88
Compact Models for Integrated Circuit Design
bending in the oxide and silicon. Since metal is an equipotential region, there
is no band bending in metal.
For p-type silicon and aluminum metal MOS capacitor (Al-SiO 2 -Si) system,
Φ
Φ
m
s
< , therefore, energy bands bend downward in the oxide and silicon near
the surface as shown in Figure 3.3a. And, there is an abrupt transition in E c and
E v levels at the material interfaces. The metal and semiconductor work function
difference (Φ m –Φ s ) causes a potential drop in oxide and near the silicon surface
due to band bending. A typical potential drop in oxide is about 0.4 V. This potential drop depends on the doping level in silicon and can be supported since no
current flows through oxide. The values shown in Figure 3.3a for the band bending in the oxide and silicon are obtained by assuming that the oxide is an ideal
insulator without any charges. We can compensate for this band bending by
applying an external voltage V fb = (Φ m –Φ s ), which caused the band bending in
the first place. V fb is referred to as the flat band voltage and the band structure for
an MOS capacitor at flat band condition is shown in Figure 3.3b.
Thus, the condition for flat band voltage at the Si/SiO 2 interface is given by
V fb
m
s
ms
=
−
≡
Φ
Φ Φ
(3.5)
where:
Φ ms is the work function difference between the gate electrode and bulk
silicon (in units of volts)
Then for an Al-SiO 2 -pSi system,
q
q
q
q
q
E q
ms
m
s p
m
s
g
B
Φ
Φ
Φ
Φ
=
−
=
−
+
+
χ
φ
2
(3.6)
Considering the values shown in Figure 3.2a–c, we get
Φ ms
B
p
= −
+
(
)
0 51
.
φ
for type silicon
-
(3.7)
Since φ B ≅ 0 29
. V for substrate concentration N b = 1 × 10 15 cm –3 ; therefore, Φ ms
is a negative number. Similarly, for an Al-SiO 2 -nSi system,
q
q
q
q
q
E q
ms
m
s n
m
s
g
B
Φ
Φ
Φ
Φ
=
−
=
−
+
−
χ
φ
2
(3.8)
Φ ms
B
n
= −
−
(
)
0 51
.
φ
for -type silicon
(3.9)
Equation 3.9 shows that Φ ms for MOS capacitors with an n-type silicon is also a
negative number for N b < 1 × 10 18 cm –3 . Since in advanced CMOS technologies,
the channel is undoped or lightly doped, Φ ms is always negative. This work function difference causes band bending when the materials are brought in contact.
For degenerately doped polysilicon gate electrode, the band structure for
an MOS capacitor system is shown in Figure 3.4.
Compact Models for Integrated Circuit Design
bending in the oxide and silicon. Since metal is an equipotential region, there
is no band bending in metal.
For p-type silicon and aluminum metal MOS capacitor (Al-SiO 2 -Si) system,
Φ
Φ
m
s
< , therefore, energy bands bend downward in the oxide and silicon near
the surface as shown in Figure 3.3a. And, there is an abrupt transition in E c and
E v levels at the material interfaces. The metal and semiconductor work function
difference (Φ m –Φ s ) causes a potential drop in oxide and near the silicon surface
due to band bending. A typical potential drop in oxide is about 0.4 V. This potential drop depends on the doping level in silicon and can be supported since no
current flows through oxide. The values shown in Figure 3.3a for the band bending in the oxide and silicon are obtained by assuming that the oxide is an ideal
insulator without any charges. We can compensate for this band bending by
applying an external voltage V fb = (Φ m –Φ s ), which caused the band bending in
the first place. V fb is referred to as the flat band voltage and the band structure for
an MOS capacitor at flat band condition is shown in Figure 3.3b.
Thus, the condition for flat band voltage at the Si/SiO 2 interface is given by
V fb
m
s
ms
=
−
≡
Φ
Φ Φ
(3.5)
where:
Φ ms is the work function difference between the gate electrode and bulk
silicon (in units of volts)
Then for an Al-SiO 2 -pSi system,
q
q
q
q
q
E q
ms
m
s p
m
s
g
B
Φ
Φ
Φ
Φ
=
−
=
−
+
+
χ
φ
2
(3.6)
Considering the values shown in Figure 3.2a–c, we get
Φ ms
B
p
= −
+
(
)
0 51
.
φ
for type silicon
-
(3.7)
Since φ B ≅ 0 29
. V for substrate concentration N b = 1 × 10 15 cm –3 ; therefore, Φ ms
is a negative number. Similarly, for an Al-SiO 2 -nSi system,
q
q
q
q
q
E q
ms
m
s n
m
s
g
B
Φ
Φ
Φ
Φ
=
−
=
−
+
−
χ
φ
2
(3.8)
Φ ms
B
n
= −
−
(
)
0 51
.
φ
for -type silicon
(3.9)
Equation 3.9 shows that Φ ms for MOS capacitors with an n-type silicon is also a
negative number for N b < 1 × 10 18 cm –3 . Since in advanced CMOS technologies,
the channel is undoped or lightly doped, Φ ms is always negative. This work function difference causes band bending when the materials are brought in contact.
For degenerately doped polysilicon gate electrode, the band structure for
an MOS capacitor system is shown in Figure 3.4.