3.4 as
shown in the following example.
Let us consider a p-type silicon with N b = N a = 1 × 10 15 cm –3 at room temperature 300 K so that v kT ≅ 0.0259 V. Then using n i = 1.45 × 10 10 cm –3 , we can show
from Equation 3.4 that the value of f B ≅ 0.29 V. Now, considering q s
χ = 4 05
. eV
and E g = 1.12 eV for silicon, we get from Equation 3.2, qΦ sp ≡ qΦ s ≅ 4.90 eV. For
aluminum, qΦ m = 4.1 eV; therefore, for a p-type silicon, Φ m < Φ s , that is, the
energy required to free an electron from the p-type silicon is higher than the
energy required to free an electron from aluminum.
In order to calculate Φ s for polysilicon gate, it is assumed that the polysilicon is degenerately doped so that the Fermi energy lies at the band edges,
that is, E f is at E c for an n-type polysilicon and E f is at E v for a p-type polysilicon. For nanoscale CMOS (complementary metal-oxide-semiconductor)
technology, work function engineering is used to achieve the target value
of metal gate work function [4]. The work functions of commonly used gate
material for IC technology are shown in Table 3.1 [5,6].
Now, let us consider the energy bands of three materials shown in
Figure 3.2a–c are brought in contact to form an MOS capacitor system. It can be
shown that when different materials are in contact with each other, the work