192
Compact Models for Integrated Circuit Design
Now, from Gauss’s law we can show that
E
E
Q
K
E
Q
K
x
x
i
si
x
b
si
1
2
0
2
0
−
=
=
ε
ε
and
(5.47)
where:
Q i and Q b are the inversion and bulk-charge densities, respectively, due to
the applied V gs
Substituting for E x1 and E x2 from Equation 5.47 into Equation 5.46, we can show
E
K
Q Q
eff
si
i
b
=
+
1
1
2
0
ε
(5.48)
In order to represent both electrons and holes, the general expression for the
effective electric field is expressed as
E
K
Q Q
eff
si
i
b
=
+
(
)
1
0
ε
η
(5.49)
where:
the constant η = 1/2 for electrons and η = 1/3 for holes [35–37]
The measured μ eff versus E eff plots show a universal behavior independent of
doping concentration at high effective vertical electrical fields and dependence on the substrate doping concentration and interface charge at low
effective vertical electric fields as shown in Figure 5.9a.
The experimentally observed universal mobility behavior is due to the
relative contributions of different scattering mechanisms [38,39] set by the
strength of vertical electrical fields as shown in Figure 5.9b. As shown in
Figure 5.9b, μ eff is determined by Coulomb scattering of the ionized impurities and oxide charges, phonon scattering due to thermal vibration, and
surface roughness scattering at the Si/SiO 2 interface. At high vertical electric
fields, surface roughness scattering dominates as the carrier confinement is
close to the interface, resulting in a decrease of μ eff with the increase of E eff as
observed in Figure 5.9a.
The deviation from the universal behavior observed in Figure 5.9a, particularly in the heavily doped substrates at low effective electric fields, is
due to the ionized impurity scattering, Coulomb scattering, and phonon
scattering. At low effective vertical electric fields, Q i is low and < result, the ionized impurity scattering and Coulomb scattering by ionized
impurities and oxide charges become dominant scattering mechanisms in
Compact Models for Integrated Circuit Design
Now, from Gauss’s law we can show that
E
E
Q
K
E
Q
K
x
x
i
si
x
b
si
1
2
0
2
0
−
=
=
ε
ε
and
(5.47)
where:
Q i and Q b are the inversion and bulk-charge densities, respectively, due to
the applied V gs
Substituting for E x1 and E x2 from Equation 5.47 into Equation 5.46, we can show
E
K
Q Q
eff
si
i
b
=
+
1
1
2
0
ε
(5.48)
In order to represent both electrons and holes, the general expression for the
effective electric field is expressed as
E
K
Q Q
eff
si
i
b
=
+
(
)
1
0
ε
η
(5.49)
where:
the constant η = 1/2 for electrons and η = 1/3 for holes [35–37]
The measured μ eff versus E eff plots show a universal behavior independent of
doping concentration at high effective vertical electrical fields and dependence on the substrate doping concentration and interface charge at low
effective vertical electric fields as shown in Figure 5.9a.
The experimentally observed universal mobility behavior is due to the
relative contributions of different scattering mechanisms [38,39] set by the
strength of vertical electrical fields as shown in Figure 5.9b. As shown in
Figure 5.9b, μ eff is determined by Coulomb scattering of the ionized impurities and oxide charges, phonon scattering due to thermal vibration, and
surface roughness scattering at the Si/SiO 2 interface. At high vertical electric
fields, surface roughness scattering dominates as the carrier confinement is
close to the interface, resulting in a decrease of μ eff with the increase of E eff as
observed in Figure 5.9a.
The deviation from the universal behavior observed in Figure 5.9a, particularly in the heavily doped substrates at low effective electric fields, is
due to the ionized impurity scattering, Coulomb scattering, and phonon
scattering. At low effective vertical electric fields, Q i is low and < result, the ionized impurity scattering and Coulomb scattering by ionized
impurities and oxide charges become dominant scattering mechanisms in